An Information Theoretic Approach to Optimal
Amplifier Operation
Nicole M. Nelson and Pamela A. Abshire
Electrical and Computer Engineering/Institute for Systems Research
University of Maryland, College Park MD, 20742 USA
nmnelson,pabshire@umd.edu
function of input signal power. We compare the theoretical
capacity with an empirical capacity computed from
measurements of noise and transfer functions.
Abstract— Amplifier performance can be severely degraded by
the presence of intrinsic physical noise sources. We analyze the
information transfer rates for simple and wide-range
operational transconductance amplifiers (OTAs) using the
principles of information theory. Frequency transfer
characteristics and input-referred noise of both simple and
wide-range OTAs are modeled using process dependent noise
parameters to determine the frequency ranges of operation
that provide minimum noise and maximum information
capacity. The theoretical predictions of channel capacity are
compared with empirical estimates of capacity determined
using transfer characteristics and output noise obtained
experimentally. The amplifiers have been fabricated in a
commercially available 0.5µm process.
I.
II.
Many physical systems can be modeled as Gaussian
channels corrupted by noise. Noise is assumed to be
contributed additively at the output of the channel,
independently of the input signal. The capacity of the
information channel is obtained by maximizing the mutual
information between the input and output. For a Gaussian
channel subject to input signal power constraint P, the
capacity increases linearly with bandwidth F and
logarithmically with signal-to-noise ratio according to the
classical expression
INTRODUCTION
C = F log 2 (1 +
As CMOS devices and supply voltages scale down,
fundamental noise limits remain the same causing a decrease
in the achievable signal to noise ratio. Noise is therefore a
significant source of performance limitation in integrated
circuits. A measure of circuit performance may be obtained
by applying the methods of information theory to analog
circuits. In prior work [1-3], methodology was formulated to
determine the efficiency of information transfer rates through
analog circuits by modeling the circuits as Gaussian channels
and determining the information capacity from circuit
models of input-referred noise. The theoretical models were
validated by comparison to capacity computed from
measured transfer functions and noise. In [1] these methods
were applied to a wide-range operational transconductance
amplifier (OTA).
P
N
) bits
sec
(1)
where C is the capacity of the Gaussian channel and N is the
noise power. In the case of colored Gaussian noise, the
capacity is given by
C = max
2
S ( f ):σ < P
∫ log (1 +
2
S( f )
N( f )
) df
(2)
where the input signal power spectral density S(f) that
maximizes the information transfer and provides capacity is
determined using the water-filling technique in which the
input signal power is allocated to the frequency bands with
lowest noise spectral density, given by:
f2
Analysis in [1] modeled the input-referred noise spectral
density as a convex function, since only gate source
capacitances were considered. In this work, we consider a
more general small signal model of the MOSFET, including
gate-drain capacitances, in order to accurately determine
input-referred noise, optimal operating frequencies, and
capacity. We apply algorithms developed in [1] with a more
accurate small signal model to determine the input-referred
noise of simple and wide-range OTAs and use this
information to determine the information capacity as a
0-7803-9197-7/05/$20.00 © 2005 IEEE.
GAUSSIAN CHANNEL CAPACITY
P = ∫ S ( f ) df =
f1
f2
∫ (υ − N ( f ) )
+
df
(3)
f1
Figure 1 illustrates the technique: the input-referred noise of
a simple OTA is shown, along with the signal power
allocation (shaded area) that allows the channel to transmit
information at the channel capacity.
13
Figure 3. Measured noise of NMOS transistor (W=1750 µm
L=7µm) in a commercially available 0.5 µm process.
Figure 1. Input-referred noise of simple OTA. Shaded area is the
signal allocation that provides maximum information transmission.
A. Noise Sources in a MOS Transistor
The four main sources of noise in a MOS transistor
include: thermal noise due to the thermal excitation of charge
carriers in a conductor; flicker noise due to electron traps
caused by imperfections in the silicon-silicon dioxide
interface; shot noise generated by the channel current; and
gate current noise caused by thermal fluctuations in the
channel. Gate current noise is negligible, and shot noise is
caused by the same mechanism as thermal noise [3].
