“Statistical Electronics”
Noise Processes in RF Integrated Circuits
(Oscillators and Mixers)
Donhee Ham
Harvard RF and High-Speed IC
& Quantum Circuits Laboratory
copyright © 2003 by Donhee Ham
Donhee Ham
Outline
1. Introduction – “Statistical Electronics”
2. Phase Noise in Oscillators
“Virtual Damping and Einstein Relation in Oscillators”
3. Noise in Mixers
“Cyclostationary Noise in CMOS Switching Mixers”
4. Stochastic Resonance and RF Circuits
5. Soliton Electronics
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RF IC Design Challenges – I: Design Constraints
Circuit idea
Circuit analysis –
hand calculation
Simulation
(Circuits and EM)
Parasitic
extraction
PCB design
IC Layout
IC Characterization
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RF IC Design Challenges – II: Physical Constraints
* This slide best suits digital silicon CMOS technology.
1. Active devices
–
–
Trade-offs among speed, overhead, gain, & breakdown
voltage
Poor noise performance
2. Passive devices
–
–
Skin effect  loss & noise
Conductive substrate  loss & noise
3. Cross talk
–
e.g. “unintentional” injection
locking through
conductive substrate
4. Poor ground reference
5. and more…
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Noise in RF Receivers
S
• Shannon’s Theorem : C  B  log 2 (1  )
N
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Signal Path Noise
NF2  1 NF3  1
NFtot  NF1 

 
G1
G1G2
(Friis equation)
• NF quantifies the degradation in SNR in the receiver.
• NF sets the lower end of receiver dynamic range.
• NF of a receiver is practically always dominated by NF of its front-end.
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Frequency-Reference Noise (Phase Noise)
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Noise in Front-End Circuits
Mixers & Oscillators
Front End
1. Nonlinear and/or time-varying systems
Rich dynamics complicates the noise
processes.
LNA
- LTI system
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2. Currently available noise models
They have greatly helped designers better
understand noise processes in oscillators and
mixers. However, they assume rather
phenomenological standpoints and a more
fundamental yet intuitive understanding is still
needed. Proper physical understanding could
lead to deeper design insight.
(e.g.) Trade-off between voltage swing and
phase noise in oscillators is often not best
understood.
Chronology of Oscillator Phase Noise Study
1.
Mathematical & physical ground work
•
•
Kubo (1962), Stratonovich (1967) – Essential understanding.
Lax (1967) – Comprehensive and general mathematical-physics
analysis of phase noise (hard to beat!).
Leeson (1966) – Phenomenological, yet insightful tuned-tank
2.
electrical oscillator phase noise model.
3.
CAD-oriented approaches – Kartner, Demir et al.
4.
Recent circuit design-oriented approaches
•
•
•
•
•
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McNeill – Jitter study in ring oscillators.
Razavi – Q-based phase noise modeling.
Rael & Abidi – Phase noise factor calculation.
Hajimiri & Lee – General study of time-varying effects; first account
for the interaction between cyclostationary noise and impulse
sensitivity and its impact on phase noise.
and many more…(omitted here not due to technical insignificance
but due to space limitation.).
* Many important other works on phase noise are omitted in this slide due only to space limitation.
Physics of Noise I – Einstein (1905)
“Brownian Motion”
x (t )  2 Dt
2
Einstein Relation
1 kT
D 
 m
(D: diffusion constant)
Fluctuation-Dissipation Theorem
• Fluctuation: microscopic description of thermal motions of liquid molecules
• Dissipation: macroscopic average of thermal motions of liquid molecules
• Fluctuation and dissipation are of the same physical origin, and in thermal
equilibrium, they balance each other out.
• When the fluctuation-dissipation balance is reached (equilibrium),
1 2
1
mv  kT
2
2
(energy equipartition)
Donhee Ham
v
2
kT

