Modelling Colored Noise under Large-Signal Conditions
by
Nikhil M. Kriplani
A dissertation submitted to the Graduate Faculty of
North Carolina State University
in partial satisfaction of the
requirements for the Degree of
Doctor of Philosophy
Electrical Engineering
Raleigh
2005
Approved By:
Dr. Griff L. Bilbro
Dr. W. Rhett Davis
Dr. Michael B. Steer
Chair of Advisory Committee
Dr. Douglas W. Barlage
ABSTRACT
KRIPLANI, NIKHIL M. Modelling Colored Noise under Large-Signal Conditions. (Under
the direction of Professor Michael B. Steer).
A time-domain simulation approach to modelling colored noise in electrical circuits
is described. This approach tries to place minimal restrictions on the magnitude and the
nature of the noise present in a circuit in an effort to capture the effects of nonlinear interactions between signal and noise. The approach uses the mathematical theory of nonlinear
dynamics and chaos to produce stochastic-looking series using simple deterministic iterative
rules or maps. The characteristics of these series can be modified easily to produce a large
range of spectral characteristics. The advantage of using the chaotic maps approach is that
modifying the spectral characteristics usually requires the tweaking of a small number of
parameters. This is in contrast to more traditional time-series-based approaches to noise
generation which require a large number of parameters to accurately model the characteristics of common sources of noise found in electrical circuits. The validity of this approach
to modelling is tested by implementing a unified deterministic and stochastic framework of
equations in a high dynamic range simulator. The resulting stochastic system of equations
describing a nonlinear noisy network are setup and solved assuming the Stratonovich interpretation. Simulated results are compared with measured results using two representative
circuits. The first circuit is a varactor-tuned voltage-controlled oscillator and simulated
phase noise at the output of the circuit is compared with measured values. The second
circuit is a low-noise X-band MMIC power amplifier and the effect of noise on the amplification of this device is investigated. Gain versus input power curves are generated
in simulation when the circuit is fed with large levels of input noise and contrasted with
measurement. Both these cases demonstrate that this approach to the modelling of large
levels of noise is valid and perhaps even essential in order to accurately predict the effects
of having non-negligible levels of noise in an electronic circuit.
ii
To my parents, who provided me with the opportunity to make something of my life.
To my fate, for allowing it to happen.
iii
Biography
Nikhil M. Kriplani was born in the city of Mumbai, India in 1977. He obtained his Bachelor’s
degree in Electronics and Telecommunications Engineering at the Maharashtra Institute of
Technology in Pune, India in 1999 and his Master’s degree in Electrical Engineering from
North Carolina State University at Raleigh, NC, U.S.A in 2002. His (unordered) interests
include sports, mathematics, acoustics, philosophy, writing, healthy living and searching for
the purpose of his existence.
iv
Acknowledgements
I would like to thank Dr. G. Bilbro, Dr. D. W. Barlage, Dr. Wm. R. Davis and Dr. M.
B. Steer for serving on my committee. In particular, I would like to thank Dr. M. B. Steer
for the doses of motivation, guidance and funding for as long as I have been in the Ph.D.
program. It still does seem quite amazing that it has all worked out this way, and needless
to say, it would have been impossible without his generous influence. I would like to acknowledge the camaraderie between my fellow graduate student corps, Aaron Walker, Sonali
Luniya, Mark Buff, Jayesh Nath, Frank Hart, Alan Victor and Wonhoon Jang and thank
them for all the light moments, the seemingly endless discussions on unrelated, seemingly
meaningless but seemingly interesting topics and most importantly, sharing the experience
of sailing in this same boat together. Finally, I am grateful to my wife, Larisa, for giving
up a part of her life and supporting me during the final stages of my work.
v
Contents
List of Figures
viii
List of Tables
xi
List of Abbreviations
xii
1 Introduction
1.1 Motivations and Objectives . . . . . . .
1.2 Unleashing Chaos . . . . . . . . . . . . .
1.3 Establishing a Framework . . . . . . . .
1.4 Circuit Simulator Noise Analysis Review
1.5 Original Contributions . . . . . . . . . .
1.6 Overview . . . . . . . . . . . . . . . . .
1.7 Publications . . . . . . . . . . . . . . . .
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2 Literature Review
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . .
2.3 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 1/f (Flicker) Noise . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Scale Invariance . . . . . . . . . . . . . . . . . . . . .
2.5.2 Stationarity and Gaussianity . . . . . . . . . . . . . .
2.5.3 The Empiricism of Hooge . . . . . . . . . . . . . . . .
2.5.4 Number Fluctuations of Charge and Mobility . . . . .
2.5.5 Surface Effect or Bulk effect . . . . . . . . . . . . . . .
2.5.6 Dependence on Mean Voltage, Current and Resistance
2.5.7 The Effect of Temperature . . . . . . . . . . . . . . .
2.6 1/f (Flicker) Noise Models . . . . . . . . . . . . . . . . . . .
2.6.1 Surface Trapping Model . . . . . . . . . . . . . . . . .
2.6.2 Transmission line 1/f noise . . . . . . . . . . . . . . .
2.6.3 Pole Placement . . . . . . . . . . . . . . . . . . . . . .
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vi
2.7
2.8
2.6.4 Fractional Noises . . . . . . . . . . . .
2.6.5 Power-Law Shot Noise . . . . . . . .
2.6.6 1/f Noise from Chaos . . . . . . . . .
2.6.7 Self-Organized Criticality (and others)
2.6.8 Phase Noise . . . . . . . . . . . . . . .
Simulation of Noise in Circuits . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . .
3 Stochastic Differential Equations
3.1 Introduction . . . . . . . . . . . .
3.2 Basic Theory . . . . . . . . . . .
3.3 Scalar SDEs . . . . . . . . . . .
3.3.1 A Simple Example . . . .
3.3.2 Equivalence of the Itô and
3.4 Vector SDEs . . . . . . . . . . .
3.5 Itô v/s Stratonovich Forms . . .
3.6 Summary . . . . . . . . . . . . .
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37
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Stratonovich forms .
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59
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4 Nonlinear Dynamics, Chaos and Intermittency
4.1 Introduction . . . . . . . . . . . . . . . . . . . . .
4.2 Nonlinear Dynamics and Chaos . . . . . . . . . .
4.2.1 Fixed Points . . . . . . . . . . . . . . . .
4.2.2 Periodic Points . . . . . . . . . . . . . . .
4.2.3 Neutral Fixed Points . . . . . . . . . . . .
4.2.4 Bifurcation Theory . . . . . . . . . . . . .
4.2.5 Sliding into Chaos . . . . . . . . . . . . .
4.3 Generating White Noise . . . . . . . . . . . . . .
4.4 Intermittency and Flicker Noise . . . . . . . . .
4.4.1 Basic Theory . . . . . . . . . . . . . . . .
4.4.2 Characteristics of Intermittency . . . . . .
4.4.3 Nonlinear Intermittent Functions . . . . .
4.5 The Logarithmic Intermittent Map . . . . . . . .
4.6 Summary . . . . . . . . . . . . . . . . . . . . . .
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69
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5 Implementation of Stochastic Framework
5.1 Introduction . . . . . . . . . . . . . . . . . .
5.2 An Acquaintance with f REEDATM . . . .
5.3 Transient Analysis in f REEDATM . . . . .
5.3.1 Linear Network . . . . . . . . . . . .
5.3.2 Nonlinear Network . . . . . . . . . .
5.3.3 Formulation of the Error Function .
5.3.4 Conversion to Algebraic Form . . . .
5.4 Implementation of Transient Noise Analysis
5.4.1 Enhanced Device Models . . . . . .
5.4.2 Nonlinear Maps . . . . . . . . . . . .
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90
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vii
5.5
5.4.3 Noisy Error Function . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
111
6 Noise in a Voltage-Controlled Oscillator
6.1 Introduction . . . . . . . . . . . . . . . . .
6.2 A Varactor Voltage-Controlled Oscillator
6.3 Simulation and Validation . . . . . . . .
6.4 Summary . . . . . . . . . . . . . . . . . .
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112
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7 Noise and Amplification
7.1 Introduction . . . . . .
7.2 Setup and Verification
7.3 Further Investigations
7.4 Summary . . . . . . .
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8 Conclusions and Future Work
133
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Bibliography
137
A Essential Itô and Stratonovich
150
B Implementing Infinite R-C Transmission Line Models
154
C Source Code
C.1 Noise-enabled npn-BJT . . . . . . . .
C.2 Noise-enabled p-n Junction Diode . .
C.3 Noise-enabled Curtice-Cubic MESFET
C.4 Noise-enabled Resistor . . . . . . . .
C.5 White Noise Voltage Source . . . . .
C.6 The Parker-Skellern Model . . . . . .
C.7 The OML MESFET model . . . . . .
C.8 The Ziggurat Technique . . . . . . . .
C.9 The Logistic Map Noise Generator . .
C.10 The Logarithmic Map Noise Generator
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D f REEDATM Netlists
213
D.1 The Varactor-tuned VCO circuit . . . . . . . . . . . . . . . . . . . . . . . . 213
D.2 The X-band MMIC Circuit Netlist . . . . . . . . . . . . . . . . . . . . . . . 215
viii
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
3.1
4.1
4.2
4.3
4.4
4.5
4.6
Thermal noise equivalent circuits. . . . . . . . . . . . . . . . . . . . . . . .
Thermal noise voltages and currents in equilibrium at T0 . . . . . . . . . . .
Lumped RC transmission line excited by a white noise current source from
[28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A 1/f γ noise generator from [65]. . . . . . . . . . . . . . . . . . . . . . . .
Digital 1/f γ noise generator from [66]. . . . . . . . . . . . . . . . . . . . .
1D Brownian motion realization. . . . . . . . . . . . . . . . . . . . . . . . .
1D fBm realization with H = 0.2. . . . . . . . . . . . . . . . . . . . . . . .
1D fBm realization with H = 0.8. . . . . . . . . . . . . . . . . . . . . . . .
A general shot noise generator from [80]. . . . . . . . . . . . . . . . . . . .
A power law impulse response function with β = 1/2. . . . . . . . . . . . .
A general 1-D dynamical system consisting of a nonlinear function and a
recursive loop from [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The bifurcation diagram for the logistic map. . . . . . . . . . . . . . . . . .
Output spectrum with g=0.925. . . . . . . . . . . . . . . . . . . . . . . . . .
Output spectrum with g=0.975. . . . . . . . . . . . . . . . . . . . . . . . . .
Output spectrum with g=0.9975. . . . . . . . . . . . . . . . . . . . . . . . .
PSD for f1 (x), f2 (x) and f3 (x). . . . . . . . . . . . . . . . . . . . . . . . . .
SOC power law of the distribution of cluster sizes. . . . . . . . . . . . . . .
Uncertainty in carrier frequency due to phase noise. . . . . . . . . . . . . .
A typical oscillator configuration. . . . . . . . . . . . . . . . . . . . . . . . .
A typical plot of the phase noise of an oscillator versus offset from the carrier.
17
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55
Differences in the solution of the SDE in Eqn. (3.21) assuming the Itô and
Stratonovich interpretations, with X0 = 0, k = 2 and α = 1. . . . . . . . .
65
The logistic map, with λ = 4. . . . . . . . . . . . . . . . . .
An example orbit of the logistic map, with x0 = 0.2. . . . . .
The logistic map with λ = 4 and its fixed points, as indicated.
The quadratic map with c = 0 and its fixed points. . . . . .
The quadratic map with c = −1 and prime-period 2 orbit. .
Different types of neutral fixed points. . . . . . . . . . . . . .
71
71
72
73
73
74
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ix
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.1
6.2
6.3
6.4
6.5
6.6
6.7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Quadratic map with c = 0.4 and with no fixed points. . . . . . . . . . . . .
Quadratic map with c = 0.25 and with one fixed point. . . . . . . . . . . .
Quadratic map with c = 0.0 and with two fixed points, one attracting and
one repelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bifurcation diagram for the quadratic map. . . . . . . . . . . . . . . . . . .
Correlation plot of the logistic map with λ = 4. . . . . . . . . . . . . . . .
Spectrum of the logistic map with λ = 4. . . . . . . . . . . . . . . . . . . .
The logarithmic map, with β = 0.000005. . . . . . . . . . . . . . . . . . . .
Sample realization of the logarithmic map, with β = 0.000005. . . . . . . .
Correlation plot of the logarithmic map with β = 0.000005. . . . . . . . . .
Spectrum of the logarithmic map with β = 0.000005. . . . . . . . . . . . .
Connections between a heterogeneous collection of elements. . . . . . . . .
A partitioned network of linear and nonlinear elements and sources. . . . .
Large-signal Gummel-Poon BJT circuit. . . . . . . . . . . . . . . . . . . . .
The Gummel-Poon BJT model, along with noise sources. . . . . . . . . . .
Large-signal model for a p-n junction diode. . . . . . . . . . . . . . . . . .
Noise-enabled p-n junction diode model. . . . . . . . . . . . . . . . . . . .
Noiseless Curtice Cubic large-signal model. . . . . . . . . . . . . . . . . . .
Noise-enabled Curtice Cubic large-signal model. . . . . . . . . . . . . . . .
Partitioned network which now contains contributions from transient noise
sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulated output of the VCO at terminal (B) indicating a frequency of oscillation of 45 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Varactor-tuned VCO schematic, from [144]. . . . . . . . . . . . . . . . . . .
Dependence on VCO oscillation frequency v/s bias. . . . . . . . . . . . . .
Degradation of phase noise at varactor v/s bias. . . . . . . . . . . . . . . .
Phase noise comparison between data and experiment with bias voltage at
0 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phase noise comparison between data and experiment with bias voltage at
6 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phase noise comparison between data and experiment with bias voltage at
12 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Layout of the two-stage X-band MMIC. . . . . . . . . . . . . . . . . . . . .
Measurement setup for the X-band MMIC amplifier. . . . . . . . . . . . .
Comparison between measured curves of gain with no input noise and noise
maintained at −20 dBc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power transferred from the center frequency into sidebands. . . . . . . . .
Comparison between simulated curves of gain with no input noise and noise
maintained at −20 dBc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Degradation of the output sinusoid due to noise. . . . . . . . . . . . . . . .
Comparing simulated gain obtained with a 20 ps delay and no delay with
measured gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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76
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79
82
82
87
87
88
88
92
93
97
101
102
104
106
107
110
113
114
115
116
117
118
118
121
121
122
123
124
124
130
x
7.8
Distorted output voltage with a 20ps delay. . . . . . . . . . . . . . . . . . .
131
xi
List of Tables
5.1
5.2
5.3
5.4
5.5
5.6
BJT model parameters in f REEDATM . . . . . . . . . . . . . .
BJT model parameters in f REEDATM continued. . . . . . . . .
Noisy diode model parameters in f REEDATM . . . . . . . . . .
Noisy Curtice Cubic model parameters in f REEDATM . . . . .
Noisy resistor model parameters in f REEDATM . . . . . . . . .
White noise voltage source model parameters in f REEDATM .
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98
99
103
105
108
109
6.1
6.2
BJT model parameters values as used in the VCO circuit. . . . . . . . . . .
Diode model parameters values as used in the VCO circuit. . . . . . . . . .
116
116
7.1
7.2
7.3
7.4
MESFET model parameters in for X-band MMIC. . . . . . . . . . .
Parker-Skellern model parameters in f REEDATM . . . . . . . . . . .
Parker-Skellern model parameters used in the X-band MMIC netlist.
OML model parameters used in the X-band MMIC netlist. . . . . .
123
126
128
129
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xii
List of Abbreviations
AR
Auto-Regressive
ARIMA Auto-Regressive Integrated Moving Average
ARMA Auto-Regressive Moving-Average
BJT
Bipolar Junction Transistors
BM
Brownian Motion Process
CAD Computer-Aided Design
CDF Cumulative Distribution Function
fBM
fractional Brownian Motion
HOT Highly Optimized Tolerance
IC
Integrated Circuit
JFET Junction Field-Effect Transistor
KCL Kirchhoff’s Current Law
KVL Kirchhoff’s Voltage Law
LPTV Linear Periodically Time-Varying
LSSS Large Signal Small Signal
LTI
linear time-invariant
xiii
LTV
Linear Time-Varying
MA
Moving-Average
MMIC Microwave Monolithic Integrated Circuit
MNAM Modified Nodal Admittance Matrix
MOSFET Metal Oxide Semiconductor Field-Effect Transistor
OO
Object-Oriented
PDF Probability Distribution Function
pHEMT pseudomorphic-High-Electron-Mobility-Transistor
PMF Probability Mass Function
PSD
Power Spectral Density
RF
Radio Frequency
SDE
Stochastic Differential Equations
SDIC Sensitive Dependence on Initial Conditions
SOC
Self-Organized Criticality
SPICE Simulation Program with Integrated Circuit Emphasis
WSS Wide-Sense Stationary
1
Chapter 1
Introduction
1.1
Motivations and Objectives
Noise in electrical circuits presents a practical limit on the performance of electrical circuits
and systems. The sources of noise are numerous and in most cases, their origins are well
known and several models for these sources exist in the literature. In all cases, the models
for representing noise are random in nature and the only characterization of these sources of
noise is statistical. Noise sources can be broadly classified as thermal, shot and flicker type.
Thermal noise is associated with the random motion of carriers in a material and the extent
of the motion is proportional to the resistance of the material and its temperature. Shot
noise is generally found in junction semiconductors, although it was originally observed in
vacuum tubes and its existence is attributed to the motion of charges across a junction
formed by joining two semiconductor materials with opposite charge concentrations. The
origin of flicker noise, also referred to as 1/f noise where f is the frequency, is still somewhat
ambiguous in the sense that there is no general consensus or agreement. Flicker noise is
a more general form of a power law noise or a 1/f α noise where α is considered to vary
between 0 and 2. Its various manifestations are found not just in electrical circuits but in
a wide spectrum of scientific situations which is perhaps the biggest reason for there being
no unified explanation on the origins of such a source of noise. Although there are several
other types of noise sources that one can encounter, this works deals with the modelling of
2
the afore-mentioned sources of noise in the time domain.
Traditionally, noise analysis is performed in the frequency domain and is commonly referred to as AC noise analysis. Computer-Aided Design (CAD) tools such as the
circuit simulator SPICE, which are invaluable aids in linear Integrated Circuit (IC) design,
all implement an AC noise analysis. Most linear ICs operate under small-signal conditions
which requires the establishment of an operating point, which once fixed, is not expected
to change during operation. This allows the development of Linear Time-Invariant (LTI)
network models of the circuit elements. This approach makes it convenient to analyze such
circuits in the frequency domain with the use of a rich and well-established numerical analysis theory from linear algebra. Accurate and efficient implementation of these techniques
are the major reason for adoption of SPICE and SPICE-like simulators by every analog and
RF engineer.
In spite of the wide-spread use of SPICE, there are some limitations when it comes
to simulating circuits which cannot be classified by an LTI or small-signal representation.
An example of such a circuit would be an electrical oscillator which not only has variable
operating points and large signal swings, but requires nonlinear effects to be able to start and
sustain the oscillation phenomenon. Another example is a high-power amplifier which not
only has large signal levels, but interactions with out-of-band noise has pronounced effects
on performance. The LTI approach is inadequate for modelling noise in such circuits. Also,
increasing density in ICs and reduced supply voltages result in a decrease in signal-to-noise
ratio and the interaction of signal and noise becomes more important. The only way to
model these effects is by performing a transient noise analysis without restrictions on signal
level, nonlinearity and operating point.
This work concentrates on modelling noise in the time domain with the use of
Stochastic Differential Equations (SDEs) and efficient chaotic noise generators which can
model transient noise with a wide variety of spectral characteristics. It requires a modification of the system of equations describing the circuit to include noisy, random additive
and multiplicative terms, a circuit simulator with a high dynamic range to be able to satisfactorily test these concepts and example real-world circuits to be able to check for the
accuracy and efficiency of the implementation. The noise analysis has been implemented in
the circuit simulator f REEDATM and two examples have been used to verify the accuracy
of modelling and simulation: a varactor-tuned voltage-controlled oscillator which has largesignal excursions and a X-band low-noise Microwave Monolithic Integrated Circuit (MMIC)
3
power amplifier which has a significant amount of nonlinear interactions between signal and
noise at high power levels. Most importantly, the experimental validation indicates that
the physical nature of noise, in particular flicker noise, is adequately modelled using the
approach developed here.
1.2
Unleashing Chaos
Effective modelling of noise requires noise sources that can efficiently capture the characteristics of noise as they appear in measurable phenomena. As mentioned in the previous
section, the three main sources of noise in electrical circuits are thermal noise, shot noise and
flicker noise. Thermal and shot noise have white spectral characteristics meaning that all
their spectral components are assigned equal weight. Time-domain models for white noise
are abundant and either use extremely efficient pseudo-random noise generators1 or use
variations of time series that have Auto-Regressive (AR), Moving-Average (MA) or AutoRegressive Moving-Average (ARMA) forms as found in the classic work by Box, Jenkins
and Reinsel [1].
Flicker noise has characteristics that are more “interesting”. There have been
several approaches to try and find a model that can sufficiently mimic the various and
sometimes contradictory properties of flicker noise with exponent one, also called 1/f noise.
For example, 1/f noise has equal power in all decades of frequency. This has led researchers
to question as to how low in frequency the flicker effect can persist and as to whether there
should be infinite power at zero frequency. Another question often raised is if a 1/f process
is a Gaussian process. Some studies have claimed that it must be Gaussian while others
have shown it cannot be so because a Gaussian process would imply a flatenning of the
frequency response at low frequencies. This cannot be true if the process is to have equal
power for all frequency decades. Issues of this nature and several others are explored in
the next chapter. Just as with white noise, variations of AR, MA and ARMA models, also
called Auto-Regressive Integrated Moving Average (ARIMA) models, do exist for flicker
noise. The problem with using these models for flicker noise is often associated with the
concept of memory. Intuitively, the memory of a stochastic process is a measure of the
inter-relatedness of a sample of the process at a particular instant with every other sample
1
http://www.random.org
4
of the process. Often this is characterized by the generation of the correlation plot of
the process. Flicker noise with exponent zero (white noise) is termed memoryless and
the correlation plot of ideal white noise has a single impulse at the origin. This implies
a complete lack of correlation between the samples of a white noise sequence and that
each sample exists independent of every other sample of the process. Flicker noise with
exponent two (brown noise) is termed a low-memory process. This means that although
there is some memory associated with the samples of a brown noise process, the amount
of correlation is small and the correlation plot in this instance fades exponentially. AR,
MA and ARMA models generally tend to produce such exponentially decaying behavior.
Flicker noise with exponent one (1/f noise) exhibits what is termed long memory. In this
case, the correlation plot decays at a rate that is slower than exponential and for the case
of ideal 1/f noise, it never decays completely. Examples of slow decay rates are polynomial
rates and logarithmic rates. ARIMA time series models can be made to have slower decay
rates than the exponential rate but this comes at a cost. They typically require a large
number of parameters in order to get a “sufficiently slow” rate of decay and this number
tends to infinity as the stochastic process approaches ideal 1/f noise. Regardless, ARIMA
models have been very popular and are widely used in physics, economics and engineering.
The mathematical theory of chaos is associated with nonlinear dynamical systems
and in its simplest form, it uses nonlinear 1-D iterative functions or maps that are characterized by a very small number of parameters, typically less than four. Nonlinear dynamics
is a theory that essentially watches these nonlinear systems evolve with time. It has been
observed that it is relatively straightforward to be able to produce rich and complex dynamical behavior from a very simple set of underlying rules that repeat in an iterative
fashion. Different sets of behavior can be obtained with different values of parameters and
initial conditions of the map. One can imagine this arrangement as being akin to a feedback network where the processing starts at some value of initial condition and the output
that is processed at the current point in time serves as the input at the next instant of
time. The chaos phenomenon is generally associated with what is popularly called “sensitive dependence on initial conditions” (SDIC) and although an accepted definition is more
wide-ranging (see Chapter 4) SDIC is a convenient way to understand the concept. SDIC
is a compact method of saying that no matter how close two unequal initial conditions are,
when passed through a nonlinear chaotic network the corresponding outputs will continue to
diverge away from each other. Viewing output sequences that result from chaotic networks
5
(examples are provided in Chapter 4) provides a feel for the complexity that is possible by
using basic deterministic rules in a repetitive fashion. The single most important advantage
of chaotic maps is that one can produce a large array of stochastic behavior by using a
small number of parameters. This is a frugal approach to modelling complicated behavior
and the economy in the number of parameters allows for easier parameter tweaking in order
to accurately approximate the desired real-life characteristic. As it turns out, there is a
certain class of maps called intermittent maps, which under the right conditions can produce stochastic behavior that closely match the behavior of 1/f noise. The intermittency
phenomenon is essentially the presence of bursty behavior in the output sequence and is
characterized by alternating periods of low activity and high activity. The length of these
periods is random and if allowed to run for a long time, an intermittent sequence can be
shown to possess long memory - the essential requirement of 1/f noise. An example of
such a map is Holland’s logarithmic map [137] wherein the memory characteristics have a
logarithmic decay and it is controlled by a single parameter. The features of this map and
the requirements for any intermittent map to display long memory properties is explained
in detail in Chapter 4.
1.3
Establishing a Framework
Traditional nonlinear circuit simulation is performed in the time domain and it usually
requires the setting up of integro-differential equations using a combination of KCL and
KVL that suitably describes the circuit under consideration. This system of equations
describing the circuit models is then discretized in time and integrated numerically and
there exist a large collection of algorithms to determine the solution of this system at the
next step in time based on information present at the current step. This is the approach
used in the circuit simulator SPICE and has proved to be a numerically robust approach
to circuit simulation.
This approach however, is not without its limitations. The device models are
intrinsically tied to the routines that analyze the circuits they are included in. This means
that making even a minor change to an analysis routine would require a corresponding
change in every device supported in the simulator. With a large number of devices in
the simulator catalog and with increasing device complexity, this process tends to become
6
cumbersome and error-prone. The circuit simulator f REEDATM is one of the few circuit
simulators that uses a modular approach to modelling and simulation. It relies heavily on
Object-Oriented (OO) concepts which enables it to form a clean separation between data
and algorithm. In other words, the device models are no longer a part of the simulator
analysis routine and both analysis and model exist as individual entities. These entities are
tied together by an abstract framework that makes use of several key OO principles such
as container classes, encapsulation and overloading as is detailed in [2].
The establishment of this framework is an important building block to being able
to incorporate sources of noise into the simulator. The separation of element and analysis
facilitates rapid incorporation and testing of models for noise and allows the model designer
to focus almost exclusively on the modelling process. Noise sources can either be treated
as separate voltage or current sources in which case they serve as individual elements in a
circuit or they can be embedded inside a pre-existing noiseless model. In either case, the
basic solution algorithm requires no updates and changes and insertion of these stochastic
elements transforms the ordinary differential system of equations to a stochastic differential
system of equations. It can be argued that modelling any real-life phenomenon must include
some unpredictability and with the appropriate framework for simulation in place, the
majority of the intellectual focus can be exerted on handling the intricacies related to
the development, deployment and performance of the underlying stochastic models. This
framework flexibility permits a parallel or simultaneous simulation of both deterministic
and stochastic phenomena while preserving the advantages of having an unmodified list of
recipes, or solution routines, to solve this enhanced system of equations. On account of
these changes, the simulator must now be able to track a larger variation of signals at each
node in the circuit and therefore must have a high dynamic range. This was shown to be
true for f REEDATM in [147].
1.4
Circuit Simulator Noise Analysis Review
This section presents a qualitative review on the popular techniques of simulating noise in
circuit simulators. The essential assumptions and methodologies behind each technique is
explained and their strengths and weaknesses are highlighted.
The LTI approach to simulating noise in circuits is a frequency-domain approach.
7
It constructs small-signal equivalent networks for the devices in the circuit and their sources
of noise. With this method, a nonlinear circuit is assumed to have a time-invariant steady
state operating point which is maintained for the entire duration of the analysis. The nonlinear circuit is then linearized about this operating point and an LTI transfer function
describing the circuit is determined. The noise sources in the network are assumed uncorrelated and the effect of each of these sources on the output is individually computed
for each output frequency of interest. The noise analysis is nothing but a computation of
the transfer functions between the input noise sources and the output for the frequencies
of interest. This method relies on the theory of interreciprocal adjoint networks and was
introduced in [102]. Precise details of this technique are provided in Section 2.7.
A common technique of analyzing noise in circuits in the frequency domain with
a periodically varying operating point is the Large Signal Small Signal (LSSS) technique or
the conversion matrix technique, [87]. Here the nonlinear circuit is assumed to be driven
by a large sinusoidal signal and another smaller signal. The response of the circuit to this
smaller signal is desired. The circuit is first analyzed assuming the large signal input only,
usually by the harmonic balance method. The nonlinear elements are then linearized about
the steady-state solution, small-signal time-varying equivalents of the circuit are obtained
and a small-signal analysis is performed. The frequency domain state variables in a timevarying circuit element are determined by finding the so-called conversion matrix for the
network. The conversion matrix converts the input frequency into corresponding spectral
lines or sidebands at the output thereby providing a way to determine the contribution of
each input signal component on the output spectrum. If the small signal is assumed to
be noise, then the conversion matrix technique can be used to determine the contribution
of each noise component on the output spectrum, [88]. This technique, however, requires
the number of output sidebands to be fixed and increasing numbers of sidebands sends
the system closer to instability, [90]. In RF circuits, there are typically a large number of
nonlinear elements which are represented by complex semiconductor device equations. In
this case, harmonic balance simulation techniques can become unreliable, [91].
There are several techniques for simulating noise in the time domain and the underlying motivation for all of them is to be able to simulate circuits that are not operating
under LTI conditions. Instead, the circuits are assumed to be operating under Linear Periodically Time-Varying (LPTV) conditions and it has been found that it is possible to
predict the response of time-variant circuits such as electrical oscillators more accurately
8
with this procedure. Using this procedure, a nonlinear circuit is assumed forced with periodic excitations and to have a periodic steady state solution. This solution can be obtained
by different techniques, [92]. The nonlinear circuit is then linearized about this periodic
solution and a time-varying transfer function of the circuit is set up in order to enable the
computation of the output variables. This procedure assumes that the noise processes are
cyclostationary – in other words, the moments of the stochastic processes describing noise
are periodic. With these assumptions, [93] formed discretized versions of the time-varying
transfer function, associated a fixed number of time points with each output frequency of
interest and approximated the derivatives numerically. As compared with the LTI case in
which the time-invariant system of equations must be solved only once at each frequency
of interest, the time-varying system of equations must now be solved for every time point
corresponding to each output frequency. Speed and problem size optimizations associated
with this technique have been shown to produce better results, [94]. The disadvantages of
these methods is that it requires the signal and noise to be periodic in nature and it is only
feasible to calculate the first-order moment of the cyclostationary output. Evaluation of
higher-order moments is prohibitive. In [95], a harmonic-balance based frequency-domain
method was introduced which calculates not just the first but also the second-order moment
of the cyclostationary output. The method calculates the steady state output with periodic
excitations using harmonic balance and the LPTV system of equations is obtained after
linearization of this steady-state solution. The Fourier coefficients of the periodically timevarying transfer function from the noise sources to the output of the system are calculated
by assuming a fixed number of harmonics and solving the linear system of equations with
preconditioned iterative techniques. This technique is similar to the LSSS noise analysis
except that the authors focus on solving the problem with numerically efficient techniques.
The approach used in [96] can be thought of the time-domain version of the approach in [95] with the difference being that it can handle a more general Linear TimeVarying (LTV) network configuration. The authors numerically calculate the mean and
autocorrelation functions, which are time-domain characterizations, of the output variables
of the circuit thereby obtaining a second-order characterization of the random processes
that constitute the variables in the circuit. The equations describing the nonlinear circuit
are set up in the time domain and are treated as SDEs but the sources of noise are assumed
to have an additive effect only and multiplicative effects of noise and state-dependent circuit
variables are not considered. Performing a Fourier transform on the autocorrelation of these
9
output variables provides, in general, a time-varying power spectral density.
The author in [101] focuses on the problem of phase noise in oscillators and treats
the equation for an oscillator in the time domain with sources of noise. The equation
is linearized in the time domain and the transversal and tangential components of the
limit cycle corresponding to the periodic solution of the nonlinear differential equation
are computed. The transversal component contributes amplitude noise and the tangential
component contributes phase noise. A transient transfer function is then set up and the
contribution of these noise components on the output is determined. The technique used in
[97] uses a similar approach of obtaining the phase noise of a linearized oscillator equation
but uses an Impulse Sensitivity Function (ISF). This function aids in the calculation of
the contribution of the phase noise at the output of the system from each source of noise
present in the circuit. The authors in [97] show how to derive the ISF for a simple oscillator
circuit but the ISF will differ for different circuit configurations and it is not clear how
this approach can be extended to any configuration. For both methods in [97] and [101],
the approach of considering that the effect of noise sources on the output phase noise can
be decomposed into orthogonal components is shown to be inadequate, [98], in the sense
that they do not account for the fact that phase deviations can grow indefinitely with time
and that amplitude deviations cause phase deviations. A similar argument can be made
against the technique proposed in [104] which generates pseudo-random time domain noise
generators and uses them to generate models for circuits elements. The noise sources are of
the form of a sum of a fixed number of sinusoids which have random amplitude and phase
components, each of which is a coefficient that must be determined by empirical means.
Once the coefficients have been determined, a traditional transient analysis is performed
on the circuit. It is also unclear as to how many components for the amplitude and phase
coefficients must be selected and what their respective values should be.
The work in [98] is based on the work in [101] but improves the modelling descriptions of the effects of perturbation on a steady-state periodic circuit. In particular, the
perturbed oscillator state can have amplitude and phase deviations and while the amplitude deviations are always assumed small, the phase deviations can grow without bound. As
before, there is still a separation between the amplitude and phase components. However,
while amplitude can still be handled using linear perturbative analysis, the phase deviations
are represented by a nonlinear differential equation that cannot, in general, be linearized.
The authors show that the phase deviation increases linearly with time and develop numer-
10
ical techniques to solve for such conditions. This work can be seen as bringing together the
techniques in [95] and [96] and extending it to include linearly increasing phase error. An
recent analytical review of all the time and frequency-based approaches to treating phase
noise is provided in [106].
There is tremendous improvement in the understanding of the effects of noise in
circuits but some gaps still remain. Most the techniques highlighted above apply to the
phase noise problem in periodic circuits where the noise is assumed to be small in the sense
that it does not affect the steady state operation (either time-invariant or time-variant) of
the circuit. Also, effects of multiplication of noise with random state-dependent variables
are neglected. This can be an important consideration because an instantaneous value of
noise, as in the case of flicker noise, can be large enough such that when multiplied with a
deterministic signal, a shift in the bias point of the circuit can result. Another issue is the
modelling of sources of colored noise in a circuit simulator. Several techniques for modelling
these sources have been developed in theory and in practical circuits (see Section 2.6 for
a more detailed exposition) but implementing them as sources of colored noise in a circuit
simulator can be a challenge. An example of the difficulty is quantified in Appendix B.
1.5
Original Contributions
This work represents a first attempt at using chaotic maps to model colored noise in electronic circuits. The chaotic noise generators are shown to be able to produce spectral
characteristics that are very similar to white and 1/f noise and it requires the tuning of
only a single parameter. This is a parsimonious approach to modelling complex stochastic
behavior and as is shown in the later chapters of this work, this approach works quite well
when applied to real circuits. The descriptions and properties of the white noise generating
map are provided in Section 4.3 and for the flicker noise generating map in Section 4.4. The
procedure for obtaining the noise sequences are based on iterative deterministic rules and
the SDIC property of chaotic sequences ensures that two chaotic sequences starting at initial
values that differ by arbitrarily small amounts will be different from each other at nearly
every point. The use of this technique enables one to effectively harness the accuracy of
the transient simulation approach and avoid the problems associated with random number
generators in a Monte Carlo type simulation.
11
Another contribution of this work is the implementation of a transient noise analysis in a full-fledged electronic circuit simulator. It requires the extension of the robust
simulation framework found in f REEDATM to allow for a concurrent deterministic and
stochastic analysis of circuits that consist of transient sources of colored noise, details of
which can be found in Section 5.4. It neither imposes a restriction on the nature of the
noise itself nor does it assume that the noise will have a specific form of interaction with
the deterministic elements. In other words, multiplicative effects of noise and signal are
considered in addition to additive effects and the fact that instantaneous magnitudes of the
noise components when multiplied with signal can be large enough to cause changes in the
bias point of the circuit is considered. In addition, this work interprets the system of SDEs
in the sense of Stratonovich [115], and it is argued in Section 3.5 that this interpretation
is the only appropriate one when multiplicative effects are to be modelled appropriately.
Previous attempts at using SDEs to model noise, [96], [99], [100], [101], all make use of the
Itó assumption. The Stratonovich approach is found to be accurate when modelling circuits that have significant components of flicker noise and non-negligible levels of nonlinear
interaction between the deterministic and stochastic terms. The intention is to develop and
make use of a modelling process that approaches the workings of a physical circuit.
Chapter 6 presents comparisons of simulation of phase noise of a circuit that
contains large signal swings at several nodes of the circuit. Different levels of bias conditions
are considered. The circuit is a varactor-tuned voltage-controlled oscillator and contains
active nonlinear elements. It is shown that once the flicker noise parameters for the active
devices are set for one level of bias, the same parameters can be used to predict the phase
noise for other bias levels. A match is also obtained between between simulation and
measurement for a higher level of bias that causes to the varactor in the circuit to approach
the breakdown condition. For this match, it is shown that a scaling coefficient larger than
unity is required for the shot noise of the varactor.
Chapter 7 contains simulation runs that investigates the effect of large levels of
input noise on the power gain of an X-band high power low-noise amplifier. The simulation
approach developed in this work is found to be well-suited to analyze such circuits as
indicated by a comparison of simulation runs with measured results. This chapter also
presents, for the first time, measured results that indicate a reduction in power gain in the
presence of large levels of noise. These levels of noise are generated digitally at baseband
and upconverted to the carrier frequency at X-band in the signal generator. With further
12
simulations that include appropriate values of time delays inside the device models, it is
demonstrated that the reduction in power gain in the presence of noise can be approximated
more accurately.
1.6
Overview
Chapter 2 is a concise review of a large body of literature on noise in systems. The attempts of various researchers to provide physical explanations for the noise phenomenon
has been explored and a brief overview about these efforts is provided. Also reviewed are
the approaches to modelling noise and in particular, flicker noise. The chapter ends with a
mathematical explanation of how noise is traditionally analyzed in simulators.
Chapter 3 is an introduction to the theory of SDEs and deals with common forms
of modelling with and interpreting SDEs. In particular, the chapter explains the origins of
the Itô form and the modified Stratonovich form for interpreting SDEs. It illustrates the
differences in results that can occur when using the two forms to solve the same differential
equation and also suggests a solution to this quandary by providing an intuitive look at
some of the assumptions behind using each form of SDE.
Chapter 4 highlights the theory of nonlinear dynamics and chaos by first providing
the appropriate background to the theory and embellishing the theory with small doses of
intuition and graphics. The chapter ends with explaining how a certain property of chaotic
functions, namely intermittency, can be used to generate noisy sequences with spectral
characteristics that resemble colored noise as found in electronic circuits and elsewhere.
Chapter 5 provides details of the implementation of the parallel deterministic and
stochastic framework to analyze transient noise in the circuit simulator f REEDATM . It
provides details of the changes that were made to the linear and nonlinear device elements
in f REEDATM to include white and flicker sources of noise and effects of these changes on
the existing simulator framework.