Therefore we consider only flicker and thermal noise. Both
components are referred to the input of the transistor (the
gate) by dividing by the gain, giving:
(a)
Vi 2
∆f
4 kT γ
gm
+ Kf
ID
Af
g m 2Cox L2 f β
(4)
where kT is the thermal energy, L is the device length, Cox is
the oxide capacitance, ID is the channel current, gm is the
transconductance, and Kf , Af and β are process dependent
constants. γ is a constant equal to 1/2 for subthreshold and
2/3 for above threshold operation. The process dependent
parameters Kf , Af and β were extracted according to
experimental procedures developed in [4,5] using an Agilent
4395A Network/Spectrum/Impedance Analyzer. Equation
(4) can be rewritten as
(b)
Figure 2. a) Simple OTA. b) Wide-Range OTA.
log
III.
=
( ) = log (
Vi 2
∆f
K f ID
Af
Cox Leff 2 gm 2
) − β log ( f )
(5)
The noise spectral density measured for an NMOS
transistor of size W =1750 µm and L=7µm is shown in
Figure 3. This noise data has been fitted using least-squares
methods to determine noise parameters Kf=10-24.49V2/F and
β=1. Since fairly large changes in the drain current are
necessary to produce measurable change in the spectral
density for the chosen sizes, Af was assumed to be 1.
INPUT-REFERRED NOISE MODEL
Circuit diagrams for the simple and wide-range
operational transconductance amplifiers under consideration
are shown in Figure 2 (a) and (b), respectively. The transfer
functions and input-referred noise for both simple and widerange OTAs are modeled and used to determine the
information capacity. Both circuits are considered to be
composed of a cascade of analog transistors each with its
own transfer function and equivalent input noise sources.
14
Figure 4. Calculated input-referred noise for the simple and widerange OTAs at several bias currents.
Figure 5. Measured input-referred noise for the wide-range
OTA at bias current of 10µA.
B. Wide-Range OTA
The input-referred noise of the wide-range OTA (Figure
2b) was derived in the same manner as the simple OTA
above assuming balanced transconductances gm1=gm2,
gm3=gm4=gm5=gm6 and gm7=gm8 and equal output resistances.
For a wide-range OTA fabricated in the same 0.5µm process,
these transfer characteristics yield a low pass function with a
cut-off frequency around 10KHz and open loop gain of 40 at
low frequencies. Figures 4 and 5 show the theoretical and
measured input-referred noise of the wide-range OTA as a
function of frequency, respectively for several values of the
bias current (Ibias=0.1, 1, 10µA) in Figure 4 and for
Ibias=10µA in Figure 5. The information capacity was
calculated as a function of average input signal power via
waterfilling and is shown in Figure 6 below.
B.
Effect of Gain and Feedback
For a system composed of i analog blocks each with its
own noise source Vni and gain Ai in each stage the inputreferred noise is given by
2
Vin 2 = Vn12 + VAn 22 +
1
Vn 32
A12 A22
+ ... +
Vni 2
A12 A2 2 ... Ai−12
(6)
For ideal feedback the equivalent input noise generators
may be moved unchanged outside the feedback loop without
affecting circuit noise performance
CALCULATED RESULTS
IV.
A. Simple OTA
The small signal transfer function for the simple OTA
shown in Figure 2a was determined using a small signal
model that includes the effects of both gate-drain and gatesource capacitances. All calculations assume balanced
transconductances gm4=gm1 and gm3=gm2 and equal output
resistances. The transfer function is given by:
Vo1
V+
=
Vo 2
V−
=
( sC
( sCgd 1 − gm1 )( sCgd 2 − gm 2 )
1
gd 2 + ro
)( sC
2
gd 1 + ro + g m 3 + 2 sC gs 2
For both OTAs the input-referred noise is not a simple
convex function as reported by [1]. The total input-referred
noise depends on the transfer function which in turn depends
on the MOSFET parasitic capacitances. Considering only
gate-source capacitances leads to the erroneous conclusion
that the noise increases without bound as a function of
frequency. Including the gate-drain capacitances does not
significantly change the lower frequency noise spectrum, but
at high frequencies the Miller effect causes the slope of the
transfer function to decrease, which in turn causes the noise
spectral density to deviate from the expected f2 slope at high
frequencies. In fact, the model predicts that the noise
spectrum approaches an asymptotic value as frequency
increases (see Figure 4).
