ABSTRACT 
 
 
 
Title of Thesis:            INFORMATION CAPACITY AND POWER EFFICIENCY 
                               IN OPERATIONAL TRANSCONDUCTANCE AMPLIFIERS 
                                             
Suvarcha Malhotra, Master of Science, 2003 
 
Thesis directed by:      Professor Pamela A. Abshire 
                                    Department of Electrical and Computer Engineering 
 
 
 
Information capacity is a fundamental and quantitative bound on the ability of a physical 
system to communicate information. The capacity depends only on the physical 
properties of the channel, such as bandwidth, noise, and constraints on the signal values; 
it does not depend on specific tasks for which the channel may be used. Real analog 
systems possess intrinsic physical noise such as thermal noise and flicker noise and 
inevitably suffer degradation of information content. We investigate the information 
transmission and information-power efficiency of an Operational Transconductance 
Amplifier (OTA).  We present empirical results for the information capacity of an 
integrated OTA and compare these results with our theoretical model. We notice a 
significant increase in information content if the system is operated in spectral regions 
with higher frequency and lower noise level.  
INFORMATION CAPACITY AND POWER EFFICIENCY IN OPERATIONAL 
TRANSCONDUCTANCE AMPLIFIERS 
 
by 
Suvarcha Malhotra 
 
 
 
 
      Thesis submitted to the Faculty of the Graduate School of the 
      University of Maryland, College Park in partial fulfillment 
of the requirements for the degree of   
Master of Science 
2003 
 
 
 
 
 
 
 
 
Advisory Committee: 
                 Professor Pamela A. Abshire, Chair 
                 Professor Ralph Etienne-Cummings 
                 Professor Timothy Horiuchi 
                 Professor Elisabeth Smela  
   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
?Copyright by 
Suvarcha Malhotra 
2003 
 
 
 
 
 
 
 
 
 
 
 ii
 
 
 
 
 
 
 
 
 
 
?To the best parents in the world; mine? 
 
 
 
 
 
 
 
 
 
 
 
 
 iii
ACKNOWLEDGEMENTS 
I would like to thank my advisor and mentor Dr. Pamela Abshire for her intellectual 
support, patient guidance and mentorship. Her vision and constant encouragement were 
the biggest motivating factors for this work. I thank Dr.Ralph Etienne-Cummings, Dr. 
Timothy Horiuchi and Dr. Elisabeth Smela for serving on my committee. 
 I would like to thank my colleagues at the IBIS Lab for a healthy and supportive 
research environment. I particularly wish to thank Makesh Loganathan for working with 
me closely on numerous projects. Special thanks to Somasekhar Bangalore Prakash for 
numerous discussions ranging from technical glitches to food fads. I would like to thank 
Honghao Ji who had an answer for every technical problem I ever asked him. My thanks 
to Jean- Marie Launstein, Natasha Reeves, Jack Chaiyupatumpa  for working with me on 
cell recording experiments. 
 My heartfelt thanks to my friend Sachin Arora for his emotional and intellectual 
support and patient companionship. I wish to thank Mainak Sen for his optimism and 
belief in me and for his constant encouragement through times good and bad. My thanks 
to my friend and room mate Prabha Ramachandran for being the cheerful face at home 
and making me see the good in all situations. 
 I wish to thank my family for their love, support and encouragement and taking 
pride in me and my work. My Dad for being my role model and always teaching me 
lessons of life in his own special way. My Mom; my pillar of strength for enduring 
numerous hardships to get me where I am today. My brother for always pushing me in 
my academic pursuits and the numerous pep talks. My grandparents for their love and 
blessings. 
 iv
TABLE OF CONTENTS 
List of Figures????????????????????????????    vii 
 
1 INTRODUCTION ................................................................................................... 1 
2 EXTRA-CELLULAR RECORDINGS FROM MUSCLE CELLS.........................   5 
2.1 Electrical Activity is Action Potentials of the Cells ....................................... 6 
 2.2 Experimental Setup for Recording from Cells................................................ 7 
 2.3 Cell Recordings............................................................................................... 10 
 2.4 Conclusion ...................................................................................................... 10  
3 OPERATIONAL TRANSCONDUCTANCE AMPLIFIER (OTA) .......................  12 
 3.1 Operation......................................................................................................... 14 
 3.2 Small Signal Analysis..................................................................................... 14 
  3.2.1 Open Loop Gain.................................................................................. 15 
3.2.2 AC Analysis........................................................................................ 16 
3.2.3 Gain Bandwidth .................................................................................. 17 
3.2.4 Maximum Output Current................................................................... 17 
3.2.5 Slew Rate ............................................................................................ 18 
 3.3 DC Analysis.................................................................................................... 18 
  3.3.1    Input Common Mode Range............................................................... 18 
              3.3.1.1    Minimum Input Voltage ...................................................... 19 
              3.3.1.2    Maximum Input Voltage Range........................................... 19 
  3.3.2    Output Voltage Range......................................................................... 20 
  3.3.3    Static Power Dissipation..................................................................... 21 
 