m
Physics of Noise II – Nyquist (1928)
“4kTR Noise”
4kTR noise is a
special case of the
fluctuation-dissipation
theorem.
P  kTf
4kTR noise
4kTR noise is analogous- nay,
essentially equivalent to blackbody radiation following Planck
radiation law in the classical
regime.
Donhee Ham
energy equipartition
(2 degrees of freedom
– electric & magnetic)
or,
counting of the
resonance modes for a
given bandwidth
Physics of Noise III
“Brownian Motion in RC Circuits”
Steady-state probability
distribution function (PDF) of the
voltage, v, across the capacitor
“Fluctuation-dissipation balance”
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“Energy equipartition”
• Donhee Ham, Statistical Electronics: Noise Processes in Integrated Communication
Systems, PhD dissertation, California Institute of Technology, 2002.
Bridging the Gap …
Design-Oriented Approaches
Physics-Based Approaches
Statistical
physics
CAD-Oriented Approaches
Donhee Ham
Electrical
Circuits
“Statistical Electronics”
• Circuit engineering
• Integrated circuits
• Statistical physics
• Thermodynamics
Nonlinear Devices
Quantum Devices
Transistors
Meso/nano scale
devices
Time-Varying
Circuits
Autonomous
Circuits
(Oscillators)
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Driven
Circuits
(Mixers)
Stochastic
Resonance
Noise-enhanced
heterodyning, phase
synchronization
Distributed
Circuits
Lossy transmission lines,
noise waives, etc.
Communication
Systems
PLLs, Frequency
Synthesizers
CDRs
Outline
1. Introduction – “Statistical Electronics”
2. Phase Noise in Oscillators
“Virtual Damping and Einstein Relation in Oscillators”
3. Noise in Mixers
“Cyclostationary Noise in CMOS Switching Mixers”
4. Stochastic Resonance and RF Circuits
5. Soliton Electronics
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Self-Sustained Oscillator
Equivalent model for tuned-tank oscillators
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Oscillator Phase Noise
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Ensemble of Identical Oscillators
Same
initial
phase
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Phase Diffusion - I
v ( t )  v 0 cos( w 0 t  f )
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< f (t ) >  2 D t
2
Diffusion
Constant
Phase Diffusion - II
v ( t )  v 0 cos( w 0 t  f )
< f (t ) >  2 D t
2
Diffusion
Constant
most probable state
• D (diffusion constant) : rate of entropy increase
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Virtual damping and Einstein relation in oscillators,” IEEE JSSC, March 2003.
Virtual Damping
Phase Diffusion Constant
< f 2 (t ) >  2 D t
“ Virtual Damping Rate ”
L{ w } 
D
• 1 GHz, -121dBc/Hz at 600kHz offset : D = 5.6
Donhee Ham
w0
2D
(w ) 2
 10  9
• Donhee Ham and Ali Hajimiri, “Virtual damping and Einstein relation in oscillators,” IEEE JSSC, March 2003.
Experimental Virtual Damping – I
Virtual Damping Measurement Setup
Centre frequency :
5 MHz
Donhee Ham
Ensemble average on 512
Waveforms triggered at the
same phase initially
• Donhee Ham and Ali Hajimiri, “Virtual damping and Einstein relation in oscillators,” IEEE JSSC, March 2003.
Experimental Virtual Damping – II
Damping Rate : D
Centre frequency
: 5 MHz
Ensemble average
on 512 waveforms
triggered at the
same phase initially
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Virtual damping and Einstein relation in oscillators,” IEEE JSSC, March 2003.
Experimental Virtual Damping – III
L{ w} 
Donhee Ham
2D
2

