INFORMATION CAPACITY AND POWER EFFICIENCY IN OPERATIONAL
TRANSCONDUCTANCE AMPLIFIERS
Makeswaran Loganathan, Suvarcha Malhotra, Pamela A. Abshire
Electrical and Computer Engineering /Institute for Systems Research
University of Maryland, College Park, Maryland 20742 USA
{makeshl,suvarcha,pabshire}@glue.umd.edu
ABSTRACT
Noise limits information transmission in analog systems.
Real analog systems possess intrinsic physical noise such
as thermal noise and flicker noise and inevitably suffer
degradation of information content. This loss in
information can be reduced if the analog system operates
at frequencies where noise is minimal. Using principles of
low power circuit design and information theory, we
present a method to estimate the information transfer rates
of analog systems and to determine the maximum
theoretical limit. We have applied our method to an
operational transconductance amplifier (OTA) and show
that measured data corroborates our analytical predictions.
We find a significant increase in information content if the
system is operated in spectral regions with lower noise.
1.
INTRODUCTION
The transmission of data through analog systems is
corrupted by the intrinsic thermal and flicker noise in
these systems. The degradation can be significant when
signal power and noise power are comparable. Noise in
the input can lead to a considerable loss of information
while noise at the output is less prohibitive once signal
power has been boosted by amplification. Special care
must be given to reduce noise at the input stage,
particularly for low power signals. The aim of this work is
to explore the implications of information theory in
designing and optimizing microelectronic circuits. Several
authors [1-4] have reported efforts in this direction.
In many applications, the frequency range of operation
of the analog system is specified by the spectral power
content of the input signal, determined by the environment
and input statistics for particular applications. Examples
of such systems include amplifiers recording signals from
sensors and imagers acquiring images from natural
scenes. The power of these signals is often concentrated at
low temporal frequencies, which leads to system designs
that attenuate all higher frequencies. However we know
from basic noise theory that semiconductor device noise is
concentrated at these low frequencies, and hence the
design of these sensor systems considerably reduces the
signal to noise ratio (SNR) resulting in a reduction in the
;‹,(((
information content. If we operate these amplifiers at
frequencies where their noise level is minimized, then we
achieve an increase in SNR and consequently the
information content.
In an analog circuit, flicker and thermal noise
components of the MOSFETs are the dominant noise
sources. The input referred noise can be calculated from
transfer function analysis, and the entire circuit can be
modeled as an information transfer channel corrupted by
colored Gaussian noise. The efficiency of information
transfer through this circuit can be maximized by
concentrating signal power in spectral regions where the
channel noise is minimal. As the input referred noise is
typically cup-shaped, the water filling technique [5]
provides the most efficient distribution of signals in
channels with colored Gaussian noise. Just as water
distributes itself in a vessel, the power in a given system is
allocated to frequency bands starting from the spectral
region with lowest noise and then spilling over to the
noisier parts of the channel. We apply this algorithm to
obtain the ideal frequency range of operation of an
Operational Transconductance Amplifier (OTA).
2. OPTIMAL SIGNAL ALLOCATION
In this section, we consider a system composed of n
analog blocks 1,2 … n with transfer functions A1, A2…An
and equivalent input noise sources V1, V2 …Vn as shown
in Fig. 1. The noise from each block is assumed to be
composed of a flicker noise source and a white Gaussian
noise source. We treat the entire system as a colored
Gaussian channel and determine the frequency range of
operation for maximum information transfer rate.
Fig.1 System composed of n blocks with transfer functions A1,
A2…An and equivalent input noise sources V1, V2 …Vn.
2.1. Low Frequency Analysis
At low frequencies, we assume the frequency transfer
function of each of the blocks to be perfectly flat. The
input referred noise of the system is given by (1) and
shown in Fig. 2(a). We assume that there is a brick wall
,
,6&$6
filter at the output of the system that blocks all
frequencies above frequency Fmax to restrict attention to
the low frequency case.
V2
V 22
(1)
...... 2 2 n
2
A1
A1 A2 ...... An21
The capacity C is the maximum information transfer
rate of a channel [6], given by
V in2
V1 2 f2
C
max2
S ( f ):V P
³ log
f1
2
§
S( f ) ·
¨1 ¸ df
© N( f ) ¹
operating the system at frequencies below the dominant
pole (f1). Fig. 3(c) shows the most efficient signal
allocation when the system is operated at frequencies
below f1, and Fig. 3(d) shows the most efficient signal
allocation for unrestricted operation. In section 3, we
perform a similar analysis for a practical amplifier.
(2)
where S(f) and N(f) are signal and noise power spectral
densities and the maximization is over all the signals such
that the average signal power is less than the power
constraint P. The frequency range (f2-f1) defines the
bandwidth.