 v
 3.4 Characterization .............................................................................................. 21 
  3.4.1 Gain and Phase Margins..................................................................... 21 
  3.4.2 Input Offset Voltage ........................................................................... 22 
   3.4.2.1    Random offset...................................................................... 22 
   3.4.2.2 Systematic offset.................................................................. 22 
  3.4.3 Total Harmonic Distortion.................................................................. 22 
  3.4.4 Noise Sources in OTA........................................................................ 23 
   3.4.4.1 MOSFET Thermal Noise..................................................... 23 
   3.4.4.2 Flicker Noise........................................................................ 24 
   3.4.4.3 Representation of Noise in Circuits ..................................... 26 
    3.5   Conclusion ......................................................................................................... 27  
4 INFORMATION CAPACITY IN OTA ..................................................................  28 
4.1 Introduction..................................................................................................... 29 
4.2    System Topology ............................................................................................ 33 
 4.2.1 Low Frequency Analysis .................................................................... 33 
 4.2.2 High Frequency Analysis.................................................................... 34 
 4.2.3 Noise at the output .............................................................................. 35 
 4.2.4 Effect of Feedback on Input Noise ..................................................... 36 
4.3 Practical Amplifier Configurations................................................................. 36 
 4.3.1 Operational Transconductance Amplifier........................................... 37 
 4.3.2 Custom Amplifier ............................................................................... 44 
4.4 Experimental Results ...................................................................................... 45 
4.5 Discussion....................................................................................................... 50 
 vi
4.6 Conclusion ...................................................................................................... 50 
5    BIT-ENERGY.......................................................................................................... 51 
 5.1 Capacity per unit Signal Power....................................................................... 52 
 5.2 Capacity per unit Total Power ........................................................................ 53 
      5.3 Conclusions..................................................................................................... 54 
6 NOISE EFFICIENCY FACTOR............................................................................. 56 
 6.1 Noise Efficiency Factor for a bipolar transistor.............................................. 56 
 6.2 Capacity of BJT and OTA .............................................................................. 57 
 6.3 Conclusions..................................................................................................... 59 
7 CONCLUSION........................................................................................................  61 
      7.1 Future Work.................................................................................................... 63 
         7.1.1 Encoding/Decoding Scheme............................................................... 63 
         7.1.2 Range of Analytical Model................................................................. 64 
BIBLIOGRAPHY?????????????????????????.   66 
  
 
 
 
 vii
LIST OF FIGURES 
 
 
 