Chapter 6 simulates a varactor-tuned voltage-controlled oscillator with a resonant
frequency of 45 MHz. Simulated phase noise is compared with measured results at different levels of bias conditions in an effort to verify the validity of this modelling approach.
Flicker noise parameters for the nonlinear devices are determined by fitting simulation and
measurement for one value of bias and using the same values, it is shown that it is possible
13
to predict the phase noise response at another bias level.
Chapter 7 presents the modelling of the effect on gain of a high-power low-noise
MMIC amplifier when fed in with a composite signal consisting of a carrier at 10 GHz and
large levels of noise. Curves of gain versus input power are generated and simulated results
are compared with results obtained from measurement. The MMIC amplifier consists of
a pair of pseudomorphic-High-Electron-Mobility-Transistors (pHEMTs) connected back to
back and is modelled in simulation using the Curtice model [148] for a MESFET. In order to
model the distortive effects of noise at high levels of input power more accurately, the delayenabled Curtice model is used in simulation and preliminary comparisons with measurement
are presented.
Finally Chapter 8 summarizes this work and provides suggestions for future research in this interesting field of nonlinear circuit analysis and computer-aided design.
1.7
Publications
1. N. M. Kriplani, S. R. Luniya and M. B. Steer, “Capturing the Effect of Noise on
Amplification,” Microwave Comp. Lett., submitted, Nov. 2005.
2. N. M. Kriplani, D. P. Nackashi, C. J. Amsinck, N. H. Di Spigna, M. B Steer, P. D.
Franzon, R. L. Rick, G. C. Solomon and J. R. Reimers, “Physics-Based Molecular
Device Model in a Transient Circuit Simulator,” Jnl. Chem. Phys., submitted Sep.
2005.
3. Frank P. Hart, Nikhil M. Kriplani, Sonali R. Luniya, Carlos E. Christoffersen and
Michael B. Steer, “Streamlined Circuit Model Development with f REEDATM and
ADOL-C,” The 4th International Conference on Automatic Differentiation, July 2004.
4. M. B. Steer, C. Christoffersen, S. Velu and N. Kriplani, “Global Modeling of RF and
Microwave Circuits,” Mediterranean Microwave Conference Digest, June 2002.
14
Chapter 2
Literature Review
2.1
Introduction
This chapter is a survey of the vast amounts of literature available on the history of the
understanding of noise and noise processes. This chapter is by no means comprehensive but
it makes an attempt to mention a fair number of references on the subject ranging from the
historical origins of noise to some more recent approaches to noise modelling. Section 2.2
introduces mathematical definitions and formulas related to random variables and random
processes that will be helpful in reading the rest of the chapter. Section 2.3 is a brief
introduction to thermal noise based on thermodynamical principles followed by Section 2.4
which is an introduction to shot noise. Section 2.5 and Section 2.6 respectively survey
theories of the physical origin of flicker noise and various mathematical models to describe
flicker noise processes. Finally, Section 2.7 reviews the original method of analyzing noise
in electronic circuit simulators.
2.2
Mathematical Preliminaries
A random process is a family of random variables {X(t), t ∈ T } defined on a given probability space indexed by the parameter t, denoting time, where t varies over the index set T .
The Cumulative Distribution Function (CDF) for a fixed time t1 of this random process is
15
defined as
FX (x1 ; t1 ) = P {X(t1 ) ≤ x1 }
(2.1)
where P (.) denotes the probability function from the sample space Ω into the unit interval
on the real line. FX (x1 ; t1 ) forms the first-order distribution of X(t). Likewise, given t1
and t2 , the joint CDF or the second-order distribution of the random process is given by
FX (x1 , x2 ; t1 , t2 ) = P {X(t1 ) ≤ x1 , X(t2 ) ≤ x2 }.
(2.2)
In general, for an n-th order distribution,
FX (x1 , . . . , xn ; t1 , . . . , tn ) = P {X(t1 ) ≤ x1 , . . . , X(tn ) ≤ xn }.
(2.3)
The Probability Mass Function (PMF) for a discrete random process is given by
pX (x1 , . . . , xn ; t1 , . . . , tn ) = P {X(t1 ) = x1 , . . . , X(tn ) = xn }
(2.4)
and the Probability Distribution Function (PDF) for a continuous random process is expressed in terms of its CDF as
∂ n FX (x1 , . . . , xn ; t1 , . . . , tn )
.
∂x1 . . . ∂xn
fX (x1 , . . . , xn ; t1 , . . . , tn ) =
The mean of a random process or its first-order moment is defined by

 P∞ x(t)p (x, t) discrete random variable
X
−∞
µX (t) = E[X(t)] =
R
 ∞ x(t)pX (x, t)dx continuous random variable
−∞
(2.5)
(2.6)
provided the sum and the integrand exist1 . Likewise, the nth-order moment is defined as

 P∞ xn (t)p (x, t) discrete random variable
X
−∞
n
(2.7)
E[X (t)] =
R
 ∞ xn (t)pX (x, t)dx continuous random variable
−∞
The autocorrelation function is a method to determine the relationship between values of
a function separated by different instants of time. For a random variable, it is given by
RX (t, s) = E[X(t)X(s)].
(2.8)
When the random variable is discrete, the one-dimensional autocorrelation function of a
random sequence of length N is expressed as
RX (i) =
N
−1
X
j=0
1
are absolutely summable
xj xj+i
(2.9)
16
where i is the so-called lag parameter. It is a way of expressing the relationship between
values of the random sequence for different values of the lag parameter. The autocovariance
function of X(t) is given by
KX (t, s) = E[{X(t) − µX (t)}]E[{X(t) − µX (t)}]
= RX (t, s) − µX (t)µX (s)
(2.10)
while the variance or dispersion is given by
σX (t) = E[{X(t) − µX (t)}]2 = KX (t, t).
(2.11)
A random process X(t) is stationary in the strict sense if
FX (x1 , . . . , xn ; t1 , . . . , tn ) = FX (x1 , . . . , xn ; t1 + τ, . . . , tn + τ )
(2.12)
∀ti ∈ T , i ∈ N. If the random process is Wide-Sense Stationary (WSS), then it is stationary
with order 2. This means that in most cases only its first and second moments, i.e. the
mean and autocorrelation, are independent of time. More precisely, we have
E[X(t)] = µ
RX (t, s) = E[X(t)X(s)]
= RX (|s − t|)
= RX (τ ).
(2.13)
Note that the mean is constant for a WSS process and the autocorrelation is dependent only
the time difference τ . A random process that is not stationary to any order is nonstationary
and its moments are explicitly dependent on time.
A Gaussian random process is a continuous random process with PDF of the form
µ
¶
1
−µ(x, t)2
fX (x, t) = p
exp
(2.14)
2σ(t)2
2πσ(t)2
where µ(x, t) represents the mean of the random process and σ(t) represents the variance.
A normal random process is a special case of a Gaussian random process in that it has
a mean of zero and a variance of unity. A Poisson random process is a discrete random
process with parameter λ(t) > 0 has a PMF given by
pX (k) = P (X(t) = k) = eλt
(λt)k
k!
(2.15)
17
where λ(t) is generally time dependant and the mean and variance of a Poisson random
process is λ(t). So, for a Poisson random process,
µX
= E[X(t)] = λ(t)
(2.16)
2
σX
= Var(X(t)) = λ(t).
(2.17)
A concise introduction to random variables and processes can be found in in [6]. For a
classical and near-complete treatment, refer to [7].
2.3
Thermal Noise
According to the theorem of Nyquist, thermal noise is a result of the random motion of free
electrons in a conductor which are in a state of constant thermal agitation at temperature
T . These random fluctuations result in a random current and a random voltage across the
terminals of the conductor shown schematically as in Fig. 2.1. The Power Spectral Density
R noisy
R noiseless
R noisy
Vn (t)
R noiseless
I (t)
n
Figure 2.1: Thermal noise equivalent circuits.
(PSD) of Vn (t) and In (t) are related by
SVn (f ) = R2 SIn (f ).
(2.18)
One can derive the PSD of the voltage spectrum using the principles of thermodynamics in
accordance with the developments of Nyquist in 1928 [3], which is outlined below.
18
Consider a pair of resistances R1 and R2 whose values are independent of frequency
and are in equilibrium at temperature T . They are arranged as in the circuit shown in
Fig. 2.2 and thermal noise voltages and currents are indicated therein. Assuming the self-
R (noisy)
1
R (noisy)
2
R (noiseless)
1
i1
R (noiseless)
2
i2
V (t)
n1
V (t)
n2
P12
P21
Figure 2.2: Thermal noise voltages and currents in equilibrium at T0 .
inductance and capacitance to be negligible at all frequencies, the equilibrium state of the
two resistors allows one to invoke the second law of thermodynamics which requires that
there is no net exchange of energy between the two resistors. Hence, for an arbitrary
frequency range (f, f + ∆f ) the power transferred from R1 to R2 must be the same as that
transferred from R2 to R1 independent of the location of (f, f + ∆f ). Therefore, the mean
power P 12 = P 21 where P ij is the mean power transferred from Ri to Rj for bandwidth
∆f . For voltages v1 , v2 and currents i1 , i2 for the circuit in the bandwidth ∆f then we can
write
i1 =
i2 =
v1
R1 + R2
v2
R1 + R2
P 12 = i21 R2
P 21 = i22 R1 .
(2.19)
Since P 12 = P 21 , v12 and v22 can be written in terms of the voltage spectra Sv1 (f ), Sv2 (f )
giving
Sv1 (f )R2 = Sv2 (f )R1 .
(2.20)
Setting R1 = R2 shows that the PSD would have to be independent of the specifics of the
19
resistors and the method of conduction. What remains is to find the PSD as a function
of voltage, Sv (f ). On coupling two equal resistances R together with the help of an ideal
lossless transmission line of characteristic impedance Z0 = R such that the line is matched
and reflectionless at either end, a certain mean amount of electromagnetic energy would
be stored in the line over the bandwidth ∆f in the equilibrium state. This mean energy
U = ∆k(UH + UE ), where UH is the mean magnetic energy per mode and UE is the
mean electric energy per mode and ∆k = 2L∆f /v is the number of modes in the frequency
range ∆f for a line L cm long and velocity v of propagation. Invocation of the Equipartition
Theorem permits one to assign energy of amount kB T /2 to the electric and magnetic energy
per mode. This gives UH + UE = kB T and so
U = ∆k(UH + UE ) = 2kB T L∆f /v
(2.21)
which is the one dimensional case of the Rayleigh-Jeans law for black-body radiation. Since
half the energy goes from R1 to R2 and vice-versa, the mean power (energy per second)
from R1 into R2 is
1
P 12 = U/(L/v) = ∆f kB T
2
(2.22)
with an identical expression for P 21 . From Eqn. (2.19) we have
v2
P 12
1
v12 R2
Sv (f )R2 ∆f
∆f R2 ∆f
= kB T ∆f =
=
= 1
(R1 + R2 )2
(R1 + R2 )2
(R1 + R2 )2
(2.23)
and if R1 = R2 = R, then we immediately obtain
Sv (f ) = 4kB T R.
(2.24)
The PSD of the current from Eqn. (2.18) is
SI (f ) =
4kB T
.
R
(2.25)
This result is an ideal one as it is valid for all frequencies, in other words is frequency
independent. However, it would also result in infinite power if all the frequencies were
considered. There must therefore be a limit in frequency beyond which this equation is
not valid. To do this, this expression is replaced by Planck’s expression hf (ehf /kB T − 1)−1
which gives
Sv (f ) =
4Rhf
.
ehf /kB T − 1
(2.26)
20
The critical frequency at which the spectrum will roll-off is determined by hf0 = kB T ⇒
f0 = kB T /h = 2.1 × 1010 Hz. At room temperature this is roughly 6 × 1012 Hz. Nyquist’s
theorem applies not only to resistors but to general linear passive elements. For a more
thorough examination of this, refer to Nyquist’s original work [3]. For more recent revisits
of these concepts, see [4] and [5].
2.4
Shot Noise
Shot noise was first observed by Schottky in 1926 [8] in vacuum tubes. Shot noise was the
effect of the electrons incident on the anode which occurred at random intervals where the
incidence of each electron on the anode is an individual event. A random superposition
of these individual events which are non-overlapping, forms shot noise. In a conventional
semiconductor p-n junction, the heavily doped p and n regions make a very small thermal
noise contribution and it is the emission of carriers into the depletion region that contributes
to shot noise. Every carrier that crosses the depletion region generates a pulse. This arrival
rate of pulses can be modelled by a Poisson arrival process of rate λ. At a specific time
instant t, the total charge that has crossed the depletion layer is
Q(t) = qN (t)
(2.27)
where q is the charge associated with each carrier and N (t) is the number of carriers that
have crossed until time t. The time derivative of this equation provides the expression for
current which can also be represented as a superposition of charges arriving at randomly
distributed intervals expressed as
I(t) =
X
d
Q(t) =
qδ(t − Ti )
dt
(2.28)
i
where Ti represents the random crossing interval. Assuming that the length of the Ti ’s
have a Poisson distribution, the pulse train I(t) is a Poisson pulse train. Note that the
implicit assumption is that the charge crossing is instantaneous and that successive pulses
are non-overlapping. In a more general case, one can imagine passing a pulse train through
a system having an impulse function h(t) which depends on the device under consideration.
This gives
I(t) =
X
i
qh(t − Ti ).
(2.29)
21
The mean of the pulse train is
Z
Z
∞
I(t) = E[(t)] = q
λh(t − s)ds = 2qλ
−∞
∞
h(t − s)ds = 2qλ
(2.30)
0
and defining the noise current to be the difference between the total current and the mean,
we get
˜ = I(t) − I(t).
I(t)
(2.31)
The mean of this noise current is obviously zero and its autocorrelation can be computed
as
˜ + τ /2)I(t
˜ − τ /2)]
RI˜(t, τ ) = E[I(t
Z ∞
= 4q 2 λ
h(t + τ /2 − s)h(t − τ /2 − s)ds
Z0 ∞
= 4q 2 λ
h(s + τ /2)h(s − τ /2)ds
0
= RI˜(τ )
(2.32)
which shows that shot noise is a WSS process. To calculate the PSD, we use the standard
form
SI˜(f ) = 4λq 2 |H(f )|2 .
(2.33)
From Eqn. (2.30), we have
λ=
I
2q
(2.34)
which gives for the PSD
˜
SI˜(f ) = 2q I|H(f
)|2
(2.35)
which is an expression for shot noise as a function of the mean current across the barrier. If
one considers the pulses across the barrier to be ideal Dirac delta functions, then |H(f )|2 = 1
and the PSD becomes
SI˜(f ) = 2q I˜
(2.36)
which is the PSD of a white noise process and is independent of frequency. This derivation
can be enhanced by assuming a time-varying rate for the Poisson process λ(t) and that
successive pulses have a degree of overlap, but in all cases it can be shown that the shot
noise process has a white PSD and is Gaussian [75].
22
2.5
1/f (Flicker) Noise
One way of judging the importance and impact of 1/f noise is by considering the numerous
references on the subject and the vast number of situations in which this phenomenon
manifests itself. Apart from confirming the frequency independent nature of shot noise
originally proposed by W. Schottky, J. B. Johnson observed another type of noise whose PSD
increased with decreasing frequency. Schottky [8] suggested that this effect is independent
of the shot noise effect and is a consequence of the irregularities in the properties of the
surface of the cathode which results in “flicker” of the thermionic current. Since then, this
type of low-frequency noise has been observed in a variety of engineering and scientific
situations and perhaps the only thing that is universally agreed upon, is the ubiquity of
the phenomenon of 1/f noise. The statistics of white noise (to be shown later), as seen
in its autocorrelation function, indicate a process that has no memory. On a correlation
plot, a plot which shows the degree of correlation between samples of a time-sequence,
future values of a memoryless process are completely independent of the past values. In
the frequency domain, this implies a PSD that is independent of frequency. Although no
physical process is truly white for an infinite number of frequencies, white noise is “white”
for all practical purposes. The slope of the PSD of white noise is zero and can also be
referred to as 1/f 0 noise. On the other hand, brown noise is associated with a Brownian
Motion Process (BM) defined later, and like white noise has very little long term memory.
Although there is a high correlation between successive samples of a BM, the correlation
plot decays exponentially indicating negligible long term memory properties. The PSD of
brown noise has a slope of two and is also referred to as 1/f 2 noise. Lying in the middle of
these extremes is “true” 1/f noise, some properties of which are explored beginning with
the next subsection. For now, it must be emphasized that a true 1/f noise process exhibits
long term memory character and its correlation plot, which in general, never decays. The
remainder of this work will refer to all types of noise that follow a power law as 1/f noise
irrespective of the precise value of the slope of its PSD. In general, any 1/f γ noise with
0 ≤ γ ≤ 2 is 1/f noise.
Examples of the ubiquity of 1/f noise or long memory processes include vacuum
tubes [8], height of the floods of the river Nile [9], analysis of fractal music [10], size distribution in meteorites that annually hit the earth [11], self-organized criticality and sandpile
slides [12], fragmentation processes which show a power-law distribution at a critical value
23
of tuning parameter [13], in the PSD of fluctuations in the audio power of many musical
selections like Bach’s Brandenburg Concertos [14], fluctuations in neuro-membranes [15],
sunspot numbers in a 11 or 22 year period of the solar cycle [16], in the distribution of numbers in continued fraction expansions [17] and in the realizations of nowhere-differentiable
functions like the Weierstrass-Mandelbrot function [18], to name a few, all of which exhibit
some sort of power-law behavior. A large and varied collection of such examples can be
found in [19].
2.5.1
Scale Invariance
A 1/f spectrum, as mentioned above, implies a more general form, i.e. 1/f γ where exponent
γ typically lies between 0 and 2. This makes white noise (γ = 0) and Brownian motion or
brown noise (γ = 2) subsets of the general set of 1/f γ type noises. However, a true 1/f
spectrum is one in which the exponent γ = 1 and it is characterized by the PSD
SX (f ) = c/f
(2.37)
where c is independent of frequency. Integrating Eqn. (2.37) gives power that tends to
infinity as frequency tends to zero. There is a fair amount of discussion in the literature
about the existence of the low frequency limit of this noise and it would seem likely that
there should be a plateau in the graph of a measured 1/f process. However, measurements
on operational amplifiers were conducted down to 0.5 µHz in [20] which corresponds to
one cycle in 3 weeks, but no leveling of the spectrum was found. A variance analysis of a
1/f noise source was carried out in [21] down to 3.3µHz on a pair of noisy carbon resistors
inserted in a Wheatstone bridge arrangement. The spectrum of the noise source was still
found to be approximately 1/f and the variance was found to increase logarithmically. These
results extended similar variance analysis results reported in [22] to lower frequencies.
A high frequency limit for 1/f noise depends on its intensity because at higher
frequencies, 1/f noise intensity eventually becomes lower than the thermal and shot noise
intensities which represent a lower limit of the noise present in the system. For both low and
high frequency limits, there exist plausible physical explanations as to why one should not
expect infinite noise power i.e. some form of low pass filtering in the system sets the high
frequency limit and discovery of the low frequency limit requires allocating large amounts of
time per measurement. In general it is safe to assume that there exists a reasonable upper
24
and lower bound for frequencies. Thus the integrated PSD or the power is over some finite
frequency range and is given by
1
PX (f ) =
2π
Z
f2
f1
SX (f )df.
(2.38)
An interesting perspective on this question of low-frequency and high-frequency limits of
1/f noise is provided in [23], which is based on the total number of decades of frequency
that could conceivably exist in a 1/f spectrum. The lower limit is 10−17 Hz and is based
on the age of the universe and the upper limit is 1023 Hz which corresponds to the time it
takes for light to travel the classical radius of the electron. This gives a span of around
40 decades which is currently beyond the range of any measurement equipment. The noise
level of 1/f noise, even at extremely low frequencies, is quite low and even in the unlikely
case of the 40 decade span, the noise power will be insignificant compared to the DC power
used in the biasing of a general circuit.
2.5.2
Stationarity and Gaussianity
The answer to whether 1/f noise is stationary or nonstationary is strongly related to
whether there exists a lower limit to the frequency at which the noise process exhibits
a 1/f behavior. It is generally believed that if a lower cutoff does exist, then there will be a
leveling off of the 1/f process as discussed earlier. Experimental evidence provided in [22],
[24] and [25] indicate statistical fluctuations that are unlike those that one would find in
a stationary system. This was, however, followed up by a theoretical analysis of variance
of Gaussian random noise superimposed with a 1/f spectrum and experimental analysis of
1/f noise in carbon resistors and bipolar transistors as reported in [26]. The theory was
developed with an inherent assumption of stationarity and experimental results were found
to match closely with theoretical predictions thereby validating the underlying stationarity
assumption. The authors in [27] take up the same issue and argue that the autocorrelation of a stationary random process is dependent only on time difference as indicated in
Eqn. (2.13). The PSD is defined as the Fourier transform of the autocorrelation function
and is given by
Z
SX (f ) =
∞
−∞
RX (τ ) exp(−j2πf τ )dτ.
(2.39)
25
To determine the asymptotic behavior of SX (f ) as f → 0, one uses Leibniz’s theorem to
take the derivative of Eqn. (2.39) in the limit as
Z ∞
dSX (f )
∂
lim
= lim
[RX (τ ) exp(−j2πf τ )dτ ]
f →0
f →0 −∞ ∂f
df
Z ∞
= lim
−j2πτ [RX (τ ) exp(−j2πf τ )dτ ]
f →0 −∞
Z ∞
= −j
2πτ [RX (τ )dτ ]
−∞
The property of RX (τ ) is that it is necessarily even, i.e.
R∞
−∞ τ RX (τ )dτ
(2.40)
= 0. This means
that at some frequency the derivative dSX (f )/df will vanish which would result in a flattening of the PSD curve. This contradicts the measurements in [20] and [21] as discussed
above. This leads to the suggestion that 1/f noise is the result of a nonstationary process
and the authors in [27] proceed to derive the time-dependent mean and autocorrelation
functions for a general nonstationary stochastic process. Perhaps the best way to resolve
this is to accept the analysis of Keshner in [28] wherein he shows that the nonstationary
autocorrelation function produced by his 1/f noise model can be written down as the sum
of two terms: one dependent on time and the other dependent only on time difference τ .
If the underlying assumption is that the time of observation is much shorter than the total elapsed time since the process began, then the correlation function can be considered
“almost stationary”.
Another important question is whether 1/f noise is a Gaussian process. A large
number of random processes are Gaussian or are assumed Gaussian in some limiting sense.
Gaussian random processes require only up to second-order statistics (mean and autocorrelation) for complete characterization. For non-Gaussian random processes, one needs to
take arbitrarily higher-order moments for a complete characterization of the process. The
author in [29] investigated the amplitude distribution of 1/f noise from a carbon resistor
in the frequency range 8 Hz to 10 kHz of various sample lengths and found they all exhibited Gaussian behavior. His analysis also showed that the variances of the sample sets
taken at different times showed an exponential distribution and that the 1/f noise process
displayed a weaker form of stationarity. A more comprehensive treatment can be found in
[30] wherein the author performed measurements on five different sources of 1/f noise: A.
Current noise near threshold in a MOSFET, B. A 1MΩ carbon resistor, C. Current noise in
the base-collector junction of a reverse-biased n-p-n Si transistor, D. Voltage fluctuation at
the output of a common-emitter n-p-n transistor and E. Current noise in a reverse-biased
26
p-n diode. The PSD in the range 0.03 Hz to 5 kHz were determined and were found to all
be of the 1/f type. In the case of sources A, B and C, the probability density functions
were found to be Gaussian in nature. Source D showed some deviation from a Gaussian
structure while source E was significantly different, perhaps due to the differences in the
mechanism generating the noise in these five cases. Linearity of the noise waveforms produced was measured in the following way: A string of 2N + 1 noise values were digitized
thereby producing a sequence Xn , −N ≤ n ≤ N . The digitized values were scaled by a
fixed amount to produce vn in some range about zero. The time behavior of this subset
was averaged over the entire ensemble, < vn |v0 = V0 >, where <> indicates averaging.
Note that this quantity, < vn |v0 = V0 >, is related to the autocorrelation and hence for a
linear system, < vn |v0 = V0 > should be independent of v0 while for a nonlinear process,
it should be dependent on v0 . It was found that the linearity degraded while going from
source A to E which led to the interpretation: The more Gaussian a noise source is, the
more independent it is of its starting point, i.e. v0 . This does not necessarily allow one to
deduce the linearity or the nonlinearity of the system as pointed out in [31] wherein it was
noted that the linearity or nonlinearity can only be attributed to the underlying equations
that describe the physical property and not necessarily the physical process itself. A more
recent analysis was made in [32] wherein artificially generated 1/f noise-like waveforms using some elementary nonlinear dynamical systems were considered and it was found that
the probability density functions for all these systems were distinctly non-Gaussian.
2.5.3
The Empiricism of Hooge
As a departure from the theories of the existence of 1/f noise prevailing at the time, the
authors in [42] showed that it is possible to get 1/f noise in metal films, in particular,
thin gold metal films. Based on his experiments and others reported in literature, Hooge
also proposed in [43] that 1/f noise in all homogeneous materials can be represented by an
empirical formula given by
SX (f )
αH
.
=
2
Ro
Ntot f
(2.41)
Ntot represents the total number of carriers in the specimen, Ro represents it’s mean resistance and Sxx (f ) the PSD of the resistance fluctuations. The parameter αH is the
“universal” Hooge parameter and was claimed to be equal to 2 × 10−3 . This equation satisfied, in an empirical way, several cases of resistors showing 1/f noise at room temperature.
27
This parameter has been modified several times since, each time representing a seemingly
smaller subset of materials which exhibit 1/f noise. Discrepancies have since been found by
more than a factor of 10 in the experiments in [40] where an increase in this parameter value
with increase in temperature was found. In [44] and [45], the authors took films made of
the same metal, having roughly the same resistivity and made with the same technique and
found that the parameter varied more than an order of magnitude. Hooge’s hypotheses were
based on the presence of 1/f noise in metals and could not satisfactorily provide values of
parameters for liquids. In an experiment on ionic solutions, the authors in [46], found that
parameter αH varied according to the ionic concentration of the liquid. Although Hooge’s
parameter was a grand attempt in unifying theories of 1/f noise, it is now treated mostly
as a historical anecdote which in itself, is a grand achievement.
2.5.4
Number Fluctuations of Charge and Mobility
Fluctuations in resistance that show a 1/f distribution arise due to a fluctuation in some
number: in either the number of charge carriers or in the mobility of the charge carriers. As
seen in Eqn. (2.41), Hooge’s law suggests an inverse dependence of the noise on the number
of charge carriers. This suggests that a possible mechanism for 1/f noise in continuous
homogeneous metals is a number fluctuation of the carriers. The mechanism of carrier
trapping, in which carriers are trapped in a material for varying finite time intervals, could
plausibly cause a fluctuation in the number of carriers. However, for metals, the number
of traps is very low and the carriers are unlikely to find a large number of traps which
eliminates number fluctuations as a possible mechanism of 1/f noise in metals. This, of
course, is a direct contradiction of Hooge’s law which requires an inverse dependence on the
number of carriers to be true. One can exclude trap mechanisms as a source for fluctuations
in number of carriers in metals. The situation is different for semi-conductors where it is
possible to find a large number of traps which can provide a range of lifetimes to sufficiently
account for 1/f noise. This is the basis of the model proposed by McWhorter, and is
outlined in the next section.
The alternative to fluctuation in the number of carriers is fluctuation in mobility.
In [47], 1/f noise in the open circuit thermo-electro-motive force of both intrinsic germanium
and extrinsic germanium and silicon was observed. Based on the calculations therein, it
was concluded that mobility fluctuations could be the only source of the 1/f fluctuations
28
in the Hall coefficient of the material and would also support Hooge’s law provided the
mobility fluctuation in the carriers are independent. This has been explored further in [48].
However, independent mobility fluctuations associated with individual carriers as a source
of 1/f noise would have to mean that fluctuations in mobility would have to show long
characteristic times: much longer than what is observable which is usually of the order of
picoseconds. This places a cloud of uncertainty over the theory of mobility fluctuations.
2.5.5
Surface Effect or Bulk effect
The conductance near the surface of a semiconductor depends on the magnitude of the
charge near its surface. Fluctuations in this charge will naturally lead to fluctuations in
the value of the conductance. This makes it possible to modulate the conductance of
the surface region of a semiconductor by changing the amount of charge near its surface.
When Hooge proposed his law in [43] to explain 1/f noise in metals, he believed that 1/f
noise had to be an effect of the bulk of the metal. However, papers citing the opposite
appeared, for example, in [49] wherein after measuring 1/f noise on GaAs resistors, it was
concluded that 1/f noise was a surface effect. This argument has been provided credence by
experiments where surface treatments have been found to vary the levels of 1/f noise and it
also correlates the absence of surface effects in a JFET to the low levels of 1/f noise present
there. There still remains speculation about whether 1/f noise is a result of a surface effect
or a bulk effect. In [50] it is suggested that both models are valid and one type of noise
would dominate the other depending on the device under study.
2.5.6
Dependence on Mean Voltage, Current and Resistance
From the time that voltage spectra with 1/f fluctuations were first observed in the presence
of a steady current, the dependence of 1/f noise on the mean voltage or current has been
queried. For a uniform conductor, we know that the voltage spectral density SV (f ) is
proportional to V 2 and the current spectral density SI (f ) is proportional to I 2 in the
region that Ohm’s law is obeyed. A long standing viewpoint is that noise or fluctuation is
always present in a resistor even in the absence of a mean current but the current in the
device serves simply to reveal this fluctuation, not cause it. The authors in [37] performed
experiments on continuous metal films at room temperature and found that samples of pure
29
metals and bismuth had comparable amounts of 1/f voltage noise with the power spectrum
proportional to the square of the voltage measured across the sample. They observed the
PSD of the white noise in these samples is proportional to the resistance of the sample.
The resistance of the samples themselves fluctuate and the white noise also fluctuates at
the same frequencies. They measured the PSD of the low frequency fluctuations of the
white noise, itself a fluctuating quantity. In effect, they measured the noise of the noise and
found that its spectrum was proportional to 1/f . Since this is essentially the same result
one would obtain using conventional measurement techniques, i.e. using nonzero current,
they concluded that 1/f noise is indeed produced only by fluctuation in the resistance of
the sample and the current serves to “show” this fluctuation. This experiment was also
carried out in [38] on carbon resistors and the results were confirmed. What the above
experiments illustrate is that resistance fluctuation is a possible source of 1/f noise. When
current at RF is flowing through a resistor which shows 1/f noise in the presence of a direct
current, a noise similar to 1/f noise is produced in the sidebands on each side of the center
frequency. This is commonly referred to as phase noise and it scales in proportion to the
mean-square value of the high-frequency current. This implies that there should be a high
degree of correlation in the components of this noisy sideband with the low-frequency 1/f
noise. This was confirmed in [39], who estimated the correlation coefficient to be as high as
0.95. This experiment was performed on a carbon resistor and they concluded that because
of the high degree of correlation, phase noise can be attributed to resistance fluctuations.
2.5.7
The Effect of Temperature
In [40], the authors performed measurements on temperature dependence of 1/f noise by
analyzing thin films of Cu and Ag on a sapphire substrate. Their observations were over
the frequency range of 0.2 Hz to 200 Hz and they used a high current density over this
frequency band to maximize the ratio of 1/f noise to thermal noise. They found an inverse
relationship between the exponent of the 1/f noise (γ) and temperature. γ increased by
roughly 20% at 150K compared to the value at 390K, and at a given temperature, the
thickness of the sample did not affect the amount of 1/f noise present.
The effect of the substrate on the temperature dependence was investigated in [41],
wherein the same experimental setup as in [40] was used but with both quartz and sapphire
substrates. It was discovered that above room temperature, the substrate had little effect
30
on the spectra of 1/f noise and but below room temperature, the Cu on quartz showed a
flattening effect below 300K and did not consistently reduce with reducing temperature as
was the case with Cu on sapphire. In the case of Ag, the results were unchanged from those
of [40]. This led them to propose that there are two types of 1/f noise: A Type-A noise
which weakly depends on temperature and a Type-B noise which is strongly temperature
dependent. Hence in Ag, the Type-B noise dominated to such an extent that different
substrate materials did not matter to the eventual noise-temperature dependence, but in
Cu on quartz type-A noise was high because of which the type-B noise was only present
within a specific temperature range.
Excellent references on the various resolved and mostly unresolved properties of
1/f noise can be found in several reviews: [51], [52], [53], [54], [55] and [58].
2.6
1/f (Flicker) Noise Models
There are several theoretical models in the literature that have been proposed for 1/f noise
which are mathematical in nature. While some of them use physical intuition to explain
this phenomenon, others use a broad array of mathematical functions to produce a model
for 1/f noise. Several such models are discussed below.
2.6.1
Surface Trapping Model
This model is based on the idea of traps present in a semiconductor. A free carrier is
immobilized or trapped when it falls into a recombination center (trap). When several such
carriers are trapped, it means that they are not available for conduction and as a result,
the resistance of the semiconductor is modulated. If in the simplest case, a single trap
is considered, then the kinetics of the fluctuation are characterized by a single relaxation
time or time constant τz . If this trapping process obeys a Poissonian statistic, then the
correlation of this process is purely exponential and its spectrum is Lorentzian. To see this,
first note that the PMF of a Poisson random process of rate λ is given by
pX (k) = eλt
(λt)k
k!
(2.42)
where the k’s are integers. Consider a Poisson random process X(t), with rate λ and
consider the random process Y (t) = (−1)X(t) . Thus the process Y (t) starts at Y (0) = 1
31
and flips back and forth between -1 and +1 at times ti which are Poisson is distribution.
This is also known as a semi-random telegraph signal. Following the steps as in [75] we
have,

 +1 if X(t) is even
Y (t) =
 −1 if X(t) is odd
Thus we have for probabilities,
P [Y (t) = +1] = P [X(t) is an even integer]
(λt)2
= e−λt [1 +
+ . . .]
2!
= e−λt cosh(λt)
(2.43)
P [Y (t) = −1] = P [X(t) is an odd integer]
(λt)3
+ . . .]
= e−λt [λt +
3!
= e−λt sinh(λt).
(2.44)
and
The mean of this process is given by
µY (t) = E[Y (t)] = (1)P [Y (t) = 1] + (−1)P [Y (t) = −1]
= e−λt (cosh(λt) − sinh(λt))
= e−2λt .
(2.45)
To find the autocorrelation of Y (t), we note that Y (t)Y (t+s) = 1 if there is an even number
of events in the interval (t, t + s) and Y (t)Y (t + s) = −1 if there is an odd number of events
in the interval (t, t + s). So we have,
RY (t, t + s) = EY [(t)Y (t + s)]
X
X
(λs)n
(λs)n
= (1)
e−λs
+ (−1)
e−λs
n!
n!
n even
n odd
∞
X
(−λs)n
= e−λs
n!
0
= e−λs e−λs
= e−2λs .
(2.46)
32
Thus, we see that the autocorrelation is exponential and WSS. The PSD is evaluated using
the one-sided Weiner-Khintchine theorem as
Z ∞
SY (f ) = 4
RY (t, t + s) cos(2πf s)ds
Z0 ∞
= 4
e−2λs cos(2πf s)ds
0
=
4λ
.
(2πf )2 + 4λ2
(2.47)
If the rate λ of the Poisson process is equal to the relaxation time associated with the trap,
then the PSD above can be re-written as
SY (f ) =
4τz
.
(2πf )2 + 4τz2
(2.48)
This shows that the PSD is Lorentzian: it falls off as 1/f 2 at high frequencies and levels
off at low frequencies. Considering a large number of traps each with an associated time
constant, one can assume that the spreading of these time constants follow some probability
distribution which must satisfy
Z
0
∞
p(τz )dτz = 1
and the PSD of the entire fluctuation N (t) is then given as
Z ∞
τz p(τz )
SN (f ) = 4µN (t)
dτz .
1
+
(2πf )2 τz2
0
(2.49)
(2.50)
It was suggested in [59] that if p(τz ) ∝ 1/τz in some interval τ1 ≤ τz ≤ τ2 and is zero
outside this interval then SN (f ) ∝ 1/f in the range τ2−1 ≤ f ≤ τ1−1 . It was showed in
[60] and [61] showed that 1/f noise would result if there were a superposition of several
such processes with different time constants such that each time constant was inversely
proportional to temperature: τz = τ0 exp(E/kB T ), where E is the activation energy of the
process, kB is Boltzmann’s constant, T is the temperature and τ0 represents the number
of attempts made to overcome the energy barrier E. The distribution of these activation
energies FE (E) was assumed almost constant and since p(τz )dτz = FE (E)dE one obtains
the function p(τz ) = FE (E)/(dτz /dE) = kB T FE (E)/τz . Then p(τz ) ∝ 1/τz if FE (E) is
constant.
Instead of activation, this form of p(τz ) could also be obtained by assuming a
tunnelling process as McWhorter did in [62]. He assumed that charge tunnels from the
33
surface of the semiconductor to the traps located in the oxide and the distribution of the
times of tunnelling is exponential and is given by
µ ¶
d
τz = τ0 exp
δ
(2.51)
where δ and τ0 are constants and McWhorter assumed δ ' 0.1 nm. For a distribution of
time constants between τ1 and τ2 , it follows
p(τz )dτz =
dτz /τz
.
ln(τ2 /τ1 )
Substituting this equation into Eqn. (2.50), the PSD is found to be
Z τ2
4µN (t)
1
SN (f ) =
dτz
ln(τ2 /τ1 ) τ1 1 + (2πf )2 τz2
4µN (t) [arctan(2πf τ2 ) − arctan(2πf τ1 )]
=
ln(τ2 /τ1 )
2πf
(2.52)
(2.53)
which closely approximates a 1/f spectrum over the specified spread of time constants.
Extensions of the original McWhorter form have since appeared in the literature. For
example [63] modified the WSS autocorrelation function of Eqn. (2.46) to
1
RY (t, t + s) = e−2λt (1 − e−4λt )
4
(2.54)
which is now dependent on time t and is hence nonstationary. They found that using this
modification provided for a more accurate estimate of noise in switched MOSFET circuits.
They used three examples to demonstrate this: A periodically switched transistor, a ring
oscillator and a source follower. The authors in [64] considered Heterojunction Bipolar
Transistors and formed a more generic framework of McWhorter’s theory which allows any
form of the distribution of carrier lifetimes. More specifically, the distribution function was
also a function of position of carrier in the base region. It allowed for the density of traps
to vary with respect to energy and distance. In [68], the author suggested a generalized
model to produce 1/f noise which he called a mechanical model. He perturbed his system
with random time-dependent perturbations which had a certain mean square amplitude and
lifetime probability density. This seems to eventually be equivalent to the models detailed
above.
2.6.2
Transmission line 1/f noise
The author in [28] introduced a simple RC transmission line model which when fed with
white noise produced 1/f noise at its output. The circuit is shown in Fig. 2.3. The circuit
34
R
R
White
Noise
R
C
C
R
C
RC line
Figure 2.3: Lumped RC transmission line excited by a white noise current source from [28].
consists of an infinitely long transmission line and fed with a white noise current source of
magnitude I at its input. The impedance of this line is
s
R
Z(f ) =
j2πf C
(2.55)
where R and C are the resistance and capacitance per unit length respectively. The PSD
of the voltage at the output of the line is
SX (f ) = I 2
R
.
j2πf C
(2.56)
Assuming a line of infinite length, SX (f ) is proportional to 1/f down to zero frequency. On
the other hand, if the line is of finite length, then there will exist a lower frequency below
which SX (f ) is white and its value is
flow =
1
2πRC`2
(2.57)
with ` being the length of line. Keshner went on to derive an autocorrelation function for
this model and proved that it consists of a sum of two terms: a nonstationary one and a
stationary one, making the overall function nonstationary. Keshner also showed that if the
time of observation was much smaller than the time elapsed since the system was turned
on, then this autocorrelation function could be considered almost stationary.