)
( sCgd 1 − gm1 )
( sC
1
gd 1 + ro
)
(7)
Vo = Vo1 + Vo 2
This means that the input-referred noise is no longer cup
shaped, so once the dip in the input-referred noise spectrum
has been completely filled with input signal spectral power,
further increases in the input signal power cause the capacity
to increase as if the noise were spectrally uniform. For white
noise the capacity increases linearly with the signal power.
The minimum value of the input-referred noise spectral
density shifts to higher frequencies as the bias current
increases, as shown in Figure 4. This minimum noise
which, for bias current Ibias=10uA, is a low pass function
with cut off frequency around 10KHz and open loop gain of
20 at low frequencies. Figures 1 and 4 show the measured
and theoretical input-referred noise of the simple OTA as a
function of frequency, respectively, at the bias current
Ibias=10µA. Water filling is performed on the input-referred
noise spectral density to compute its capacity as a function of
bias current and input signal power.
15
wide-range OTA uses PMOS transistors while the simple
OTA uses NMOS input transistors. For an input signal
power of 1µW the simple OTA has a theoretical capacity of
3.36×108 bits/sec while the wide-range OTA has a
theoretical capacity of 2.06×107 bits/sec, which matches the
empirical capacity of 1.45×107 bits/sec and [1] which
calculated a theoretical capacity of 3.58 × 107bits/sec.
Therefore the new model provides a tighter estimate of
capacity, and we conclude that the parasitic gate-drain
capacitances play an important role in shaping input-referred
noise and determining capacity of the overall circuit.
VI.
We analyzed the information transfer rates for simple and
wide-range operational transconductance amplifiers using
the principles of information theory. Frequency transfer
characteristics and input-referred noise of both simple and
wide-range OTAs are modeled and used to determine the
frequency ranges of operation that provide minimum noise
and maximum information. Theoretical predictions of
channel capacity are compared with empirical estimates of
capacity determined using measured transfer characteristics
and output noise. The models in the present work improve
upon previously reported work [1] in that they account for
gate-drain capacitances. This allows for tighter bounds on
the capacity which agree very well with empirical estimates.
Figure 6. Calculated capacity vs signal power for wide-range OTA.
frequency depends on the process dependent parameters Kf
and Af. Lower values of Kf can cause the noise spectral
density to exhibit a peak between the minimum and the
asymptotic regime, an interesting observation which will be
investigated further in future work. The larger the value of Af
is, the higher the frequency at which the minimum noise
occurs. Since Kf and Af are process dependent parameters,
the fabrication technology plays a crucial role in determining
noise and capacity.
V.
CONCLUSIONS
EXPERIMENTAL RESULTS
ACKNOWLEDGMENT
A chip with implementations of the simple and widerange OTAs shown in Figure 2 was fabricated in a
commercially available 0.5 µm process and used to
determine the transfer characteristics and output noise. The
output noise spectral density for an NMOS transistor shown
in Figure 3 was measured using an Agilent 4395A
Network/Spectrum/Impedance Analyzer in order to derive
process-dependent noise parameters Kf and β. The same
instrument was used to measure output noise and frequency
transfer characteristics for simple and wide-range OTAs for
several different bias currents (not shown). From these
measurements the input-referred noise spectral densities
shown in Figures 1 and 6 were computed. Finally the
empirical capacities for the simple and wide-range OTAs
were estimated from the computed input-referred noise.
We thank the MOSIS service for chip fabrication. This
material is based on work supported by the Laboratory for
Physical Sciences and by the National Science Foundation
(P.A.CAREER Award 0238061).
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[4]
Figure 6 shows the information capacity for a wide-range
OTA computed in three ways: using the model in [1], using
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data reported in this paper. The new model results in lower
values of capacity, so it provides a tighter bound on the
information transmission of the circuit. In fact at low values
of signal power the new model agrees extremely well with
capacity computed from experiment.
[5]
[6]
[7]
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simple OTA is higher than for the wide range OTA. This
result is as expected, since the input differential pair of the
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16
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