Figure                                                                  Page 
2.1    Typical action potential in a neuron...................................................................... 7 
2.2    Block diagram of set up for extra-cellular recordings .......................................... 8 
2.3    Cells adhered to a Silicon substrate ...................................................................... 8 
2.4 Passive fixture for recording from the cells..........................................................     9 
2.5 Sample of recordings from the cells ..................................................................... 10 
3.1    OTA Symbol and Equivalent Circuit.................................................................... 12 
3.2    A single stage Operational Transconductance Amplifier ..................................... 13 
3.3    OTA open loop gain schematic............................................................................. 15 
3.4    OTA AC Analysis schematic................................................................................ 16 
3.5    Input differential pair with diode connected load................................................. 18 
3.6    OTA cascaded output............................................................................................  20 
3.7    MOSFET thermal noise current source model ..................................................... 24 
3.8 MOSFET flicker noise current source model ....................................................... 25 
3.9 Total Noise Spectrum ........................................................................................... 26 
3.10 Additional gain stage at the output ....................................................................... 26 
4.1    A Gaussian Channel.............................................................................................. 29 
4.2    Water Filling ......................................................................................................... 31 
4.3    System comprising of n blocks A1, A2 ? An .....................................................  33 
4.4    (a) Input referred noise of the system, (b) Ideal distribution of Power for maximum 
information transfer rate ................................................................................................ 34 
 viii
4.5    (a) Transfer function of each of the blocks A1, A2? An, (b) Input referred noise of 
the entire system, (c) Regular use of the system, (d) Ideal distribution of Power for 
maximum information transfer rate. f1 is the most dominant pole of the system, fmin 
corresponds to the lowest noise level............................................................................. 35 
4.6    (a) A simple first order feedback system, (b) Equivalent schematic of the system 36 
4.7    (a) Operational Transconductance Amplifier with cascodes????????.   37 
4.7    (b) Noise contributions from each device in the OTA..........................................   39 
4.8 Noise Contributions of devices in the MOSFET ..................................................  40 
4.9 Ideal distribution of signal power for constant f2 and varying f1.........................  41 
4.10 Information transfer rates as a function of frequency........................................... 41 
4.11 Capacity as a function of Bias current and Signal power ..................................... 42 
4.12 Capacity as a function of Bias current and Signal power ..................................... 43 
4.13 Capacity as a function of Bias current and Signal power ..................................... 43 
4.14 Bio-amplifier......................................................................................................... 44 
4.15 (a) Block Diagram of set-up for noise and transfer function measurements ......... 45 
4.15 (b) Actual set-up for noise and transfer function measurements ........................... 46 
4.16 Measured Noise at Output ....................................................................................  47 
4.17 Measured Transfer Function.................................................................................  47 
4.18 Input Referred Noise from Experimental data......................................................  48 
4.19 Variation of capacity with varying f2 ...................................................................  49 
4.20 Comparison of actual data with model for capacity with changing f2 .................  49 
5.1 Capacity per unit Signal Power............................................................................. 53 
5.2 Capacity per unit Total Power ............................................................................. 54 
 ix
5.3 Variation of capacity metrics with power constraints...........................................  54 
6.1 Capacity of a simple Bipolar Transistor ............................................................... 58 
6.2 Capacity of OTA with Thermal noise................................................................... 58 
6.3 Capacity of OTA and BJT as a function of signal power and bias current ..........   59 
6.4 Capacity of OTA / Capacity of BJT ..................................................................... 59 
7.1 (a) System with input signal whose frequency spectrum cannot be altered (b) 
System used for maximum gain (c) System used for maximum capacity..................... 63 
7.2 Cross?over between subthreshold and above threshold ....................................... 64 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1
 
 
 
Chapter 1 
Introduction 
"No amount of experimentation can ever prove me right; a single experiment can prove 
me wrong." - Albert Einstein  
 
I present this work as an ongoing effort towards making sensor networks more efficient 
within the limited resources they have at their disposable. Tradeoffs of performance 
versus resources are everywhere, placing fundamental and practical limits on 
performance and efficiency. Highly efficient integrated systems that use minimal energy 
for maximum impact are essential for autonomous systems whose resources are precious. 
Transmitting signals is hideously inefficient in light of the inevitable power and 
bandwidth limitations in sensor networks. Judicious allocation of the resources to 
maximize the performance of these systems is the key to functional remote and 
distributed sensing applications. With the growing density of sensor networks, there is a 
need for systems that operate at low power, have low noise and are efficient in terms of 
information transfer. Several authors [6,12,21] have investigated the implications of 
information theory in designing microsystems. Abshire [2] performed an analysis of the 
information processing in the blowfly photoreceptor and applied the same principles to 
 2
studying the information efficiency of silicon photoreceptors. We use principles of 
information theory and analog circuit design to investigate the performance cost tradeoffs 
in an Operational Transconductance Amplifier (OTA), a commonly used block in 
amplification stages in sensory systems. 
 There is a whole community of designers working on making low voltage, low 
noise amplifiers for extra-cellular recordings. Harrison [9] designed a low power low 
noise amplifier for neural recordings applications. Horiuchi et al [10] designed a spike 
detector for recording from the blowfly photoreceptor. These systems have to be 
reasonably small, provide good amplification, have frequency selectivity, work at low 
voltages and have low noise. These requirements become more stringent considering that 
these circuits are commonly placed in dense arrays and need to be battery powered. We 
are greedy. We want systems that give us maximum information at minimum cost. In 
other words, we care about increasing the capacity of these systems for a fixed power 
budget. We perform detailed analysis to determine how we can maximize the information 
transfer rate of the OTA. 
 
Chapter2: This chapter is a progress report of our experiments with recording the 
electrical activity of mammalian cells extra-cellularly. Our current set-up consists of 
discrete components but we intend to use our custom OTA amplifier in the future. The 
results of these endeavors give us insight into the spectral and power content of the 
signals we are trying to measure with our amplifier. This sets the signal power budget and 
the frequency cut offs for the OTA.  
 