2.6.3
Pole Placement
1/f γ noise generators were developed in [65] by using a generic summation model that sums
N Lorentzian spectra. If the poles are placed such that the density of poles is uniformly
35
distributed with respect to the logarithm of the frequency, then SX (f ) would represent a
1/f spectrum. The resultant power spectrum is
N
2σ X fH
SX (f ) =
2 + f2
π
fH
(2.58)
h=1
and σ is the variance of the considered quantity and fH represents the frequency of the
poles. If the discrete sum was replaced by an integral, the poles fH would also assume a
continuum of values and the PSD would be expressed as
Z
2σ ∞ fH
SX (f ) =
2 + f 2 DfH dfH
π 0 fH
(2.59)
where DfH represents the distribution of poles. If DfH ∝ 1/fH , then a 1/f spectrum
is obtained. The motive behind their work was to find the appropriate number of poles
(Lorentzian spectra) and their location required to generate a 1/f distribution. They used
the circuit shown in Fig. 2.4, which consists of several parallel RC circuits that are connected
in series to feed the input of an amplifier. Each resistor in the parallel RC combinations
contributes thermal (white) noise. The PSD of the noise voltage at the output of the RC
R1
en
Vi
C1
R2
Ro
in
C2
Ci
Ri
Sv
Ao Vi
Rn
Cn
Figure 2.4: A 1/f γ noise generator from [65].
circuits SX (f ) is of the form in Eqn. (2.58) and each fH = 1/2πCRH . If en , in and Yi are
all zero, then the amplifier noise output spectrum would be SV (f ) = A2o SX (f ), where Ao is
the frequency independent voltage gain of the amplifier which is the desired form. However,
36
there are the Ri and Ci frequency dependent terms that have to be accounted for. If
Ri À
N
X
Rh
(2.60)
h=1
and
Ci ¿ C/N
(2.61)
then the effect of Ri and Ci can be neglected and it would be possible to get a 1/f response
over a wide frequency range.
In [66] was considered the finite form of the model in [28] which is nothing but a
cascade of N first-order filters having a transfer function of the form
QN
i=1 (s − s0i )
Ha (s) = A QN
i=1 (s − spi )
(2.62)
for an appropriate gain term A. The N poles spi are uniformly distributed with respect to
log f . The output spectrum of this filter with a white noise input is
So (f ) = Ha (f )2 Si (f ) ∝ Ha (f )2 .
(2.63)
The more the number of poles per frequency decade, the greater the accuracy of the 1/f
spectrum. Corsini and Saletti went on to further realize this analog model in the z-domain
to produce a digital 1/f noise generator. The transfer function in the z-domain is
QN
−1 exp(s T )
oi
i=1 1 − z
H(z) = Az QN
−1
exp(spi T )
i=1 1 − z
(2.64)
which is true because of the mapping of a pole/zero from the analog domain to a pole/zero
to the frequency domain given by s − sX → 1 − z −1 exp(sX T ), where T is the sampling
rate. The digital filter implementing this 1/f sequence is shown in Fig. 2.5.
In [67] the RC transmission line model considered two special cases depending on
the impedance of the load: a) open-circuited load and b) short-circuited load. For the open
circuit case with reflection coefficient Γ = 1 the input impedance of the line can be written
as
√
cosh(` RC)
√
ZIopen (s) = Z0 (s)
(2.65)
sinh(` RC)
and s is the usual Laplace operator and ` is the length of the line, while for the short circuit
case, the input impedance with reflection coefficient Γ = −1 can be written as
√
sinh(` RC)
√
ZIshort (s) = Z0 (s)
.
cosh(` RC)
(2.66)
37
(i)
x (n)
(i)
y (n)
−1
z
a
i
b
i
Figure 2.5: Digital 1/f γ noise generator from [66].
Using the rational series approximations for the sinh and cosh terms, the respective input
impedances can be re-expressed as
Q∞
4`2 sRC
1
n=1 1 + (2n−1)2 π 2
ZIopen (s) =
Q∞
`2 sRC
`Cs
n=1 1 + 2 2
(2.67)
n π
and
Q∞
2
1 + `nsRC
2 π2
ZIopen (s) = `R Q∞n=1
4`2 sRC
n=1 ` + (2n−1)2 π 2
(2.68)
The open circuit model acts like an integrator and hence its PSD varies as 1/f 2 . The short
circuit model does show a 1/f response but only within a limited frequency range. If the
frequency gets too high or too low, the characteristic tends to flatten out.
2.6.4
Fractional Noises
Fractional noises or fractional Brownian Motions (fBM) are random processes that form
a special class of Gaussian processes that are typically characterized by a parameter 0 <
H < 1. They differ from typical Markov processes in the sense that a long span of these
random sequences show a strong sense of dependence as made more precise in [73]. A
random process X(t), t ≥ 0 is called a Wiener process or BM if the following conditions are
satisfied:
1b. With probability 1, X(0) = 0, or the process starts at the origin and X(t) is a continuous
function of t.
38
2b. ∀t ≥ 0, s > 0, the increment X(t + s) − X(t) is normally distributed with mean µ = 0
and variance σ 2 = s. Thus,
1
P (X(t + s) − X(t) ≤ x) = √
2πs
Z
µ
x
exp
−∞
−u2
2s
¶
du.
(2.69)
3b. If 0 ≤ t1 ≤ t2 ≤ . . . ≤ t2n , the increments X(t2 ) − X(t1 ), X(t4 ) − X(t3 ), . . ., X(t2n ) −
X(t2n−1 ) are independent.
4b. The increments X(t + s) − X(t) are stationary, which means they have distributions
independent of t.
For t > 0, s > 0 the autocovariance of a BM can be derived as follows:
Var[X(t) − X(s)] = E((X(t) − X(s) − E[X(t) − X(s)])2 )
= E[((X(t) − E[X(t)]) − (X(s) − E[X(s)]))2 ]
= E((X(t) − E[X(t)])2 − 2(X(t) − E[X(t)])(X(s) − E[X(s)])
+(X(s) − E[X(s)])2 )
= Var[X(t)] − 2Cov[X(t), X(s)] + Var[X(s)].
1
(Var[X(t)] + Var[X(s)] − Var[X(t) − X(s)]).
Cov[X(t), X(s)] =
2
This gives, for the autocovariance
KX (t, s) =



1
2 [t
1
2 [t
(2.70)
+ s − (t − s)] = s t > s
+ s − (s − t)] = t
s>t
KX (t, s) = min(t, s).
(2.71)
The autocorrelation is equal to the autocovariance for a zero mean process. Thus for a BM
the autocorrelation is
RX (t, s) = KX (t, s) = min(t, s)
(2.72)
which is time-dependent. A random process is said to have a mean square derivative X 0 (t)
if
(·
lim E
²→0
¸2 )
X(t + ²) − X(t)
− X 0 (t)
= 0.
²
(2.73)
It can be shown [74] that the mean square derivative of X(t) exists if ∂ 2 RX (t, s)/∂t∂s exists.
If so, then the mean and autocorrelation of the X 0 (t) is
E[X 0 (t)] =
0
RX
=
d
0
E[X(t)] = µX (t)
dt
∂ 2 RX (t, s)
.
∂t∂s
(2.74)
(2.75)
39
Using Eqn. (2.72), we can compute

 1 t>s
∂
RX (t, s) = U (t − s) =
 0 t<s
∂s
where U (t−s) is the unit step function and is discontinuous at t = s. Thus ∂ 2 RX (t, s)/∂t∂s
does not exist and one can say that a BM does not have a mean square derivative. However,
we can still write
∂ 2 RX (t, s)
∂
= U (t − s) = δ(t − s)
∂t∂s
∂t
(2.76)
where δ(t − s) is the Dirac delta function. Thus the autocorrelation of the mean square
derivative of a BM is a delta function. This implies a complete lack of correlation between
adjacent samples of the process which is nothing but white noise. To see this in the frequency
domain, the PSD of the mean square derivative of the BM
Z ∞
0
SX (f ) =
RX 0 (τ )e−j2πf τ dτ
Z−∞
∞
=
δ(τ )e−j2πf τ dτ
−∞
= 1
(2.77)
which is independent of frequency. Note that t − s has been replaced by τ above. This
establishes that white noise can be thought of as a derivative of a BM which is an important
observation. White noise is a 1/f process with an exponent of unity. Calculating the PSD
of a BM is not so trivial since one look at Eqn. (2.72) will convince that the condition
Z ∞
kRX (τ )kdτ < ∞
(2.78)
−∞
will not be satisfied and this is a necessary condition to calculate the spectrum using the
Wiener-Khintchine theorem. However, it can be numerically shown that the spectrum of a
BM tapers off as 1/f 2 or the exponent is 2. An example of a sample realization of a 1D
BM is shown in Fig. 2.6. Brownian motion is considered scale invariant. Intuitively this
means that a sample path realization as in Fig 2.6 looks the same in a probabilistic sense
under any resolution. If the same simulation is run with a different timestep, the resulting
sample path will have identical properties. Since one can only simulate/measure with finite
precision, the normal distribution of condition 2b from above can be reformulated as
Z x
1
2
P (X(t + s) − X(t) ≤ x) = √
e−u /2ds du
(2.79)
2πds −∞
40
Brownian Motion (Wiener Process)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
0.2
0.4
0.6
0.8
1
Time
Figure 2.6: 1D Brownian motion realization.
where d t represents the position of the interval of observation of the BM. The result is that
the probability distribution function of BM is invariant in scale. For d > 0, replacing s by
d s and x by d0.5 x does not change the value of the right hand side of Eqn. (2.69).
P (X(t + s) − X(t) ≤ x) = P (X(dt + ds) − X(dt) ≤ d0.5 x)
(2.80)
Thus changing the temporal scale by a factor d and the spatial scale by a factor d0.5 results
in a process that is indiscernible from the original, giving statistical self-similarity. In terms
of PDFs, this works out as
fX (x0 = d0.5 x, t0 = dt) = d−0.5 f (x, t).
(2.81)
Brownian motion can be quite restrictive for modelling purposes mainly because it imposes
stationary and independent increments which are normally distributed. In [73] the concept
of fractional Brownian Motion (fBM) was introduced which dispenses with the condition
of independence. fBM with index H(0 < H < 1) is defined as a Gaussian process X(t)
satisfying the following conditions:
1f. With probability 1, X(0) = 0, or the process starts at the origin and X(t) is a continuous
function of t.
2f. ∀t ≥ 0, s > 0, the increment X(t + s) − X(t) is normally distributed with mean µ = 0
41
and variance σ 2 = s2α .
3f. The increments X(t + s) − X(t) are stationary which means they have distributions
independent of t.
¶
−u2
exp
du.
(2.82)
P (X(t + s) − X(t) ≤ x) = √
2s2H
2πs2H −∞
In comparison to fBM, note the difference in variance and the absence of condition 3b. An
1
Z
x
µ
4
3
2
Amplitude
1
0
−1
−2
−3
−4
−5
100
200
300
400
500
600
sequence
700
800
900
1000
Figure 2.7: 1D fBm realization with H = 0.2.
example of a sample realization of a 1D fBM process with H = 0.2 is shown in Fig. 2.7 and
with H = 0.8 is shown in Fig. 2.8. Both plots were generated using the “wfbm” command
in MATLAB. Brownian motion is a special case of fBM with H = 0.5. The distributions
of functions defined by fBM cannot have independent increments except in the Brownian
motion case of H = 0.5. The autocovariance of this process is given by
1
KX (t, s) = (t2H + s2H − |t − s|2H ).
2
Again, as a generalization of Eqn. (2.81)
fX (x0 = dH x, t0 = dt) = d−H f (x, t).
(2.83)
(2.84)
Just like white noise is the derivative of Brownian motion, a generic 1/f α noise can be
defined by the fractional derivative
dα
x(t) = w(t)
dtα
(2.85)
42
Figure 2.8: 1D fBm realization with H = 0.8.
where w(t) represents the generic 1/f α noise and α is any number between 0 and 2. From
the theory in [76], it can be shown that the solution to the fractional differential equation
in Eqn. (2.85) is given by
1
x(t) =
Γ(α/2)
Z
t
(t − τ )α/2−1 w(τ )dτ
(2.86)
0
where Γ(.) is the Gamma function. This is equivalent to a linear system driven by white
noise with an impulse response function
h(t) =
tα/2−1
Γ(α/2)
(2.87)
whose Laplace transform is 1/sα/2 thereby giving us a noise with arbitrary value of exponent.
This was suggested in [77].
There are digital models which generate Brownian motion in the z-domain by starting out
with a difference equation describing a certain process.
Xk+1 = Xk + wk
(2.88)
describes a discretized random process Xk and wk represents an independent and identically
distributed (IID) random variable with variance K∆k. This is the general random walk
43
formula as shown in [75]. Taking its z-transform yields
H(z) =
which results in the PSD
SX (f ) =
1
1 − z −1
(2.89)
K∆k
(2 sin 2πf2∆k )2
w.
(2.90)
This approximates a BM at low frequencies since sin(ω) ≈ ω for small ω. Hence the PSD is
SX (f ) =
1
K/∆k
∝ 2
2
(2πf )
f
(2.91)
which indicates a BM. In [78], the author proposed a fractional extension of this concept
and used a digital model defined by
Hf (z) =
1
(1 − z −1 )α/2
(2.92)
and proceeded to derive the discrete autocorrelation and autocovariance functions of fBM
analogous to the continuous versions discussed earlier. Hence, the PSD is also an extension
of the BM case and can be described as
K∆k
(2 sin(πf ∆k))α
SX (f ) =
(2.93)
which for small frequencies can be approximated by
SX (f ) ≈
K∆k 1−α
(2πf )α
(2.94)
giving a noise model with arbitrary exponent α. In [79], the author takes a similar approach
by taking a power series expansion of Eqn. (2.92) given as
H(z) = 1 +
α −1 α/2(α/2 + 1) −2
z +
z + ...
2
2!
(2.95)
and the coefficients of this series give the pulse response of the transfer function. The k th
numerator in Eqn. (2.95) can be expressed as
hk =
k
Y
(α/2 + n − 1)n
(2.96)
n=1
and the pulse response values can be evaluated using the recursive algorithm
h0 = 1
hk = (α/2 + k − 1)
hk−1
.
k
(2.97)
44
Another variation of Brownian Motion is a Lévy process which does not assume
a Gaussian distribution although the increments are independent and stationary. A Lévy
process is said to be stable if d−1/γ X(dt) and X(t) have the same distribution for all d > 0
and for some γ. This idea has been used to generate 1/f noise and is covered in Section 2.6.5.
2.6.5
Power-Law Shot Noise
The authors in [80] revisited the method of generation of shot noise which can be described
in the time domain by a Poisson pulse generator of rate µ that feeds an input linear timeinvariant filter having a particular impulse response whose output is shot noise. The process
is demonstrated graphically in Fig. 2.9 The amplitude distribution of shot noise approaches
Rate u
Poisson
Generator
Poisson point process
Linear Filter
h(t)
Shot Noise
Figure 2.9: A general shot noise generator from [80].
a Gaussian distribution as the rate of the Poisson process increases [81]. If this rate is
much greater than the reciprocal of the characteristic time duration of the impulse response
function, then the amplitude distribution of the resulting noise will approach a Gaussian
form. An example of such an impulse function is a decaying exponential as is shown in
Fig. 2.9 and a shot noise constructed from such an impulse response function tends to have
a Gaussian amplitude distribution as rate µ increases. Shot noise can be expressed as an
45
infinite sum of such impulse response functions as follows:
∞
X
I(t) =
h(t − tk )
(2.98)
k=−∞
where the times tk are random events and have a Poisson distribution with rate µ. The
impulse response functions can be both deterministic or stochastic although in the stochastic
case, it is unlikely to get accurate results for higher than first order statistics.
When the impulse response function is a decaying power law, the characteristic
time can become arbitrarily large or small as a consequence of which, the amplitude distribution ceases to be Gaussian. Shot noise generated as a result of using such an impulse
response function is called power-law noise. The impulse response can be described as

 Kt−β A ≤ t < B
h(K, t) =
 0
otherwise.
An example of a power law impulse response transfer function is shown in Fig. 2.10 wherein
β = 1/2. The authors then derive statistical properties for generic power law shot noise
10
9
8
Impulse Response h(t)
7
6
5
4
3
2
1
0
0
50
100
150
200
250
Time t
300
350
400
450
500
Figure 2.10: A power law impulse response function with β = 1/2.
and find that the amplitude probability density function, the autocorrelation function and
the PSD all follow a power law. Since a power law dependence indicates the presence of
46
all time scales, they conclude that this implies fractal behavior. The amplitude probability
density function follows a Lévy stable form for exponent β > 1. If the parameters A and B
are equal to 0 and ∞ respectively, then it can be shown ([80]) that the PSD of the power
law shot noise is
SI (f ) ∝
1
(2πf )α
(2.99)
where α = 2(1 − β). If α = 1, then this becomes a model for 1/f noise. In [82] a transfer
function of the form h(t) = t−1/2 was used as a model for 1/f noise and the above power-law
shot noise approach is a generalization of that, one special case (α = 1) being a model for
1/f noise.
2.6.6
1/f Noise from Chaos
g
n
Output
x
f(.)
n
x n+1
Unit Delay
Figure 2.11: A general 1-D dynamical system consisting of a nonlinear function and a
recursive loop from [32].
The authors in [32] use elementary concepts of chaotic dynamical systems to generate
sequences of 1/f noise (referred to as colored noise in [32]). They use a generic discrete
nonlinear function with a recursive loop providing unit-delay feedback as shown in Fig. 2.11.
The assumption is that the nonlinear function has no delay and the output of the function
is obtained iteratively as follows:
xn+1 = gn xn
(2.100)
where the subscript n denotes discrete time and gn is a gain factor. After a unit delay, xn+1
47
is transferred back to the input of the nonlinearity which instantaneously yields xn+2 =
gn f (xn+1 ). The function f () is chosen carefully so as to satisfy the conditions for chaos
(Chap. 10 in [33]), one of the features of which is that a chaotic function can generate
stochastic looking sequences using completely deterministic functions, see [34]. A popular
example of a chaotic nonlinear function is the logistic map expressed as
fλ (x) = λx(1 − x)
(2.101)
where λ is the parameter of this function and typically takes value 4. The plot of values
of this function on the ordinate versus the parameter values on the abscissa is called a
bifurcation diagram and for this logistic map, this is shown in Fig. 2.12. The fixed points of
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
λ
Figure 2.12: The bifurcation diagram for the logistic map.
any dynamical system fc (x) with parameter c are the solutions of the equation fc (x) = x.
For the logistic map in Eqn. (2.101), the fixed points are x = 0 and x = 1 − 1/λ. By fixed
point it means that once an orbit hits a fixed point at time n ≥ 0, it stays at the point
for all time n0 > n. If the orbit hits x = 1 at time n, then it is sent to fixed point x = 0
at time n + 1 making x = 0 an eventually fixed point. The authors consider the special
case of λ = 4g including a constant gain term and restrict the range of x in the interval
[0, 1] along the lines of [35] and the fixed points are 0 and 1 − 1/4g and the eventually fixed
point is 1. At roughly λ = 3, a pair of paths on the bifurcation diagram opens up and this
48
corresponds to a period-doubling bifurcation (g > 0.75), as in [32]. As the parameter is
increased still further, there is a seemingly endless cascade of period doubling bifurcations
√
0
until we reach the period 3 window corresponding to the parameter λ = (1 + 8)/2. A
famous theorem in [36] proves that the existence of a period 3 window in the bifurcation
0
diagram implies chaos. For values of parameter just smaller than λ above, the sequence of
values of the map would show a period of 3 but with values of parameters just greater than
0
λ , the periodicity disappears irrespective of the length of the sequence. The spectrum of
such long sequences (107 terms ) was computed in [32] by performing an FFT resulting in
X(f ) and the power spectrum was found by averaging X(f )2 over many runs, each having
a different initial condition that is not a fixed point and lies inside the interval [0, 1]. This
is expressed more precisely as
N
SX (f ) = (
1 X
)
(X(f ))2
N
(2.102)
m=1
where N different values of initial condition were chosen. The averaged PSD for the logistic
map for gain values of 0.925, 0.975 and 0.9975 are shown in Fig. 2.13, Fig. 2.14 and Fig. 2.15
respectively. The authors then made two modifications to the standard Logistic function.
The first modification of f1 (x) can be written as follows
f1 (x) = h(x(4.5 − x)/3.5) h(x) = 4x(1 − x).
(2.103)
The second modification is
f2 (x) = h(x(6 − x)/5) h(x) = 4x(1 − x)
(2.104)
while f3 (x) is the original Logistic map. The gain was now chosen as a uniform random
variable distributed between [0, 1] and was allowed to vary at each step resulting in a three
new time sequences. The PSD of all three sequences are computed as before and are plotted
in Fig. 2.16. All the spectra show some sort of 1/f low-frequency behavior although their
slopes are not exactly 1/f . The authors also computed the probability distributions of these
variable gain sequences and found that they were significantly non-Gaussian.
2.6.7
Self-Organized Criticality (and others)
In [12], it was found that certain dynamical systems with several degrees of freedom evolve
naturally into what is called Self-Organized Criticality (SOC) or self-organized critical states
49
g = .925
1
0.9
0.8
0.7
0.6
S(f)
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
600
Figure 2.13: Output spectrum with g=0.925.
g = .975
1
0.9
0.8
0.7
0.6
S(f)
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
Figure 2.14: Output spectrum with g=0.975.
600
50
g = .9975
1
0.9
0.8
0.7
S(f)
0.6
0.5
0.4
0.3
0.2
0
100
200
300
400
500
600
Figure 2.15: Output spectrum with g=0.9975.
5
Variable Gain sequence FFTs
10
4
10
3
10
S(f)
2
10
1
10
0
10
1
2
10
10
f
Figure 2.16: PSD for f1 (x), f2 (x) and f3 (x).
3
10
51
which are states which are just barely stable. As an example of a system that exhibited
self-organized criticality (SOC), they considered a simple sandpile model. If the slope of the
pile of sand was too large, then the pile is considered far from equilibrium and will collapse
soon after the slope reaches a critical value. At this critical value, the system would be
marginally or critically stable with respect to small perturbations. It was argued that the
phenomenon of 1/f noise was the dynamical response of this critically stable sandpile to
small random perturbations. Numerical simulations on cellular automata were performed
whose evolution followed some pre-defined rules. A 2-D case was considered in which the
integer variable z signifying spatial co-ordinates is updated synchronously according to
z(x, y) → z(x, y) − 4,
z(x ± 1, y) → z(x ± 1, y) + 1,
z(x, y ± 1) → z(x, y ± 1) + 1
(2.105)
and a particular site would topple if z exceeds a critical value, say K at that site. This
topple would cause the site to not only reduce in strength by 4 units, but would add one
unit to each of its neighbors. Fixed boundary conditions were used, i.e. z = 0 at the
boundaries. The system would start at some random condition z À K and would simply
evolve according to the rules above until it would stop, i.e. all the z values would be less
than K. Once this critical state would be reached, then the dynamics of the system would
be probed by applying small local perturbations. Depending on the size of the perturbation,
clusters of different sizes would form. If the size of a cluster is taken to be the number of
occupied sites, then a log–log plot of the distribution of cluster sizes or avalanches D(s)
versus s would follow a power law or in other words represent a 1/f process. A simulation
was run on a 50x50 array and the distribution of avalanche sizes was found to resemble a
power law in a first order approximation. This is shown in Fig. 2.17.
This idea has generated a lot of inter-disciplinary interest in the scientific community and has resulted in further refinement. The concept of evolution into a self-organized
critical state is called emergence when applied to the theory of complex systems. In other
words, a large dynamical system starting out from some initial condition evolves according
to a simple set of rules and finally organizes itself into what is called a complex system. A
generalized explanation of this is provided in [70]. To illustrate a cross-disciplinary application of this idea, the theory of complexity applies to the evolution of biological systems as
expounded in [71] wherein it is claimed that species have not evolved into their current state
52
9
Simulated Avalanche Data
1st order Polynomial Fit
8
7
6
Slope = −1.2
5
Avalanche
4
Magnitude
3
2
1
0
0
1
2
3
4
5
6
frequency
Figure 2.17: SOC power law of the distribution of cluster sizes.
purely by Darwinian natural selection [72] but have undergone a process of evolution, which
in a simple case resembles the evolution of a cellular automaton, to reach a critical state
only to be further refined by natural selection. The role of natural selection in evolution
of biological systems is reassessed and it is suggested that emergence and natural selection
have proceeded hand-in-hand as biological systems evolve to their current state.
A recent alternative for an explanation of the occurrence of power laws can be
found in the work in [85] wherein power law distributions are generated by a mechanism
called “Highly Optimized Tolerance” or HOT. It is shown how systems which have evolved
due to mechanisms such as natural selection (biological systems) or by virtue of good engineering practices, exhibit power laws. These systems which have evolved to a robust state
show a good tolerance to external perturbative effects. The key difference between SOC
and HOT is that in SOC, the system occupies a state that is close to its non-equilibrium
state also referred to as the “edge of chaos” and can be sent into non-equilibrium by a small
random perturbation while in HOT, the system is further away from the critical point and
is therefore resistant to small perturbative effects. Such systems are designed as a tradeoff
between factors such as yield, performance, resource overhead and sensitivity to risk.
53
2.6.8
Phase Noise
The phenomenon of phase noise is encountered in circuits containing oscillators and mixers
and is characterized as a frequency-domain phenomenon. Phase noise represents the uncertainty in determination of the carrier frequency and instead of appearing as a perfect delta
function at the carrier frequency in the frequency domain, it occupies a band of frequencies about the carrier whose PSD exhibits a power-law characteristic. This is represented
graphically in Fig. 2.18 where df represents the band of frequencies about the carrier fc . A
fc
fc
df
Figure 2.18: Uncertainty in carrier frequency due to phase noise.
typical oscillator is shown in Fig. 2.19 and its instantaneous output can be represented by
the equation
Voutput (t) = V0 cos(2πfc t + φ(t)).
(2.106)
The function φ(t) affects the phase of the output signal and its random nature gives rise
to phase noise. The PSD of the oscillator can be related to the PSD of the phase noise
according to
Soutput (f ) ≈
P
(Sφ (f + fc ) + Sφ (f − fc ))
2
(2.107)
where P is the power of the carrier. What now remains is a model for the nature of the
PSD of φ(t), Sφ (f ). The author in [86] considered an LC tank as a feedback circuit having
a bandwidth df . At frequencies close to the carrier, phase errors at the input will result
in frequency uncertainty at the output. This is a function of the sources of noise of the
amplifier and the response of the feedback network. The author quantified the fluctuations
54
due to phase noise in terms of the single sideband noise PSD. It is defined as
·
¸
Psideband (fc + df, 1Hz)
L(df ) = 10 log
Pcarrier
(2.108)
where Psideband (fc + df, 1Hz) represents the single sideband power at a frequency offset of
df from the carrier with a measurement bandwidth of 1 Hz. Note that this representation
assumes variations in amplitude are negligible compared to the variations in phase. Leeson’s
LTI phase noise model for tank oscillators predicts the following behavior for L(df ):
Amplifier
Input
Output
Feedback n/w
Figure 2.19: A typical oscillator configuration.
(
L(df ) = 10 log
"
µ
¶2 # µ
¶)
df1/f 3
2F kT
fc
1+
1+
Ps
2QL df
|df |
(2.109)
where F is an empirical parameter called the device excess noise number, k is Boltzmann’s
constant, T is the absolute temperature, Ps is the average power dissipated in the tank, fc
is the center frequency of oscillation of the tank, QL is the loaded Q, df is the frequency
offset from the carrier and df1/f 3 is the corner frequency between the 1/f 3 and 1/f 2 regions.
The typical nature of L(df ) is shown in Figure 2.20.
The author hypothesized that this PSD has two major contributions of noise: 1)
white noise near the oscillator frequency and 2) flicker noise away from the carrier which
get up-converted about the carrier due to nonlinearities in the oscillator loop.
2.7
Simulation of Noise in Circuits
The simulation technique behind most circuit simulators, the most popular and widely used
being SPICE and its variants, is a formulation of Kirchhoff’s Laws, i.e. Kirchhoff’s Current
55
L (∆ f )
1
f 3
1
f 2
noise
corner
∆f
Figure 2.20: A typical plot of the phase noise of an oscillator versus offset from the carrier.
Law (KCL) and less frequently Kirchhoff’s Voltage Law (KVL). Resistors, capacitors, inductors and voltage-controlled current sources form the basic set of network elements used
in circuit simulations. Semiconductor devices are modelled as components made up of an
interconnection of these basic elements. Given an interconnection of several such elements
in a circuit, a set of differential equations describing the network are set up by using KCL
and KVL. The most popular technique of formulating the set of equations is called Modified
Nodal Analysis and a typical system of equations can be represented as
I(x, t) +
d
Q(x) = 0
dt
(2.110)
where I(x, t) represents the memoryless elements and the independent sources while Q(x)
represents the reactive elements, L and C. The state variables are represented by x and they
consist mainly of nodal voltages. When there are stochastic forcing terms in this equation,
the state variables will also be stochastic which means in order to simulate a noise analysis,
one must simulate the statistics of the state variables, where its mean would be a first order
statistic and the autocorrelation and PSD would be a second order statistic. Usually, when
the stochastic terms are Gaussian, we need only consider up to the second order statistic
for complete characterization.
For a SPICE-like noise simulation, first the time-invariant steady state solution
of the nonlinear circuit (DC solution) is determined and this nonlinear circuit is linearized
about this point. Eqn. (2.110) is then modified and expressed as
GX + C
d
X=0
dt
(2.111)
56
where G and C represent n × n matrices and X ∈ <n . Adding a current noise source to
the above equation gives
GX + C
d
X + TN(t) = 0
dt
(2.112)
where N (t) represents random noise and T is the incidence matrix mapping the noise source
to the appropriate node. If the PSD of the noise source is known (usually assumed white),
the PSD of the state variables can be determined from Eqn. (2.112) as
GH(f ) exp(j2πf t) + C
d
H(f ) exp(j2πf t) + T exp(j2πf t) = 0.
dt
(2.113)
where Hb (f ) is the Fourier Transform of the impulse response h(t) of the system and denotes
a vector of transfer functions from the noise source to X. This leads to
GH(f ) + j2πf CH(f ) + b = 0 ⇒ (G + j2πf C)H(f ) = −T
(2.114)
This now is a system of linear equations to be solved at each frequency by standard linear
algebra routines like LU decomposition. The matrix of the PSD can then be calculated as
SX (f ) = H(f )SN (f )HT (f )∗
(2.115)
where HT (f ) is the transpose of H(f ). The PSD of a single output is usually desired, for
example the voltage difference between two nodes. This output can be written as a linear
combination of the components of the state variables as follows:
Y (t) = dT X(t)
(2.116)
where d represents a vector of constants and if only the difference between the voltage at
two nodes is desired, then d would contain only two terms, a 1 and a −1. The PSD of Y
is given as
SY (f ) = dT SX (f )d
= H(f )SN (f )HT (f )∗ d
(2.117)
and one needs to calculate the vector dT H(f ) instead of the entire matrix H(f ). From
Eqn. (2.114) we have
H(f ) = −(G + j2πf C)−1 T
(2.118)
dT H(f ) = −dT (G + j2πf C)−1 T = V(f )T N
(2.119)
and
57
where the vector V(f ) is the solution of the equation
(G + j2πf C)T V(f ) = −d.
(2.120)
Thus one needs to perform a single LU decomposition at every frequency point of the
matrix (G + j2πf C) with the right hand side set to d. This is the classical form of the AC
noise analysis first proposed in [102] and is used in the SPICE circuit simulator. Although
this method is efficient in calculating the noise response of a circuit with time-invariant
excitations, it requires considerably more work if time-varying excitations exist. In the case
of the existence of a periodic steady-state, the nonlinear circuit can be linearized about this
periodic steady state leading to a time-variant modification of Eqn. (2.112) which can be
written as
G(t)X + C(t)
d
X + TN(t) = 0
dt
(2.121)
and it is also assumed that the noise sources are cyclostationary where by cyclostationarity it
is meant that a random variable has periodic statistics, i.e. mean, autocorrelation and higher
order statistics are periodic. If only the mean and the autocorrelation are periodic, then
the process is said to be wide-sense cyclostationary. Continuing the analysis of Eqn. (2.121)
along the lines of the time-invariant case gives a time-varying PSD of state variable X and
the transfer functions would also be time-varying. Methods to generate the Fourier series
coefficients of time-varying PSD have been proposed in [103].
2.8
Summary
This chapter presented a summary of the common sources of noise in electronic circuits.
These are thermal, shot and flicker noise. The derivation of the form of thermal noise
is provided as based on thermodynamical principles. Shot noise was assumed to be a
superposition of several random processes that follow a Poissonian statistic and its spectral
density is found to be white. In the case of flicker noise, an extensive review of the various
theories of the origins of flicker noise are presented and explorations on the properties
of flicker noise are highlighted. These include the scale invariance of flicker noise, the
nonstationarity of a flicker noise process, the distribution of a flicker noise process, claims
about whether flicker noise is caused by surface effects or bulk effects, the dependence of
flicker noise on current, voltage and resistance of the material under investigation and the
58
effect of temperature on flicker noise. This is followed by a collection of approaches to model
flicker noise which includes infinite analog and digital transmission line models, fractionally
differenced Brownian motion processes, shot noise shaped by appropriate transfer functions
to produce 1/f spectral characteristics and generation of flicker-like sequences with iterative
actions on nonlinear functions. The original theory of the important phenomenon of phase
noise in electrical oscillators is reviewed followed by a description of the common SPICE-like
approach to noise analysis in the frequency domain based on the use linear time-invariant
models for circuit elements.
59
Chapter 3
Stochastic Differential Equations
3.1
Introduction
Stochastic differential equations extend ordinary differential equations to accommodate the
presence of random terms that model unpredictable real-life phenomena. Such random
phenomena are found in a variety of places such as population growth models, option pricing
in finance, electrical noise in analog circuits and in stochastic control problems, to name a
few. In general, the difference between an SDE and an ODE lies in the presence of a random
term with specific characteristics that matches (in a statistical sense) the characteristics of
the real-life process under observation. There is neither a restriction on the effect of the
insertion of this random term in the equation, i.e. it can be purely additive or it may
multiply with some deterministic term, nor on its amplitude. Section 3.2 is an elementary
mathematical introduction to the theory of SDEs and highlights the difference between an
ODE and an SDE. Section 3.3 introduces the theory of scalar SDEs more thoroughly and
explains the two common interpretations of a general SDE, i.e. the Itô and the Stratonovich
forms. Depending on the interpretation assumed, one can in general obtain different results
while solving the same SDE as illustrated in Subsection 3.3.1 while Section 3.4 extends
the theory of scalar SDEs to the vector case. Finally, Section 3.5 provides a resolution
on choosing between the Itô and Stratonovich forms while trying to model general real-life
phenomena.
60
3.2
Basic Theory
Langevin was among the first to consider SDEs for modelling the dynamics of Brownian
motion [107]. Instead of an ODE such as
dx
= a(t, x)
dt
(3.1)
he considered a noisy differential equation of the form
d
Xt = a(t, Xt ) + b(t, Xt )ξt .
dt
(3.2)
The term a(t, Xt ) is the deterministic drift coefficient while the term b(t, Xt )ξt represents the
stochastic perturbative effect. The term b(t, Xt ) is an intensity factor and the ξt s are random
processes. Based on observations, if the random processes ξt are found to have a constant
spectral density, then they are referred to as white noise processes. Since this means that
the process is assigned an equal weight to each of the Fourier frequency components, the
covariance must be a constant multiple of the Dirac delta function δt . Substituting a = 0
and b = 1 in the above equation implies that ξt is the pathwise derivative of a Brownian
motion process Bt . Hence one can write
d
Bt = ξt .
dt
(3.3)
Although the sample paths of the Brownian motion process are nowhere differentiable and
are of unbounded variation, Japanese mathematician K. Itô [112], [113], [114] was able to
provide a way out of this problem by introducing a new stochastic process, the Itô stochastic
integral, which enables Eqn. (3.2) to be written in differential form as
dXt = a(t, Xt )dt + b(t, Xt )dBt
(3.4)
or in integral form as
Z
Xt (ω) = X0 (ω) +
0
Z
t
a(s, Xs (ω))ds +
0
t
b(s, Xs (ω))dBs
(3.5)
where the second integral is not a conventional Riemann integral but an Itô stochastic
integral and is evaluated with respect to Brownian motion. The stochastic calculus so
developed requires a modified chain rule formula and convergence exists not in the pathwise
sense, but only in the mean-square sense. The following sections express these ideas in some
detail and also illustrates common usage and examples of SDEs.
61
3.3
Scalar SDEs
As mentioned already, a generic SDE can be written in differential form as
dXt = a(t, Xt ) + b(t, Xt )dBt
and is a symbolic representation of the stochastic integral equation
Z t
Z t
Xt = X0 +
a(s, Xs )ds +
b(s, Xs )dBs
t0
(3.6)
(3.7)
t0
where Bt is the Brownian motion process (Wiener process). Recall, the Brownian motion
process is a random process whose increments are Gaussian with initial value zero. It has
a zero mean E[Bt ] = 0, a mean square given by E[Bt2 ] = t, t > 0 and has independent
increments expressed as E[(Bt4 − Bt3 )][(Bt2 − Bt1 )] = 0, ∀0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 . Although
the sample paths of Brownian motion are continuous they are nowhere differentiable and
are not of bounded variation. Thus the second integral in Eqn. (3.7) does not exist in
the Riemann sense and is a stochastic integral. The first integral however is completely
deterministic and is of the conventional Riemann form.
For a suitable class of random function h : [0, T ] → < and partitions 0 = t0 < t1 <
t2 . . . < tN = T with maximum spacing ∆, the Itô integral is defined as the mean-square
(m.s.) limit of Itô sums in which the integrand is evaluated at the lower end point of the
partition tj of each sub-interval [tj , tj+1 ]. In other words
Z
0
T
h(t, ω)dBt = m.s. − l.i.m∆→0
N
−1
X
h(tj , ω)(Btj+1 − Btj ),
(3.8)
j=0
where l.i.m.∆→0 is a limit in the mean. Note the difference in this case with conventional
calculus: there is equality here only in the mean-square sense.
Evaluating this limit at points other than the starting point of each time interval, arbitrary values in this interval can be chosen. For instance, if h(t, ω) = Bt (ω) and
evaluation point τj = (1 − λ)tj + λtj+1 where 0 ≤ λ ≤ 1, the mean-square limit is equal
to 21 Bt2 − ( 12 − λ)T . The Itô integral corresponds to the case of λ = 0. R. L. Stratonovich
proposed an alternative stochastic integral [115] wherein the integrand is evaluated at the
midpoint of each interval, i.e. τj = (tn + tn+1 )/2. This is called the Stratonovich integral
62
which corresponds to λ = 1/2. So,
Z T
h(t, ω) ◦ dBt = m.s.
0
−l.i.m∆→0
N
−1
X
h((tj + tj+1 )/2), ω)(Btj+1 − Btj )
(3.9)
j=0
and the Stratonovich SDE is written in differential form as
dXt = a(t, Xt )dt + b(t, Xt ) ◦ dBt .