w
( )
Injected
current noise
PSD (A2/Hz)
Measured D
(sec-1)
PN from
measured D
(dBc/Hz)
PN from spec.
analyzer
(dBc/Hz)
2.60 x 10-15
1.02 x 104
-92.9
-93.0
4.84 x 10-15
1.56 x 104
-91.0
-90.0
9.66 x 10-15
3.53 x 104
-87.4
-86.5
2.12 x 10-14
9.30 x 104
-83.3
-81.7
6.04 x 10-14
1.90 x 105
-80.0
-79.5
• Donhee Ham and Ali Hajimiri, “Virtual damping and Einstein relation in oscillators,” IEEE JSSC, March 2003.
Linewidth Compression
“Unified View of Resonators and Oscillators”
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Virtual damping and Einstein relation in oscillators,” IEEE JSSC, March 2003.
Brownian Motion & Einstein Relation
< x (t ) >  2 D t
2
“ Einstein Relation ”
1 kBT
D 
 m
Sensitivity
(Energy
equipartition)
Friction
(Energy loss)
Donhee Ham
Einstein Relation in Oscillator Phase Noise
- Determination of Virtual Damping Rate, D -
v02 C QL
D~  B  0
1 k T w
sensitivity
Einstein
relation
friction (energy loss
and/or noise)
R
 g d 0    
1
L
Q ~
w 0C
• The virtual damping rate, D, can be also mathematically derived by solving a time-varying diffusion equation for the phase
diffusion. It’s a simple kind of math, which can be found in Donhee Ham et al, “Virtual damping and Einstein relation in
oscillators,” IEEE JSSC, March 2003. The result of the mathematical derivation perfectly agrees with the virtual damping
rate jotted down above, obtained resorting only to Einstein relation.
Donhee Ham
Anatomy of Oscillator Phase Noise
“Design Insight”
(due to virtual damping)
Einstein relation
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Virtual damping and Einstein relation in oscillators,” IEEE JSSC, March 2003.
LC Oscillator Design Example
gate poly
oxide
source
(p)
drain
(p)
n-well
p-substrate
MOS varactor
Spiral inductor
I bias
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Graphical Optimization
With a decreasing inductance,
L=Lmin= Lopt
4
start-up
amplitude
T.R.1
T.R.2
c (pF)
c (pF)
3
T.R.1
2
T.R.2
amplitude
1
start-up
0
0
20
40
60
w (μ m)
Donhee Ham
80
100
0
20
40
60
80
100
w (μ m)
• Donhee Ham and Ali Hajimiri, “Concepts and methods in optimization of integrated LC VCOs,” IEEE JSSC, June 2001.
Robust Design
6
start-up
c (pF)
4
• Solid lines
: fast corner
• Broken lines : slow corner
• Shaded region : unreliable design
T.R.1
2
T.R.2
0
0
20
40
60
80
100
w (μ m)
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Concepts and methods in optimization of integrated LC VCOs,” IEEE JSSC, June 2001.
Phase Noise Measurement
Bias Tee
Vdd
Bond
Wires
DUT
Circuit
Board
Vdd
Bias
Tee
Spectrum Analyzer
50
matching
Probe Station
1.0 mm
1.1 mm
Supply voltage
2.5 V
Current (Core)
4 mA
Center frequency
2.33 GHz
Tuning range
26 %
Output power
0 dBm
Phase noise
@ 600kHz
-121dBc/Hz
Conexant 0.35um BiCMOS
(MOS Only)
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Concepts and methods in optimization of integrated LC VCOs,” IEEE JSSC, June 2001.
Performance Comparison
Performance Metric : Power-Frequency-Tuning-Normalized (PFTN) Figure of Merit
 kT  f  2  f  2 
PFTN  10 log 
  0    tune    L{ f off }
 Psup  f off   f 0  
PFTN
0
CMOS

 20


Bipolar
CMOS/bondwire inductor
CMOS distributed
 40


CMOS/special metal layer
This work
Publications (Chronological Order)
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Concepts and methods in optimization of integrated LC VCOs,” IEEE JSSC, June 2001.
Outline
1. Introduction – “Statistical Electronics”
2. Phase Noise in Oscillators
“Virtual Damping and Einstein Relation in Oscillators”
3. Noise in Mixers
“Cyclostationary Noise in CMOS Switching Mixers”
4. Stochastic Resonance and RF Circuits
5. Soliton Electronics
Donhee Ham
MOS Switching Mixer
• Two modes of mixer operation
Hard-Switching
Soft-Switching
• C : IF port capacitance
- Mixer parasitic capacitors
- IF amplifier input capacitance
- Important design parameter
• Role of energy storing elements?
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Characteristic of Mixers
“Cyclostationary Noise”
“Noise is shaped in time.”
x(t )  n(t ) p(t )
stationary noise
cyclostationary noise
• Cyclostationary noise is periodically modulated noise.
• It results when circuits have periodic operating points.
• Its statistical averages are time-dependent.
Donhee Ham
periodic/deterministic
function
PSD of Cyclostationary Noise
1
2
S x ( f )  lim
XT ( f )  Sx ( f ;t)
T  T
T
( X T ( f )   x(t )e j 2ft dt )
0
• operationally-defined,
time-varying PSD.
• F.T. of autocorrelation.
“measurement = LTI bandpass filtering”
Donhee Ham
Cyclostationary Noise Flow in RF Systems
S ( f ;t)
S ( f ;t)
Donhee Ham
Importance of Cyclostationary Noise
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Switching mixers: theory and measurement,” Submitted to IEEE JSSC.
Theoretical Prediction of CG and NF
new
prediction
Our approach
conventional
• fIF = 10 MHz
• fLO = 300 MHz
new
prediction
Utilization of stochastic calculus
to evaluate the noise figure.
• The next 4 slides sketch the theoretical analysis which
resulted in the new predictions presented in this slide.
Further details of this theoretical analysis can be found
in Donhee Ham et al, “Switching mixers: theory and
measurement,” Submitted to IEEE JSSC.
Donhee Ham
conventional
“Optimum design
capacitance.”
Thevenin Equivalent Circuit
“time-varying filtering”
hard switching
soft switching
IF component
Non-IF components
Donhee Ham
• A. R. Shahani et al, ``A 12-mW wide dynamic range CMOS front-end for a portable GPS receiver,"
IEEE JSSC, Dec. 1997.
Deterministic Dynamics - I
• Donhee Ham et al, “Switching
mixers: theory and measurement,”
Submitted to IEEE JSSC.
Donhee Ham
Via Pseudo-beating
(pattern generation)
Deterministic Dynamics - II
• Donhee Ham et al, “Switching
mixers: theory and measurement,”
Submitted to IEEE JSSC.
“Conversion Gain Enhancement”
Bump size ~ Harmonic Richness
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Stochastic Dynamics
dvnoise gT (t )
g (t )