 3
Chapter 3: The block that we choose to optimize in this large sensing scenario is the 
Operational Transconductance Amplifier. We provide a detailed functional and 
behavioral analysis of this class A amplifier. The parameters used to characterize the 
performance of the system are: Open Loop Gain ( A
OL
), Gain Bandwidth Product (GB), 
Maximum Output Current (
IOUT
MAX
), Slew Rate (SR), Static Power Dissipation (
PSTATIC
), 
Gain/Phase Margin (GM/PM),  Offset Voltage (V
OS
), Total Harmonic Distortion (THD), 
Noise (V
ni
). We are particularly interested in the noise analysis as this is the starting point 
for doing the information transfer analysis on the OTA structure. We present a detailed 
description of the noise sources present in a MOS device and their distribution with 
frequency. The input referred noise for the OTA is determined both for the low frequency 
and the high frequency case. 
 
Chapter 4: This is the core of our research towards maximizing the capacity of the 
OTA. We use conventional small signal analysis to determine the transfer functions and 
input referred noise for the OTA. We model this noise as a colored Gaussian channel 
with a power constraint and apply the water-filling algorithm to determine the optimal 
region where we should operate this circuit to achieve maximum capacity. The idea is 
that since  noise is cup-shaped, if we start filling in signal in the region where the noise is 
minimum then we achieve a high signal-to-noise ratio and hence higher capacity. This 
happens at considerably higher frequencies. We observe that if we shift the signal to the 
higher spectral regions, we increase the capacity. We present results from this analytical 
model corroborating this approach. We have a custom amplifier fabricated in a 
commercially available 0.5 ?m technology. We perform noise and transfer function 
 4
measurements on this amplifier and apply the water filling model on the data we collect 
from these measurements. 
 
Chapter 5:  We investigate another criterion for evaluating the efficient performance 
of our system i.e. capacity per unit cost. We determine how much power is needed to 
achieve the capacity our circuit can give us. Our definition of power is two-fold: power 
constraint on the input signal, and supply requirements for the system to operate. 
 
Chapter 6: Noise Efficiency Factor (NEF) is a metric for comparing the noise 
performance of amplifiers. Since the capacity is a function of the noise performance of 
the circuit, it is only natural that we extend our model to include the NEF. We determine 
the capacity of a simple ideal bipolar transistor considering only thermal noise which is 
the definition for NEF. We compare the capacity of our system with that of a bipolar 
transistor given similar power constraints and current levels. 
 
Chapter 7: Work to be done: we notice that capacity is maximum if we operate the 
amplifier at much higher frequencies. But these amplifiers are designed for applications 
at low frequency. Is it feasible to shift the spectral content of the signal to these higher 
frequencies without spending more energy in the process? Is it possible to shift the 
bandwidth of the system to a higher frequency to maximize capacity but keep the desired 
frequency selectivity of the system?  These are some of the issues that will be dealt with 
in the future. 
 5
 
 
 
Chapter 2 
 
Extra-Cellular Recordings from Muscle 
Cells 
 
Electric current flow in the conductive extra-cellular medium around neurons during 
action potentials can be detected by means of extra-cellular microelectrodes as voltage 
'spikes'. Extra-cellular spike potentials recorded from the mammalian cells have a 
duration of between 0.2 and 20 millisecond. Their amplitudes are typically a few hundred 
micro- volts although they may vary in amplitude from the noise level of the electrode 
(several microvolts) up to several millivolts, depending on the type of cell and the quality 
of the recording system. The greatest advantage of extra cellular recording is that the 
activity of cells can be recorded without having to impale and consequently damage 
them. Most in vivo monitoring of electrical properties of the cells is done with extra-
cellular recording [25, 26]. 
 Signals picked up by extra cellular electrodes are in the microvolt range and they 
need to be amplified to be processed in more conventional electronic devices such as 
oscilloscopes, analyzers or computers. The main difficulty with extra cellular recording is 
the electrical "noise". This term refers to spontaneous voltage fluctuations, which appear 
as a thickening of the baseline when viewed on an oscilloscope at low sweep speed. The 
 6
noise may result from external interference from electrical sources in the vicinity of the 
recording set-up (mains line hum pickup) and/or from the intrinsic properties of the 
substances making up the electrode and electrical circuit (thermal noise) used to amplify 
electrode signals. Noise in the input circuit and the external medium sets a limit to the 
minimum amplitude signal that can be reliably measured. 
 