(3.10)
The symbol ◦ is used to indicate that the equation under consideration is being interpreted
in the Stratonovich sense and the Stratonovich integral equation becomes
Z t
Z t
Xt = X0 +
a(s, Xs )ds +
b(s, Xs ) ◦ dBs .
t0
(3.11)
t0
The solution of the Itô SDE in Eqn. (3.7) is a diffusion process with transition probability
density p = p(s, x; t, y) that satisfies the Fokker-Planck equation [116]
p :=
∂p
∂
1 ∂2
+
(ap) −
(σp) = 0
∂t
∂y
2 ∂y 2
(3.12)
with σ = b2 . The chain rule for Itô stochastic calculus differs from deterministic calculus
because of the inclusion of an additional term which arises because the expectation of the
squared increment of Brownian motion (E[(∆B)2 ]) on a time interval of length ∆t is equal
to ∆t. This is a first order term. Letting Yt = U (t, Xt ) where U : [t0 , T ] × < → < has
continuous second order partial derivatives and Xt is a solution of the second order Itô SDE
(Eqn. (3.7)), the differential of Yt satisfies the equation
dYt = L0 U (t, Xt )dt + L1 U (t, Xt )dBt .
(3.13)
The partial differentiation operators are defined as
L0 U =
∂U
∂U
1 ∂2U
+a
+ b2 2
∂t
∂x
2 ∂x
(3.14)
and
∂U
(3.15)
∂x
with all functions evaluated at (t, x). The Itô formula of Eqn. (3.13) can be interpreted as
L1 U = b
the Itô stochastic integral equation
Z t
Z t
Yt = Yt0 +
L0 U (s, Xs )ds +
L1 U (s, Xs )dBs
t0
t0
(3.16)
63
for t ∈ [t0 , T ]. In the case for Stratonovich calculus, the chain rule happens to be exactly
the same as for the conventional (deterministic) case. Like before, if Yt = U (t, Xt ) and Xt
is the solution of the Stratonovich SDE in Eqn. (3.10), then
dYt = L0 U (t, Xt )dt + L1 U (t, Xt ) ◦ dBt ,
(3.17)
where the partial differential operators are defined as
L0 U =
∂U
∂U
+a
∂t
∂x
(3.18)
and
∂U
.
(3.19)
∂x
This is exactly the same form as the chain rule for ODEs. A more rigorous explanation for
L1 U = b
the difference between the Itô and Stratonovich forms is provided in Appendix A.
3.3.1
A Simple Example
As an example to illustrate the difference in the Itô and Stratonovich form of the solution,
the following SDE
dXt
= a t Xt
(3.20)
dt
is solved exactly. Here at = k + αBt , and α, k are constants. Writing this equation in
differential form gives
dXt = kXt dt + αXt dBt
(3.21)
or
dXt
= kdt + αXt dBt
Xt
Integrating this equation and using the initial condition B0 = 0, we get
Z t
dXs
= kt + αBt .
0 Xs
(3.22)
(3.23)
Itô’s formula (Eqn. (3.13)) is now used to evaluate the integral above. Choose Y (t, Xt ) =
U (t, Xt ) = ln Xt and obtain
d(ln Xt ) =
=
=
1
1
1
dXt + (− 2 )(dXt )2
Xt
2 Xt
dXt
−1 2 2
−
α Xt dt
Xt
2Xt2
dXt 1 2
− α dt.
Xt
2
64
Hence
dXt
1
= d(ln Xt ) + α2 dt.
Xt
2
Using this result in the integral equation (Eqn. (3.23)) we get
ln
Xt
1
= (k − α2 )t + αBt
X0
2
or equivalently
1
Xt = X0 exp((k − α2 )t + αBt ).
(3.24)
2
This form of the solution corresponds to the Itô form. The Stratonovich form of the solution
is exactly like one would expect from classical calculus. Interpreting the SDE in Eqn. (3.21)
as being of Stratonovich form, we can rewrite it as
dXt = kXt dt + αXt ◦ dBt .
(3.25)
Xt = X0 exp(kt + αBt )
(3.26)
The solution of this equation is
which serves to emphasize two points made earlier. One, that depending on the interpretation of the SDE, two markedly different outcomes can result and two, the Stratonovich
form gives results that conform with results from classical calculus. The difference between
the two forms is demonstrated graphically in Fig. 3.1, wherein the values X0 = 0, k = 2
and α = 1 have been used. Note that both results are of the form
Xt = X0 exp(µt + αBt )
and the difference lies in the drift coefficient µt . This equivalence is taken up in the next
subsection. The process of this type is called a geometric Brownian motion and is widely
used in biology and economics.
3.3.2
Equivalence of the Itô and Stratonovich forms
As noted in the earlier subsection, the Itô and Stratonovich forms provide in general, different results to the same SDE. The difference lies in the drift coefficient as was noted in the
example above. This makes it possible to transform the solution from one form to another
by modifying the drift coefficient appropriately. For example, starting out with the Itô SDE
dXt = a(t, Xt )dt + b(t, Xt )dBt ,
(3.27)
65
50
Amplitude of random variable X
Ito solution
Stratonovich solution
25
0
0
1
0.5
1.5
time
Figure 3.1: Differences in the solution of the SDE in Eqn. (3.21) assuming the Itô and
Stratonovich interpretations, with X0 = 0, k = 2 and α = 1.
the corresponding Stratonovich SDE is
1
∂
dXt = [a(t, Xt ) − b(t, Xt ) b(t, Xt )]dt + b(t, Xt )dBt .
2
∂x
(3.28)
On the other hand, starting with the Stratonovich SDE
dXt = p(t, Xt )dt + q(t, Xt ) ◦ dBt ,
(3.29)
the corresponding the Itô SDE is
1
∂
dXt = [p(t, Xt ) + q(t, Xt ) q(t, Xt )]dt + q(t, Xt )dBt .
2
∂x
(3.30)
This means it is possible alternate between the two forms, choosing whichever form is
suitable to the application under consideration.
66
3.4
Vector SDEs
All the above arguments carry over to vector values SDEs. An N-dimensional SDE and an
M-dimensional BM process Bt = (Bt1 , . . . , BtM ) can be expressed in vector form as
dXt = a(t, Xt )dt +
M
X
bj (t, Xt )dBtj
(3.31)
bi,j (t, Xt )dBtj
(3.32)
j=1
which can be written component-wise as
dXti
=
a(t, Xti )dt
M
X
+
j=1
where i = 1, . . . n. The coefficient bi,j is the (i, j)t h component of the N × M matrix
B = [b1 | . . . |bM ] where the bj s are column vectors. The Itô chain rule for this system of
equations now becomes
0
dYt = L U (t, Xt )dt +
M
X
Lj U (t, Xt )dWtj
(3.33)
j=1
where the partial differential operators L0 , L1 , . . . Lj are given by
L0 =
M
N
N
X
∂
∂
1 X X k,j l,j ∂ 2
ak k +
b b
+
∂t
2
∂x
∂xk ∂xl
(3.34)
k,l=1 j=1
k=1
and
j
L =
N
X
bk,j
k=1
∂
∂xk
(3.35)
for j = 1, . . . , M . The corresponding Stratonovich SDE in component form is
dXti
=
ā(t, Xti )dt
+
M
X
bi,j (t, Xt ) ◦ dBtj
(3.36)
j=1
which has the same solutions as the Itô SDE with a modified drift coefficient. That is
N
āi (t, Xt ) = ai (t, Xt ) −
M
1 X X k,j
∂bi,j
b (t, Xt )
(t, Xt )
2
∂xk
(3.37)
k=1 j=1
where i = 1, . . . n. Just like in the scalar case, the system of Stratonovich equations can be
solved using classical ODE methods.
67
3.5
Itô v/s Stratonovich Forms
As noted earlier, both the Itô and the Stratonovich forms are mathematically accurate but
interpreting an SDE in either form will give different end results. This is a paradox, which
must be overcome if one is to favorably accept the results of the stochastic differential
framework. This is not the first instance in which this conundrum has been addressed.
There are several approaches that have been used to provide a justification for choosing an
interpretation. Here, we refer to the work of West et al. in [117] where they trace back the
origin of the SDE to Langevin’s equation, see Eqn. (3.2). This equation in its original form
is linear in the dependent variable of the system and the stochastic terms are purely additive. The probability distribution function of this dependent variable can be determined
by the so-called Fokker-Planck equation of evolution, [116]. Obtaining this Fokker-Planck
equation requires the use of the ordinary rules of calculus. If the Langevin equation is linear
and the stochastic term is purely additive, then the final form of the Fokker-Planck equation
obtained using Itô’s rules of calculus are identical to the form obtained using the ordinary
rules of calculus, or the form proposed by Stratonovich. However, when the fluctuations are
non-additive, as in the more general case, the two interpretations produce different results
although as shown earlier, one can go from one form to another. The key to understanding
the right approach to use requires an understanding of the underlying assumptions behind
each approach. In [117], West et al. perform a “systematic analysis of correlations.” When
the SDE involves a multiplication between the deterministic term and the stochastic term,
the differences between the two interpretations become immediately evident. One form assumes that there is a finite amount of correlation between the deterministic and stochastic
term. This is the form used by Stratonovich. The other form, or Itô’s classical form, considers zero correlation between the deterministic and stochastic components. As shown in
[117], the probability density function obtained with the Itô assumption is not normalizable
and is therefore not a valid probability density function. Starting out with the Stratonovich
assumption will, on the other hand, produce a probability density function that is normalizable. The result remains unchanged even if one assumes a non-zero correlation between the
stochastic and deterministic term which goes to zero in the limiting case, as demonstrated
by the Wong-Zakai theorem, [119]. The conclusion one can draw is that when trying to
model a “physical” system, there will in general always be a finite amount of correlation
between the deterministic and stochastic terms. The solution of a such a model can then
68
only be appropriately determined in the Stratonovich sense. The essence of these conclusions are shown to be unchanged using other methods of rationalization like using a Master
equation in [120]. This has also been validated experimentally in [118] wherein the authors
consider physical and chemical systems described by Langevin models with non-additive
fluctuations and find the Stratonovich assumption to be accurate. Further experimental
validation can be found in [121]. To allow for consideration of multiplicative effects, our
approach has therefore made use of the Stratonovich assumption.
3.6
Summary
This chapter provided a brief overview of the theory of SDEs starting from the original Langevin form. It highlights the concepts behind the formulations of the Itô and
Stratonovich forms and shows how one may obtain different results while using a particular
interpretation. The vector SDE case is also touched upon and shown to be an extension
of the scalar case. The “controversy” between the Itô and the Stratonovich forms is also
brought up. The resolution of this controversy is dependent on the nature of the process
that is being modelled with the SDE. If the interactions between the stochastic and deterministic terms in the equation cannot be neglected, then it is reasoned that the Stratonovich
form is the appropriate one, which is based both on prior theoretical and practical work.
69
Chapter 4
Nonlinear Dynamics, Chaos and
Intermittency
4.1
Introduction
The mathematical theory of nonlinear dynamics attempts to explain the evolution of processes by using nonlinear equations. Every system is intrinsically dependent on time. Some
systems show more variation with time than others. Some systems are periodic, i.e. they
exhibit repetitive behavior while other systems tend to be less affected by the march of
time. However, any system viewed long enough, will show a change in its properties or
characteristics. Technically speaking, the time-dependent characteristics of every system
falls under the rather wide umbrella of dynamical systems. A subset of these systems have
linear responses in time while the vast majority of these systems are influenced by nonlinear mechanisms. It is the aim of nonlinear dynamics to try to describe these seemingly
boundless variety of systems in a clear and precise framework of the language of mathematics. Popular examples of nonlinear dynamical systems include population dynamics, stock
market trends, pendulum motion and weather patterns.
70
4.2
Nonlinear Dynamics and Chaos
The types of systems typically considered here are nonlinear, discrete and iterative in nature
and are also referred to as maps. To illustrate this, consider an example of such a system.
A classic example for modelling the dynamics of population growth is the logistic map.
Herein, if n denotes discrete time and xn denotes the fraction of the total population at
time n, then the logistic map expresses xn+1 , the fraction of the total population at the
next unit of time, as
xn+1 = λxn (1 − xn ).
(4.1)
Here λ is a constant that depends on environmental conditions and for the logistic map,
0 < λ ≤ 4. Watching how this system works, or monitoring its evolution, is achieved by
starting off with an initial condition x0 and computing x1 using the formula above. That
completes iteration 1. For iteration 2, this x1 value is used to compute a value for x2 . This
iterative process produces a stream of values which are collectively called the orbit of the
map. This is reminiscent of the feedback system in Fig. 2.11. The logistic map by virtue
of its construction takes values that lie between 0 and 1. The map is shown graphically in
Fig. 4.1. An example of an orbit that starts at x0 = 0.2 and undergoes 100 iterations is
shown in Fig. 4.2. A preliminary impression of this figure might indicate the appearance
of some complex behavior that arises from a simple deterministic iterative rule. Indeed, it
is this fascinating property of chaotic maps that is used in the development of electronic
device noise models, as will be explained later.
4.2.1
Fixed Points
There are several types of orbits possible for a dynamical system. A fixed point x∗ is a
point that satisfies the equation f (x∗ ) = x∗ , for any nonlinear function f (x). For example,
finding the fixed points of the logistic map would require solving the equation λx(1 − x) = x
or λx(1 − x) − x = 0 for x. This gives the fixed points of the logistic map as either x∗ = 0
or x∗ = (λ − 1)/λ. Fixed points can also be determined geometrically by finding the point
of intersection of the map with the y = x line or the 45◦ line. So, for the logistic map
with λ = 4 the y = x line intersects the map at x = 0 and at x = 0.75, which can be
analytically verified to be the fixed points of this map. This graphical procedure is made
clearer in Fig. 4.3. A fixed point is called attracting if an orbit tends to approach it whereas
71
1
0.9
0.8
0.7
x
n+1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
xn
0.6
0.8
1
Figure 4.1: The logistic map, with λ = 4.
Figure 4.2: An example orbit of the logistic map, with x0 = 0.2.
72
it is called repelling if an orbit moves further away from it with increasing iteration count.
Mathematically, the nature of a fixed point can be determined by the magnitude of the
derivative of the map evaluated at the fixed point. So, if x∗ is a fixed point, then it is an
attracting fixed point if |f (x∗ )| < 1 and it is a repelling fixed point if |f (x∗ )| > 1.
1
0.9
0.8
0.7
xn+1
0.6
fixed point
at x = 0.75
0.5
0.4
0.3
0.2
0.1
0
0
fixed point
at x = 0
0.2
0.4
xn
0.6
0.8
1
Figure 4.3: The logistic map with λ = 4 and its fixed points, as indicated.
4.2.2
Periodic Points
Another kind of orbit is a periodic orbit. A point xp is periodic if f k (xp ) = xp for some
positive integer k. The smallest value of k for which this is true represents the prime period
of the orbit. As another example of a nonlinear map, consider the quadratic map given by
xn+1 = x2n + c
(4.2)
where c is the constant parameter for the map. Fig. 4.4 shows the map for c = 0.
A prime-period 2 orbit for the quadratic map with c = −1 is shown in Fig. 4.5. The orbit
starts at x0 = 0.
73
1
fixed point
at x = 1
0.8
x
n+1
0.6
0.4
fixed point
at x = 0
0.2
0
−1
−0.5
0
xn
0.5
1
Figure 4.4: The quadratic map with c = 0 and its fixed points.
Figure 4.5: The quadratic map with c = −1 and prime-period 2 orbit.
74
4.2.3
Neutral Fixed Points
A neutral fixed point is another type of fixed point in the sense that it is neither attracting
nor repelling. Therefore, if x∗ is a neutral fixed point, then the magnitude of the derivative
of the map at the neutral fixed point is unity. In other words, |f 0 (x∗)| = 1. Although
it is neither attracting nor repelling, a map having a neutral fixed point can behave in
more complicated ways in the vicinity of the neutral fixed point. This behavior can be
ascertained by finding higher order derivatives of the map evaluated at the neutral fixed
point, typically second and third order derivatives. Graphically this is depicted in Fig. 4.6
which shows general nonlinear maps wherein the intersection of the map and the y = x
line represents the neutral fixed point. The orbits behave differently in the vicinity of this
I
repel
II
attract
attract
repel
Neither attracting nor repelling
Neither attracting nor repelling
III
IV
repel
attract
attract
repel
Weakly repelling
Weakly attracting
Figure 4.6: Different types of neutral fixed points.
75
fixed point as shown in the figure. The orbits can be either attracting from one end and
repelling from the other (Types I and II), repelling from both ends (Type III) or attracting
from both ends (Type IV).
4.2.4
Bifurcation Theory
The word bifurcation itself means splitting into two. In the context of nonlinear dynamics,
the theory of bifurcation is an analytical and visual technique to observe how the dynamics
of a nonlinear map change with variation in the parameter of the map. As a visual example,
consider once again the quadratic map with parameter c expressed by the equation fc (x) =
x2 + c. When c > 0.25, there are no fixed points since the map (the parabola) does not
intersect the y = x line. This situation is depicted in Fig. 4.7 where the parameter c = 0.4
and also shows a divergent orbit originating at x = 0.
1.5
x
n+1
1
0.5
0
−2
−1.5
−1
−0.5
0
xn
0.5
1
1.5
2
Figure 4.7: Quadratic map with c = 0.4 and with no fixed points.
As the value of c reduces to 0.25, a fixed point is formed as the y = x line becomes a tangent
to the map. This is shown in the Fig. 4.8. As c decreases below 0.25, the fixed point splits
76
(bifurcates) into two fixed points. Thus a bifurcation is said to have occurred. In the case
of this map, the bifurcation is of the “tangent” type or the “saddle-node” type, the precise
definition of which will follow soon. There are other forms of bifurcations possible but apart
from the tangent bifurcation, only the so-called period-doubling bifurcation will be defined
later. The two fixed points produced as a result of the tangent bifurcation have different
properties. One of them is an attracting fixed point and the other is repelling. This is
shown in the Fig. 4.9. One of the orbits shown in the figure moves and eventually converges
to the attracting fixed point at x = 0.0 and the other orbit moves away from the fixed point
at x = 0.75 and diverges to infinity.
1.6
1.4
1.2
1
fixed point
0.8
0.6
0.4
0.2
0
−1
−0.5
0
0.5
1
Figure 4.8: Quadratic map with c = 0.25 and with one fixed point.
The dynamics of this quadratic map are more complicated than presented so far.
However, the presentation so far is meant to provide a feel for the richness in characteristics
that are possible when the parameter of a map as seemingly simple as the quadratic map
is varied.
77
1.4
1.2
1
repelling fixed
point
0.8
0.6
0.4
attracting fixed
point
0.2
0
−0.2
−1
−0.5
0
0.5
1
Figure 4.9: Quadratic map with c = 0.0 and with two fixed points, one attracting and one
repelling.
Tangent Bifurcation
For a one-parameter family of nonlinear maps, fλ (x) with continuous partial derivatives, a
tangent bifurcation is said to have occurred at parameter value λ0 and fλ0 (e) = e if there
is an open interval I and a δ > 0 such that:
1. For λ0 − δ < λ < λ0 + δ, the map has no fixed points in I.
0
2. fλ0 (e) = 1, the fixed point is neutral.
00
3. fλ0 (e) 6= 0.
4.
∂
∂λ fλ0 (e)
6= 0.
5. For λ0 < λ < λ0 + δ, the map has two fixed points, one attracting and one repelling.
Intuitively, this means that the map has no fixed points outside of the interval I
for values of λ slightly different from λ0 , has one fixed point when λ = λ0 and two fixed
points, one attracting and one repelling, for λ > λ0 . Condition 3 indicates that a map is
generally concave-up or concave-down near the fixed point, so that there is only one fixed
point when λ = λ0 and x = e.
78
Period-Doubling Bifurcation
Another common type of bifurcation is the period-doubling bifurcation which occurs in a
one-parameter family of nonlinear maps, fλ (x), with continuous partial derivatives at parameter λ = λ0 and at x = e if there is an open interval I and δ > 0 if:
1. For all λ ∈ [λ0 − δ, λ0 + δ], there is a unique fixed point e for the map in I.
0
2. fλ0 (e) = −1, the fixed point is neutral.
3. For λ0 − δ < λ ≤ λ0 , the map has no period-2 cycles in I and e is attracting (resp.
repelling).
4. For λ0 < λ < λ0 + δ, there is a unique period-2 cycle d1 and d2 in I with fλ (d1 ) = d2 .
The period-2 cycle is attracting (resp. repelling) and the fixed point e is repelling (resp.
attracting).
5. All the periodic orbits tend to the fixed point as λ → λ0 .
000
6. fλ0 (e) 6= 0.
7.
0
∂
2
∂λ (fλ0 (e))
6= 0.
Intuitively, as the parameter of the map changes, a fixed point may go from attracting to repelling and give rise to a attracting period-2 cycle. Conversely, a fixed point
may go from repelling to attracting and give rise to a repelling period-2 cycle. It is important to note that a period-2 cycle may itself undergo a period-doubling bifurcation and
give rise to a period-4 cycle. This process can continue indefinitely and give rise to a large
number of periods of the power of two.
A bifurcation diagram is a convenient way to represent all forms of bifurcation for a
particular map in the two-dimensional plot, or in the λ-x plane. For each value of λ, it plots
all the possible values that the map can take. The bifurcation diagram (the c-x plane) for
the quadratic map is shown in Fig. 4.10 with the tangent and period-doubling bifurcations
associated with this map appropriately annotated. The lone tangent bifurcation represents
the formation of a fixed point as was explained earlier and a successive cascade of perioddoubling bifurcations follow. A striking feature of this map is its self-similar structure. This
is highlighted by the square boxes in the figure. The areas inside the boxes are scaled versions
of each other and indeed, one can observe this repeating trend at smaller and smaller scales.
This is a feature of a typical bifurcation diagram. The period-doubling regions abruptly end
and give rise to what is known as the period-3 window which is then followed by a dense
79
Figure 4.10: Bifurcation diagram for the quadratic map.
80
seemingly structure-less region called the chaotic region. The significance of the period-3
window and the chaotic region are discussed next.
4.2.5
Sliding into Chaos
The bifurcation diagram depicts the path or the route to chaos. It enables an investigation
into the types of bifurcations that a map undergoes until it finally reaches the chaotic
regime. In the quadratic map considered earlier, there are successions of period-doubling
bifurcations before the relatively sparse region called the period-3 window is reached. As
the name suggests, this region is characterized by the existence of an orbit having a period
of 3. Li and Yorke in [36], as a special case of the theorem of Sarkovskii, showed that for a
map to exhibit chaos it had to have an orbit of prime-period 3. This theorem provides a way
to easily determine if a map is chaotic. All that is required is to find an orbit of period 3.
Inside the period-3 window of the quadratic map further period-doubling bifurcations occur
resulting in a period-6 orbit, and so on. Following this period-3 window is a region that is
seemingly formless. This is the chaotic region and for every map there is a critical value
of its parameter beyond which this regime is reached. Although it may seem impossible,
there are several mathematical definitions available that can define what chaos is. Given
next is the definition according to Devaney in [33]. Before that, one needs to understand
the meaning of a dense set. If X and Y are sets and Y is a subset of X, then the set Y is
dense in X if, for any point x ∈ X, there is a point y ∈ Y that is arbitrarily close to X.
The measure of closeness is provided by some sort of distance function. According to the
definition by Devaney, a dynamical system f (x) is chaotic if:
1. The periodic points of the dynamical system are dense in the set of all points.
2. The dynamical system is transitive, meaning that for any two points x and y and any
² > 0, there is a third point z within ² of x whose orbit comes within ² of y.
3. The dynamical system exhibits Sensitive Dependence on Initial Conditions (SDIC) which
means that there exists a β > 0 such that for any x and any ² > 0, there is a y within ²
of x and an integer k such that the distance between f k (x) and f k (y) is at least β. This
is another way of saying that no matter how close together two orbits start off, after k
iterations they will end up at least β apart. This is the most “popular” definition of chaos.
Chaos manifests itself in a variety of ways and one way to look at it, apart from
the bifurcation plot, is by finding a histogram plot or density plot of a dynamical system
81
in the chaotic regime. For a large enough number of iterations, the histogram will always
contain points in every bin irrespective of the width of the bin. Manifestations of chaos
have been observed in engineering, for example in Chua’s circuit [122]. Applications of this
theory are also emerging like in the field of communications theory as in the work of Pecora
and Caroll in [123], Cuomo, Oppenheim and Strogatz in [124], Chen and Yao in [125] and
Kocarev and Parliz in [126], image watermarking in the work of Nikolaidis and Pitas in
[127] and in the modelling of internet packet traffic flows in the work of Mondragin, Pitts
and Arrowsmith in [128] and Erramilli, Singh and Pruthi in [129].
4.3
Generating White Noise
There are several chaotic functions that have been used to successfully generate a white
noise sequence or a sequence with delta correlation. The Chebyshev map was used in [130],
the Bernoulli map in [131] and several other examples found in [34]. In this work the
logistic map with parameter λ = 4 is used as a source of white noise and a realization
of the map produces a discrete sequence of numbers that are meant to represent a white
noise process. This map has been defined in Eqn. (4.1). To demonstrate that this map is a
suitable generator of white noise, the correlation plot of a sequence generated by this map
is shown in Fig. 4.11. From the definition in Eqn. (2.9), the correlation plot is a measure
of the relationship between different values of a random variable for different values of lag.
This figure clearly shows the delta correlation properties that one expects from a good white
noise generator.
The spectrum of this noise source is obtained by performing the Fourier transform
operation on the autocorrelation sequence and the result is shown in Fig. 4.12. Excellent
techniques of generating white noise with pseudo-random numbers with large periods also
exist, for example [132]. This approach has been used inside some of the circuit models in
f REEDATM as white noise generators. These details are provided in the next chapter.
4.4
Intermittency and Flicker Noise
The phenomenon of intermittency was reported by Pomeau and Manneville in [133] while
observing turbulence in fluids wherein sudden transitions between stable periodic states and
82
1
Autocorrelation
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
Lag parameter (i)
Figure 4.11: Correlation plot of the logistic map with λ = 4.
Normalized Power
0.1
0.01
1
10
Frequency (Hz)
Figure 4.12: Spectrum of the logistic map with λ = 4.
100
83
chaotic states was observed. A feature of any intermittent function is that it exhibits regular
laminar phases separated by intermittent bursts and uses simple nonlinear iterative deterministic maps to produce complex stochastic-looking series. In the sub-sections that follow,
a single parameter intermittent map is used to generate sequences with long memory characteristics but before that, some basic definitions are concepts are reviewed. An overview
of the theory if intermittency can be found in reviews articles in [134] and [135]. The next
few subsections present the mathematical basics of intermittent functions, describe typical
properties and explore maps that display intermittency. In particular, the logarithmic map
uses only a single parameter but still has very desirable memory characteristics. This makes
it suitable for modelling noise in electronic devices and circuits.
4.4.1
Basic Theory
Let fλ (x) : J → J be a parametric family of maps of parameter λ representing a chaotic
nonlinear process where J is a closed interval, generally [0, 1]. A realization of the discrete
time sequence produced by iteration on the nonlinear map is denoted by xn , n = 0, 1 . . .,
where n denotes units of time. Let the parameter λ have a particular value, i.e. λe at which
the map fλ (x) has a neutral fixed point at xe . So, at xe for parameter λe , the map has a
value fλe (xe ). The neighborhood of the neutral fixed point xe is given by
Nxδ (xe ) = {x : x ∈ [xe − δ, xe + δ]}
(4.3)
and the analogous neighborhood of the parameter value λe corresponding to the neutral
fixed point is given by
Nλδ = {λ : λ ∈ [λe − δ, λe + δ]}
(4.4)
Recall, that for any map a fixed point of the map is the solution of the equation fλ (x) = x.
This fixed point is said to be attracting if the following condition holds:
0
|fλ (x)|(λe ,xe ) | < 1
(4.5)
For a fixed point to be repelling
0
|fλ (x)|(λe ,xe ) | > 1
(4.6)
and for a fixed point to be neutral
0
|fλ (x)|(λe ,xe ) | = 1.
(4.7)
84
Neutral fixed points can be broadly classified into three categories: a) Weakly attracting,
b) Weakly repelling and c) Attracting/Repelling fixed point. By weakly repelling we mean
that the orbit of the map in the vicinity of the fixed point gradually tends to drift away
from the fixed point. More precisely, a fixed point is weakly repelling if there exists a δ > 0
such that for all x ∈ Nxδ , the map fλ (x) can be represented as
fλ (x) = x + F (x),
(4.8)
where F (x) satisfies the following properties:
1. F (x) is positive on the right-hand side of the neutral fixed point excluding the neutral
fixed point.
2. F (x) is negative on the left-hand side of the neutral fixed point excluding the neutral
fixed point.
3. F (x) is differentiable on Nxδ0 (xe ) and
4. F (λ, x) → o(|x − xe |) as x → xe .
For the generation of long memory intermittent sequences, only maps with a weak repelling
fixed point can be used, [136].
4.4.2
Characteristics of Intermittency
According to [136], a Type-II repelling fixed point intermittency for a parametric family of
maps fλ (x) with a unique weakly repelling fixed point xe at parameter value λe satisfies
the following conditions:
(1) There exists a parameter value λ = λ0 such that the map fλe (x) exhibits a unique
neutral fixed xe .
(2) There exists a collection {Ij : j = 1, . . . , m} with m > 1 of closed intervals with disjoint
interiors satisfying J = ∪m
j=1 Ij , for all (λ, x) ∈ Nλδ ×J, the map is a continuous function of x
and λ in the interior of each Ij and for λ = λe , the map satisfies the irreducibility condition,
p(i)
which means that for each i = 1, . . . , m, one can find and p(i) such that fλ (Ii ) ⊃ J. The
irreducibility condition is another way of saying that the orbit of the map will eventually
leave J.
(3) There exists a j ∗ and a δ > 0 such that
(i) xe ∈ Ij ∗
(ii) Nxδ (xe ) ⊂ Ij ∗
85
(iii) For all λ ∈ Nλδ (λe ) and x ∈ Nxδ (xe ), fλ (x) can be written as
fλ (x) = (λ − λe ) + x + F (λ, x)
(4.9)
(iv) F (λ, x) > 0 is a positive function of x if x > xe and F (λ, x) < 0 is a negative function
of x is x < xe .
(v) F (λ, x) is differentiable for all (λ, x) ∈ Nλδ (λe ) × Nxδ (xe ) and
∂
F (λ, x)|(λe ,xe ) = 0
∂λ
(4.10)
F (λ, x) = o(|x − xe |),
(4.11)
and
as x → xe .
(4) For all x ∈
/ Nxδ (xe ) and all λ ∈ Nλδ (λe ) the map is uniformly expanding, meaning that
|fλ (x)| > 1.
(5) For λ 6= λe , there exists a repelling fixed point in Nxδ (xe ) but no attracting fixed point.
In [137], Eqn. (4.9) takes the form
fλ (x) = x + x2 F (λ, x)
(4.12)
with limx→0 xF (λ, x) = 0 and limx→0 F (λ, x) = ∞.
4.4.3
Nonlinear Intermittent Functions
In accordance with Eqn. (4.9), an example of a map that exhibits intermittency is the
Polynomial Map defined on J = [0,1] as:

 x(1 + 2a xa ) if 0 ≤ x ≤ 1/2
fλ (x) =
 2x − 1
if 1/2 < x ≤ 1
(4.13)
The long-memory behavior of the map can be ascertained by computing its correlation
function. In this case, although the correlation function cannot be found exactly, a bound
for it has been found in [138]. It is given as
Rf,g (n) ≤ Bn1−1/a .
(4.14)
86
It can be shown that this map satisfies all the conditions in the definition in the previous
section and it shows long memory if a ∈ (1/2, 1).
In accordance with Eqn. (4.12), an example of a map that exhibits intermittency is the
Logarithmic Map defined on J = [0,1] as:

 x(1 + Y (β)x log(1/x)1+β ) if 0 ≤ x ≤ 1/2
fβ (x) =
 2x − 1
if 1/2 < x ≤ 1
(4.15)
where Y (β) = 2(log 2)−(1+β) is chosen to ensure that limx→1/2− fβ (x) = 1. It can be shown
[137] that the rate of decay of correlation of this map is bounded as
R(n) ≤ B(log n)−β .
(4.16)
This is said to have a logarithmic mixing rate which can be made as slow as desired by
varying the value of β. The logarithmic mixing rate is “slower” than the polynomial rate
and this makes it a desirable generator of 1/f noise.
4.5
The Logarithmic Intermittent Map
The intermittent map of Eqn. (4.15) is used to generate a chaotic sequence with power-law
(long memory) properties. The description of the map over the unit interval is shown in
Fig. 4.13. Iterations on this map produce an intermittent time sequence that shows distinct
regions of laminar and chaotic behavior, thereby emphasizing its intermittent structure.
This is shown in Fig. 4.14. The output from this map can be thought of as a “signal”
with low-frequency content, a 1/f signal. This map shows the ability to generate a 1/f like frequency-domain response with a single parameter β. For the value of β used here
(0.000005), the covariance of the map has been shown ([137]) to decay at a logarithmic
rate which is slower than an exponential rate. The correlation plot of this map is shown in
Fig. 4.15 and it makes the slow decay property evident, particularly in comparison to the
white noise case of Fig. 4.11. This spectrum of this signal is shown in Fig. 4.16 where a
million points were used to generate the corresponding time sequence.
This slow rate or long memory translates to a 1/f response in the frequency
domain. It is much harder to get a long-memory characteristic in the frequency domain
with exponential rates which are conventionally produced by auto-regressive (AR) sequences
87
1
0.9
0.8
0.7
y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 4.13: The logarithmic map, with β = 0.000005.
1
0.9
0.8
map output
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1000
2000
3000
4000
5000
6000
sequence n
7000
8000
9000
10000
Figure 4.14: Sample realization of the logarithmic map, with β = 0.000005.
88
1
0.9
0.8
Autocorrelation
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
80
60
100
Lag parameter (i)
Figure 4.15: Correlation plot of the logarithmic map with β = 0.000005.
Normalized Power
1
0.1
0.01
0.1
1
10
100
Frequency (Hz)
Figure 4.16: Spectrum of the logarithmic map with β = 0.000005.
89
or auto-regressive moving-average (ARMA) sequences. In both cases, a large number of
parameters are required to get the desired response. In the case of the logarithmic map,
a 1/f -like response is produced with a single parameter and a large variety of responses
can be produced by tweaking β. This is one of the most important advantages of using
chaotic maps. When compared with the infinite transmission line models of Section 2.6.2,
this approach is also more suited to implementation in a circuit simulator. This has been
detailed in Appendix B.
4.6
Summary
This chapter presented a brief introduction to nonlinear dynamics and the meaning of chaos
in a mathematical and graphical framework. After mentioning the various types of possible
fixed points, bifurcation diagrams were used to understand the common types of bifurcations found in dynamical systems. The phenomenon of intermittency was introduced and
its properties were mentioned. In particular, a class of intermittent maps having certain
properties were highlighted and the logarithmic map was introduced. This map has a desirable property of producing a long-memory sequence or a 1/f -like response in the frequency
domain using a single parameter. This is also more amenable to implementation in a circuit
simulator environment.
90
Chapter 5
Implementation of Stochastic
Framework
5.1
Introduction
The theories of Chapter 3 and Chapter 4 are brought to life by implementing them in the
circuit simulator f REEDATM the details of which are provided in this chapter. Section 5.2
is a brief introduction to f REEDATM and touches upon the main underlying principles
upon which f REEDATM is built along with some of the support libraries it requires and
the model catalog that it supports. Section 5.3 is an introduction to the basic fixed timestep routine in f REEDATM and explains how a general circuit is split up into its linear and
nonlinear parts and how the error function is formulated and analyzed. Section 5.4 extends
the basic transient analysis to include the noise generator components present in a circuit
and highlights the differences in the error function as a result of these additions.
5.2
An Acquaintance with f REEDATM
The circuit simulator f REEDATM is a state-variable based simulator which is developed
on object-oriented principles [139] that allows for a clean separation of data and algorithm.
91
This permits a device model developer to be ignorant of the details of the implementation
of the algorithm used to analyze the circuit. Likewise, the algorithm developer can be
fairly uninvolved in the model-creation process and still produce a robust analysis routine.
f REEDATM uses several off-the-shelf libraries to perform several computing/mathematical
tasks. As of this writing, f REEDATM uses a combination of the BLAS 1 and SuperLU 2
libraries to perform linear algebra and sparse matrix operations respectively. It uses the
FFTW 3 library to perform fast-fourier transform operations and the NNES 4 library to
solve nonlinear systems of equations. It uses the package ADOL-C 5 to automatically compute derivatives for its device models with a minimal execution penalty and with a high
degree of accuracy. This reduces the size of the model code by several degrees and allows
the developer to focus on the numerical characteristics of the model than be bogged down
by manually programming and debugging the derivatives, which has been known to be a
time-consuming process. f REEDATM has a long list of models to choose from. These include conventional electronic models like resistors, capacitors and inductors, voltage and
current sources, semiconductor diodes, BJT, JFET and several levels of MOSFET transistors. There are also microwave elements such as gyrators, circulators, microwave diodes
and MESFETs. There is also support for behavioral models like analog mixers, filters and
Foster’s canonical N-port formulation. There are some exotic models such as models for
molecular devices, in particular molecular diodes. It can also read in table-based models
such as the IBIS model or Y-matrix port information. New models are constantly under
heavy development and this catalog is ever-expanding.
f REEDATM supports several analysis types ranging from a basic fixed-time step
analysis which is explained in detail in the next section, to more complex SPICE-like variable
time step routines. There is also support to perform conventional operating point or DC
analysis as well as the steady-state frequency-domain based Harmonic Balance analysis.
The approach to f REEDATM can be thought of as equation-based process modelling as
opposed to the older sequential modular approach to process modelling in the sense of [140].
This enables f REEDATM to to implement models that are not just restricted to electronic
devices but become a universal circuit simulator with a generic model formulation system
1
www.netlib.org/blas
http://crd.lbl.gov/ xiaoye/SuperLU/
3
www.fftw.org
4
www.netlib.org/opt
5
http://www.math.tu-dresden.de/ adol-c/
2
92
that can use any appropriate variable as a state variable. This allows for a heterogeneous
nature of variables to be present in a single error function. In other words, it allows for
the simulation of different types of devices connected together in useful configurations. A
generic example is shown in Fig. 5.1, which connects spatially-distributed elements to linear
and nonlinear electronic and thermal elements.
SPATIALLY
DISTRIBUTED
CIRCUIT
LINEAR
NONLINEAR
CIRCUIT
NETWORK
LINEAR NETWORK
THERMAL
NETWORK
Figure 5.1: Connections between a heterogeneous collection of elements.
5.3
Transient Analysis in f REEDATM
f REEDATM implements a fixed time step transient analysis routine tran2. The idea is to
convert the differential equations describing the circuit into an algebraic system of nonlinear
equations using time marching integration methods. The circuit is partitioned into separate
groups which contain linear elements and sources in one group and nonlinear elements in
the other group. This is represented schematically in Fig. 5.2. To formulate the error
function, the nonlinear elements are replaced by nonlinear voltage or current sources. For
every nonlinear element, one terminal is assumed to be the reference and the element is
replaced by a set of sources connected between the remaining terminals and the reference
terminal. In general, current sources are preferred over voltage sources because they are
amenable to nodal analysis.