vn (t )  T veff ,n (t )
dt
C
C
synchronized
Langevin
Equation
v(t )v(t   )
Fourier
Transform
S ( f IF , t )
measurement
(time-average)
S ( f IF , t )
S ( f IF , t )  S0 ( f IF )
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Switching mixers: theory and measurement,” Submitted to IEEE JSSC.
Mixer Test Chip and Board
Test capacitors
(to be laser-trimmed)
MOS switching
mixer core
0.7 mm
Post amplifier
1.2 mm
Conexant 0.35um BiCMOS Chip
(MOS Only)
Donhee Ham
Assembled printed-circuit board for
the chip test
• Donhee Ham and Ali Hajimiri, “Switching mixers: theory and measurement,” Submitted to IEEE JSSC.
Mixer Measurement Setup
LODC
LO
RF
RFDC



• Test On-Chip Capacitors
• Cut by Laser Trimming


IF
AMP
Mixer
NF
Noise Diode
(HP Noise Source)
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Switching mixers: theory and measurement,” Submitted to IEEE JSSC.
Measurement Results - I
“First observation of cyclostationary noise effects”
Measurement Result
Theoretical Prediction
NF(dB)
NF(dB)
Hard Switching
Hard Switching
Soft Switching
Soft Switching
IF capacitor (fF)
Donhee Ham
IF capacitor (fF)
• Donhee Ham and Ali Hajimiri, “Switching mixers: theory and measurement,” Submitted to IEEE JSSC.
Measurement Results - II
“First observation of C.G. enhancement and NF degradation”
Donhee Ham
• Donhee Ham and Ali Hajimiri, “Switching mixers: theory and measurement,” Submitted to IEEE JSSC.
Outline
1. Introduction – “Statistical Electronics”
2. Phase Noise in Oscillators
“Virtual Damping and Einstein Relation in Oscillators”
3. Noise in Mixers
“Cyclostationary Noise in CMOS Switching Mixers”
4. Stochastic Resonance and RF Circuits
5. Soliton Electronics
Donhee Ham
When Noise Plays a Creative Role…
Brownian Motor
Electrical Brownian Motor
T1
T1
T2
thermal noise
Other Example
T2
- Dithering in A/D converters
Stochastic Resonance
SNR
 U 
Escape rate ~ exp 
 kT 
1.
Stochastic
Resonance
2.
3.
U
Donhee Ham
T
Noise-enhanced
heterodyning
Noise-induced
phase sync.
Noise-enhanced
linearization
Outline
1. Introduction – “Statistical Electronics”
2. Phase Noise in Oscillators
“Virtual Damping and Einstein Relation in Oscillators”
3. Noise in Mixers
“Cyclostationary Noise in CMOS Switching Mixers”
4. Stochastic Resonance and RF Circuits
5. Soliton Electronics
Donhee Ham
Ultra-Fast Nonlinear Electronics
“Soliton Electronics”
NLTL
Positive
active feedback
time
?
Pulse train generator
“Soliton oscillator”: analogous to pulse lasers (e.g. femto-second lasers).
Donhee Ham
Harvard RF and High-Speed IC
& Quantum Circuits Lab
http://www.deas.harvard.edu/~donhee
donhee@deas.harvard.edu
•
•
•
•
•
Wireless communication circuits (RF IC)
Wireline communication circuits (high-speed IC)
Statistical & soliton electronics
Quantum devices and circuits
UWB communication circuits
Donhee Ham
Acknowledgement
•
Caltech (Ali Hajimiri, Michael Cross, Chris White, Ichiro Aoki, Hui
Wu, Behnam Analui, Hossein Hashemi, Yu-Chong Tai, P. P.
Vaidyanathan, and David Rutledge.),
•
IBM T. J. Watson (Mehmet Soyuer, Dan Friedman, Modest Oprysko,
and Mark Ritter),
•
Analog Devices (Larry DeVito),
•
IBM Fishkill (J.O. Plouchart and Noah Zamdmer),
•
Conexant Systems (Currently, Skyworks Inc. and Jazz
Semiconductor.),
•
Lee Center, ONR, and NSF,
•
Paul Horowitz (Harvard).
Donhee Ham