2.1 Electrical Activity is Action Potentials of the Cells
The action potential is a large depolarization signal [14]. It can reach up to 110 mV in 
amplitude; it lasts only 1 millisecond and can be conducted at rates of 1 to 100 m/s. A 
sample is shown in Fig 2.1. The action potential is a stereotyped all or none signal. Its 
amplitude and duration does not depend on the amplitude and duration of the input signal 
(the synaptic or the receptor potential). Once the firing threshold has been reached the 
action potential is initiated. Unlike input potentials, which spread passively and decrease 
in amplitude with distance, the action potential does not decay as it travels along the axon 
to the terminal of the neuron (this distance can be up to 1meter) [14]. 
 Since the action potential is an all or none signal, a single action potential can not 
carry information about the amplitude of the input signal. The information of the input 
potential amplitude is not lost. The integrative component of the cell transforms the 
amplitude of the input signal into the frequency of an action potential train. In turn, the 
duration of the input signal is determined by the period during which action potentials are 
generated. 
 7
 
Fig. 2.1 Typical action potential in a neuron [14] 
2.2 Experimental Setup for Recording from Cells 
Utilizing traditional photolithographic processes to produce planar arrays of 
microelectrodes has proven to be of immense value to the neuroscience community. 
There are numerous reports describing the plating of various cell types on planar 
microelectrode arrays including neurons and cardiac muscle cells [25, 26]. Cardiac cells 
in culture under proper media and environmental conditions exhibit spontaneous and 
synchronous action potential activity throughout the culture. By analyzing the signals 
generated by the spontaneous culture using a microelectrode array, Whittington et al [25] 
were able to monitor the continuous electrophysiological state of the culture and its 
variation in space and time. 
 We [20] perform a first run of recordings from Bovine Aortic Smooth Muscle 
Cells (BAOSMC), which are easy to culture and are known to have electrical activity. 
Our test set-up shown in Fig 2.2 had the necessary amplification and filtering blocks. A 
silicon substrate patterned with rectangular gold pads (die) is used as the microelectrode 
array for the extra-cellular recordings. The conventional aluminum pads are plated with 
gold since gold is a bio-compatible substance and is mostly non-reactive and non- 
 8
corrosive [8]. The cells are left over-night to let them adhere to the pads. Healthy cells 
spread out, elongate and attach to the pads as shown in Fig 2.3. We [20] examine the test 
die under the microscope to determine the number of cells adhering to each pad. 
 
 
Fig. 2.2 Block diagram of set up for extra-cellular recordings [20] 
 
Fig. 2.3 Cells adhered to a silicon substrate  
The signals received from the cells are very small in magnitude (?V range), so small that 
surrounding noise can frequently obscure these signals. Also, these signals are confined 
to a small bandwidth. We need a set up to filter out the environmental noise (~ 60 Hz) 
and the noise above the frequencies of interest (above 7 KHz). Since the signal 
amplitudes lie in micro/milli volts, we need an amplification stage to boost the signal. In 
order to minimize external noise pickup and interference, experimental set-up has to be 
correctly shielded and grounded. Overall shielding from noise was provided by placing 
 9
test fixture in a die cast aluminum box with a closeable lid. The wire from the electrode 
to the output was kept as short as possible, less then 2" (5 cm). Line-powered equipment 
was kept as far away from the site of recording as possible. Everything near the 
preparation is grounded to a single point.  
 The test die is packaged in a DIP package and a vector board as shown in Fig. 2.4 
was made for the measurements.  
 
Fig. 2.4 Passive fixture for recording from the cells 
A thick well of conductive epoxy was put around the die to isolate the packaging material 
from the cells and also to protect the bond wires. The components on the board include 
an amplifier and a band pass filter. The amplifier receives the signal from the cells and 
magnifies it 10 times. This amplification is applicable to the noise associated with the 
signal as well. This amplified signal is then given to a band pass filter, since we want to 
remove the low frequency 60 Hz noise and the high frequency (above 7 KHz) noise. The 
band pass filter can also be used to amplify the filtered signal by a factor of 10. The 
amplifier we used is a high precision low noise amplifier OPA227. The amplifier was 
 10
tuned to give a gain of 10.The filter we used is the Maxim 274 continuous time active 
filter consisting of four independent cascadable second order sections. We designed each 
of these sections to be Butter Worth band pass filters centered at 1.5 kHz and having a 
pass band between 500 Hz and 5 kHz with a midband gain of 10. The recorded output is 
100 times amplified. 
2.3 Cell Recordings 
To get an estimate of the noise associated with the set up and the growth medium, we 
recorded from the growth medium and de-ionised water before recording from the cells. 
Shown in Fig 2.5 is a sample of our recordings where the noise shown (between the blue 
lines) is an average of our recordings without cells. 
 