93
5.3.1
Linear Network
To formulate the MNAM of the linear section, two matrices G and C of equal size nm are
created, where nm represents the total number of non-reference nodes in the circuit and
the number of additional required variables [141]. The time-dependent contributions from
fixed sources and the nonlinear elements are inserted into a vector s of length nm on the
right hand side. All conductors and frequency-independent MNAM stamps arising in the
formulation will be entered in G, whereas capacitor and inductor values and other values
associated with dynamic elements will be stored in matrix C. The linear system so obtained
can be written as
Gu(t) + C
du(t)
= s(t).
dt
(5.1)
Here u is the vector of nodal voltages and required currents. The source vector s can be
split into two parts, sf and sv such that s = sf + sv . The sf vector contains contributions
from the independent sources in the circuit while the sv vector has the currents that are
injected into the linear network from the nonlinear network.
v1
I1
v2
I2
v3
I3
Linear network
I1
Nonlinear
device
Vnl(1,2)
I3
Vnl(2,3)
and sources
...
...
vn-1
I n-1
vn
In
...
Nonlinear
v(n-1,n)
Inl(n)
device
Figure 5.2: A partitioned network of linear and nonlinear elements and sources.
94
5.3.2
Nonlinear Network
The nonlinear voltages and currents are expressed as a function of the state variable x as
vN L (t) = u[x(t),
dm x(t)
dx(t)
,...,
, xD (t)]
dt
dt
(5.2)
iN L (t) = w[x(t),
dx(t)
dm x(t)
,...,
, xD (t)].
dt
dt
(5.3)
The error function of an arbitrary circuit is developed using connectivity information which
is described by an incidence matrix and the constitutive relations describing the nonlinear
elements. This connectivity information is placed inside the incidence matrix T. If the
number of columns of T is nm and the number of rows is equal to the number of state
variables ns , then in each row a +1 is entered in the column corresponding to the positive
terminal of the nonlinear element in that row and −1 in the column corresponding to the
negative terminal of the nonlinear element in that row. This means that each row of T has
at most two nonzero elements and the number of nonzero elements cannot exceed 2ns . Let
vL (t) be the vector of port voltages of the nonlinear elements which can be calculated from
the nodal voltages of the linear network as
vL (t) = Tu(t).
(5.4)
Also, the current inserted into the linear network is given by
sv (t) = TT iN L (t).
5.3.3
(5.5)
Formulation of the Error Function
The error function is formulated at the interface between the linear elements and the nonlinear elements/sources. In general, this is a reduced number of variables when compared
with having to solve for every state variable in the circuit. The error functions combines
the contributions from the linear and nonlinear sides as
Gu(t) + C
du(t)
= sf (t) + TT iN L (t).
dt
(5.6)
The error function itself is defined as
f(t) = vL (t) − vN L (t) = Tu(t) − vN L (t) = 0.
(5.7)
95
The size of the error function is equal to the number of state variables present in the
nonlinear part of the circuit and this represents the minimum number of variables that are
required to solve this equation. In the case of most microwave circuits, there are usually an
equal number of linear and nonlinear elements and this approach will reduce the number
of variables to be solved for as compared to the case when the voltage at every node is a
variable.
5.3.4
Conversion to Algebraic Form
The differential equations above are converted into nonlinear algebraic voltage and current
equations and are expressed in discretized time as
0
vN L (xn ) = u[xn , xn , . . . , x(m)
n , xD,n ]
(5.8)
0
iN L (xn ) = w[xn , xn , . . . , x(m)
n , xD,n ]
0
(5.9)
0
where xn = x(tn ), xn = x (tn ), (xD,n )i = xi (tn − τi ) and tn is the current value of time and
τi is a fixed value of time delay. Eqn. (5.6) can be written in discretized form as
0
Gun + Cun = sf,n + TT iN L (xn ).
(5.10)
0
Using time-marching integration approximations, the vector un can be calculated as
0
un = aun + bn−1
(5.11)
0
where bn−1 and un have the same dimensions. This value of un can now be used in
Eqn. (5.10) to obtain
Gun + C[aun + bn−1 ] = sf,n + TT iN L (xn ).
(5.12)
un = [G + aC]−1 [sf,n − Cbn−1 + TT iN L (xn )].
(5.13)
Solving for un gives
Compared with Eqn. (5.7), the discretized version of the error function can be written as
f(xn ) = ssv,n + Msv iN L (xn ) − vN L (xn ) = 0
(5.14)
ssv = T[G + aC]−1 [sf,n − Cbn−1 ]
(5.15)
where
96
and
Msv = T[G + aC]−1 TT .
(5.16)
This error function can be written more compactly as
f(xn ) = vL (xn ) − vN L (xn ) = 0.
(5.17)
The above equation represents a system of equations of order ns where ns is the number of
common nodes between the linear part of the circuit and the nonlinear part, or the number
of state variables in the nonlinear portion of the circuit.
5.4
Implementation of Transient Noise Analysis
This section provides details of the changes that were required to be made in order to
simulate noise in f REEDATM . In particular, changes to the linear and nonlinear models
were made which enhanced the large-signal deterministic models into models that contain
transient sources of noise. The details of these changes are provided in this section. The
effects of these changes are reflected in the error function, which is now dependent on
stochastic state variables describing the circuit under consideration. As a result, the error
function is now a noisy error function as shown in Subsection 5.4.3.
5.4.1
Enhanced Device Models
Several linear and nonlinear device models in f REEDATM were modified to include random
voltage and current terms based on known sources of noise associated with the device. What
follows are descriptions of the changes made to the npn-BJT model, the p-n junction diode,
the MESFET model, the resistor and a white noise generating voltage source.
Gummel-Poon npn-BJT
The BJT model in f REEDATM is based on the Gummel-Poon [142] model which is similar to
the large-signal model in the SPICE circuit simulator. The Gummel-Poon model addressed
drawbacks of the original Ebers-Moll model. The most important of those are secondorder effects such as decrease in current gain at low-current levels and high-level injection
into the base region. Low-current drop in gain results from increased base current due to
97
recombination effects that reduces current gain. The effect of base-width modulation, which
results in a change of the collector-base and emitter-base current was included. High-level
injection effects, which accounts for the injection of a significant amount of minority carriers
into the base region was also considered as was the dependence of base resistance on current
[56]. The large-signal equivalent circuit of the BJT is shown in Fig. 5.3.
RC
Ccx
IBC
CBC
RBB
C
ICC
βr
B
CBE
IBE
IEC
βf
IC
RE
E
Figure 5.3: Large-signal Gummel-Poon BJT circuit.
In f REEDATM , this model consists of 56 parameters and is implemented as a
four-terminal device with 3 state variables. They are the voltage of the base with respect
to emitter (Vbe ), the voltage of the base with respect to the collector (Vbc ) and the voltage
of the collector with respect to the substrate (Vcs ). The outputs associated with the model
are 3 voltages, namely the collector, base and emitter voltages with respect to substrate
and 3 currents, namely the currents flowing in the collector, base and emitter terminals.
The parameters used in the model along with the default values are listed in Table 5.1 and
Table 5.2.
What follows is a summary of the equations that model the characteristics of the
device. The parameters of the model as they appear in the equations are written in a
fixed-width font. The model work in both forward and reverse modes. The base current
consists of two components – a base-emitter current and a base-collector current expressed
as
ib = ibe + ibc .
(5.18)
98
Parameter
is
bf
nf
vaf
ikf
ise
ne
br
nr
var
ikr
isc
nc
re
rb
rbm
irb
rc
eg
cje
vje
mje
cjc
vjc
mjc
xcjc
fc
tf
xtf
vtf
itf
tr
xtb
xti
tre1
tre2
trb1
trb2
Description
transport saturation current
ideal maximum forward beta
forward current emission coefficient
forward early voltage
corner for forward-beta high current roll-off
base-emitter leakage saturation current
base-emitter leakage emission coefficient
ideal maximum reverse beta
reverse current emission coefficient
reverse early voltage
corner for reverse-beta high current roll off
base collector leakage saturation current
base-collector leakage emission coefficient
emitter ohmic resistance
zero bias base resistance
minimum base resistance
current at which rb falls to half of rbm
collector ohmic resistance
bandgap voltage
base emitter zero bias p-n capacitance
base emitter built in potential
base emitter p-n grading factor
base collector zero bias p-n capacitance
base collector built in potential
base collector p-n grading factor
fraction of cbc connected internal to rb
forward bias depletion capacitor coefficient
ideal forward transit time
transit time bias dependence coefficient
transit time dependency on vbc
transit time dependency on ic
ideal reverse transit time
forward and reverse beta temperature coefficient
is temperature effect exponent
re temperature coefficient (linear)
re temperature coefficient (quadratic)
rb temperature coefficient (linear)
rb temperature coefficient (quadratic)
Default
1e−16
100
1
0
0
0
1.5
1
1
0
0
0
2
0
0
0
0
0
1.11
0
0.75
0.33
0
0.75
0.33
1
0.5
0
0
0
0
0
0
3
0
0
0
0
Table 5.1: BJT model parameters in f REEDATM .
Units
A
–
–
V
A
A
–
–
–
V
A
A
–
Ω
Ω
Ω
A
Ω
eV
F
V
–
F
V
–
–
–
S
–
V
A
S
–
–
–
–
–
–
99
Parameter
trm1
trm2
trc1
trc2
tnom
t
cjs
mjs
vjs
area
ns
iss
beta
kf
alpha
Description
rbm temperature coefficient (linear)
rbm temperature coefficient (quadratic)
rbm temperature coefficient (linear)
rbm temperature coefficient (quadratic)
nominal temperature
operating temperature
collector substrate capacitance
substrate junction exponential factor
substrate junction built in potential
current multiplier
substrate p-n coefficient
substrate saturation current
exponent for chaotic map
scaling factor for flicker noise
power for dependence of flicker noise on current
Default
0
0
0
0
300
300
0
0
0.75
1
1
0
0.000005
1
2
Units
–
–
–
–
K
K
–
–
–
–
–
–
–
–
–
Table 5.2: BJT model parameters in f REEDATM continued.
This can also be written as
ibf
ibr
+ ile +
+ ilc .
bf
br
The ideal forward diffusion current is expressed as
µ
µ
¶
¶
Vbe
ibf = is exp
−1
nf vth
ib =
(5.19)
(5.20)
where vth = kT /q. k is Boltzmann’s constant, T is the temperature of operation of the
device and q is the charge of an electron. The base-emitter recombination current ile is
expressed as
µ
µ
¶
¶
Vbe
ile = ise exp
−1 .
(5.21)
nf vth
The reverse diffusion current is a function of the base-collector voltage and is written as
µ
µ
¶
¶
Vbc
ibr = is exp
−1
(5.22)
nf vth
and the base-collector recombination current is written as
¶
¶
µ
µ
Vbc
−1 .
ilc = ise exp
nf vth
(5.23)
The equation for the collector-emitter current can be determined from the above defined
terms as
Ice =
(Ibe − Ile )bf − (Ibc − Ilc )br
KQB
(5.24)
100
where the term KQB is used to model the effect of forward or reverse current roll-off which
can occur due to base-width modulation and high-level current injection effects into the
base. KQB can take two forms depending on whether the parameter ikf or ikr is set. If
ikf is set (forward diffusion), then
KQB
0.5
=
1 − TVAF − TVAR
Ã
r
1+
4(Ibe − Ile )bf
1+
ikf
!
whereas if ikr is set (reverse diffusion) then
Ã
!
r
0.5
4(Ibc − Ilc )bf
KQB =
1+ 1+
.
1 − TVAF − TVAR
ikr
(5.25)
(5.26)
In the above TVAF = Vbc /var and TVAF = Vbe /var. The resistance of the base is a non-ideal
resistance and is a function of KQB . It is written as
µ
¶
rb − rbm
1
rbm +
.
Rbb =
area
KQB
(5.27)
The model in f REEDATM also includes the contribution to the output currents from the
capacitors shown in Fig. 5.3 but the equations are not included here.
To include noise sources in the above model, thermal, shot and flicker noise sources
are considered. Based on the development in [55] and [56], when the transistor is in the
forward active region, minority carriers diffuse and drift across the base into the basecollector region. They undergo acceleration when the enter the collector-base depletion
region because of the field existing there and are swept into the collector. This is a random
process and is a source of shot noise in the collector. Recombination effects in the baseemitter region and carrier injection from the base into the emitter are also random processes
contributing to a shot effect in the base current and emitter current respectively. The
parasitic resistors in the BJT contribute thermal noise modelled as current noise sources.
For the collector, base and emitter resistances respectively, these noise sources are expressed
as
r
it,rc =
s
it,rb =
r
it,re =
2kT
ξc
rc
(5.28)
2kT
ξb
Rbb
(5.29)
2kT
ξe
re
(5.30)
101
where ξc , ξb and ξe are sequences of white noise generated by the Logistic map of Section 4.3.
Three sources of shot noise currents are considered, proportional to the collector-emitter,
base-emitter and base-collector currents respectively. These are written as
is,ce =
is,be =
is,bc =
p
p
p
q|ice |ξce
(5.31)
q|ibe |ξbe
(5.32)
q|ibc |ξbc .
(5.33)
Although there is more than one known source of flicker noise in BJTs [83], it has been
shown that the dominant source of flicker noise can be represented by a single current source
between the base and emitter terminal and is a function of the base-emitter recombination
current [84]. The flicker current noise source can be expressed as
p
if,be = kf |ile |alpha ξf
(5.34)
where ξf is a flicker noise sequence generated by the Logarithmic map in Section 4.4 and α
controls the dependence of the flicker noise component on the non-ideal base current. By
default, it is set to 2.
RC
Ccx
B
C BC
I BC
C
i s,bc
RBB
i t,rc
i s,be
i t,rb C
BE I
BE
i f,be
IC
i s,ce
RE
i t,re
E
Figure 5.4: The Gummel-Poon BJT model, along with noise sources.
Apart from alpha, these changes add two additional parameters to the BJT element. The
first parameter, beta, allows control over the slope of the flicker characteristic in the frequency domain and its default value is set to 0.000005. The other parameter, kf, allows
102
for scaling of the amplitude of the flicker noise process and its default value is set to 0.001.
Adding these noise sources to the noiseless Gummel-Poon BJT model produces the noisy
equivalent large-signal circuit shown in Fig. 5.4. The non-ideal diodes of Fig. 5.3 are omitted
here for simplicity. The underlying assumption of the model in f REEDATM is that all the
sources of noise are uncorrelated with one another. This can be justified by considering each
intrinsic noise-generating processes within the device to be an independent process. The
f REEDATM source code for the noise-enabled npn-BJT model is provided in Appendix C.1.
Diode
The diode model in f REEDATM is based on the SPICE diode and is implemented as a twoterminal device with one state variable, i.e. the voltage across the device. The large-signal
model of the noiseless diode consists of a current source in parallel with a capacitance, both
of which are in series with a parasitic resistance. This is shown in Fig. 5.5. It consists of
15 parameters which are listed in Table 5.3.
Anode
Rs
I
Cd
d
Cathode
Figure 5.5: Large-signal model for a p-n junction diode.
The current flowing through the diode, Id is modelled as a exponential function
that depends on the voltage Vd across it. It is expressed as
Id = is(eVd /nVth − 1) − Ib .
(5.35)
where Vth = kT /q. The current Ib is the current in the diode in breakdown mode. It is
103
Parameter
is
n
bv
bv
fc
cj0
vj
m
tt
area
charge
rs
beta
kf
alpha
Description
saturation current
emission coefficient
current magnitude at the reverse breakdown voltage
breakdown voltage
coefficient for forward-bias depletion capacitance
zero-bias depletion capacitance
built-in junction potential
PN junction grading coefficient
transit time
area multiplier
use charge-conserving model
series resistance
exponent for chaotic map
scaling factor for chaotic noise
power for dependence of flicker noise on current
Default
1e−14
1
1e−10
0
0.5
0
1
0.5
0
1
true
0
0.000005
1
1
Units
A
–
A
V
–
F
V
–
s
–
–
Ω
–
–
Table 5.3: Noisy diode model parameters in f REEDATM .
written as

 0
Ib =
 ibv e(Vd +bv)/nVth
Vd ≥ (1 + bv)
Vd < (1 + bv).
(5.36)
The diode model in f REEDATM calculates charge between the diode terminals and computes the resulting current as a derivative of charge. It also has an option (the charge
parameter) which if unset uses the original diffusion capacitance of the SPICE model to
compute time-varying currents. These equations are not provided here. The noise model
for the diode adds a thermal current source for the parasitic resistance, a shot noise current
source and a flicker noise current source that is dependent on the current flowing through
the diode [56]. The enhanced diode model including noise sources is shown in Fig. 5.6. The
thermal, shot and flicker noise models respectively are written as
r
2kT
ξt
it,rs =
rs
p
is,d =
qId ξs
q
alpha
if,d = kf Id
ξf
(5.37)
(5.38)
(5.39)
where the parameter kf is the scaling coefficient for flicker noise and alpha controls the
dependence of the flicker noise component on the current in the diode. The parameter beta
104
Anode
Rs
i t,rs
i f,d
i s,d
I
Cd
d
Cathode
Figure 5.6: Noise-enabled p-n junction diode model.
controls the slope of the generated 1/f characteristic. As before, the random variables ξt
and ξs , representing thermal and shot noise processes are generated using the Logistic map
and ξf representing a flicker noise process is generated using the Logarithmic map. The
source code for this diode model in f REEDATM is provided in Appendix C.2.
MESFET
There are several MESFET models in f REEDATM , namely the Curtice Cubic model [148],
the Materka model [149], the Parker- Skellern [150] model and the model proposed by Ooi,
Ma and Leong in [151] henceforth referred to as the OML model. The Curtice Cubic model
was enhanced by including sources of noise as described below. It consists of two state
variables, the voltage across the gate-source junctions and the voltage across the gate-drain
junctions. The large-signal equivalent circuit without noise sources for the Curtice Cubic
MESFET is shown in Fig. 5.7. The noise-enabled model consists of 32 parameters as listed
in Table 5.4.
The Curtice Cubic model computes drain-source current Ids as a polynomial function of input voltage, Vin . This current is expressed as
Ids = (a0 + a1 Vin + a2 Vin2 + a3 Vin3 ) tanh (gama Vds )
(5.40)
where a third-order polynomial with coefficients a0, a1, a2 and a3 is used. The values of
these coefficients are generally obtained from measured data. To be able to include the effect
105
Parameter
a0
a1
a2
a3
beta
vds0
gama
vt0
cgs0
cgd0
is
n
ib0
nr
td
vbi
fcc
vbd
tnom
avt0
bvt0
tbet
tm
tme
eg
m
xti
tj
area
map beta
kf
af
Description
drain saturation current for VGS = 0
coefficient for Vin
Vin coefficient for Vin3
coefficient for Vin3
dependence on Vds
Vds at which beta was measured
slope of drain characteristic in the linear region
voltage at which the channel current is forced to 0
gate-source barrier capacitance for VGS = 0
gate-drain barrier capacitance for VGS = 0
diode saturation current
diode ideality factor
breakdown current parameter
breakdown ideality factor
channel transit time
built-in junction potential
forward-bias depletion capacitance coefficient
breakdown voltage
reference temperature
pinch-off voltage linear temperature coefficient
pinch-off voltage quadratic temperature coefficient
beta power law temperature coefficient
Ids linear temperature coefficient
Ids power-law temperature coefficient
Barrier height at 0 ◦ K
grading coefficient
diode saturation current temperature exponent
junction temperature
area multiplier
exponent for chaotic map
scaling factor for flicker noise
flicker noise exponent
Default
0.1
0.05
0
0
0
0.4
1.5
1e−10
0
0
0
1
0
10
0
0.8
0.5
1e10
293
0
0
0
0
0
0.8
0.5
2
293
1
0.000005
1
1
Table 5.4: Noisy Curtice Cubic model parameters in f REEDATM .
Units
A
A/V
A/V2
A/V3
1/V
V
1/V
V
F
F
A
−
A
−
secs
V
V
V
◦K
1/◦ K
1/◦ K2
1/◦ K
1/◦ K
1/◦ K2
eV
−
−
◦K
−
−
−
−
106
i DG
C DG
RD
D
G
R IN
i
i DS
GS
RDS
CDS
C
GS
RS
S
Figure 5.7: Noiseless Curtice Cubic large-signal model.
of the increase of pinchoff voltage with drain-source voltage, the transit time (parameter
td) associated with the FET is considered in the formulation of the input voltage Vin as
Vin = Vgs (t − td)[1 + vds0 + Vds ].
(5.41)
Curtice in [148] found that this time-delay is dependent on the value of the input voltage to
the device. In f REEDATM , this delay effect is modelled and is a parameter in the netlist
describing the circuit under consideration. So, once this parameter is set to a non-zero
value, the data structure associated with the input voltage is stored not just for the current
time instant but also for the assigned delayed value of input voltage. This allows memory
effects to be included in the simulation framework. Nonlinear capacitors associated with
the MESFET as shown in Fig. 5.7 are also implemented in the model as described in [148]
and is not repeated here.
To model the effects of noise in the MESFET, the main sources of noise considered
([56], [57]) are thermal noise associated with the parasitics of the MESFET, thermal noise
associated with the channel resistance of the device and flicker noise which is a function
of the drain-source current. These main sources of noise are shown in Fig. 5.8. The shot
noise associated with a MESFET is assumed to be generated from the gate-leakage current
[56] and is not modelled here. The thermal noise associated with the drain and source
107
i DG
i t,rd
C DG
RD
D
G
R IN
i
i DS
RDS
GS
CDS
i t,ch + i f,ch
C
GS
RS
i t,rs
S
Figure 5.8: Noise-enabled Curtice Cubic large-signal model.
resistances respectively is modelled as
r
2kT
ξt
rd
r
2kT
=
ξt
rs
it,rd =
(5.42)
it,rs
(5.43)
where ξt represents a white noise generated by the Logistic map of Section 4.3. The model
for the thermal noise of the channel is as found in [57] and is expressed as
It,ch =
where
1 + η + η2
4kT
beta(Vgs − vto)
ξt
3
1+η

 1−
η=
 0
Vds
Vgs −vto
Vds ≤ Vgs − vto
Vds > Vgs − vto
and the flicker noise of the device is modelled as
q
af ξ
if,ch = kf Ids
f
(5.44)
(5.45)
(5.46)
where kf is an amplitude scaling coefficient for flicker noise and af is the flicker noise exponent. The ξf are random variables representing a flicker noise process as generated in
Section 4.4 and the parameter map beta controls the slope of the generated 1/f characteristic. The source code for the noise-enabled Curtice Cubic MESFET element is available in
Appendix C.3.
108
Parameter
res
temp
kth
Description
Resistance value
Temperature
Noise scaling coefficient
Default
1000
300
1
Units
Ω
K
–
Table 5.5: Noisy resistor model parameters in f REEDATM .
Resistor
The resistor model in f REEDATM is a linear model that consists of a single parameter,
i.e. the value of the resistance. The noise-enabled resistor was implemented as a nonlinear
element with its state-variable being the voltage across its terminals. In the case of the
resistor, the only noise source considered is thermal noise. Therefore, in addition to deterministic output of the resistor, there is a temperature-dependent thermal noise current
contribution expressed as
r
it,r =
2kT
ξr
R
(5.47)
where R is the value of the resistance, T is the temperature and k is Boltzmann’s constant.
The parameters associated with this resistor are shown in Table 5.5.
The random processes ξr are assumed white and are generated using the Logistic
map of Section 4.3. The source code for the resistor is provided in Appendix C.4.
White Noise Generating Voltage Source
A white noise voltage generator is implemented in f REEDATM as a linear source of Gaussian
distributed random variables. The parameters for this element are listed in Table 5.6.
The model can generate Gaussian random variables at some DC offset, it can provide a
user-specified delayed sequence, different scales of random variables and allows for nonnormalized Gaussian random variables by adjusting the mean and variance to a desired
value.
The white noise sequences are generated using the Ziggurat technique, [132]. The
essential source code to generate a white noise sequence with the Ziggurat technique has
been used from [132] but has been modified to fit into the f REEDATM framework. The
source code for the voltage source is listed in Appendix C.5. The Ziggurat implementation
109
Parameter
vo
td
mean
variance
kn
Description
Offset value
Delay time
Mean of the random variable
Variance of the random variable
Scaling coefficient
Default
0
0
0
1
1
Units
V
secs
–
–
–
Table 5.6: White noise voltage source model parameters in f REEDATM .
in f REEDATM is provided in Appendix C.8.
5.4.2
Nonlinear Maps
The nonlinear chaotic maps from Section 4.3 and Section 4.4, namely the Logistic map
and the Logarithmic map are implemented in f REEDATM as separate classes which can
be instantiated inside any element. These maps can work in conjunction with a transient
analysis only and during the instantiation process, information about the time step and
stop time must be passed. Each of the maps uses this information to dynamically allocate
memory large enough to hold a sequence that is as long as the transient simulation itself.
After this instantiation, the map chooses a random starting point in its domain and proceeds
to generate an orbit whose characteristics are controlled by a single parameter. For the
Logistic map, this parameter is λ and its default value is set to 4. For the Logarithmic
map, the parameter is β and the default value is 0.000005. Each element generates one
map for each intrinsic noise source. The SDIC property of chaos ensures that the noise
sources realized in this way will have different instantaneous values. This ensures that the
different noise sources for the devices are uncorrelated. The source code for the Logistic
map generator is provided in Appendix C.9 and that for the Logarithmic map is provided
in Appendix C.10.
5.4.3
Noisy Error Function
The examples above show how changes to individual linear and nonlinear elements can be
made. On account of the OO structure of f REEDATM , making a change to an element
requires no changes to be made to the numerical solver routines. The analysis routines are
110
not aware of specific details about the elements under consideration and strive to minimize
the error function presented to them by an abstract framework implemented in the simulator. The partitioning of a generic circuit in f REEDATM happens just like in the noiseless
case but the contributions of the noise-enabled elements now appear at inter-facial nodes
as is shown in Fig. 5.9. As is apparent from the figure, the nonlinear elements now include
contributions from internal noisy current or voltage sources. Effects of linear noisy voltage
sources also appear at the interface as in the case of the white noise generating voltage
source model. The error function is now transformed to
Linear
elements
and
sources
Vξ
I1
V2
I2
V3
I3
Vn−1
In−1
Vn
In
Nonlinear
device
Vnl
Noise
voltage
Nonlinear
device
Inl
Noise
current
Figure 5.9: Partitioned network which now contains contributions from transient noise
sources.
f(xξ,n ) = vL (xξ,n ) − vN L (xξ,n )
(5.48)
and contains a mixture of deterministic terms and stochastic noise terms that can be both
additive and multiplicative. These stochastic terms need not necessarily be small signal
as they may produce significant instantaneous components upon interaction with large
deterministic signals in a circuit. If the resulting system of stochastic differential equations
is interpreted in the Stratonovich sense, it can be solved by the same techniques used to solve
the system of equations represented in Eqn. (5.17). This allows for easy implementation
of a concurrent deterministic and stochastic framework and enables the use of pre-existing
robust and established numerical solution techniques.
It is also possible to modify the transient analysis routine to include random terms
directly into the error function instead of modifying individual element models. This procedure was considered earlier and although it was quicker to deploy in practice, it was
111
discarded in favor of enabling noise in individual device elements. The reason for this is
that modifying the analysis directly is too generic and unintuitive and does not correspond
to sources of noise as known in electronic devices. To determine the noise characteristics
correctly, it is essential to use noise models that are particular to the devices in the circuit
under consideration.
5.5
Summary
This chapter provides implementation details of the establishment of the stochastic framework in the circuit simulator f REEDATM . The method of this implementation keeps the
basic transient analysis framework of f REEDATM unchanged and essentially extends the
deterministic framework to include random sources of noise. This approach is made possible due to the object-oriented structure of f REEDATM and in particular, due to the fact
that there is a clear distinction between element routines and analysis routines. This also
permits the resulting stochastic system of differential equations to be analyzed with existing
integration techniques, i.e. as a consequence of using the Stratonovich interpretation. If the
fact that handling stochastic signals may require minor modifications to the convergence
parameters of the solver or changing the solver routine itself, it can be accomplished by
preserving the existing structure of data.
112
Chapter 6
Noise in a Voltage-Controlled
Oscillator
6.1
Introduction
This chapter presents simulation runs compared with measured results of an varactortuned voltage controlled oscillator circuit, the details of which are presented in Section 6.2.
The circuit uses nonlinear devices like bipolar junction transistors and varactors which are
modelled as reverse-biased diodes. Section 6.3 explains the procedure required to extract
flicker noise parameters for the nonlinear devices in the circuit and presents comparisons
between simulation and measurement for different values of input bias.
6.2
A Varactor Voltage-Controlled Oscillator
This section presents a varactor-tuned voltage controlled oscillator circuit designed to oscillate at a resonant frequency between 45 and 55 MHz. The schematic of the oscillator is
shown in Fig. 6.2. The point marked (A) is the output of the resonator and the transformers
shown inside a box, represent the feedback path. The transistors Q1 and Q2 are emittercoupled npn-BJT transistors and are chosen to provide sufficient gain so as to maintain
113
oscillation. The voltage source, VSS, is used to bias the varactor which provides the requisite voltage-dependent capacitance to the resonator. The voltage swing at the resonator
is fairly large, and can be as high as 40 V p-p. The output is measured at point (B). The
output at this terminal is a square wave-like signal and a snapshot of the simulated output
voltage versus time is shown in Fig. 6.1. This circuit has been used from [144].
Output (volts)
1
0
-1
-2
1.5e-06
1.6e-06
1.55e-06
1.65e-06
Time (seconds)
Figure 6.1: Simulated output of the VCO at terminal (B) indicating a frequency of oscillation of 45 MHz.
6.3
Simulation and Validation
As is detailed in [144], the oscillator phase noise, which is a single-sided power spectral
density of phase fluctuation, was measured using the Agilent Model 4352S Phase Noise
test set. External sources of noise were prevented from having a major impact on the
measurements by use of appropriate shielding and the circuits were operated on battery.
Single-sided phase noise measurements were taken at frequency offsets varying from 100 Hz
to 100 KHz at bias conditions of 0V, 6V and 12V. According to measurement, the frequency
of oscillation of the VCO is dependent on bias. This dependence is reproduced from [144]
and is shown in Fig. 6.3. A measurement was also made of the degradation of phase noise
observed at the varactor terminal with respect to the varactor bias voltage. The reason
for these measurements is to observe the point when the varactor goes into the breakdown
+
− VSS
L=10uH
L=10uH
Varactor
C = 100pF
C=47pF
C=47pF
(A)
Q1
feedback
R=47
L=10uH
L=10uH
Q2
C=1nf
C=1nF R=1k
R=10
R=3.9K
C=1nF
C=1nF
+ VS
−
V=12V
R=1000
(B)
114
Figure 6.2: Varactor-tuned VCO schematic, from [144].
115
60
VCO Oscillation Frequency (MHz)
59
58
57
56
55
54
53
52
0
2
4
6
8
10
Tuning Voltage (V)
Figure 6.3: Dependence on VCO oscillation frequency v/s bias.
region and its effect on phase noise at the varactor terminals. This plot reproduced from
[144] is shown in Fig. 6.4. As is evident from the figure, once the bias voltage increases above
5 V, the phase noise at the varactor begins to degrade and beyond the 10 V mark, there is a
noticeable increase of phase noise indicative of the beginning of the breakdown mechanism.
Once the bias voltage crosses 18 V, the breakdown effect is apparent as indicated by the
sudden vertical spike in the graph of Fig. 6.4.
The models for the elements used in the circuit were modified in f REEDATM to
include sources of noise as detailed in Section 5.4. In particular, the noise-enabled npn-BJT
and resistor elements were used with the reverse-biased p-n junction diode that models
the varactor in the circuit. The important parameter values used for the BJT model are
shown in Table 6.1 while the diode parameters are shown in Table 6.2. The f REEDATM
compatible netlist used for this simulation is provided in Appendix D.1.
To perform the simulation, values were required to be set for the flicker noise
scaling parameter kf for both the BJT and diode. This is because the mechanism of 1/f
noise generation in a semiconductor device is not unique, unlike thermal and shot noise,
which makes it difficult to set a global value for kf that will work with any device. Numerous
simulations were performed in order to find a value of kf for both the BJT and the diode
that will fit well to experimentally measured values of phase noise. Initially, a value was
116
Phase noise (dBc/Hz)
-80
-100
-120
1
10
Tuning voltage (V)
Figure 6.4: Degradation of phase noise at varactor v/s bias.
BJT parameters
bf = 255.9
ikf = 0.2847
mjc = 0.3416
nr = 1.0
tr = 46.91e-9
xtf = 3.0
br = 6.092
is = 14.34e-15
mje = 0.377
rb = 10.0
vaf = 74.03
kf = 1e-2
cjc = 7.306e-12
ise = 14.34e-15
nf = 1.0
rc = 1.0
vtf = 1.7
beta = 0.000005
cje = 22.01e-12
itf = 0.6
ne = 1.307
tf = 411.1e-12
xtb = 1.5
Table 6.1: BJT model parameters values as used in the VCO circuit.
Diode parameters
is = 1.365p
m = 0.4261
ibv = 10.0e-6
rs = 1
vj = 0.75
kf = 1e-4
n=1
fc = 0.5
beta = 0.000005
cj0 = 14.93e-12
bv = 25
Table 6.2: Diode model parameters values as used in the VCO circuit.
117
0
Simulation
Measurement
Phase noise (dBc/Hz)
-50
-100
-150
100
1k
10 k
100 k
frequency offset (Hz)
Figure 6.5: Phase noise comparison between data and experiment with bias voltage at 0 V.
determined while trying to fit to phase noise measurements with 0 V bias. Once this value
was determined and a suitable match to measurement was obtained, the same value of kf
for the BJT and diode was used in a simulation with the 6 V bias condition to obtain a
match to the corresponding phase noise measurements at 6 V. The values of kf for the
BJT with 0 V bias was determined to be 1 × 10−2 , while for the diode kf = 1 × 10−4 was
used. It has been shown before [145] that varactors contribute minimal 1/f noise to the
overall phase noise response and this explains the low value of kf used for the diode. The
main contribution of flicker noise in this circuit comes from the active elements. The BJT
and diode contribute thermal noise associated with their resistive parasitics and shot noise
associated with their junctions and the external resistors in the circuit contribute thermal
noise. The inductors and capacitors are assumed ideal and hence have no internal sources
of noise [146].
The results of comparison between simulated and measured curves for the cases of
0 V and 6 V bias conditions are shown in Fig. 6.5 and Fig. 6.6 respectively. This approach
was also attempted for the 12 V bias condition and the match was initially not good. To
understand why, it is informative to look at Fig. 6.4 which reveals an increased level of
phase noise associated with the varactor at 12 V. However, further simulations were run
118
0
Simulation
Measurement
Phase noise (dBc/Hz)
-50
-100
-150
100
1k
10 k
100 k
frequency offset (Hz)
Figure 6.6: Phase noise comparison between data and experiment with bias voltage at 6 V.
0
Simulation
Measurement
Phase noise (dBc/Hz)
-50
-100
-150
100
1k
10 k
100 k
frequency offset (Hz)
Figure 6.7: Phase noise comparison between data and experiment with bias voltage at 12 V.
119
but this time the shot noise scaling component of the varactor, Ksh , was increased from
unity and after several attempts, a value to 1×104 was found to provide an improved match
between simulated runs and measured curves. This is indicated in Fig. 6.7. The correctness
of this latter empirical fitting procedure for shot noise is questionable given the known
fundamental nature of shot noise. The simulation was still performed and results presented
because considering increased levels of shot noise in the varactor when it is approaching
breakdown mode is valid and is a possible approach for obtaining a match of simulated
curves from f REEDATM with measured data. Although this does not necessarily validate
the simulation approach or change fundamental concepts about shot noise, it might open
new avenues for modelling noise in devices when they are on the threshold of breakdown.
6.4
Summary
This chapter describes the simulation of phase noise in a circuit which has nonlinear interactions between large values of signal and noise. It does not require that the magnitude of
the noise be negligibly small compared to the signal. A varactor-tuned VCO circuit is set
up for simulation with appropriate time-domain noise models for the devices in the circuit.
Simulations for phase noise are carried out and compared with measured results. Parameter values for flicker noise scaling coefficients for the BJT and the diode are determined
after performing several simulations with 0 V bias and trying to match simulated runs with
measured data. Once these parameter values are found, they remain unchanged for the
simulation with 6 V bias and a close match between data and experiment is obtained. It
is found that phase noise can be determined quite accurately for a fairly large range of frequency offsets. Higher values of bias drive the varactor close to breakdown and this causes
increased levels of phase noise. A match between simulation and experiment is obtained
in this case as well but requires artificial adjustments to the amplitude of shot noise. It is
reasoned that this scaling is justified since the varactor is on the threshold of breakdown
when the reverse bias is set at 12 V.
120
Chapter 7
Noise and Amplification
7.1
Introduction
This chapter explores the effect of high levels of noise, superimposed on a carrier, on the
gain-compression characteristics of a linear X-band MMIC driver amplifier. A series of measurements are performed each with increasing levels of input power and the corresponding
gain-compression curves are generated. These curves are compared with simulated results
obtained from f REEDATM . Even though total noise power may be large, perturbations
from time-point to time-point can be very small and high simulation precision is required
to track signals and separate stochastic from deterministic results after each time step and
each nonlinear iteration. This is possible because of the high dynamic range of f REEDATM
[147], and the existence of a large catalog of nonlinear models.
7.2
Setup and Verification
The power amplifier considered is an X-band pHEMT MMIC amplifier (Filtronic LMA411)
whose layout diagram is shown in Fig. 7.1. The MMIC operates between 8.5 and 14 GHz
and has an output power of +14 dBm at 1 dB gain compression and +17 dBm at 3 dB
gain compression. At 10 GHz, the noise figure is 2 dB. The circuit consists of 2 cascaded
pHEMT stages each in a class-A common source configuration. The device was biased at
121
a DC voltage of 6 V and a current of 94 mA. The measurement setup used is shown in
Figure 7.1: Layout of the two-stage X-band MMIC.
o/p
i/p
GPIB Bus
Agilent E8267C
Signal Generator
DUT − LMA411 MMIC
dual pHEMT LNA
Spectrum Analyzer
Figure 7.2: Measurement setup for the X-band MMIC amplifier.
Fig. 7.2. A white noise sequence was generated digitally and transferred via a GPIB bus to
the Agilent E8267C signal generator which was set to a carrier frequency of 10 GHz. The
signal generator is capable of accepting externally generated signals in a band of 100 MHz
about its center frequency. This procedure creates a composite signal consisting of a 10 GHz
sinusoid and a synthesized 100 MHz wide band-limited white noise sequence. The white
noise sequence was generated using the randn function in Mathworks’s MATLAB which
122
currently uses the Ziggurat technique [132] for generating white sequences, [157]. This
approach is known to generate white noise samples with a very large period, roughly 21492 ,
thus ensuring that the random numbers so generated will be truly random for almost all
lengths of datasets of interest.
The amplitude of the composite signal varied from −20 dBm to 5 dBm at 10 GHz
in steps of 1 dBm and of the gain of the amplifier was measured at 10 GHz with two different
input conditions: one with no noise at the input and second with the Signal-to-Noise Ratio
(SNR) maintained at −20 dBc. A plot of measured gain versus input power with no input
noise is compared with the plot of measured gain with noise level maintained at −20 dBc
and is shown in Fig. 7.3.
20
Noiseless case measurement
-20 dBc noise measurement
Power gain (dBm)
18
16
14
12
10
-20
-15
-10
-5
0
5
Input power (dBm)
Figure 7.3: Comparison between measured curves of gain with no input noise and noise
maintained at −20 dBc.