To transmit a signal satisfying the average power
constraint through this system with maximum efficiency,
the water-filling approach tells us that we should start
filling at Fmax and add power at progressively lower
frequencies until we reach the total power content in the
signal. This result follows from the observation that input
referred noise spectral density is a monotonically
decreasing function of frequency. The distribution of the
signal power for maximum information transfer rate is
shown in Fig. 2(b).
(a)
(b)
Fig.2 (a) Input referred noise of the system; (b) Ideal distribution
of signal power for maximum information transfer rate.
2.2 .High Frequency Analysis
In the general case, the transfer functions of these
analog blocks vary with frequency. We consider a cascade
of low pass blocks, with transfer functions A1, A2 ... An for
each of the blocks as shown in Fig. 3(a). The noise
contribution from each block is assumed to be similar to
that shown in Fig. 2(a). The input referred noise of the
system is calculated using (1) and sketched in Fig. 3(b).
For efficient use of this system, we should first
allocate signal power at the minimum of the input referred
noise spectrum and continue adding signal power at
neighboring lower and higher frequencies (where the
noise is higher) until we have allocated the maximum
signal power. This results in signal power being
concentrated at frequencies above the dominant pole of
the system, in contrast with the traditional practice of
(a)
(b)
(c)
(d)
Fig. 3 (a) Transfer functions A1, A2… An; (b) Input referred
noise of the entire system; (c) Optimal allocation of signal power
for operation below the dominant pole of the system; (d)
Optimal distribution of signal power when system operation in
not restricted in bandwidth. f1 is the dominant pole, and fmin is
the frequency corresponding to the lowest noise level.
2.3. Contributions from output stages and feedback
The methodology described above can be easily
extended to incorporate the effects of noise introduced
elsewhere in the system or configurations which affect
system transfer properties such as feedback. Additional
system noise is modeled as a white or colored Gaussian
signal depending on its spectrum. If the noise is
significantly smaller than the system noise, then it does
not affect the performance of the system. If they are
comparable, the effective input referred noise of the
system should include this noise too, and then water
filling should be performed on the total input referred
noise. If the system under consideration is used in a
feedback network that introduces noise at the output, then
the system capacity and power efficiency decrease; an
analysis can be found in [7].
3. PRACTICAL AMPLIFIER CONFIGURATIONS
3.1. Operational Transconductance Amplifier
We derive input referred noise for a cascoded OTA [8]
(Fig. 4), and then perform water filling on the input
referred noise spectral density to compute capacity. We
model the noise sources by flicker and thermal noise [9].
The input referred noise for a MOS transistor is given by:
,
V n2
§ KF * I DAF
( 4 kT J g m ) ¨¨
© C ox * f
· § 1 ·
¸¸ * ¨¨ 2 ¸¸
¹ © gm ¹
where kT is thermal energy, gm is the transconductance, J
is a constant equal to 1/2 for sub threshold and 2/3 for
above threshold operation, KF is a process dependent
constant on the order of 10-25 V2F, ID is the current level
and Cox is the gate oxide capacitance per unit area. The
first term in (3) corresponds to the thermal noise and the
second term to the flicker noise. At low frequencies,
assuming perfectly matched devices, the input referred
noise for the circuit is given by (4):
2
2
2
V0
(3)
2
§g ·
§g ·
§g ·
§g ·
2
2Vn21 4¨ m3 ¸ Vn23 2¨ m6 ¸ Vn26 ¨ mN ¸ VnN2 ¨ mP ¸ VnP2 (4)
VinOTA
© gm1 ¹
© gm1 ¹
© gm1 ¹
© gm1 ¹
In order to minimize noise we must maximize gm1 and
minimize gm3, gm5, gmN and gmP. However we cannot
arbitrarily reduce the sizes of M3 and M4 as that reduces
the matching and the phase margin. Decreasing the bias
current reduces the flicker noise, but at the cost of a lower
slew rate. Bias current and aspect ratios of the different
transistors must be chosen to satisfy these conflicting
constraints under the available resources (size, power,
etc.) allocated to the system.
V 01 V 02
(5)
Cn and Cgdn denote the gate to source and the gate to drain
capacitance of each transistor. We determine capacity of
this analytical OTA model as a function of the bandwidth
of system operation. Results are computed using
MATLAB and plotted in Fig. 5. We assume that signal
power has maximum standard deviation 0.71mV, and that
the signal is restricted between a lower cutoff frequency f1
and an upper cutoff frequency f2. In one case we fix f1 at
10 Hz, and vary f2 to find the optimal signal allocation for
different values of f2. As f2 varies from 1 kHz to 10 MHz,
we find an increase in information rate by a factor of
104dB! Alternately we fix f2 at 1010 Hz and vary f1. With
f1 fixed the capacity increases as we increase f2, and with f2
fixed the capacity increases as we decrease f1. We find
the optimal frequency band for operation of this circuit to
be around the second pole, in contrast to regular system
operation below the first pole.