Fig. 2.5 Sample of recordings from the cells 
 
2.4 Conclusion 
We [20] perform the extra-cellular measurements of the stimulated and spontaneous 
action potentials of the mammalian cells. We accomplish two objectives: we prove the 
 11
electrically activity of the BAOSMC and also determine amplitudes and bandwidth of the 
action potentials of these cells. This gives useful information about the amplification and 
filtering limits for the amplifier design. 
 Currently, as shown our [20] set-up is made up of discrete components. The next 
stage of recordings will be done with cells placed on the custom OTA amplifier.  
 
 12
 
 
 
Chapter 3 
Operational Transconductance Amplifier (OTA) 
In order to optimize a system it is imperative to analyze it completely. We present a 
detailed description of the structure of OTA and the various characterization parameters. 
This is the first step towards formulating the model for information capacity for a 
particular system. The schematic symbol and equivalent circuit model for an Operational 
Transconductance Amplifier (OTA) are shown in Fig 3.1 [7,13,19].  
 
Fig. 3.1 OTA Symbol and Equivalent Circuit 
The OTA converts a differential input voltage to an output current relative to a 
transconductance gain parameter Gm=i
o
/v
i
. Ideally, the input and output resistances are 
infinite (R
i
 =R
o
=?) such that i
i
=i
Ro
=0 and the output current is absorbed solely by the 
load. The conventional OTA is classified as a class A amplifier and is capable of 
generating maximum output currents equal to the bias current applied. The equivalent 
circuit model indicates that the transconductance amplifier generates an output current 
 13
(i
o
) proportional to an input voltage (v
i
) based on the transconductance gain G
m
. The open 
circuit voltage gain of the conventional OTA model in Fig 3.3 is given by A=G
m
 R
o
. 
A conventional, one stage, CMOS, Operational Transconductance Amplifier (OTA) 
configuration is shown in Fig 3.2 
 
Fig. 3.2 A single stage Operational Transconductance Amplifier 
The OTA employs a differential input pair and three current mirrors. The differential 
input pair is comprised of transistors M
1,2
. The differential pair is biased by M
B1,B2
. 
Mirrors formed by M
3,5
 and M
4,7
 reflect currents generated in the differential pair to the 
output shell. The current generated by the mirror of M
3,5
 is then reflected to the output via 
the mirror formed by M
6,8
. The mirror gain factor, K, indicates the gain in mirrors formed 
by M
3,5
 and M
4.7
. Cascoding transistors M
casN
,
casP
 are biased by V
casN
 and V
casP
 
respectively  and provide increased gain via increased (cascoded) output resistance.  
 14
 The conventional OTA is differentiated from other amplifiers by the fact that its 
only high impedance node is located at the output terminal. The OTA [13] does not 
employ an output buffer and is therefore, only capable of driving capacitive loads.  
 
3.1 Operation 
The conventional OTA uses a differential pair in conjunction with three current mirrors 
to convert an input voltage into an output current. Common mode signals (V
in1
=V
in2
) are, 
ideally, rejected. For a common mode input voltage, the currents are constant and will be: 
i
d1
=i
d2
=I
BIAS
/2 and i
out
=0. A differential input signal will generate an output current 
proportional to the applied differential voltage based on the transconductance of the 
differential pair. The conventional OTA is only capable of producing an output current 
with a maximum amplitude equal to the bias current in the output shell. For this reason, 
the conventional OTA is a referenced as a class A structure capable of producing 
maximum signal currents equal to that of the bias current applied. Slew rate (SR) is 
directly proportional to the maximum output current and is defined as the maximum rate 
of change of the output voltage. For a single stage amplifier, the slew rate is the output 
current divided by the total load capacitance. The conventional OTA therefore suffers the 
consequence that high speed requires large bias currents which translates to large static 
power dissipation.  
 
3.2 Small Signal Analysis 
We perform a detailed small signal analysis for determining the transfer function, the 
bandwidth and noise transfer functions for the OTA information capacity model. 
 15
3.2.1 Open loop gain 
Fig 3.3 will be used for the open loop gain analysis. The output current, in terms of the 
mirror gain factor (K), is given by: 
21out d d
iKiKi=?
                                                                                                                 (1)       
where:  
1112 22
,
dmindmin
igVigV==
                                                                                                      (2) 
 
Fig. 3.3 OTA open loop gain schematic 
Assuming: g
m1
=g
m2
, and substituting (2) in (1) 
1,2 1 2
()
out m in in
iKgVV=?
                             (3) 
This indicates the transconductance gain of the OTA is given by:  
1,2mm
GKg=
                                                                                                                          (4) 
The output resistance is a cascode resistance and is given by: 
78
||
out mN oN o mP oP o
R grr grr=
                                                                                                   (5) 
 16
Combining (3) and (5), the output voltage is then given by:  
1,2 1 2 7 8
()(|)
out out out m in in mN oN o mP oP o
viR KgVVgrrgrr== ?
                                                                   (6) 
and the open loop gain is:  
1,2 7 8
(||)
out
OL m mN oN o mP oP o
in
v
AKggrgr
v
==
                                                                                    (7) 
3.2.2 AC Analysis 
 