In the plot the total noise power in the 100 MHz noise bandwidth is specified. High levels
of noise suppress the effective power gain of the amplifier with respect to the carrier. The
suppressed power at the center frequency is dissipated into frequency bands around 10
GHz. This is represented schematically in Fig. 7.4. It should be noted that a noise level at
−20 dBc means that total input noise is maintained at a constant level of 20 dB below the
carrier.
A similar plot of simulated power gain versus input with no input noise is com-
Power
123
f
fc
Figure 7.4: Power transferred from the center frequency into sidebands.
Device 1
a0 = 0.09910
beta = 0.01865
vbi = 0.8
nr = 1.2
Device 2
a0 = 0.1321
beta = 0.03141
vbi = 1.5
n = 1.2
a1 = 0.08541
gama = 0.8293
cgd0 = 3f
t = 1e-12
a2 = -0.0203
vds0 = 6.494
cgs0 = 528.2f
vbd = 12
a3 = -0.015
vt0 = -1.2
is = 3e-12
kf = 1e-9
a1 = 0.1085
gama = 0.7946
cgd0 = 4e-15
t = 1e-12
a2 = -0.04804
vds0 = 5.892
cgs0 = 695.2f
vbd = 12
a3 = -0.03821
vt0 = -1.2
is = 4e-12
kf = 1e-9
Table 7.1: MESFET model parameters in for X-band MMIC.
pared with the plot of simulated power gain with noise level maintained at −20 dBc and
is shown in Fig. 7.5. Just as in the case of MATLAB, the white noise sequence was generated in f REEDATM using the Ziggurat technique, the source code of which is provided in
Appendix. C.8. The pHEMTs were modelled using the Curtice Cubic model [148] and the
parameters for the polynomial coefficients for the cubic model were obtained from Filtronic
Corporation 1 . These parameters are listed in Table 7.1. The simulation uses the noiseenabled version of the Curtice Cubic MESFET as described in Section 5.4. The netlist used
for this simulation is provided in Appendix D.2.
To observe the degradative effect of noise on the purity of the output waveform, a
1
http://www.filtronic.co.uk
124
20
Noiseless case simulation
-20 dBc noise simulation
Power gain (dBm)
18
16
14
12
-20
-15
-10
-5
0
5
Input power (dBm)
Figure 7.5: Comparison between simulated curves of gain with no input noise and noise
maintained at −20 dBc.
2
Output voltage (volts)
1
0
-1
-2
8.0 n
8.2 n
8.4 n
8.6 n
8.8 n
Time (secs)
Figure 7.6: Degradation of the output sinusoid due to noise.
125
snapshot of the transient output is shown in Fig. 7.6 with noise level maintained at −20 dBc.
This figure shows distortion in the amplitude and shape of the output sinusoid with the
input power level set at 0 dBm. It is important to note that this distortion is due to both
saturation effects and noise interference.
The simulated power gain curve in the high-noise level case shows a comparatively
smaller degree of compression compared to the measured results of Fig. 7.3. It is important
to be able to understand the reason for this discrepancy. The procedure used to uncover
the reason for this difference is the topic of the next section and represents an interesting
avenue for future research.
7.3
Further Investigations
As seen in the previous section, simulated curves of power gain versus input power do not
show as much compression at high power levels in the presence of noise as do measured
curves. This section presents a summary of the attempts made to find a resolution to this
problem.
The first attempt focuses on improvements to the Curtice Cubic model. Since the
release of the Curtice Cubic model, there have been several newer models for the MESFET
that have appeared in the literature that aim to have better large signal behavior in the
saturation and cutoff regions and better model behavior in the knee regions of the largesignal characteristics. Some examples are the Parker-Skellern (PS) model [150], Triquint’s
own model (TOM) 2 and more recently, the MESFET model by Ooi, Ma and Leong (OML)
[151]. The TOM model already exists in f REEDATM while the PS and OML models were
implemented. The PS model takes a different approach compared to the Curtice Cubic
model in the sense that it models drain current as a power-law function of effective gate
and drain voltages. The parameter table for the model is provided in Table 7.2.
The drain-source current is expressed as
Ids =
id
1 + delta P
where
q
id = beta Vgt [1 − (
2
www.triquint.com
Vdt q
) ]
Vgt
(7.1)
(7.2)
126
Parameter
acgam
beta
cgs
cds
delta
fc
hfeta
hfe1
hfe2
hfgam
hfg1
hfg2
ibd
is
lfgam
lfg1
lfg2
mvst
n
p
q
rs
rd
taud
taug
vbd
vbi
vst
vto
xc
xi
z
tj
tnom
afac
lam
Description
capacitance modulation
linear region transconductance scale
zero-bias gate-source capacitance
zero-bias drain-source capacitance
thermal reduction coefficient
forward bias capacitance parameter
high-frequency vgs feedback parameter
hfgam modulation by vgd
hfgam modulation by vgs
high-frequency vgd feedback parameter
hfgam modulation by vsg
hfgam modulation by vdg
gate-junction breakdown current
gate-junction saturation current
low-frequency feedback parameter
lfgam modulation by vsg
lfgam modulation by vdg
sub-threshold modulation
gate-junction ideality factor
linear region power law exponent
saturated region power law exponent
source ohmic resistance
drain ohmic resistance
relaxation time for thermal reduction
relaxation time for gamma feedback
gate junction breakdown voltage
gate junction potential
sub-threshold potential
threshold voltage
capacitance pinch-off reduction factor
saturation knee potential factor
knee transition parameter
device temperature
nominal temperature
gate width scale factor
channel length modulation
Default
0
10−4
0
0
0
0.5
0
0
0
0
0
0
0
10−14
0
0
0
0
1
2
2
0
0
0
0
1
1
0
-2.0
0
1000
0.5
300
300
1
0
Units
–
A.V−Q
F
F
1/W
–
–
1/V
1/V
–
1/V
1/V
A
A
–
1/V
1/V
1/V
–
–
–
Ω
Ω
sec
sec
V
V
V
V
–
–
–
◦K
◦K
–
–
Table 7.2: Parker-Skellern model parameters in f REEDATM .
127
and P is the power dissipated at the drain-source junction and is expressed as
P = id Vds .
(7.3)
Near pinchoff, the drain current reduces exponentially with respect to gate potential and
will increase when gate-breakdown occurs. To model this effect correctly, gate potential Vgt
is expressed as
Vgt
µ
µ
= mvst(1 + mvst Vds ) ln 1 + exp
vgst
vst(1 + mvst Vds )
¶¶
.
(7.4)
where
0
0
vgst = vgs − vto − γl Vgd − γh (Vgd − Vgd ) − ηh (Vgs − Vgs )
(7.5)
and
0
0
0
0
0
0
γl = lfgam − lfg1 Vgs + lfg2 Vgd
γh = hfgam − hfg1 Vgs + hfg2 Vgd
ηh = hfeta − hfe1 Vgd + hfe2 Vgs
0
dVgs
dt
dVgd
= Vgd − taug
.
dt
Vgs = Vgs − taug
0
Vgd
(7.6)
(7.7)
(7.8)
(7.9)
(7.10)
The intrinsic drain terminal potential Vdt is controlled by parameter z which ensures a
smoother knee region of the large signal input-output characteristic and is expressed as
q
q
√
√
1
2 − 1
2 .
Vdt =
(vdp 1 + z + Vsat )2 + zVsat
(vdp 1 + z − Vsat )2 + zVsat
(7.11)
2
2
This formulation introduces an effective drain potential term vdp which has a range of [0, ∞]
and gets mapped into [0, Vsat ], the range of Vdt . When vdp = 0, Vdt ≈ 0 and for large vdp ,
Vdt = Vsat . The vdp itself is expressed as a function of parameters p and q as
µ
¶p - q
Vgt
p
vdp = Vds
q vbi − vto
(7.12)
and the drain voltage in the saturation is written as
Vsat = xi(vbi − vto)
Vgt
.
xi(vbi − vto) + Vgt
(7.13)
The capacitance model used is the one proposed in [152] and is not reproduced here. The
source code for this model is provided in Appendix C.6.
128
Device 1
beta = 0.01865
vbi = 0.8
Device 2
beta = 0.03141
vbi = 1.5
vbd = 12
cgd = 3f
q = 1.2
cgs = 528.2f
vt0 = -1.2
is = 3e-12
vbd = 12
cgd = 4e-15
q = 1.2
cgs = 695.2f
vt0 = -1.2
is = 4e-12
Table 7.3: Parker-Skellern model parameters used in the X-band MMIC netlist.
This model was substituted for the Curtice Cubic in the netlist for the MMIC
circuit (Appendix D.2). An example set of parameters for this model are provided in [150]
and although they don’t exactly match the output characteristics of the circuit while using
the Curtice model, the results are still comparable. The non-default parameter values
are provided in Table 7.3. Upon running the netlist with the new PS model, the results
both with and without noise unfortunately do not indicate any changes when compared to
simulations with the original Curtice model. In other words, the model does not seem to
predict any noticeable reduction in gain when the circuit is fed with noise as compared to
when the circuit is fed with sinusoidal input alone.
The PS model was replaced with the OML model [151] which shares many characteristics with the Curtice model. It also uses a third order polynomial to describe the
output characteristics of a MESFET power amplifier. The difference lies in the fact that
while the Curtice model uses a polynomial function of gate-source voltage, the OML model
uses a polynomial function of the effective gate-source voltage, Veff . The output current
according to the OML model is then given as
3
2
Ids = a1 Veff
+ a2 Veff
+ a3 Veff .
(7.14)
The parameters a1 , a2 and a3 and the effective voltage are expressed as
a1 = b1 Vds
a2 = b4 q
a3 = b5 q
Vdseff
2
1 + Vdseff
Vdseff
2
1 + Vdseff
(7.15)
(7.16)
(7.17)
129
Device 1,2
b1 = -0.7437
b5 = 21.8151
vto = -1.2
b2 = 2.8974
g = 0.2339
b3 = 4.4187
gamma = 0.12
b4 = 15.329
delta = 0.04
Table 7.4: OML model parameters used in the X-band MMIC netlist.
2
b2 Vds + b3 Vds
1 + g Vgst
q
³
´
1
2 + delta2
=
Vgst + Vgst
2
Vdseff =
Veff
Vgst = Vgs − vto + gamma Vds
(7.18)
(7.19)
(7.20)
where b1, b2, b3, b4, b5, g, gamma, delta and vto are parameters of the model. The
capacitance equations of the Curtice model was retained in this model as were the related
parameters in the netlist. The source code for this model is available in Appendix C.7.
Using parameter values indicated in Table 7.4, the Curtice model was replaced by the OML
model. However, just like in the case of the PS model, there is no significant change between
simulations of the case of a pure sinusoidal input and a sine input superimposed with noise.
On the assumption that higher-order polynomials are required to be able to accurately predict saturation effects, the f REEDATM model VccsPoly was considered. The
VccsPoly model is a behavioral model that formulates output current as a polynomial function of arbitrary order N of input voltage. It requires the coefficients of the polynomial
to be supplied by the user and contains an optional gain parameter. The method used to
select these coefficients and the order of the polynomial chosen was somewhat arbitrary
and required a large number of trials. The basic methodology was to consider the minimum
polynomial order of 3 with decreasing values of higher-order coefficients. For the case of
third order and greater, the even order coefficients were set to zero and consecutive odd
order terms had opposite signs. There is no apparent technique to choose the values of these
coefficients and up to 11 order polynomials with varying gain constants were considered.
The results produced were unsatisfying and there was still no discernable difference between
the compression characteristics in the absence and presence of input noise. This suggests
that there is perhaps another mechanism at work that is causing a reduction in power in
the presence of noise.
130
20
Curtice model with 20 ps delay
Measured data
Curtice model with no delay
Power gain (dBm)
18
16
14
12
10
-1
0
1
2
3
4
5
Input power (dBm)
Figure 7.7: Comparing simulated gain obtained with a 20 ps delay and no delay with
measured gain.
The f REEDATM framework makes it possible to handle time delays inside the
model of an element. The Curtice Cubic model in f REEDATM has a parameter that allows
the user to set a specific value of time delay in the netlist. This time delay value, τ , makes
the Curtice model compute the output current as a function of voltage that is τ time units
in the past. Choosing the appropriate time delay depends on the channel transit time of
the active devices in the circuit and the elements surrounding it. Although it is possible
to obtain a fairly accurate estimate of the time delay [153], a first attempt to selecting an
appropriate value was made with a trial-and-error approach. For values of time delay of less
than 20 ps, there was not an appreciable decrease in gain between the noiseless and noisy
cases but there was a noticeable difference with delay set at 20 ps for both amplifiers. A
plot showing the drop in gain with input power varying from 0 – 5 dBm with the delay set
at 20 ps was compared with the measured gain characteristics in the presence of noise as
shown in Fig. 7.3 and the simulated gain characteristics in the presence of noise as shown
in Fig. 7.5. This plot is shown in Fig. 7.7. As seen in this figure, the power gain dropoff
obtained with a delay of 20 ps is quite sharp. At 0 dBm, it is comparable with the gain
value of the simulated case with no delay but at 5 dBm it is comparable with the measured
result. At powers lower than 0 dBm, simulations using this delay value produces results
than can differ quite significantly from the measured results. This might lead one to suggest
131
4
3
Output voltage (volts)
2
1
0
-1
-2
-3
8.0 n
8.2 n
8.4 n
8.6 n
8.8 n
Time (secs)
Figure 7.8: Distorted output voltage with a 20ps delay.
that the effects of delay mechanisms, if validated, is dependent on input power which is not
unreasonable and it originally reported by Curtice, [148]. A snapshot of the output voltage
taken with the input power set at 0 dBm shows considerably more distortion compared with
the output of Fig. 7.6 and this is shown in Fig. 7.8. It is known that power amplifiers can
exhibit strong delays and this can have serious effects on performance, particularly at high
input powers as measured in [154], quantified in [155] and modelled in [156]. Although the
investigations in this section are rudimentary, they present an interesting avenue for future
research in the understanding of the effects of interactions of noise with large input signals
on the RF performance of amplifiers.
7.4
Summary
The major result of this chapter is verification that effects of high levels of noise can be
modelled in a transient circuit simulator. The major change required is to separately track
deterministic and stochastic signals at each node in a circuit. Then at each node instead
of performing the solution of an ordinary differential equation, the solution of a stochastic
differential equation, discretized in the conventional manner, is required. Since stochastic
132
signals have very small variations relative to the expected deterministic signal levels it is
crucial that the transient simulator achieve high dynamic range. The approach is verified by
using nonlinear elements for the circuit devices in a high dynamic range simulator. Measured
and simulated plots for the gain of the amplifier versus input power level are compared in the
cases of no input noise and high level noise. The results also demonstrate an increased level
of compression in the presence of larger levels of noise and this effect has also been captured
by simulation, albeit not to the same extent as measurement. Further investigations were
carried out to try and determine an effective way to model this discrepancy and it was found
that advanced FET models and higher-order polynomial models did not have any impact on
the simulated results in the presence of noise. Including fixed time delays into the element
descriptions was found to be able to approximately capture the effect of reduction of power
gain in the presence of noise. However, the lack of accuracy leaves several unanswered
questions such as to whether the amount of delay is dependent on input power and whether
the simulation including delays effectively captures the underlying distortive mechanism in
the presence of noise. Exploring these questions is an important undertaking in order to be
able to characterize the effects of amplifiers operating under large signal conditions in the
presence of noise.
133
Chapter 8
Conclusions and Future Work
This work details the modelling of sources of colored noise in the time domain in a high
dynamic-range circuit simulator. Colored noise sources were generated using nonlinear
iterative chaotic maps with special properties and were inserted as elements of the circuit
simulator. The existing deterministic simulator framework was then extended to stochastic
and with appropriate justifications, was solved assuming the Stratonovich interpretation
for SDEs. The approach was verified by comparing simulated and measured results of real
circuits that operate under large-signal conditions.
8.1
Summary
Modelling sources of noise in electrical circuits and the effects of interactions of noise and
large signals is important from the point of view of design of RF circuits that constantly
shrinking in size and increasing in density. This places constraints on signal to noise ratios
and increases the required dynamic range. It is no longer adequate to describe noise sources
in circuits in the frequency domain where the levels of noise are small and the effects of
noise can be assumed additive only. This work is an attempt to model sources of noise in
the time domain and to include not just additive effects but effects of multiplication of noise
with large values of deterministic signal. Instantaneous levels of noise multiplied with large
signal can have an impact on the performance of the circuit and capturing these effects
134
adequately is essential. As was shown in Chapter 7, large levels of noise combined with
high input power levels can have a serious impact on the performance of an power amplifier
operating at X-band. The modelling approach should therefore place minimal restrictions
on the levels of the noise present in the circuit and the nature of the interactions between
noise and signal.
The first main requirement for modelling noise in the time domain requires the
circuits to be represented by Stochastic Differential Equations. While the use of SDEs is
not new, previous attempts at using SDEs to model noise use the Itô form to interpret
SDEs. This form is more suitable to the derivation of mathematical proofs related to SDEs
on account of its non-anticipatory nature but requires a new calculus to solve a system of
SDEs. This form is adequate when the sources of noise are purely additive and the noise
processes are perfectly white. The Stratonovich form, on the other hand, is more suitable
for modelling noise processes that can be colored and when there are multiplicative effects
between noise and state-dependent deterministic terms. While the Stratonovich form is not
as amenable to derivations of mathematical proofs, numerical solutions of SDEs with the
Stratonovich interpretation can be determined with the conventional rules of calculus. All
factors make the use of the Stratonovich interpretation very desirable when solving circuit
simulator problems that involve SDEs.
The second main requirement for simulating noise in circuit simulators is the availability of statistically accurate time domain noise generators. The mathematical theory of
chaos provides examples of such generators that use simple deterministic iterative rules to
generate sequences of noise using a small number of parameters. This frugal use of parameters is desirable both from an implementation and usage point of view as it requires
little effort for the model developer to include sources of noise in an electronic device model
and for the user who can quickly deploy the model without spending too long tuning the
characteristics of the noise generator. Any chaotic sequence exhibits Sensitive Dependence
on Initial Conditions which ensures that any two sequences starting at initial conditions
that may differ by arbitrarily small amounts will have different values at almost every point
of the sequence. Using chaotic maps, the Logistic map for white noise and the Logarithmic map for flicker noise, ensures that different noise sequences generated will be unique.
This differs from traditional pseudo-random numbers which do not produce unique random
numbers for every sequence length.
The addition of these noise sources into the simulation framework of the f REEDATM
135
simulator required modifications to the existing noiseless linear and nonlinear device elements. The OO nature of the existing framework allows for both the addition of stochastic
terms to current deterministic models and creation of new sources of noise without having
to make changes to other parts of the simulator. Making changes to the elements allows the
modelling of noise in a device based on the physics of the device. The simulator framework
then considers these noise-enabled elements and sets up a stochastic system of equations
and minimizes a noisy error function assuming the Stratonovich interpretation.
The validity of this modelling approach is demonstrated with two circuit configurations. The first configuration is a varactor-tuned voltage-controlled oscillator which has
high signal level excursions and nonlinear interactions between signal and noise. Simulated
runs are fitted with measured phase noise results for one value of bias and the flicker coefficients for the devices in the circuit are obtained. Using these coefficients, a match is
obtained between simulated runs and measured phase noise for another level of bias. The
second configuration models a low-noise X-band MMIC amplifier fed in with a carrier at
10 GHz and large levels of white noise. Gain plots in both measured and simulated cases
are compared with measurement to observe the effect of noise on amplification and it is
found that noise has a discernible effect on the amplification properties at high input power
levels. In particular, there seem to be larger levels of suppression in the presence of noise.
The reasons for the existence of this discrepancy warrant further research and preliminary
investigations seem to suggest that this distortive effect of noise can be more effectively captured if the amplifier model includes significant levels of possibly power-dependent amounts
of time delay.
8.2
Further Research
This work is a first attempt at using chaotic maps to model sources of noise in electrical
circuits in a full-fledged electronic circuit simulator using SDEs and the Stratonovich interpretation. This presents a wide array of possibilities for future research. For example,
such simulation techniques could be used to think about noise as a more integral part of
the basic circuit during the design phase. Models for circuit devices usually contain deterministic terms and noise sources that are added to the circuit which are usually assumed
to be white or linear or are considered to have values that are small enough such that they
136
can separated from the deterministic terms. With noise sources now intrinsically tied to
the devices under consideration, it permits a more physical approach to modelling. For
example, when designing an oscillator circuit as in Chapter 6, it is now easier to get a truer
estimate of the phase noise at any node in the circuit. This can allow for an incremental
approach to design based on both device operation and noise interactions. Not only does
this apply to a wider range of circuits, it also provides the designer with a guideline as to
which part of the circuit contributes most to the overall noise response.
Another avenue exists in the development of device models that can take advantage
of this noise analysis setup and try and reproduce results that are associated with highly
nonlinear behavior. As was mentioned in Chapter 7, one of the important reasons that
the effect of large levels of noise could not be completely captured was because of the lack
of accurate delay mechanisms for the Curtice Cubic model of a MESFET. Being able to
model these nonlinearities more effectively will allow one to extract maximum benefit of
these transient noise simulation abilities.
With the use of classical calculus to solve stochastic problems on account of the
Stratonovich assumption, more attention can be devoted to finding numerical algorithms
based on classical arguments that work well in the stochastic environment. A drawback
of time-domain based numerical integration is that for certain problems, the amount of
time involved to solve a particular problem can potentially become quite large. While
the extent of this problem is reduced with improvements in processor speed and newer
computer architectures it can also be alleviated with improved time-stepping algorithms
that can reduce the time required to find a solution.
With an understanding of the memory characteristics of flicker noise or in general,
any power-law noise, there are several newer approaches that can be investigated to model
these phenomena that use a small number of parameters and still successfully capture
the behavior of the underlying complex noise mechanisms. Efforts such as Self-Organized
Criticality and Highly-Optimized Tolerance have already been mentioned in Chapter 2.
Promising research avenues for noise generation are, among others, in the field of cellular
automata [158], wavelets [159] and neural networks [160] all of which can be thought of
as complex systems [161]. It will be interesting to see whether these approaches can add
value to the current state-of-the-art or whether they will produce a paradigm shift in the
understanding and modelling of real-world systems.
137
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150
Appendix A
Essential Itô and Stratonovich
This appendix provides the mathematical theory to explain the difference between solutions
of an SDE when interpreted in the sense of Itô and Stratonovich. The main focus is to try
and develop an understanding of the meaning of the stochastic integral
Z T
gdB
(A.1)
0
where g represents a wide class of stochastic processes and B is the BM process. For
all possible sample paths, the BM process has infinite variation and is non-differentiable.
Therefore it is not possible to treat this integral as an ordinary integral, i.e. an integral
in the Riemann sense. This means that a new procedure is required to study stochastic
integrals with possibly random integrands. It is still desirable to use the Riemann sum
approach to treat these stochastic integrals or in other words, construct a Riemann sum
approximation and then pass to limits. This procedure is explained using the example
Z T
BdB.
(A.2)
0
Definitions
1. For interval [0, T ], a partition P of [0, T ] is a finite collection of points in [0, T ]
P := {0 = t0 < t1 < . . . < tm = T }.
(A.3)
2. The mesh size of P , ∆, is the maximum value that a interval can take.
∆ := max0≤k≤m−1 |tk+1 − tk |.
(A.4)
151
3. For a fixed 0 ≤ λ ≤ 1 and a given partition P , let
τk := (1 − λ)tk + λtk+1
(A.5)
where (k = 0, . . . , m − 1). The Riemann sum approximation of the integral in (A.2) is
written as
R = R(P, λ) :=
m−1
X
B(τk )(B(tk+1 ) − B(tk )).
(A.6)
k=0
It remains to find out what happens to this sum when ∆ → 0.
Lemma 1 (Quadratic Variation) Let [a, b] be an interval in [0, ∞]. Suppose
P n := {a = tn0 < tn1 < . . . < tnmn = b}
are partitions of [a, b] with ∆n → 0 as n → ∞. Then
mX
n −1
(B(tnk+1 ) − B(tnk ))2 → b − a
k=0
in L2 (.) as n → ∞.
Proof: Let Qn :=
Pmn −1
k=0
(B(tnk+1 ) − B(tnk ))2 . Then
Qn − (b − a) =
mX
n −1
[(B(tnk+1 ) − B(tnk ))2 − (tnk+1 − tnk )].
k=0
Squaring both sides and taking expected values
2
E[(Qn − (b − a)) ] =
mX
n −1 m
n −1
X
E([(B(tnk+1 ) − B(tnk ))2 − (tnk+1 − tnk )]
j=0
n
[(B(tj+1 ) −
k=0
B(tnj ))2 − (tnj+1 − tnj )]).
A BM process has independent increments and for k 6= j, the expected value of the cross
terms goes to 0. For k = j
E[(Qn − (b − a))2 ] =
mX
n −1
E[(Yk2 − 1)2 (tnk+1 − tnk )2 ]
k=0
where
Yk = Ykn :=
B(tnk+1 ) − B(tnk )
pn
tk+1 − tnk
152
is a Normal random variable with zero mean and a variance of unity. For some constant C
E[(Qn − (b − a))2 ] ≤ C
mX
n −1
(tnk+1 − tnk )2
k=0
n
≤ C∆ (b − a) → 0
as n → ∞. ¤
This shows that a stochastic integral can be approximated to a Riemann sum only in the
mean-square sense. Now given this result, what remains is to determine the limit of these
Riemann sums. This is the subject of the next Lemma.
Lemma 2 For partition P n of [0, T ] and 0 ≤ λ ≤ 1 fixed, define
Rn :=
mX
n −1
B(τkn )(B(tnk+1 ) − B(tnk )).
k=0
Then in the limit in L2 (.),
lim Rn =
n→∞
B(T )2
1
+ (λ − )T.
2
2
That is,
E[(Rn −
B(T )2
1
− (λ − )T )2 ] → 0.
2
2
Proof: Starting with
Rn :=
mX
n −1
B(τk )(B(tnk+1 ) − B(tnk ))
k=0
=
mn −1
B 2 (T ) 1 X
−
(B(tnk+1 ) − B(tnk ))2
2
2
| k=0
{z
}
:=X
+
mX
n −1
|k=0
(B(τkn ) − B(tnk ))2
{z
}
:=Y
+
mX
n −1
|k=0
(B(tnk+1 ) − B(tnk ))(B(τkn ) − B(tnk )),
{z
:=Z
}
153
we use the Lemma of Quadratic Variation to obtain X → T /2 and Y → λT as n → ∞. For
P n −1
n
n
n
n
term Z, let J = m
k=0 (B(tk+1 ) − B(tk ))(B(τk ) − B(tk )).
E[J 2 ] =
=
≤
mX
n −1
E([B(tnk+1 − B(τkn )]2 )E([B(τkn ) − B(tnk )]2 )
k=0
mX
n −1
(1 − λ)(tnk+1 − tnk )λ(tnk+1 − tnk )
k=0
mX
n −1
λ(1 − λ)T ∆n → 0.
k=0
and hence Z → 0 and n → ∞. The property of independent increments of BM has been
used. Combining the expressions for X, Y and Z establishes the Lemma. ¤
The importance of this Lemma lies in the fact that it proves that the final result of evaluating
a stochastic integral is dependent on λ. Itô used λ = 0 which corresponds to the start of
each interval whereas Stratonovich used λ = 0.5 which corresponds to the mid-point of each
interval. Itô’s definition gives the solution
Z T
B 2 (T ) T
−
BdB =
2
2
0
(A.7)
which is not what one would expect from conventional calculus. Using Stratonovich’s definition gives the solution
Z
T
BdB =
0
B 2 (T )
2
(A.8)
which is the expected result if one were to consider the solution according to conventional
calculus. In the general case, it is therefore apparent that using different definitions will
result in different solutions to a stochastic integral and the Stratonovich form is the only
form that will always provide a result consistent with conventional calculus.
154
Appendix B
Implementing Infinite R-C
Transmission Line Models
A popular way of synthesizing 1/f noise with lumped circuit elements has been considered
in Section 2.6.2, [28], and is a circuit implementation of the model originally proposed in
[69]. It consists of a cascade of a number of R-C sections driven by a white noise source
each having a different time constant. The idea behind this model to include a large
number time constants of different orders to effectively capture the long memory behavior
associated with a flicker noise process. Attempts were made in [65] and [67], among others,
to implement digital versions of the infinite R-C transmission line by taking a finite number
of sections, where each section contributes a pole to the rational transfer function of the
line. The number of sections to consider depends on the maximum error from an ideal
1/f characteristic desired and the number of decades for which the 1/f sequence is to be
generated. Difficulties with implementation in a circuit simulator have been mentioned in
[98] and [105] and it is the focus of this appendix to try and quantify the difficulties involved
by implementing these R-C sections in a circuit simulator.
Based on the approach in [67], to effectively model a 1/f response for two decades
with a relative error of 5% requires 14 R-C sections. For the same relative error, modelling
the response for two and a half decades requires 100 sections. For the duration of this
155
appendix, number of sections N = 20 is considered. Although implementation of the model
in [28] requires lesser sections, the model in [67] has the advantage of not requiring a
modification in the values of the existing R and C components with the addition of more
sections. According to [67], the locations of the poles corresponding to each section is
expressed as
pn =
−(2n − 1)2 π 2
4τ
(B.1)
where τ is the largest time constant of the sections and n varies from 1 to N . There are three
cases considered here, each with varying values of R and C and the synthesized sections
are inserted into the varactor-tuned VCO circuit in Chapter 6. Case 1 fixes the values of
each capacitor to be equal to 1nF which is equivalent in value to most of the capacitors
in the VCO circuit. Based on this value of the capacitors and the poles, the values of the
resistors is calculated and found to vary between a range of approximately 400 MΩ and 250
KΩ. Inserting these sections into the input of the VCO circuit causes a problem with the
factoring of the admittance matrix. This can happen when the matrix is ill-conditioned,
which can happen when a large range of values is present in locations in the matrix. Illconditioning is a serious problem and it makes it impossible to factor the matrix which is
essential requirement to solving any linear algebra problem. Ill-conditioning is quantified
by the condition number of a matrix which should generally be small, i.e. in the range
of 1–10, [162]. After inserting this 20 section line into the VCO circuit, the admittance
matrix cannot be factored by the numerical algebra library used in f REEDATM and the
condition number is computed as roughly equal to 1040 . It is important to note that this
number is approximate and different routines to factor the admittance matrix will yield
slightly different values but such large numbers come about due to the disparity between
the smallest and largest component in the circuit.
Case 2 fixes the value of the resistor as 100Ω which is equivalent to the values of the
other resistors in the VCO circuit and based on the values of the pole locations, computes
the values of the capacitors to vary between a maximum of roughly 4 mF and a minimum of
roughly 2 µF. Inserting this into the VCO circuit again produces an ill-conditioned system
with a condition number of approximately 1040 .
While Case 1 and Case 2 are extreme cases, a middle approach to selecting the
values of the R and C components is possible. Case 3 sets the upper limit for the R element
to 1 KΩ and the C element to 10 nF which corresponds to the time constant of 1 second.
156
For successively reducing values of time constant, the R and C values will keep reducing.
The initial values of R and C are chosen so that most of the successive R-C values will be
equivalent to the R and C values in the VCO circuit in the hope of obtaining a smaller
condition number. The minimum value obtained for the resistors is roughly R = 25Ω
while that for the capacitors is roughly C = 0.2nF. Inserting these values into the VCO
circuit produces a condition number of approximately 1039 which once again emphasizes
the ill-conditioned nature of the admittance matrix.
Under some circumstances, it might be possible to find an optimum set of values
for the elements of the section but it is not straight-forward and is dependent on the circuit
under consideration. In the case of chaotic noise generators, setting up the flicker noise
process is considerably simpler, does not produce ill-conditioning and is independent of the
circuit under consideration. As a final note, this appendix considers the effect of adding a
single R-C section to an existing circuit. For a circuit with several sources of flicker noise,
this requires insertion of several R-C sections. This appendix does not consider the effect of
having a larger number of unknowns in a circuit as a result of these additional R-C sections
but it is reasonable to assume that the effect will be detrimental as the size of the problem
keeps expanding.