Fig. 5. OTA capacity when signal power is allocated optimally
between a lower cutoff frequency f1 and an upper cutoff
frequency f2: (a) capacity as a function of f1 (fixed f2); (b)
capacity as a function of f2 (fixed f1).
Fig. 4. OTA with cascoded output.
g m1 g m 2 ; g m 3
gm4
gm5
gm7; gm6
g m8
At high frequencies the gate-source capacitances and the
Miller effect become important factors in the noise
analysis [10]. Assuming that the output resistances of the
transistors M7, M8, MN and MP are all equal to r0, the
transfer function for the OTA is shown in (5).
V 01
Vin1
g m 1 r0
2
§
2C 3
¨1 s
gm3
©
1
*
g m 6 r0
·
§
¸
¨ 2 C 6 C gd 7 (1 2
¹ 1 s¨
gm6
¨
©
g
V 02
V in 2
m1
r0
2
§
s ¨ 2C
©
1
3
(1 g
g
m 3
2
r0
)C
gd 7
·
¸
¹
§ 1 g mP r0 ·
*¨
¸
· © 1 r0 sC L ¹
¸
¸
¸
¹
§ 1 g m N r0 ·
*¨
¸
© 1 r0 s C L ¹
Fig. 6 is a 3-dimensional plot of the maximum
information transfer rates as a function of bias current and
total signal power. The capacity increases as signal power
increases, and increases as bias current increases. It was
anticipated that capacity would increase with signal
power; the result that capacity increases with bias current
is not as obvious. As bias current increases the flicker
noise and thermal noise increase in magnitude, which
tends to reduce capacity; at the same time the frequency
response of the amplifier extends to higher frequencies,
which tends to decrease input-referred noise and increase
capacity. The frequency effect dominates, so altogether
an increase in bias current increases the capacity of the
OTA; however, the increase with signal power appears to
be a stronger effect than the increase with bias current.
m 3
,
Fig. 6. OTA capacity as a function of bias current, signal power.
4. RESULTS
We determine information capacity and optimal signal
allocation for the OTA from measured noise and transfer
characteristics. We compare these empirical results with
predictions from the analytical model presented in Sec. 3.
transfer functions and noise spectra, which may include
sources of noise in addition to those explicitly modeled.
In this case the analytical results serve as an upper bound
for the empirical capacity.
As shown in Fig. 7, both empirical and analytical
capacities increase with the upper cutoff frequency f2. As
discussed in Sec. 3 and predicted in Fig. 6, the capacity is
higher for larger bias currents when the allocated
bandwidth is not restricted (i.e., the higher frequency
ranges shown in Fig. 7). In general the analytical and
empirical results are in good agreement and show similar
trends as a function of upper cutoff frequency and bias
current.
5. CONCLUSIONS
Practical analog systems suffer a reduction in information
as a signal propagates through multiple stages of a system.
This reduction can be minimized if the system is operated
in regions where noise is minimal. This suggests alternate
strategies for information-efficient sensing such as signal
chopping or modulation at the input stage to transfer
information content of the signal to a higher frequency for
optimal use of the circuit as an information channel.
6. REFERENCES
Fig. 7. OTA capacity estimated from empirical data (“data”) and
from an analytical model (“model”) as a function of the upper
cutoff frequency f2 when the lower cutoff frequency f1 is fixed.
Results are shown for three bias currents: 10 PA, 15PA, 20PA.
We measure noise power spectra and transfer
functions for a cascoded OTA in capacitive feedback
configuration with low frequency gain of 100 at three bias
currents (10PA, 15PA, 20PA). The input referred noise
of the entire amplifier is approximately the same as the
input referred noise of the OTA [8], so we estimate the
OTA capacity from the input referred noise of the
amplifier.
We use median filtering to smooth the estimates of
input referred noise spectra, then apply the waterfilling
method (Sec. 2) under the assumptions that the lower
cutoff frequency f1 is fixed at 100 Hz and that signal
power has maximum standard deviation 10 mV. Fig. 7
shows the capacity as a function of the upper cutoff
frequency f2 at three bias levels. The analytically derived
capacity exceeds the empirically estimated capacity in all
cases. The empirical results are computed from measured
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7. ACKNOWLEDGEMENTS
We thank the MOSIS service for providing chip fabrication.
These circuits will be used for teaching a bioelectronics course
at the University of Maryland. We thank the National Science
Foundation for support of this work through award 0225489. We
thank Timothy K. Horiuchi and Honghao Ji for technical
discussions concerning amplifier design.
,