Fig. 3.4 OTA AC Analysis schematic 
The gain bandwidth of the single stage OTA is limited mainly by the low impedance, 
high frequency poles at nodes A and B, in conjunction with the high impedance, low 
frequency pole at the output node. The following analysis will define the high frequency 
pole and will assume nodes A and B are equivalent nodes in terms of resistance and 
parasitic capacitance (M
1
=M
2
, M
3
=M
4
= M
5
=M
6
). The resistance at nodes A and B is 
dominated by the diode connected resistance (1/g
m
) of M
3
, 
4
 and is given by:  
,01,2
3,4 3,4
11
AB
mm
Rr
g g
=?
                                                                                                         (8) 
The parasitic capacitance at A and B is given by:  
 17
, 1,2 3,4 5,7 5,7 3,4 5,7A B gd gs gs gd gs gs
CC C C C C C=+++?+
                                                               (9) 
The pole at A and B is  
                                                                                      (10)                        
 
 
The output node capacitance is dominated by the load capacitance (C
L
) and is: 
L Lout gdN gdP
CCCCC?=++
                                                                                               (11) 
Combining (10) and (11), the dominant pole/bandwidth of the OTA is given by:  
3
78
11
22(|)
pout dB
out out L mN oN o mP oP o
f f
CR Cgrr grr??
== =
                                                               (12) 
Equation (10) indicates the high frequency pole f
PA,B
 is inversely proportional to C
gs3,4
. 
The bandwidth of the conventional OTA is given in Equation (12) and is inversely 
proportional to the load capacitance (C
L
).  
3.2.3 Gain Bandwidth 
Equations (7) and (12) are combined for the gain bandwidth (GB) product: 
1,2
2
m
L
Kg
GB
C?
=
                                                                                                             (13) 
This relation indicates that the GB is directly proportional to the mirror gain factor K. 
3.2.4 Maximum Output Current 
The maximum output current of the conventional OTA is limited by the bias current and 
is given by:  
MAX
out BIAS
IKI=
                                                                                                                      (14) 
 
3,4
,
,, 3,4
1
22(2)
m
pA B
AB AB gs
g
f
CR C??
==
 18
3.2.5 Slew Rate 
The slew rate is given by  
MAX
out BIAS
LL
IKI
SR
CC
==
                                                                                                             (15) 
The slew rate increases linearly with I
BIAS
 and the current mirror gain factor K. 
 
3.3 DC Analysis 
3.3.1 Input Common Mode Range 
The common mode range (CMR) is defined as the range of voltage (V
IN
MIN
, V
IN
MAX
) for 
which the input differential pair will remain in saturation. This range is determined by the 
amplifier structure, transistor sizes, and bias current. For the differential input stage with 
diode connected loads, the minimum and maximum input voltages can be found by 
analysis of Fig 3.5. 
 
Fig. 3.5 Input differential pair with diode connected load 
 
 19
3.3.1.1 Minimum Input Voltage 
The minimum input voltage can be expressed as:  
11 1
()
MIN SAT
IN DD DSMB DS THP
VVV VV=? + +
                                                                                         (16) 
and substituting:  
2
SAT D
DS
IL
V
WKP
=
                                                                                                                   (17) 
The minimum input voltage becomes:  
11
1
21
2
MIN BIAS MB BIAS
INDD THP
MB P P
IL IL
VV V
WK WK
=+ + +
                                                                            (18) 
where V
THP1
 is body effected and may be larger than the zero bias threshold voltage. The 
minimum input voltage is therefore limited by the V
DS, SAT
 drop requirements across M
1
, 
M
B1
 and a threshold drop across M
1
. The minimum input voltage is inversely 
proportional to the widths of transistors M
1
, M
B1
 and directly proportional to the bias 
current. To reduce V
IN
MIN
, bias current must be reduced or the widths of the input 
transistors must be increased.  
3.3.1.2 Maximum Input Voltage Range 
The maximum input voltage can be expressed:  
31
MAX
INSSGSTHP
VVVV=+ +
                                                                                                       (19) 
and with substituting V
GS3
, it becomes 
 