157
Appendix C
Source Code
C.1
Noise-enabled npn-BJT
//---------// BjtnpnN.h
//---------// Spice BJT model with charge conservation model
// with transient noise sources
//
//
//
0
NCollector
//
|
//
|
//
| /
//
|/
//
NBase 0----|
//
|
//
|
//
|
//
|
//
0
NEmitter
//
//
Author: Nikhil Kriplani
//
#ifndef BjtnpnN2_h
158
#define BjtnpnN2_h 1
#include
#include
#include
#include
"../analysis/TimeDomainSV.h"
"../fRandn.h"
"../fChaos.h"
"../fLogis.h"
class BjtnpnN2 : public AdolcElement
{
public:
BjtnpnN2(const string& iname);
~BjtnpnN2();
static const char* getNetlistName()
{
return einfo.name;
}
// Do some local initialization
virtual void init() throw(string&);
private:
// Generic state variable evaluation routine
virtual void eval(adoublev& x, adoublev& vp, adoublev& ip);
// Element information
static ItemInfo einfo;
// Number of parameters of this element
static const unsigned n_par;
// Parameter variables
double is, bf, nf, vaf, ikf, ise, ne, br, nr, var, ikr, isc;
double nc, re, rb, rbm, irb, rc, eg, cje, vje, mje, cjc, vjc, mjc;
double xcjc, fc, tf, xtf, vtf, itf, tr, xtb, xti,
double tre1, tre2, trb1, trb2;
double trm1, trm2, trc1, trc2, tnom, t, cjs, mjs,
double vjs, area, ns, iss;
double ctime;
159
double tstep, tstop, kth, ksh, beta, kf, alpha;
fChaos * fch;
fLogis * flg;
// temporary variables to store random numbers
double tmpThermal1, tmpThermal2, tmpThermal3;
double tmpThermal4, tmpThermal5, tmpThermal6;
double tmpShot1, tmpShot2, tmpShot3;
double tmpFlicker;
// Parameter information
static ParmInfo pinfo[];
};
#endif
//-----------// BjtnpnN2.cc
//-----------#include "../network/ElementManager.h"
#include "../network/AdolcElement.h"
#include "BjtnpnN2.h"
const unsigned BjtnpnN2 :: n_par = 57;
ItemInfo BjtnpnN2::einfo =
{
"bjtnpnn2",
"Gummel-Poon model with noise",
"Original model by Senthil Velu, modified by Nikhil Kriplani",
DEFAULT_ADDRESS"elements/Bjtnpn2.h.html",
"2005_07_31"
};
// Parameter Information
ParmInfo BjtnpnN2 :: pinfo[] =
{
{"is","Transport saturation current (A)", TR_DOUBLE, false},
{"bf","Ideal maximum forward beta", TR_DOUBLE, false},
{"nf","Forward current emission coefficient", TR_DOUBLE, false},
{"vaf","Forward early voltage (V)", TR_DOUBLE, false},
{"ikf","Forward-beta high current roll-off knee current (A)", TR_DOUBLE,
false},
160
{"ise","Base-emitter leakage saturation current (A)", TR_DOUBLE, false},
{"ne","Base-emitter leakage emission coefficient", TR_DOUBLE, false},
{"br","Ideal maximum reverse beta", TR_DOUBLE, false},
{"nr","Reverse current emission coefficient", TR_DOUBLE, false},
{"var","revers early voltage (V)", TR_DOUBLE, false},
{"ikr","Corner for reverse-beta high current roll off (A)", TR_DOUBLE,
false},
{"isc","Base collector leakage saturation current (A)", TR_DOUBLE,
false},
{"nc","Base-collector leakage emission coefficient", TR_DOUBLE, false},
{"re","Emitter ohmic resistance (W)", TR_DOUBLE, false},
{"rb","Zero bias base resistance (W)", TR_DOUBLE, false},
{"rbm","Minimum base resistance (W)", TR_DOUBLE, false},
{"irb","Current at which rb falls to half of rbm (A)", TR_DOUBLE, false},
{"rc","Collector ohmic resistance (W)", TR_DOUBLE, false},
{"eg","Badgap voltage (eV)", TR_DOUBLE, false},
{"cje","Base emitter zero bias p-n capacitance (F)", TR_DOUBLE, false},
{"vje","Base emitter built in potential (V)", TR_DOUBLE, false},
{"mje","Base emitter p-n grading factor", TR_DOUBLE, false},
{"cjc","Base collector zero bias p-n capacitance (F)", TR_DOUBLE, false},
{"vjc","Base collector built in potential (V)", TR_DOUBLE, false},
{"mjc","Base collector p-n grading factor", TR_DOUBLE, false},
{"xcjc","Fraction of cbc connected internal to rb", TR_DOUBLE, false},
{"fc","Forward bias depletion capacitor coefficient", TR_DOUBLE, false},
{"tf","Ideal forward transit time (S)", TR_DOUBLE, false},
{"xtf","Transit time bias dependence coefficient", TR_DOUBLE, false},
{"vtf","Transit time dependency on vbc (V)", TR_DOUBLE, false},
{"itf","Transit time dependency on ic (A)", TR_DOUBLE, false},
{"tr","Ideal reverse transit time (S)", TR_DOUBLE, false},
{"xtb","Forward and reverse beta temperature coefficient",
TR_DOUBLE, false},
{"xti","IS temperature effect exponent", TR_DOUBLE, false},
{"tre1","RE temperature coefficient (linear)", TR_DOUBLE, false},
{"tre2","RE temperature coefficient (quadratic)", TR_DOUBLE, false},
{"trb1","RB temperature coefficient (linear)", TR_DOUBLE, false},
{"trb2","RB temperature coefficient (quadratic)", TR_DOUBLE, false},
{"trm1","RBM temperature coefficient (linear)", TR_DOUBLE, false},
{"trm2","RBM temperature coefficient (quadratic)", TR_DOUBLE, false},
{"trc1","RC temperature coefficient (linear)", TR_DOUBLE, false},
{"trc2","RC temperature coefficient (quadratic)", TR_DOUBLE, false},
{"tnom","Nominal temperature (K)", TR_DOUBLE, false},
{"t","temperature (K)", TR_DOUBLE, false},
{"cjs", "Collector substrate capacitance", TR_DOUBLE, false},
{"mjs", "substrate junction exponential factor", TR_DOUBLE, false},
161
{"vjs", "substrate junction built in potential", TR_DOUBLE, false},
{"area","Current multiplier", TR_DOUBLE, false},
{"ns","substrate p-n coefficient",TR_DOUBLE,false},
{"iss","Substrate saturation current",TR_DOUBLE,false},
{"tstep", "time step for transient analysis", TR_DOUBLE, true},
{"tstop", "stop time for transient analysis", TR_DOUBLE, true},
{"kth", "thermal noise scaling factor", TR_DOUBLE, false},
{"ksh", "shot noise scaling factor", TR_DOUBLE, false},
{"beta", "exponent for chaotic map", TR_DOUBLE, false},
{"kf", "scaling factor for chaotic noise", TR_DOUBLE, false},
{"alpha", "power for dependence of flicker noise on current",
TR_DOUBLE, false}
};
BjtnpnN2::BjtnpnN2(const string& iname) :
AdolcElement(&einfo, pinfo, n_par, iname)
{
// Set default parameter values
paramvalue[0] = &(is = 1e-16);
paramvalue[1] = &(bf = 100.);
paramvalue[2] = &(nf = one);
paramvalue[3] = &(vaf = zero);
paramvalue[4] = &(ikf = zero);
paramvalue[5] = &(ise = zero);
paramvalue[6] = &(ne = 1.5);
paramvalue[7] = &(br = one);
paramvalue[8] = &(nr = one);
paramvalue[9] = &(var = zero);
paramvalue[10] = &(ikr = zero);
paramvalue[11] = &(isc = zero);
paramvalue[12] = &(nc = 2.);
paramvalue[13] = &(re = zero);
paramvalue[14] = &(rb = zero);
paramvalue[15] = &(rbm = zero);
paramvalue[16] = &(irb = zero);
paramvalue[17] = &(rc = zero);
paramvalue[18] = &(eg = 1.11);
paramvalue[19] = &(cje = zero);
paramvalue[20] = &(vje = 0.75);
paramvalue[21] = &(mje = 0.33);
paramvalue[22] = &(cjc = zero);
paramvalue[23] = &(vjc = 0.75);
paramvalue[24] = &(mjc = 0.33);
paramvalue[25] = &(xcjc = one);
162
paramvalue[26] =
paramvalue[27] =
paramvalue[28] =
paramvalue[29] =
paramvalue[30] =
paramvalue[31] =
paramvalue[32] =
paramvalue[33] =
paramvalue[34] =
paramvalue[35] =
paramvalue[36] =
paramvalue[37] =
paramvalue[38] =
paramvalue[39] =
paramvalue[40] =
paramvalue[41] =
paramvalue[42] =
paramvalue[43] =
paramvalue[44] =
paramvalue[45] =
paramvalue[46] =
paramvalue[47] =
paramvalue[48] =
paramvalue[49] =
paramvalue[50] =
paramvalue[51] =
paramvalue[52] =
paramvalue[53] =
paramvalue[54] =
paramvalue[55] =
paramvalue[56] =
setNumTerms(4);
&(fc = 0.5);
&(tf = zero);
&(xtf = zero);
&(vtf = zero);
&(itf = zero);
&(tr = zero);
&(xtb = zero);
&(xti = 3.);
&(tre1 = zero);
&(tre2 = zero);
&(trb1 = zero);
&(trb2 = zero);
&(trm1 = zero);
&(trm2 = zero);
&(trc1 = zero);
&(trc2 = zero);
&(tnom = 300.);
&(t = 300.);
&(cjs = zero);
&(mjs = zero);
&(vjs = 0.75);
&(area = one);
&(ns = one);
&(iss = zero);
&(tstep = 1.0e-9);
&(tstop = 1.0e-6);
&(kth = 1.0);
&(ksh = 1.0);
&(beta = 0.005);
&(kf = 1.0);
&(alpha = 2.);
//Set Flags
setFlags(NONLINEAR | ONE_REF | TR_TIME_DOMAIN);
setNumberOfStates(3);
// initialize the random no. generators
fch = new fChaos;
flg = new fLogis;
}
void BjtnpnN2::init() throw(string&)
{
163
//create tape
IntVector var2(3);
var2[0] = 0;
var2[1] = 1;
var2[2] = 2;
IntVector novar;
DoubleVector nodelay;
createTape(var2, var2, var2, novar, nodelay);
// initialize the random number generators
int points = (int)(tstop/tstep) + 1;
fch->setSize(points);
fch->setBeta(beta);
fch->generateMap();
flg->setSize(points);
flg->generateMap();
// variable to hold the current time
ctime = getCurrentTime();
}
BjtnpnN2::~BjtnpnN2()
{
delete flg;
delete fch;
}
//evaluate function
void BjtnpnN2::eval(adoublev& x, adoublev& vp, adoublev& ip)
{
//x[vbe,vbc,vcjs,dvbe,dvbc,dvcjs,d2vbe,d2vbc,d2vcjs]
//x[0] : Vbe
//x[1] : Vbc
//x[2] : Vcjs
//x[3] : dvbe/dt
//x[4] : dvbc/dt
//x[5] : dvcjs/dt
//x[6] : d2vbe/dt
//x[7] : d2vbc/dt
//x[8] : d2vcjs/dt
//vp[0] : Vcjs ip[0] : Ic
//vp[1] : Vbjs ip[1] : Ib
164
//vp[2] : Vejs ip[2] : Ie
adouble Ibe, Ibc, Ice, Ibf, Ile, Ibr;
adouble Ilc, kqb, kqbtemp;
adouble cbej, cbet, cbcj, cbct, ibe, ibc;
adouble vth = (kBoltzman * tnom) / eCharge;
if (!isSet(&rbm))
rbm = rb;
//dc current equations
Ibf = is * (exp(x[0]/nf/vth) - one);
Ile = ise * (exp(x[0]/ne/vth) - one);
Ibr = is * (exp(x[1]/nr/vth) - one);
Ilc = isc * (exp(x[1]/nc/vth) - one);
Ibe = Ibf / bf + Ile;
Ibc = Ibr / br + Ilc;
adouble Ibf1 = (Ibe - Ile) * bf;
adouble Ibr1 = (Ibc - Ilc) * br;
adouble kqbtem = zero;
if (ikf)
kqbtem = 4.*(Ibf1/ikf);
if (ikr)
kqbtem += 4.*(Ibr1/ikr);
kqbtemp = sqrt(one + kqbtem);
adouble tempvaf = zero;
if (vaf)
tempvaf = x[1] / vaf;
adouble tempvar = zero;
if (var)
tempvar = x[0] / var;
kqb = 0.5 * (one / (one - tempvaf - tempvar)) * (one + kqbtemp);
Ice = (Ibf1 - Ibr1) / kqb;
// Charge base-collector
condassign(cbcj, x[1]-fc*vjc,
cjc*pow(one-fc, -one-mjc) * (one-fc*(one + mjc) + mjc*x[1]/vjc),
cjc*pow(one-x[1]/vjc, -mjc));
cbct = tr * is * exp(x[1]/nr/vth)/(nr*vth);
ibc = area * (cbct + xcjc*cbcj) * x[4];
//Current Base-Emitter
165
condassign(cbej,x[0]-fc*vje,
cjc*pow(one-fc,-one-mjc)*(one-fc*(one+mje)+mje*x[0]/vje),
cje*pow(one-x[0]/vje,-mje));
adouble tZ = one;
if (vtf)
tZ = exp(.69*x[1]/vtf);
cbet=is*exp(x[0]/nf/vth)*tf*(one+xtf*tZ)/(nf*vth);
ibe = area * (cbet + cbej) * x[3];
adouble Tibe,Tibc;
Tibe = Ibe + ibe;
Tibc = Ibc + ibc;
adouble rb1;
adouble vbx,cbx,ibx;
rb1 = (rbm + (rb - rbm) / kqb) / area;
vbx = x[1] + (Tibe + Tibc) * rb;
condassign(cbx,vbx-fc*vjc,
(one-xcjc)*cjc*pow(one-fc,-one-mjc)*(one-fc*(one+mjc)+mjc*vbx/vjc),
(one-xcjc)*cjc*pow(one-vbx/vjc,-mjc));
adouble dibc,dibe,dIbc,dIbe;
dibc = area*(cbct+xcjc*cbcj) * x[7];
dibe = area*(cbet+cbej) * x[6];
dIbe = is/bf * exp(x[0]/nf/vth) * (x[3]/nf/vth)
+ ise * exp(x[0]/ne/vth) * (x[3]/ne/vth);
dIbc = is/br * exp(x[1]/nr/vth) * (x[4]/nr/vth)
+ isc * exp(x[1]/nc/vth) * (x[4]/nc/vth);
ibx = cbx * (x[3] + rb*(dibc+dibe+dIbc+dIbe));
adouble Ijs;
adouble cjjs, ijs;
Ijs = area * iss * (exp(x[2]/ns/vth) - one);
condassign(cjjs, x[2] - zero,
cjs*(one+mjs*x[2]/vjs),
cjs*pow(one-x[2]/vjs,-mjs));
ijs = Ijs + area*(cjjs * x[5]);
166
adouble iShot1, iShot2, iShot3;
adouble iFlicker;
adouble iThermal1, iThermal2, iThermal3;
adouble vThermal1, vThermal2, vThermal3;
double TEMP = 300.0;
ip[0] = Ice - Tibc - ibx - ijs; // collector current
ip[1] = ibx + Tibc + Tibe;
// base current
ip[2] = -(Ice + Tibe - ijs);
// emitter current
vp[0] = x[2] + rc*ip[0];
// Collector Substrate Voltage
vp[1] = x[2] + x[1] + ip[1]*rb;
// Base Substrate Voltage
vp[2] = x[1] - x[0] + x[2] + ip[2]*re; // Emitter Substrate Voltage
// If we are the next time step, then get new
// random values
if (getCurrentTime() > ctime)
{
ctime = getCurrentTime();
int mav = (int)(ctime / tstep);
// calculate the shot noise component and the
// flicker noise component
//double xiShot1 = frn->RNOR(0,1);
double xiShot1 = flg->getMapValue(mav);
double xiShot2 = flg->getMapValue(mav);
double xiShot3 = flg->getMapValue(mav);
tmpShot1 = xiShot1;
tmpShot2 = xiShot2;
tmpShot3 = xiShot3;
// shot noise component Ice
iShot1 = ksh * sqrt(eCharge * fabs(Ice)) * xiShot1;
// shot noise component Ibe
iShot2 = ksh * sqrt(eCharge * fabs(Ibe)) * xiShot2;
// shot noise component ibc
iShot3 = ksh * sqrt(eCharge * fabs(Ibc)) * xiShot3;
double xiFlicker = fch->getMapValue(mav);
tmpFlicker = xiFlicker;
iFlicker = kf * sqrt(pow(fabs(Ile),alpha)) * xiFlicker;
// rc, rb, re contributes thermal noise components
//double xiThermal1 = frn->RNOR(0,1);
double xiThermal1 = flg->getMapValue(mav);
167
double
double
double
double
double
xiThermal2
xiThermal3
xiThermal4
xiThermal5
xiThermal6
=
=
=
=
=
flg->getMapValue(mav);
flg->getMapValue(mav);
flg->getMapValue(mav);
flg->getMapValue(mav);
flg->getMapValue(mav);
tmpThermal1 = xiThermal1;
tmpThermal2 = xiThermal2;
tmpThermal3 = xiThermal3;
tmpThermal4 = xiThermal4;
tmpThermal5 = xiThermal5;
tmpThermal6 = xiThermal6;
iThermal1 = kth * sqrt(2.0 * kBoltzman * TEMP / rc) * xiThermal1;
iThermal2 = kth * sqrt(2.0 * kBoltzman * TEMP / rb1) * xiThermal2;
iThermal3 = kth * sqrt(2.0 * kBoltzman * TEMP / re) * xiThermal3;
vThermal1 = kth * sqrt(2.0 * kBoltzman * TEMP * rc) * xiThermal4;
vThermal2 = kth * sqrt(2.0 * kBoltzman * TEMP * rb1) * xiThermal5;
vThermal3 = kth * sqrt(2.0 * kBoltzman * TEMP * re) * xiThermal6;
}
else if ((getCurrentTime() == ctime) && (ctime > 0.0))
// use the old random values
{
iShot1 = ksh * sqrt(eCharge * fabs(Ice)) * tmpShot1;
iShot2 = ksh * sqrt(eCharge * fabs(Ibe)) * tmpShot2;
iShot3 = ksh * sqrt(eCharge * fabs(Ibc)) * tmpShot3;
iFlicker = kf * sqrt(pow(fabs(Ile),alpha)) * tmpFlicker;
iThermal1
iThermal2
iThermal3
vThermal1
vThermal2
vThermal3
=
=
=
=
=
=
kth
kth
kth
kth
kth
kth
*
*
*
*
*
*
sqrt(2.0
sqrt(2.0
sqrt(2.0
sqrt(2.0
sqrt(2.0
sqrt(2.0
*
*
*
*
*
*
kBoltzman
kBoltzman
kBoltzman
kBoltzman
kBoltzman
kBoltzman
*
*
*
*
*
*
TEMP
TEMP
TEMP
TEMP
TEMP
TEMP
/
/
/
*
*
*
rc) * tmpThermal1;
rb1) * tmpThermal2;
re) * tmpThermal3;
rc) * tmpThermal4;
rb1) * tmpThermal5;
re) * tmpThermal6;
}
// thermal current contributions
adouble dr = rc*(rb1 + re) + rb1*re;
adouble inc = (re*vThermal1 - (rc + re)*vThermal2 - rc*vThermal3)/dr;
adouble inb = (re*vThermal2 - (rb1 + re)*vThermal1 - rb1*vThermal3)/dr;
// Final collector current
ip[0] += iShot1 + inc - iShot3;
// Final base current
168
ip[1] += iShot2 + iShot3 + inb + iFlicker;
// Final emitter current
ip[2] += -iFlicker - iShot1 - iShot2;
}
C.2
Noise-enabled p-n Junction Diode
//-----// DN2.h
//-----// Spice Diode model with transient noise elements
//
//
//
|\ |
//
0 o------| >|------o 1
//
|/ |
// anode
cathode
//
//
//
Author:
//
Carlos E. Christoffersen
//
Mete Ozkar
// Noise terms added by Nikhil Kriplani
//
#ifndef _DN2_h
#define _DN2_h 1
#include
#include
#include
#include
"../analysis/TimeDomainSV.h"
"../fRandn.h"
"../fChaos.h"
"../fLogis.h"
class DN2 : public AdolcElement
{
public:
DN2(const string& iname);
~DN2();
static const char* getNetlistName()
{
169
return einfo.name;
}
// Do some local initialization
virtual void init() throw(string&);
private:
virtual void eval(adoublev& x, adoublev& vp, adoublev& ip);
// Some constants
double v1;
double ctime;
// Element information
static ItemInfo einfo;
// Number of parameters of this element
static const unsigned n_par;
// Parameter variables
double is, n, ibv, bv, fc, cj0, vj, m, tt, area, rs;
bool charge;
double tstep, tstop, kth, ksh, beta, kf, alpha;
//fRandn * frn;
fChaos * fch;
fLogis * flg;
// temporary variables to store random numbers
double tmpThermal1, tmpShot, tmpFlicker;
// Parameter information
static ParmInfo pinfo[];
};
#endif
//------// DN2.cc
//------#include "../network/CircuitManager.h"
170
#include "../network/AdolcElement.h"
#include "DN2.h"
// Static members
const unsigned DN2::n_par = 19;
// Element information
ItemInfo DN2::einfo =
{
"dn2",
"Spice diode model (conserves charge)with transient noise elements",
"Carlos E. Christoffersen / Nikhil Kriplani",
DEFAULT_ADDRESS"elements/DN2.h.html",
"2005_07_28"
};
// Parameter information
ParmInfo DN2::pinfo[] =
{
{"is", "Saturation current (A)", TR_DOUBLE, false},
{"n", "Emission coefficient", TR_DOUBLE, false},
{"ibv", "Current magnitude at the reverse breakdown voltage (A)",
TR_DOUBLE, false},
{"bv", "Breakdown voltage (V)", TR_DOUBLE, false},
{"fc", "Coefficient for forward-bias depletion capacitance",
TR_DOUBLE, false},
{"cj0", "Zero-bias depletion capacitance (F)", TR_DOUBLE, false},
{"vj", "Built-in junction potential (V)", TR_DOUBLE, false},
{"m", "PN junction grading coefficient", TR_DOUBLE, false},
{"tt", "Transit time (s)", TR_DOUBLE, false},
{"area", "Area multiplier", TR_DOUBLE, false},
{"charge", "Use charge-conserving model", TR_BOOLEAN, false},
{"rs", "Series resistance (ohms)", TR_DOUBLE, false},
{"tstep", "time step for transient analysis", TR_DOUBLE, true},
{"tstop", "stop time for transient analysis", TR_DOUBLE, true},
{"kth", "thermal noise scaling factor", TR_DOUBLE, false},
{"ksh", "shot noise scaling factor", TR_DOUBLE, false},
{"beta", "exponent for chaotic map", TR_DOUBLE, false},
{"kf", "scaling factor for chaotic noise", TR_DOUBLE, false},
{"alpha", "power for dependence of flicker noise on current",
TR_DOUBLE, false}
};
DN2::DN2(const string& iname) :
171
AdolcElement(&einfo, pinfo, n_par, iname)
{
// Set default parameter values
paramvalue[0] = &(is = 1e-14);
paramvalue[1] = &(n = one);
paramvalue[2] = &(ibv = 1e-10);
paramvalue[3] = &(bv = zero);
paramvalue[4] = &(fc = .5);
paramvalue[5] = &(cj0 = zero);
paramvalue[6] = &(vj = one);
paramvalue[7] = &(m = .5);
paramvalue[8] = &(tt = zero);
paramvalue[9] = &(area = one);
paramvalue[10] = &(charge = true);
paramvalue[11] = &(rs = zero);
paramvalue[12] = &(tstep = 1.0e-9);
paramvalue[13] = &(tstop = 1.0e-6);
paramvalue[14] = &(kth = one);
paramvalue[15] = &(ksh = one);
paramvalue[16] = &(beta = 0.000005);
paramvalue[17] = &(kf = one);
paramvalue[18] = &(alpha = one);
// Set flags
setFlags(NONLINEAR | ONE_REF | TR_TIME_DOMAIN);
// initialize the random no. generators
fch = new fChaos;
flg = new fLogis;
}
void DN2::init() throw(string&)
{
if (charge)
{
// Add one terminal
Circuit* cir = getCircuit();
unsigned tref_id = getTerminal(1)->getId();
unsigned term_id1 = cir->addTerminal(getInstanceName()
+ ":extra");
cir->connect(getId(), term_id1);
// Connect an external 1K resistor (not really needed if using the
// augmentation network)
unsigned newelem_id =
cir->addElement("res", getInstanceName() + ":res");
172
cir->connect(newelem_id, tref_id);
cir->connect(newelem_id, term_id1);
Element* elem = cir->getElement(newelem_id);
double res = 1e3;
elem->setParam("r", &res, TR_DOUBLE);
elem->init();
// Set the number of terminals
setNumTerms(3);
// Set number of states
setNumberOfStates(2);
// create tape
IntVector var(2);
var[0] = 0;
var[1] = 1;
IntVector dvar(1,1);
createTape(var, dvar);
// initialize the random number generators
int points = (int)(tstop/tstep) + 1;
fch->setSize(points);
fch->setBeta(beta);
fch->generateMap();
flg->setSize(points);
flg->generateMap();
// variable to hold the current time
ctime = getCurrentTime();
}
else
{
// Set the number of terminals
setNumTerms(2);
// Set number of states
setNumberOfStates(1);
// create tape
IntVector var(1,0);
createTape(var, var);
// initialize the random number generators
int points = (int)(tstop/tstep);
fch->setSize(points);
173
fch->setBeta(beta);
fch->generateMap();
// variable to hold the current time
ctime = getCurrentTime();
}
}
DN2::~DN2()
{
delete flg;
delete fch;
}
void DN2::eval(adoublev& x,
adoublev& vp, adoublev& ip)
{
double alfa = eCharge / n / kBoltzman / 300.; // tnom = 300K
double v1 = log(5e8 / alfa) / alfa; // normal is .5e9
double k3 = exp(alfa * v1);
// x[0]: x as in Rizolli’s equations
adouble vd,id;
// Diode voltage
condassign(vd, v1 - x[0],
x[0] + zero,
v1 + log(one + alfa*(x[0]-v1))/alfa);
// Static current
condassign(id, v1 - x[0],
is * (exp(alfa * x[0]) - one),
is * k3 * (one + alfa * (x[0] - v1)) - is);
// subtract the breakdown current
ip[0] = id - ibv * exp(-alfa * (vd + bv));
adouble iShot = 0.0;
adouble iFlicker = 0.0;
adouble iThermal = 0.0;
adouble vThermal = 0.0;
double TEMP = 300.0;
// If we are the next time step, then get new
// random values
if (getCurrentTime() > ctime)
174
{
ctime = getCurrentTime();
int mav = (int)(ctime / tstep);
// calculate the shot noise component and the
// flicker noise component
double xiShot = flg->getMapValue(mav);
tmpShot = xiShot;
iShot = ksh * sqrt(eCharge * fabs(ip[0])) * xiShot;
double xiFlicker = fch->getMapValue(mav);
tmpFlicker = xiFlicker;
iFlicker = kf * sqrt(pow(fabs(ip[0]),alpha)) * xiFlicker;
// rs contributes a thermal noise component
double xiThermal1 = flg->getMapValue(mav);
tmpThermal1 = xiThermal1;
iThermal = kth * sqrt(2.0 * kBoltzman * TEMP / rs) * xiThermal1;
}
else if ((getCurrentTime() == ctime) && (ctime > 0.0))
// use the old random values
{
iShot = ksh * sqrt(eCharge * fabs(ip[0])) * tmpShot;
iFlicker = kf * sqrt(pow(fabs(ip[0]),alpha)) * tmpFlicker;
iThermal = kth * sqrt(2.0 * kBoltzman * TEMP / rs) * tmpThermal1;
}
// add the noise current contributions
ip[0] += iThermal + iShot + iFlicker;
if (charge)
{
// x[1]: q
// x[2]: dq/dt
// Form the additional error function
adouble qvj;
double km;
if (isSet(&cj0))
{
condassign(qvj, fc * vj - vd,
vj * cj0 / (one - m) * (one - pow(one - vd / vj, one - m)),
cj0 * pow(one - fc, - m - one) *
(((one - fc * (one + m)) * vd + .5 * m * vd * vd / vj) -
175
((one - fc * (one + m)) * vj * fc + .5 * m * vj * fc * fc)) +
vj * cj0 / (one - m) * (one - pow(one - fc, one - m)));
km = vj * cj0 / (one - m) * 1e1;
}
else
{
qvj = zero;
km = 1e-12;
}
if (isSet(&tt))
{
qvj += tt * ip[0];
km += tt * 1e-2;
}
// Add capacitor current
ip[0] += x[2] * km;
ip[1] = - ip[0];
vp[1] = qvj / km - x[1];
vp[0] = vd + ip[0] * rs + vp[1];
// scale the current according to area.
ip[0] *= area;
ip[1] *= area;
}
else
{
adouble dvd_dx;
condassign(dvd_dx, v1 - x[0],
one,
one / (one + alfa*(x[0]-v1)));
adouble cd;
if (isSet(&cj0))
condassign(cd, fc * vj - vd,
cj0 * pow(one - vd / vj, -m),
cj0 * pow(one - fc, - m - one) *
((one - fc * (one + m)) + m * vd / vj));
else
cd = zero;
if (isSet(&tt))
cd += alfa * tt * ip[0];
// Add capacitor current
ip[0] += cd * dvd_dx * x[1];
vp[0] = vd + ip[0] * rs;
176
// scale the current according to area.
ip[0] *= area;
}
}
C.3
Noise-enabled Curtice-Cubic MESFET
//----------// MesfetCN.h
//----------// Curtice cubic MESFET model
// with noise elements (transient analysis only)
//
//
Drain 2
//
o
//
|
//
|
//
|---+
//
|
// Gate 1 o-----|
//
|
//
|---+
//
|
//
|
//
o
//
Source 3
//
//
//
Author: Carlos Christofferssen, Nikhil Kriplani
#ifndef MesfetCN_h
#define MesfetCN_h 1
#include "../analysis/TimeDomainSV.h"
#include "../fRandn.h"
#include "../fChaos.h"
class MesfetCN : public AdolcElement
{
public:
MesfetCN(const string& iname);
177
~MesfetCN() {}
static const char* getNetlistName()
{
return einfo.name;
}
// Do some local initialization
virtual void init() throw(string&);
private:
virtual void eval(adoublev& x,
adoublev& vp, adoublev& ip);
// Some constants
double k2, k3;
double delta_T, tn, Vt, k1, k4, k5, k6, Vt0, Beta, Ebarr, EbarrN, Nn;
double Is, Vbi;
// Element information
static ItemInfo einfo;
// Number of parameters of this element
static const unsigned n_par;
// Parameter variables
double a0, a1, a2, a3, beta, vds0, gama, vt0;
double cgs0, cgd0, is, n, ib0, nr;
double t, vbi, fcc, vbd, area;
double tnom, avt0, bvt0, tbet, tm, tme, eg, m, xti, tj;
double tmpFlicker, tmpThermal1;
double ctime;
double tstep, tstop, kth, ksh, map_beta, kf, af;
fRandn * frn;
fChaos * fch;
// Parameter information
static ParmInfo pinfo[];
};
#endif
//------------
178
// MesfetCN.cc
//-----------#include "../network/ElementManager.h"
#include "../network/AdolcElement.h"
#include "MesfetCN.h"
// Static members
const unsigned MesfetCN::n_par = 34;
// Element information
ItemInfo MesfetCN::einfo = {
"mesfetcn",
"Intrinsic noisy MESFET using Curtice-Ettemberg cubic model",
"Carlos E. Christoffersen, Nikhil Kriplani",
DEFAULT_ADDRESS"elements/MesfetCN.h.html",
"2005_12_14"
};
// Parameter information
ParmInfo MesfetCN::pinfo[] = {
{"a0", "Drain saturation current for Vgs=0 (A)", TR_DOUBLE, false},
{"a1", "Coefficient for V1 (A/V)", TR_DOUBLE, false},
{"a2", "Coefficient for V1^2 (A/V^2)", TR_DOUBLE, false},
{"a3", "Coefficient for V1^3 (A/V^3)", TR_DOUBLE, false},
{"beta", "V1 dependance on Vds (1/V)", TR_DOUBLE, false},
{"vds0", "Vds at which BETA was measured (V)", TR_DOUBLE, false},
{"gama", "Slope of drain characteristic in the linear region (1/V)",
TR_DOUBLE, false},
{"vt0", "Voltage at which the channel current is forced to be
zero for Vgs<=Vto (V)", TR_DOUBLE, false},
{"cgs0", "Gate-source Schottky barrier capacitance for Vgs=0 (F)",
TR_DOUBLE, false},
{"cgd0", "Gate-drain Schottky barrier capacitance for Vgd=0 (F)",
TR_DOUBLE, false},
{"is", "Diode saturation current (A)", TR_DOUBLE, false},
{"n", "Diode ideality factor", TR_DOUBLE, false},
{"ib0", "Breakdown current parameter (A)", TR_DOUBLE, false},
{"nr", "Breakdown ideality factor", TR_DOUBLE, false},
{"t", "Channel transit time (s)", TR_DOUBLE, false},
{"vbi", "Built-in potential of the Schottky junctions (V)",
TR_DOUBLE, false},
{"fcc", "Forward-bias depletion capacitance coefficient (V)",
TR_DOUBLE, false},
179
{"vbd", "Breakdown voltage (V)", TR_DOUBLE, false},
{"tnom", "Reference Temperature (K)", TR_DOUBLE, false},
{"avt0", "Pinch-off voltage (VP0 or VT0) linear temp. coefficient (1/K)",
TR_DOUBLE, false},
{"bvt0",
"Pinch-off voltage (VP0 or VT0) quadratic temp. coefficient (1/K^2)",
TR_DOUBLE, false},
{"tbet", "BETA power law temperature coefficient (1/K)", TR_DOUBLE,
false},
{"tm", "Ids linear temp. coeff. (1/K)", TR_DOUBLE, false},
{"tme", "Ids power law temp. coeff. (1/K^2)", TR_DOUBLE, false},
{"eg", "Barrier height at 0.K (eV)", TR_DOUBLE, false},
{"m", "Grading coefficient", TR_DOUBLE, false},
{"xti", "Diode saturation current temperature exponent", TR_DOUBLE,
false},
{"tj", "Junction Temperature (K)", TR_DOUBLE, false},
{"area", "Area multiplier", TR_DOUBLE, false},
{"tstep", "time step for transient analysis", TR_DOUBLE, true},
{"tstop", "stop time for transient analysis", TR_DOUBLE, true},
{"map_beta", "exponent for chaotic map", TR_DOUBLE, false},
{"kf", "scaling factor for chaotic noise", TR_DOUBLE, false},
{"af", "flicker noise exponent", TR_DOUBLE, false}
};
MesfetCN::MesfetCN(const string& iname)
: AdolcElement(&einfo, pinfo, n_par, iname)
{
// Set default parameter values
paramvalue[0] = &(a0 = .1);
paramvalue[1] = &(a1 = .05);
paramvalue[2] = &(a2 = zero);
paramvalue[3] = &(a3 = zero);
paramvalue[4] = &(beta = zero);
paramvalue[5] = &(vds0 = 4.);
paramvalue[6] = &(gama = 1.5);
paramvalue[7] = &(vt0 = -1e10);
paramvalue[8] = &(cgs0 = zero);
paramvalue[9] = &(cgd0 = zero);
paramvalue[10] = &(is = zero);
paramvalue[11] = &(n = one);
paramvalue[12] = &(ib0 = zero);
paramvalue[13] = &(nr = 10.);
paramvalue[14] = &(t = zero);
180
paramvalue[15]
paramvalue[16]
paramvalue[17]
paramvalue[18]
paramvalue[19]
paramvalue[20]
paramvalue[21]
paramvalue[22]
paramvalue[23]
paramvalue[24]
paramvalue[25]
paramvalue[26]
paramvalue[27]
paramvalue[28]
paramvalue[29]
paramvalue[30]
paramvalue[31]
paramvalue[32]
paramvalue[33]
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
&(vbi = .8);
&(fcc = .5);
&(vbd = 1e10);
&(tnom = 293.);
&(avt0 = zero);
&(bvt0 = zero);
&(tbet = zero);
&(tm = zero);
&(tme = zero);
&(eg = .8);
&(m = .5);
&(xti = 2.);
&(tj = 293.);
&(area = one);
&(tstep = 1e-12);
&(tstop = 10e-9);
&(map_beta = 0.000005);
&(kf = one);
&(af = one);
// Set the number of terminals
setNumTerms(3);
// Set flags
setFlags(NONLINEAR | ONE_REF | TR_TIME_DOMAIN);
// Set number of states
setNumberOfStates(2);
frn = new fRandn;
fch = new fChaos;
}
void MesfetCN::init() throw(string&)
{
k2 = cgs0 / sqrt(one - fcc);
k3 = cgd0 / sqrt(one - fcc);
delta_T = tj - tnom;
tn = tj / tnom;
Vt = kBoltzman * tj / eCharge;
k5 = n * Vt;
k6 = nr * Vt;
Vt0 = vt0 * (one + (delta_T * avt0) + (delta_T * delta_T * bvt0));
181
if (tbet)
Beta = beta * pow(1.01, (delta_T * tbet));
else
Beta = beta;
Ebarr = eg -.000702 * tj * tj / (tj + 1108.);
EbarrN = eg -.000702 * tnom * tnom / (tnom + 1108.);
Nn = eCharge / 38.696 / kBoltzman / tj;
Is = is * exp((tn - one) * Ebarr / Nn / Vt);
if (xti)
Is *= pow(tn, xti / Nn);
Vbi = vbi * tn -3. * Vt * log(tn) + tn * EbarrN - Ebarr;
k1 = fcc * Vbi;
k4 = 2. * Vbi * (one - fcc);
// initialize the random number generators
int points = (int)(tstop/tstep) + 1;
fch->setSize(points);
fch->setBeta(map_beta);
fch->generateMap();
// variable to hold the current time
ctime = getCurrentTime();
// create tape
IntVector var(2);
var[0] = 0;
var[1] = 1;
IntVector dvar(2);
dvar[0] = 0;
dvar[1] = 1;
IntVector d2var;
IntVector tvar(1,0);
DoubleVector delay(1, t);
createTape(var, dvar, d2var, tvar, delay);
}
void MesfetCN::eval(adoublev& x,
adoublev& vp, adoublev& ip)
{
// x[0]: vgs
// x[1]: vgd
// x[2]: dvgs/dt
// x[3]: dvgd/dt
// x[4]: vgs(t-tau)
182
// vp[0]: vgs , ip[0]: ig
// vp[1]: vds , ip[1]: id
// Assign known output voltages
vp[0] = x[0];
vp[1] = x[0] - x[1];
adouble ids, igd, igs, itmp, vx, cgs, cgd;
// static igs current
igs = Is *(exp(x[0] / k5) - one) - ib0 * exp(-(x[0] +vbd) / k6);
// Calculate cgs, including temperature effect.
condassign(cgs, k1 - x[0],
cgs0 / sqrt(one - x[0] / Vbi), k2 * (one + (x[0] - k1) / k4));
cgs *= (one + m * (0.0004 * delta_T + one - Vbi / vbi));
// Calculate the total current igs = static + dq_dt
igs += cgs * x[2];
// static igd current
igd = Is * (exp(x[1] / k5) - one) - ib0 * exp(-(x[1] + vbd) / k6);
// Calculate cgd, including temperature effect.
condassign(cgd, k1 - x[1],
cgd0 / sqrt(one - x[1] / Vbi), k3 * (one + (x[1] - k1) / k4));
cgd *= (one + m * (0.0004 * delta_T + one - Vbi / vbi));
// Calculate the total current igd = static + dq_dt
igd += cgd * x[3];
// Calculate ids. Include temperature effects.
vx = x[4] * (one + Beta * (vds0 - vp[1]));
itmp = (a0 + vx*(a1 + vx*(a2 + vx * a3)))* tanh(gama * vp[1]);
condassign(ids, (itmp * vp[1]) * (x[0] - Vt0), itmp, zero);
if (tme && tm)
ids *= pow((1 + delta_T * tm), tme);
adouble eta, iFlicker, iThermal1;
condassign(eta, vp[1] - x[0] - Vt0, zero, 1 - vp[1]/(x[0] - Vt0));
// If we are the next time step, then get new
183
// random values
if (getCurrentTime() > ctime)
{
ctime = getCurrentTime();
int mav = (int)(ctime / tstep);
// calculate the flicker noise component
double xiFlicker = fch->getMapValue(mav);
tmpFlicker = xiFlicker;
iFlicker = kf * sqrt(pow(fabs(ids),af)) * xiFlicker;
// Thermal noise is associated with channel resistance
double xiThermal1 = frn->RNOR(0,1);
tmpThermal1 = xiThermal1;
iThermal1 = sqrt(4.0/3.0 * kBoltzman * tj * Beta *
(x[0] - Vt0) * (1 + eta + eta*eta)/(1 + eta)) * xiThermal1;
}
else if ((getCurrentTime() == ctime) && (ctime > 0.0))
// use the old random values
{
iFlicker = kf * sqrt(pow(fabs(ids),af)) * tmpFlicker;
iThermal1 = sqrt(4.0/3.0 * kBoltzman * tj * Beta *
(x[0] - Vt0) * (1 + eta + eta*eta)/(1 + eta)) * tmpThermal1;
}
ids += iThermal1 + iFlicker;
// Calculate the output currents
ip[0] = (igd + igs) * area;
ip[1] = (ids - igd) * area;
}
C.4
Noise-enabled Resistor
//-----// RN2.h
//-----#ifndef RN2_h
#define RN2_h 1
#include "../analysis/TimeDomainSV.h"
#include "../fLogis.h"
#include "../network/AdolcElement.h"
184
class RN2 : public AdolcElement
{
public:
RN2(const string& iname);
~RN2();
static const char* getNetlistName()
{
return einfo.name;
}
// Do some local initialization
virtual void init() throw(string&);
private:
virtual void eval(adoublev& x, adoublev& vp, adoublev& ip);
// Some constants
double v1;
double ctime;
// Element information
static ItemInfo einfo;
// Number of parameters of this element
static const unsigned n_par;
// Parameter variables
double res, temp, tstep, tstop, kth;
fLogis * flg;
// temporary variables to store random numbers
double tmpThermal1;
// Parameter information
static ParmInfo pinfo[];
};
#endif
185
// -----// RN2.cc
//------#include "../network/ElementManager.h"
#include "../network/Element.h"
#include "../network/AdolcElement2.h"
#include "../network/CircuitManager.h"
#include "../analysis/TimeMNAM.h"
#include "../analysis/TimeDomainSV.h"
#include "RN2.h"
// Static members
const unsigned RN2::n_par = 5;
// Element information
ItemInfo RN2::einfo =
{
"rn2",
"RN2",
"Nikhil Kriplani",
DEFAULT_ADDRESS"elements/RN2.h.html",
"2005_09_22"
};
// Parameter information
ParmInfo RN2::pinfo[] =
{
{"res", "Resistance value (Ohms)", TR_DOUBLE, false},
{"temp", "Temperature (K)", TR_DOUBLE, false},
{"tstep", "time step for transient analysis", TR_DOUBLE, true},
{"tstop", "stop time for transient analysis", TR_DOUBLE, true},
{"kth", "thermal noise scaling factor", TR_DOUBLE, false}
};
RN2::RN2(const string& iname) : AdolcElement(&einfo, pinfo, n_par, iname)
{
// Set parameters
paramvalue[0] = &(res);
paramvalue[1] = &(temp = 300.0);
paramvalue[2] = &(tstep = 1.0e-9);
paramvalue[3] = &(tstop = 1.0e-6);
paramvalue[4] = &(kth = 1.0);
186
// Set the number of terminals
setNumTerms(2);
// Set number of states
setNumberOfStates(1);
// Set flags
setFlags(NONLINEAR | ONE_REF | TR_TIME_DOMAIN);
// initialize the random no. generator
flg = new fLogis;
}
void RN2::init() throw(string&)
{
// create tape
IntVector var(1,0);
createTape(var, var);
int points = (int)(tstop/tstep) + 1;
flg->setSize(points);
flg->generateMap();
ctime = getCurrentTime();
}
RN2::~RN2()
{
delete flg;
}
void RN2::eval(adoublev& x, adoublev& vp, adoublev& ip)
{
// x[0]: resistor voltage
vp[0] = x[0];
ip[0] = x[0]/res;
adouble iThermal = 0.0;
if (getCurrentTime() > ctime)
{
187
ctime = getCurrentTime();
int mav = (int)(ctime / tstep);
// res contributes a thermal noise component
double xiThermal1 = flg->getMapValue(mav);
tmpThermal1 = xiThermal1;
iThermal = kth * sqrt(2.0 * kBoltzman * temp / res) * xiThermal1;
}
else if ((getCurrentTime() == ctime) && (ctime > 0.0))
// use the old random values
{
iThermal = kth * sqrt(2.0 * kBoltzman * temp / res) * tmpThermal1;
}
// add the noise current contributions
ip[0] += iThermal;
}
C.5
White Noise Voltage Source
//-----// Vwn.h
//-----// This is a gaussian-distributed random noise source
// (transient analysis only)
//
//
n1
+ --- n2
//
o----(
)-----o
//
--// This element behaves as a short circuit for AC/HB analysis.
#ifndef Vwn_h
#define Vwn_h 1
#include "../fRandn.h"
class Vwn : public Element
{
public:
Vwn(const string& iname);
188
~Vwn();
static const char* getNetlistName()
{
return einfo.name;
}
// This element adds equations to the MNAM
virtual unsigned getExtraRC(const unsigned& eqn_number,
const MNAMType& type);
virtual void getExtraRC(unsigned& first_eqn, unsigned& n_rows) const;
// fill
virtual
virtual
virtual
MNAM
void fillMNAM(FreqMNAM* mnam);
void fillMNAM(TimeMNAM* mnam);
void fillSourceV(TimeMNAM* mnam);
// State variable transient analysis
virtual void svTran(TimeDomainSV *tdsv);
virtual void deriv_svTran(TimeDomainSV *tdsv);
private:
// row assigned to this instance by the FreqMNAM
unsigned my_row;
// Time for the integer number of periods before current time
double int_per_time;
// Element information
static ItemInfo einfo;
// Number of parameters of this element
static const unsigned n_par;
// Parameter variables
double vo, td, mean, variance, kn;
// Parameter information
static ParmInfo pinfo[];
// Dervied variables initialized at runtime
189
fRandn * frn;
};
#endif
//------// Vwn.cc
//------#include "../network/ElementManager.h"
#include "../analysis/FreqMNAM.h"
#include "../analysis/TimeMNAM.h"
#include "../analysis/TimeDomainSV.h"
#include "Vwn.h"
// Static members
const unsigned Vwn::n_par = 5;
// Element information
ItemInfo Vwn::einfo = {
"vwn",
"White Noise voltage source",
"Nikhil Kriplani"
DEFAULT_ADDRESS"elements/Vwn.html",
"2005_10_19"
};
// Parameter information
ParmInfo Vwn::pinfo[] =
{
{"vo", "Offset value (V)", TR_DOUBLE, false},
{"td", "Delay time (s)", TR_DOUBLE, false},
{"mean", "Mean of the white noise random variable", TR_DOUBLE, false},
{"variance", "Variance of the white noise random variable",
TR_DOUBLE, false},
{"kn", "Scaling coefficient", TR_DOUBLE, false}
};
Vwn::Vwn(const string& iname) : Element(&einfo, pinfo, n_par, iname)
{
// Set default parameter values
paramvalue[0] = &(vo = zero);
paramvalue[1] = &(td = zero);
190
paramvalue[2] = &(mean = zero);
paramvalue[3] = &(variance = 1.0);
paramvalue[4] = &(kn = 1.0);
// Set the number of terminals
setNumTerms(2);
// Set flags
setFlags(LINEAR | ONE_REF | TR_TIME_DOMAIN | SOURCE);
// Set number of states
setNumberOfStates(1);
my_row = 0;
frn = new fRandn;
}
Vwn::~Vwn()
{
delete frn;
}
unsigned Vwn::getExtraRC(const unsigned& eqn_number, const MNAMType& type)
{
// Keep the equation number assigned to this element
my_row = eqn_number;
// Add one extra RC
return 1;
}
void Vwn::getExtraRC(unsigned& first_eqn, unsigned& n_rows) const
{
assert(my_row);
first_eqn = my_row;
n_rows = 1;
}
void Vwn::fillMNAM(FreqMNAM* mnam)
{
assert(my_row);
// Ask my terminals the row numbers
mnam->setOnes(getTerminal(0)->getRC(), getTerminal(1)->getRC(), my_row);
// This element behaves as a short circuit for AC/HB analysis.
191
// Since the source vector is assumed to be initialized to zero,
// we do not need to fill it if freq != frequency.
return;
}
void Vwn::fillMNAM(TimeMNAM* mnam)
{
assert(my_row);
// Ask my terminals the row numbers
mnam->setMOnes(getTerminal(0)->getRC(), getTerminal(1)->getRC(), my_row);
return;
}
void Vwn::fillSourceV(TimeMNAM* mnam)
{
const double& ctime = mnam->getTime();
double e = zero;
double xiWhite = frn->RNOR(mean,variance);
if (ctime < td)
e = vo;
else
e = vo + kn * xiWhite;
mnam->setSource(my_row, e);
return;
}
void Vwn::svTran(TimeDomainSV* tdsv)
{
// Calculate voltage
double& e = tdsv->u(0);
if (tdsv->DC())
e = vo;
else
{
const double & ctime = tdsv->getCurrentTime();
double xiWhite = frn->RNOR(mean, variance);
if (ctime < td)
192
e = vo;
else
e = vo + kn * xiWhite;
}
// Scale state variable for numerical stability
tdsv->i(0) = tdsv->getX(0) * 1e-2;
}
void Vwn::deriv_svTran(TimeDomainSV* tdsv)
{
tdsv->getJu()(0,0) = zero;
tdsv->getJi()(0,0) = 1e-2;
}
C.6
The Parker-Skellern Model
//----------// MesfetPS.h
//----------// MESFET PS model - The Parker-Skellern model
//
//
Drain 2
//
o
//
|
//
|
//
|---+
//
|
// Gate 1 o-----|
//
|
//
|---+
//
|
//
|
//
o
//
Source 3
//
//
//
Author: Nikhil M. Kriplani
#ifndef MesfetPS_h
#define MesfetPS_h 1
class MesfetPS : public AdolcElement
193
{
public:
MesfetPS(const string& iname);
~MesfetPS() {}
static const char* getNetlistName()
{
return einfo.name;
}
// Do some local initialization
virtual void init() throw(string&);
private:
virtual void eval(adoublev& x,
adoublev& vp, adoublev& ip);
// Element information
static ItemInfo einfo;
// Number of parameters of this element
static const unsigned n_par;
double Vt;
// Parameter variables
double acgam, area, beta, cgd, cgs, delta;
double fc, hfeta, hfe1, hfe2, hfgam, hfg1, hfg2;
double ibd, is, lfgam, lfg1, lfg2, mvst, n, p, q;
double rs, rd, taud, taug, vbd, vbi, vst, vto, xc, xi, z;
double tj, tnom, afac, lam;
// Parameter information
static ParmInfo pinfo[];
};
#endif
//-----------// MesfetPS.cc
//------------
194
#include "../network/ElementManager.h"
#include "../network/AdolcElement.h"
#include "MesfetPS.h"
// Static members
const unsigned MesfetPS::n_par = 37;
// Element information
ItemInfo MesfetPS::einfo = {
"mesfetps",
"Intrinsic MESFET using the Parker-Skellern model",
"Nikhil Kriplani",
DEFAULT_ADDRESS"elements/MesfetPS.h.html",
"2005_12_15"
};
// Parameter information
ParmInfo MesfetPS::pinfo[] = {
{"acgam", "Capacitance modulation", TR_DOUBLE, false},
{"area", "Area multiplier", TR_DOUBLE, false},
{"beta", "Linear region transconductance scale", TR_DOUBLE, false},
{"cgd", "Zero-bias gate-drain capacitance", TR_DOUBLE, false},
{"cgs", "Zero-bias gate-source capacitance", TR_DOUBLE, false},
{"delta", "Thermal reduction coefficient", TR_DOUBLE, false},
{"fc", "Forward bias capacitance parameter", TR_DOUBLE, false},
{"hfeta", "high-frequency vgs feedback parameter", TR_DOUBLE, false},
{"hfe1", "HFGAM modulation by vgd", TR_DOUBLE, false},
{"hfe2", "HFGAM modulation by vgs", TR_DOUBLE, false},
{"hfgam", "High-frequency vgd feedback parameter", TR_DOUBLE, false},
{"hfg1", "HFGAM modulation by vsg", TR_DOUBLE, false},
{"hfg2", "HFGAM modulation by vdg", TR_DOUBLE, false},
{"ibd", "Gate-junction breakdown current", TR_DOUBLE, false},
{"is", "Gate-junction saturation current", TR_DOUBLE, false},
{"lfgam", "Low-frequency feedback parameter", TR_DOUBLE, false},
{"lfg1", "LFGAM modulation by vsg", TR_DOUBLE, false},
{"lfg2", "LFGAM modulation by vdg", TR_DOUBLE, false},
{"mvst", "Sub-threshold modulation", TR_DOUBLE, false},
{"n", "gate-junction ideality factor", TR_DOUBLE, false},
{"p", "linear region power law exponent", TR_DOUBLE, false},
{"q", "saturated region power law exponent", TR_DOUBLE, false},
{"rs", "source ohmic resistance", TR_DOUBLE, false},
{"rd", "drain ohmic resistance", TR_DOUBLE, false},
{"taud", "relaxation time for thermal reduction", TR_DOUBLE, false},
{"taug", "relaxation time for gamma feedback", TR_DOUBLE, false},
195
{"vbd", "gate junction breakdown voltage", TR_DOUBLE, false},
{"vbi", "gate junction potential", TR_DOUBLE, false},
{"vst", "sub-threshold potential", TR_DOUBLE, false},
{"vto", "threshold voltage", TR_DOUBLE, false},
{"xc", "Capacitance pinch-off reduction factor", TR_DOUBLE, false},
{"xi", "Saturation knee potential factor", TR_DOUBLE, false},
{"z", "knee transition parameter", TR_DOUBLE, false},
{"tj", "device temperature", TR_DOUBLE, false},
{"tnom", "nominal temperature", TR_DOUBLE, false},
{"afac", "gate width scale factor", TR_DOUBLE, false},
{"lam", "channel length modulation", TR_DOUBLE, false}
};
MesfetPS::MesfetPS(const string& iname)
: AdolcElement(&einfo, pinfo, n_par, iname)
{
// Set default parameter values
paramvalue[0] = &(acgam = zero);
paramvalue[1] = &(area = one);
paramvalue[2] = &(beta = 1.0e-4);
paramvalue[3] = &(cgd = zero);
paramvalue[4] = &(cgs = zero);
paramvalue[5] = &(delta = zero);
paramvalue[6] = &(fc = 0.5);
paramvalue[7] = &(hfeta = zero);
paramvalue[8] = &(hfe1 = zero);
paramvalue[9] = &(hfe2 = zero);
paramvalue[10] = &(hfgam = zero);
paramvalue[11] = &(hfg1 = zero);
paramvalue[12] = &(hfg2 = zero);
paramvalue[13] = &(ibd = zero);
paramvalue[14] = &(is = 1.0e-14);
paramvalue[15] = &(lfgam = zero);
paramvalue[16] = &(lfg1 = zero);
paramvalue[17] = &(lfg2 = zero);
paramvalue[18] = &(mvst = zero);
paramvalue[19] = &(n = one);
paramvalue[20] = &(p = 2.0);
paramvalue[21] = &(q = 2.0);
paramvalue[22] = &(rs = zero);
paramvalue[23] = &(rd = zero);
paramvalue[24] = &(taud = zero);
paramvalue[25] = &(taug = zero);
196
paramvalue[26]
paramvalue[27]
paramvalue[28]
paramvalue[29]
paramvalue[30]
paramvalue[31]
paramvalue[32]
paramvalue[33]
paramvalue[34]
paramvalue[35]
paramvalue[36]
=
=
=
=
=
=
=
=
=
=
=
&(vbd = one);
&(vbi = one);
&(vst = zero);
&(vto = -2.0);
&(xc = zero);
&(xi = 1000);
&(z = 0.5);
&(tj = 300.0);
&(tnom = 300.0);
&(afac = one);
&(lam = zero);
// Set the number of terminals
setNumTerms(3);
// Set flags
setFlags(NONLINEAR | ONE_REF | TR_TIME_DOMAIN);
// Set number of states
setNumberOfStates(2);
}
void MesfetPS::init() throw(string&)
{
// account for area scaling
beta *= area;
cgd *= area;
cgs *= area;
ibd *= area;
is *= area;
delta = delta/area;
rs = rs/area;
rd = rd/area;
Vt = kBoltzman * tj / eCharge;
// create tape
IntVector var(2);
var[0] = 0;
var[1] = 1;
IntVector novar;
DoubleVector nodelay;
createTape(var, var);
}
197
void
{
//
//
//
//
//
//
//
MesfetPS::eval(adoublev& x, adoublev& vp, adoublev& ip)
x[0]: vgs
x[1]: vgd
x[2]: dvgs/dt
x[3]: dvgd/dt
x[4]: vgs(t-tau)
vp[0]: vgs , ip[0]: ig
vp[1]: vds , ip[1]: id
// Assign known output voltages
vp[0] = x[0];
vp[1] = x[0] - x[1];
// Parker-Skellern MESFET model
adouble vgs_bar = x[0] - taug * x[2];
adouble vgd_bar = x[1] - taug * x[3];
adouble gamma_lf = lfgam - lfg1*vgs_bar + lfg2*vgd_bar;
adouble gamma_hf = hfgam - hfg1*vgs_bar + hfg2*vgd_bar;
adouble eta_hf = hfeta - hfe1*vgd_bar + hfe2*vgs_bar;
adouble vgst = x[0] - vto - gamma_lf*vgd_bar gamma_hf*(x[1] - vgd_bar) eta_hf*(x[0] - vgs_bar);
adouble Vst = vst * (1.0 + mvst * vp[1]);
adouble vgt = Vst * log(1.0 + exp(vgst / Vst));
adouble vsat = xi*(vbi - vto)*vgt / (xi*(vbi - vto) + vgt);
adouble vdp = vp[1] * (p/q) * pow(vgt/(vbi - vto), p-q);
adouble
+
+
vdt = 0.5 * sqrt(pow(vdp*sqrt(1.0 + z) + vsat, 2)
z*vsat*vsat)
0.5 * sqrt(pow(vdp*sqrt(1.0 + z) - vsat, 2)
z*vsat*vsat);
adouble id = afac * beta * pow(vgt, q) *
(1.0 - pow(1.0 - vdt/vgt, q));
adouble P = id * vp[1]; // Thermal modulation effects neglected;
198
adouble ids = id / (1.0 + delta/afac * P);
// Capacitance model
adouble igs, igd, Cgs, Cgd, Cds, Cm, m;
double alpha = xi*(vbi - vto)/(2.0 * (xi + 1.0));
m = 0.5 * (1.0 - vp[1] / sqrt(vp[1]*vp[1] + alpha*alpha));
adouble ve = x[0] + m * sqrt(vp[1]*vp[1] + alpha*alpha) + acgam*vp[1];
adouble vn = ve + 0.5*((ve - vto)*(xc - 1.0)
+ sqrt(pow(ve - vto, 2) * pow(xc - 1.0, 2) + 0.04));
adouble qgd = afac*cgd*(x[1] - m*sqrt(vp[1]*vp[1] + alpha*alpha)
+ acgam * vp[1]);
adouble qgs;
condassign(qgs, vn - fc*vbi, afac*cgs*vbi*(2.0*(1.0-sqrt(1.0-fc)) +
(vn/vbi-fc)/sqrt(1.0-fc) +
pow(vn/vbi-fc,2)/(4.0*pow(1-fc,1.5))),
2.0*afac*cgs*vbi*(1.0-sqrt(1.0 - vn/vbi)));
double c_gd = afac * cgd;
adouble cgs0;
adouble dvn_dve = 0.5 * (xc + 1.0 +
(pow(1.0-xc,2)*(ve-vto))/(pow(1.0-xc,2)
* pow(ve-vto,2) + 0.04));
condassign(cgs0, vn - fc*vbi, afac*cgs/sqrt(1.0-fc) *
(1.0 + 0.5*(vn/vbi -fc)/(1.0-fc)) * dvn_dve,
afac*cgs/sqrt(1.0 - vn/vbi) * dvn_dve);
Cgd = c_gd + m*(cgs0 - c_gd) - acgam*(cgs0 + c_gd);
Cgs = cgs0 + m*(c_gd - cgs0) - acgam*(cgs0 + c_gd);
Cds = acgam*((1.0 - m)*cgs0 + m*c_gd)
+ m*(m - 1.0)*(cgs0 + c_gd + 2.0*(qgd-qgs)/sqrt(vp[1]*vp[1]
+ alpha*alpha));
Cm = -acgam * (cgs0 + c_gd);
igs = afac*(is*(exp(x[0]/Vt)-1.0) - ibd*(exp(-x[0]/vbd) - 1.0));
igs += Cgs * x[2];
igd = afac*(is*(exp(x[1]/Vt)-1.0) - ibd*(exp(-x[1]/vbd) - 1.0));
igd += Cgd * x[3];
199
// Calculate the output currents
ip[0] = (igd + igs) * area;
ip[1] = (ids - igd) * area;
}
C.7
The OML MESFET model
//-----------// MesfetOML.h
//-----------// MESFET OML model (from Microwave Optical Tech Letters,
// Vol 29, no 4, p. 226, 2001.)
//
//
Drain 2
//
o
//
|
//
|
//
|---+
//
|
// Gate 1 o-----|
//
|
//
|---+
//
|
//
|
//
o
//
Source 3
//
//
//
Author: Nikhil M. Kriplani
#ifndef MesfetOML_h
#define MesfetOML_h 1
class MesfetOML : public AdolcElement
{
public:
MesfetOML(const string& iname);
~MesfetOML() {}
200
static const char* getNetlistName()
{
return einfo.name;
}
// Do some local initialization
virtual void init() throw(string&);
private:
virtual void eval(adoublev& x,
adoublev& vp, adoublev& ip);
// Some constants
double k2, k3;
double delta_T, tn, Vt, k1, k4, k5, k6, Vt0, Beta, Ebarr, EbarrN, Nn;
double Is, Vbi;
// Element information
static ItemInfo einfo;
// Number of parameters of this element
static const unsigned n_par;
// Parameter variables
double b1, b2, b3, b4, b5, gamma;
double gee, vt0, delta, cgs0, cgd0;
double is, n, ib0, nr, t, vbi, fcc;
double vbd, tnom, avt0, bvt0, tm, tme;
double eg, m, xti, tj, area;
// Parameter information
static ParmInfo pinfo[];
};
#endif
//------------// MesfetOML.cc
//------------#include "../network/ElementManager.h"
#include "../network/AdolcElement.h"
#include "MesfetOML.h"
201
// Static members
const unsigned MesfetOML::n_par = 29;
// Element information
ItemInfo MesfetOML::einfo = {
"mesfetoml",
"Intrinsic MESFET using OML model",
"Nikhil Kriplani",
DEFAULT_ADDRESS"elements/MesfetOML.h.html",
"2005_12_15"
};
// Parameter information
ParmInfo MesfetOML::pinfo[] = {
{"b1", "fitting parameter 1", TR_DOUBLE, false},
{"b2", "fitting parameter 2", TR_DOUBLE, false},
{"b3", "fitting parameter 3", TR_DOUBLE, false},
{"b4", "fitting parameter 4", TR_DOUBLE, false},
{"b5", "fitting parameter 5", TR_DOUBLE, false},
{"gamma", "Vds dependence on pinch-off potential", TR_DOUBLE, false},
{"gee", "Dependence of gate-bias on knee voltage", TR_DOUBLE, false},
{"vt0", "Voltage where the channel current is forced to 0", TR_DOUBLE,
false},
{"delta", "dependence of Veff on Vgs", TR_DOUBLE, false},
{"cgs0", "Gate-source Schottky barrier capacitance for Vgs=0 (F)",
TR_DOUBLE, false},
{"cgd0", "Gate-drain Schottky barrier capacitance for Vgd=0 (F)",
TR_DOUBLE, false},
{"is", "Diode saturation current (A)", TR_DOUBLE, false},
{"n", "Diode ideality factor", TR_DOUBLE, false},
{"ib0", "Breakdown current parameter (A)", TR_DOUBLE, false},
{"nr", "Breakdown ideality factor", TR_DOUBLE, false},
{"t", "Channel transit time (s)", TR_DOUBLE, false},
{"vbi", "Built-in potential of the Schottky junctions (V)",
TR_DOUBLE, false},
{"fcc", "Forward-bias depletion capacitance coefficient (V)", TR_DOUBLE,
false},
{"vbd", "Breakdown voltage (V)", TR_DOUBLE, false},
{"tnom", "Reference Temperature (K)", TR_DOUBLE, false},
{"avt0", "Pinch-off voltage (VP0 or VT0) linear temp. coefficient (1/K)",
TR_DOUBLE, false},
{"bvt0", "Pinch-off voltage (VP0 or VT0) quadratic temp. coefficient
(1/K^2)", TR_DOUBLE, false},
202
{"tm", "Ids linear temp. coeff. (1/K)", TR_DOUBLE, false},
{"tme", "Ids power law temp. coeff. (1/K^2)", TR_DOUBLE, false},
{"eg", "Barrier height at 0.K (eV)", TR_DOUBLE, false},
{"m", "Grading coefficient", TR_DOUBLE, false},
{"xti", "Diode saturation current temperature exponent", TR_DOUBLE,
false},
{"tj", "Junction Temperature (K)", TR_DOUBLE, false},
{"area", "Area multiplier", TR_DOUBLE, false}
};
MesfetOML::MesfetOML(const string& iname)
: AdolcElement(&einfo, pinfo, n_par, iname)
{
// Set default parameter values
paramvalue[0] = &(b1 = -0.7437);
paramvalue[1] = &(b2 = 2.8974);
paramvalue[2] = &(b3 = 4.4187);
paramvalue[3] = &(b4 = 15.329);
paramvalue[4] = &(b5 = 21.8151);
paramvalue[5] = &(gamma = 0.0378);
paramvalue[6] = &(gee = 0.2339);
paramvalue[7] = &(vt0 = -1.2262);
paramvalue[8] = &(delta = 0.1222);
paramvalue[9] = &(cgs0 = zero);
paramvalue[10] = &(cgd0 = zero);
paramvalue[11] = &(is = zero);
paramvalue[12] = &(n = one);
paramvalue[13] = &(ib0 = zero);
paramvalue[14] = &(nr = 10.);
paramvalue[15] = &(t = zero);
paramvalue[16] = &(vbi = .8);
paramvalue[17] = &(fcc = .5);
paramvalue[18] = &(vbd = 1e10);
paramvalue[19] = &(tnom = 293.);
paramvalue[20] = &(avt0 = zero);
paramvalue[21] = &(bvt0 = zero);
paramvalue[22] = &(tm = zero);
paramvalue[23] = &(tme = zero);
paramvalue[24] = &(eg = .8);
paramvalue[25] = &(m = .5);
paramvalue[26] = &(xti = 2.);
paramvalue[27] = &(tj = 293.);
203
paramvalue[28] = &(area = one);
// Set the number of terminals
setNumTerms(3);
// Set flags
setFlags(NONLINEAR | ONE_REF | TR_TIME_DOMAIN);
// Set number of states
setNumberOfStates(2);
}
void MesfetOML::init() throw(string&)
{
k2 = cgs0 / sqrt(one - fcc);
k3 = cgd0 / sqrt(one - fcc);
delta_T = tj - tnom;
tn = tj / tnom;
Vt = kBoltzman * tj / eCharge;
k5 = n * Vt;
k6 = nr * Vt;
Vt0 = vt0 * (one + (delta_T * avt0) + (delta_T * delta_T * bvt0));
Ebarr = eg -.000702 * tj * tj / (tj + 1108.);
EbarrN = eg -.000702 * tnom * tnom / (tnom + 1108.);
Nn = eCharge / 38.696 / kBoltzman / tj;
Is = is * exp((tn - one) * Ebarr / Nn / Vt);
if (xti)
Is *= pow(tn, xti / Nn);
Vbi = vbi * tn -3. * Vt * log(tn) + tn * EbarrN - Ebarr;
k1 = fcc * Vbi;
k4 = 2. * Vbi * (one - fcc);
// create tape
IntVector var(2);
var[0] = 0;
var[1] = 1;
IntVector novar;
DoubleVector nodelay;
createTape(var, var);
}
void MesfetOML::eval(adoublev& x, adoublev& vp, adoublev& ip)
{
204
//
//
//
//
//
//
//
x[0]: vgs
x[1]: vgd
x[2]: dvgs/dt
x[3]: dvgd/dt
x[4]: vgs(t-tau)
vp[0]: ugs , ip[0]: ig
vp[1]: uds , ip[1]: id
// Assign known output voltages
vp[0] = x[0];
vp[1] = x[0] - x[1];
adouble ids, igd, igs, itmp, vx, cgs, cgd;
// static igs current
igs = Is *(exp(x[0] / k5) - one) - ib0 * exp(-(x[0] +vbd) / k6);
// Calculate cgs, including temperature effect.
condassign(cgs, k1 - x[0],
cgs0 / sqrt(one - x[0] / Vbi), k2 * (one + (x[0] - k1) / k4));
cgs *= (one + m * (0.0004 * delta_T + one - Vbi / vbi));
// Calculate the total current igs = static + dq_dt
igs += cgs * x[2];
// static igd current
igd = Is * (exp(x[1] / k5) - one) - ib0 * exp(-(x[1] + vbd) / k6);
// Calculate cgd, including temperature effect.
condassign(cgd, k1 - x[1],
cgd0 / sqrt(one - x[1] / Vbi), k3 * (one + (x[1] - k1) / k4));
cgd *= (one + m * (0.0004 * delta_T + one - Vbi / vbi));
// Calculate the total current igd = static + dq_dt
igd += cgd * x[3];
// Calcutate Ids as a function of Veff
adouble a1 = b1 * vp[1];
adouble Vgst = x[0] - vt0 + gamma * vp[1];
adouble Vdseff = (b2 * vp[1] + b3 * vp[1]*vp[1]) / (1 + gee * Vgst);
adouble a2 = b4 * (Vdseff / sqrt(1 + Vdseff*Vdseff));
adouble a3 = b5 * (Vdseff / sqrt(1 + Vdseff*Vdseff));
adouble Veff = 0.5 * (Vgst + sqrt(Vgst*Vgst + delta*delta));
205
itmp = a1*Veff*Veff*Veff + a2*Veff*Veff + a3*Veff;
condassign(ids, (itmp * vp[1]) * (x[0] - Vt0), itmp, zero);
if (tme && tm)
ids *= pow((1 + delta_T * tm), tme);
// Calculate the output currents
ip[0] = (igd + igs) * area;
ip[1] = (ids - igd) * area;
}
C.8
The Ziggurat Technique
This is a technique to produce gaussian random variables with a user specified mean and
variance. This code has been slightly modified from its original form in [132] to fit into the
f REEDATM framework and is provided below.
//--------// fRandn.h
//--------#ifndef _FRANDN_H_
#define _FRANDN_H_
class fRandn
{
public:
fRandn();
~fRandn() { };
unsigned long SHR3();
double UNI();
double nfix();
void zigset(unsigned long jsrseed);
double RNOR(double mean, double st_dev);
private:
};
#endif
206
//---------// fRandn.cc
//---------#include "fRandn.h"
#include <iostream>
using namespace std;
#include <cmath>
#include <ctime>
#include <cstdlib>
static unsigned long jz, jsr=123456789;
static long hz;
static unsigned long iz, kn[128];
static double wn[128], fn[128];
const int N = 128;
// Initialize the random no. in the generator.
// A fREEDA element will initialize a variable of the
// fRandn class in it’s own init() function.
fRandn::fRandn()
{
srand(time(0));
unsigned long seed = rand();
zigset(seed);
}
unsigned long fRandn::SHR3()
{
jz = jsr;
jsr ^= (jsr << 13);
jsr ^= (jsr >> 17);
jsr ^= (jsr << 5);
return jz + jsr;
}
// create a Uniform r.v.
double fRandn::UNI()
{
return 0.5 + (signed)SHR3() * .2328306e-9;
}
double fRandn::nfix()
{
207
const float r = 3.442620f;
double x, y;
for(;;)
{
x = hz * wn[iz]; // iz==0, handles the base strip
if(iz == 0)
{
do
{
x = -log(UNI()) * 0.2904764;
y = -log(UNI());
} while (y + y < x*x); // .2904764 is 1/r
return (hz > 0)? r+x : -r-x;
}
// iz > 0, handle the wedges of other strips
if(fn[iz] + UNI() * (fn[iz-1] - fn[iz]) < exp(-.5*x*x))
return x;
// initiate, try to exit for(;;) for loop
hz = SHR3();
iz = hz & N-1;
if(abs(hz) < kn[iz])
return (hz * wn[iz]);
}
}
// set a seed for the ziggurat generator
void fRandn::zigset(unsigned long jsrseed)
{
const double m1 = 2147483648.0;
double dn = 3.442619855899;
double tn = dn;
double vn = 9.91256303526217e-3;
double q;
int i;
jsr ^= jsrseed;
q = vn/exp(-.5*dn*dn);
kn[0] = (dn/q)*m1;
kn[1] = 0;
wn[0] = q/m1;
wn[N-1] = dn/m1;
fn[0] = 1.;
fn[N-1] = exp(-.5*dn*dn);
for(i = N-2; i >= 1; i--)
208
{
dn = sqrt(-2.*log(vn/dn + exp(-.5*dn*dn)));
kn[i+1] = (dn/tn)*m1;
tn = dn;
fn[i] = exp(-.5*dn*dn);
wn[i] = dn/m1;
}
}
// get a single normal random variable
// Call this function inside the eval function of
// every element, which is called at every time step.
// This gives you a randn variable at every time step.
double fRandn::RNOR(double mean = 0, double st_dev = 1)
{
hz = SHR3();
iz = hz & N-1;
double x = (abs(hz) < kn[iz]) ? hz*wn[iz] : nfix();
return x*st_dev + mean;
}
C.9
The Logistic Map Noise Generator
//--------// fLogis.h
//--------#ifndef _FLOGIS_H_
#define _FLOGIS_H_
// The class which generates an intermittent chaotic time
// series that has flicker characteristics
class fLogis
{
public:
fLogis();
~fLogis();
inline void setSize(int sz) { size = sz; }
inline double getMapValue(int i) { return Map[i]; }
void generateMap();
private:
double * Map;
209
int size;
};
#endif
//---------// fLogis.cc
//---------#include "fLogis.h"
#include <iostream>
using namespace std;
#include <ctime>
#include <cmath>
#include <cstdlib>
fLogis::fLogis()
{
// Do nothing
}
void fLogis::generateMap()
{
srand(time(0));
double xt = (double)rand()/RAND_MAX;
const int iter = size; // 1e7
Map = new double[iter];
// generate the map
for (int j = 0; j < iter; j++)
{
Map[j] = 4.0*xt*(1-xt);
xt = Map[j];
}
// make the
// subtract
for (int i
Map[i]
sequence component’s mean about zero
0.5 from every value
= 0; i < iter; i++)
-= 0.5;
}
fLogis::~fLogis()
{
210
delete [] Map;
}
C.10
The Logarithmic Map Noise Generator
//--------// fChaos.h
//--------#ifndef _FCHAOS_H_
#define _FCHAOS_H_
// The class which generates an intermittent chaotic time
// series that has flicker characteristics
class fChaos
{
public:
fChaos();
~fChaos();
inline void setSize(int sz) { size = sz; }
inline void setBeta(double bt) { beta = bt; }
inline double getMapValue(int i) { return Map[i]; }
void generateMap();
private:
double * Map_temp;
double * Map;
double beta;
int size;
};
#endif
//---------// fChaos.cc
//---------#include "fChaos.h"
#include <iostream>
using namespace std;
#include <ctime>
#include <cmath>
#include <cstdlib>
211
fChaos::fChaos()
{
// Do nothing
}
void fChaos::generateMap()
{
srand(time(0));
double xt = (double)rand()/RAND_MAX;
const int iter = 2*size; // 1e7
//const int iter = 2*nsteps;
const int need = size;
// iterate the vector Map_temp and finally
// save the required values in Map
Map_temp = new double[iter];
Map = new double[need];
// generate the map
for (int j = 0; j < iter; j++)
{
if (xt <= 0.5)
Map_temp[j] = xt +
2*pow(log(2.0),(beta-1))
* xt*xt * pow(abs(log(xt)),(beta+1));
else
Map_temp[j] = 2*xt - 1;
xt = Map_temp[j];
}
// save the number of values specified by the value of "need"
for (int i = 0; i < need; i++)
Map[i] = Map_temp[iter - i];
// make the
// subtract
for (int i
Map[i]
flicker component’s mean about zero
0.5 from every value
= 0; i < need; i++)
-= 0.5;
}
fChaos::~fChaos()
212
{
delete [] Map;
delete [] Map_temp;
}
213
Appendix D
f REEDATM Netlists
D.1
The Varactor-tuned VCO circuit
* The Varactor-tuned VCO circuit
.options method=3 maxit=1000000 jupdm=2
vsource:vs 1 0 vdc=0.0
vsource:vin1 7 0 vdc=12.0
vsource:vin2 12 0 vdc=12.0
dn2:d1 3 2 tstop=10m tstep=1n + is=1.365p rs=1.0 n=1.0
cj0=14.93e-12 + m=0.4261 vj=0.75 fc=0.5 bv=25.0 ibv=10.0e-6 +
ksh=1 kth=1 kf=1e-4 beta=0.000005
dn2:d2 0 2 tstop=10m tstep=1n + is=1.365p rs=1.0 n=1.0
cj0=14.93e-12 + m=0.4261 vj=0.75 fc=0.5 bv=25.0 ibv=10.0e-6 +
ksh=1 kth=1 kf=1e-4 beta=0.000005
dn2:d3 3 2 tstop=10m tstep=1n + is=1.365p rs=1.0 n=1.0
cj0=14.93e-12 + m=0.4261 vj=0.75 fc=0.5 bv=25.0 ibv=10.0e-6 +
ksh=1 kth=1 kf=1e-4 beta=0.000005
dn2:d4 0 2 tstop=10m tstep=1n + is=1.365p rs=1.0 n=1.0
cj0=14.93e-12 + m=0.4261 vj=0.75 fc=0.5 bv=25.0 ibv=10.0e-6 +
ksh=1 kth=1 kf=1e-4 beta=0.000005
214
dn2:d5 3 2 tstop=10m tstep=1n + is=1.365p rs=1.0 n=1.0
cj0=14.93e-12 + m=0.4261 vj=0.75 fc=0.5 bv=25.0 ibv=10.0e-6 +
ksh=1 kth=1 kf=1e-4 beta=0.000005
dn2:d6 0 2 tstop=10m tstep=1n + is=1.365p rs=1.0 n=1.0
cj0=14.93e-12 + m=0.4261 vj=0.75 fc=0.5 bv=25.0 ibv=10.0e-6 +
ksh=1 kth=1 kf=1e-4 beta=0.000005
bjtnpnn2:q1 8 5 9 0 tstop=10m tstep=1n + bf=255.9 br=6.092
cjc=7.306e-12 cje=22.01e-12 + ikf=0.2847 is=14.34e-15
ise=14.34e-15 itf=0.6 + mjc=0.3416 mje=0.377 nf=1.0 ne=1.307
nr=1.0 + rb=10.0 rc=1.0 tf=411.1e-12 tr=46.91e-9 vaf=74.03 +
vtf=1.7 xtb=1.5 xtf=3.0 ksh=1 kth=1 kf=1e-2
+ beta=0.000005
bjtnpnn:q2 10 6 9 0 tstop=10m tstep=1n + bf=255.9 br=6.092
cjc=7.306e-12 cje=22.01e-12 + ikf=0.2847 is=14.34e-15
ise=14.34e-15 itf=0.6 + mjc=0.3416 mje=0.377 nf=1.0 ne=1.307
nr=1.0 + rb=10.0 rc=1.0 tf=411.1e-12 tr=46.91e-9 vaf=74.03 +
vtf=1.7 xtb=1.5 xtf=3.0 ksh=1 kth=1 kf=1e-2
+ beta=0.000005
rn2:r1
rn2:r2
rn2:r3
rn2:r4
rn2:r5
6 0 res=1k tstop=10m tstep=1n kth=1
6 7 res=3.9k tstop=10m tstep=1n kth=1
10 12 res=50 tstop=10m tstep=1n kth=1
13 0 res=1k tstop=10m tstep=1n kth=1
11 0 res=47 tstop=10m tstep=1n kth=1
c1
c2
c3
c4
c5
c6
1 0 c=1000p
3 4 c=47p
4 0 c=22p
6 0 c=1000p
12 0 c=1000p
10 13 c=100p
l1
l2
l3
l4
l5
l6
5 6 l=30n
4 5 l=200n
12 8 l=15n
1 2 l=20u
3 0 l=20u
9 11 l=20u
* mutual inductance
k:k1 0 coupling=0.9 l1="l1" l2="l3"
215
k:k2 0 coupling=0.9 l1="l2" l2="l3"
k:k3 0 coupling=1.0 l1="l1" l2="l2"
.tran2 tstop=10m tstep=1n im=0 opt=1 nst=9.99999m
.out plot term 13 vt in "vco_output.out"
.end
D.2
The X-band MMIC Circuit Netlist
NETLIST FOR FILTRONIC SOLID STATE MMIC LNA
.options f0=10e9 jupdm=0
.model m_line1 tlinp4 (z0mag = 95.7 k=7.55 fscale=10e9 alpha=773
+ nsect=20 fopt=10e9 tand=0.006)
.model m_line2 tlinp4 (z0mag= 81.9 k=7.73 fscale=10e9 alpha=78
+ nsect=20 fopt=10e9 tand=0.006)
.model m_line3 tlinp4 (z0mag = 76.2 k=7.82 fscale=10e9 alpha= 156
+ nsect=20 fopt=10e9 tand=0.006)
c1 2 3 6e-12
tlinp4:t1 3 0 0 0 model="m_line1" length = 1194u
tlinp4:t2 3 0 4 0 model="m_line2" length = 183u
mesfetcn:m1 42 51 62 A0=0.09910 A1=0.08541 A2=-0.0203 A3=-0.015
+ BETA=0.01865 GAMA=0.8293 VDS0=6.494 VT0=-1.2 VBI=0.8
+ CGD0=3f CGS0=528.2f IS=3e-12 NR=1.2 T=1e-12 vbd=12
+ kf=1e-9 tstep=1ps tstop=10ns
mesfetcn:m2 172 192 182 A0=0.1321 A1=0.1085 A2=-0.04804 A3=-0.03821
+ BETA=0.03141 GAMA=0.7946 VDS0=5.892 VT0=-1.2 VBI=1.5
+ CGD0=4e-15 CGS0=695.2f IS=4e-12 N=1.2 T=1e-12 vbd=12
+ kf=1e-9 tstep=1ps tstop=10ns
rn:rg1 41 42 res=0.83 tstep=2ps tstop=20ns
rn:rd1 5 51 res=0.83 tstep=2ps tstop=20ns
rn:rs1 61 62 res=0.33 tstep=2ps tstop=20ns
l:lg1 4 41 l=7e-12
l:ls1 6 61 l=11e-12
tlinp4:t3 6 0 8 0 model="m_line2" length=391u
tlinp4:t4 6 0 7 0 model="m_line2" length=401u
216
c:c_via4 7 0 c=17e-12
rn:r_via4 7 0 res=6 tstep=2ps tstop=20ns
c:c_via3 8 0 c=17e-12
tlinp4:t5 5 0 9 0 model="m_line2" length=102u
tlinp4:t6 9 0 10 0 model="m_line1" length=368u
rn:r1 10 11 res=10.53 tstep=2ps tstop=20ns
rn:r2 11 12 res=24.93 tstep=2ps tstop=20ns
c:c_s6 11 0 c=17e-12
tlinp4:t7 9 0 13 0 model="m_line2" length=33u
c:c2 13 14 c=2e-12
tlinp4:t8 14 0 15 0 model="m_line1" length=705u
tlinp4:t9 14 0 0 0 model="m_line2" length=419u
tlinp4:t10 14 0 17 0 model="m_line2" length=58u
rn:rg2 171 172 res=0.63 tstep=2ps tstop=20ns
rn:rd2 191 192 res=0.63 tstep=2ps tstop=20ns
rn:rs2 181 182 res=0.25 tstep=2ps tstop=20ns
l:lg2 17 171 l=16e-12
l:ld2 19 191 l=11e-12
l:ls2 18 181 l=11e-12
c:c_via8 18 0 c=17p
rn:r_via8 18 0 res=5 tstep=2ps tstop=20ns
tlinp4:t11 19 0 20 0 model="m_line1" length=138u
c:cfb 20 21 c=4.28e-12
rn:rfb 21 22 res=237.4 tstep=2ps tstop=20ns
.ref 0
l:lfb 22 15 l=1.268n int_res=9.55
tlinp4:t12 19 0 23 0 model="m_line1" length=313u
l:lp 23 24 l=1.268n int_res=9.55
rn:rpad 24 29 res=24 tstep=2ps tstop=20ns
vsource:v2 29 0 vdc=6
c:c_via12 24 0 c=17e-12
tlinp4:t13 19 0 25 0 model="m_line3" length=229u
c:cload 25 26 c=6e-12
217
rn:r50 26 0 res=50 tstep=2ps tstop=20ns
vsource:vin 666 0 f=10.0e9 vdc=0.0 vac=0.1 phase=-90
vwn:vwhite 2 666 kn=0.01
vsource:v1 12 0 vdc=6
.tran2 tstep=1ps tstop=10ns im=0 nst=4n
.out plot term 26 vt in "mmic.out"
.end