3
31
3
MAX BIAS
INSS THNTHP
N
IL
VV V V
WKP
??
=+ + +
??
??
                                                                                 (20) 
Again, V
THP1
 is body effected and will be larger than anticipated. In this case, the body 
effect actually increases input range by contributing to V
IN
MAX
. These results indicate the 
bias current must be reduced and the width of M
3
 must be increased to increase V
IN
MAX
. 
 20
The maximum input voltage is, therefore, only limited by a V
GS
 drop across M
3
. For this 
reason, the input voltage range is typically limited by V
IN
MIN
. The common mode range 
of the PMOS differential pair is capable of swinging further in the negative direction than 
the positive direction.  
3.3.2 Output Voltage Range  
The output voltage range is defined as (V
OUT
MAX
, V
OUT
MIN
) which represents the 
maximum output swing available. The output range of the conventional OTA is reduced 
due to cascoding at the output shown in Fig 3.6 
 
Fig. 3.6 OTA cascoded output 
The output voltage range is given as:  
,8 ,
MAX
out DD DS SAT DS SATP
VVV V=? ?
                                                                                                (21) 
,7 ,
MIN
out SS DS SAT DS SATN
VVV V=+ +
                                                                                                 (22) 
 
 21
 
 
3.3.3 Static Power Dissipation 
The static power dissipation (P
STATIC
) is the product of the sum of the currents flowing 
through the current sources or sinks with the power supply voltages and is given by: 
,1 ,1 ,2 ,5 ,7
()[ ]
STATIC DD SS D MB D M D M D M D M
PVVIIIII=? ++++
                                                             (23) 
and in terms of I
BIAS 
()(2)
STATIC DD SS BIAS
PVVI K=? +
                                                                                 (24)                         
An increase in the I
BIAS
 will increase the SR and GB of the conventional OTA at the cost 
of increased area and static power dissipation and a decrease in phase margin.  
 
3.4 Characterization 
3.4.1 Gain and Phase Margins 
The application of negative feedback requires analysis of the open loop gain. Some 
circuits will cause a phase shift in the input signal large enough that the feedback 
becomes positive (adds to the input), resulting in an unstable system [7]. Stability 
requires a phase shift in the feedback signal less than 180? for open loop gain values 
larger than 0 dB. This requirement necessitates the definition of two measures of 
stability: gain margin (GM) and phase margin (PM). Gain and phase margin parameters 
can be measured via analysis of the open loop AC response simulation. The gain margin 
is defined as the difference (in dB) in the gain at a phase of -180? and unity gain. Design 
guidelines typically specify a GM greater than 10 dB. The phase margin is defined as the 
difference (in degrees) in the phase at unity gain and -180?. The phase margin should be 
 22
greater than 45? with an optimum, critically damped, value of 60? [3]. For PM values less 
than 60? the system is under-damped, and the transient response will indicate increased 
slew rate. For PM values greater than 60? the system is over-damped, and the transient 
response will indicate decreased slew rate. Phase margin depends on the relative position 
of the high frequency pole f
PA
 (Fig 3.4) and the gain bandwidth (unity gain frequency). 
The position of the high frequency pole location is therefore directly related to the phase 
margin.  
3.4.2 Input Offset Voltage  
Ideally, if both inputs of the OTA are grounded, the output voltage should be zero [7]. 
Practically, a nonzero output voltage (offset) will be present and is due to random and 
systematic errors.  
3.4.2.1. Random Offset  
Random errors are due to mismatches in the input stage as a result of fabrication 
including (but not limited to) : threshold voltage differences and geometric differences. 
Random errors can be estimated via Monte Carlo simulations. 
3.4.2.2 Systematic Offset  
Systematic errors are inherent to the design. Systematic errors can be the result of non-
symmetries in the OTA design, creating voltage and current mismatches. The systematic 
offset can be determined via simulation and will be evident in the DC sweep simulation 
as the offset from the zero-zero intercept where the input voltage and output voltage 
should both equal zero.  
3.4.3 Total Harmonic Distortion 
Ideally, the output of an amplifier is a replica of the input signal scaled by the gain A. 
 23
The gain for large signal inputs is dependent on the input signal amplitude [7]. For a 
purely sinusoidal input signal:  
() sin( )
in M
Vt V wt=
                                                                                                              (25)              
The non-ideal output signal of an amplifier can be expressed as: 
12
( ) sin( ) sin(2 ) .... sin( )
OUT M M n M
VtaV taV t aV nt? ??=+ +
                                                     (26)          
where the desired output is a fundamental a
1
V
M
 sin(?t) and ideally a
2
 through a
n
 are zero. 
3.4.4 Noise sources in OTA 
Analog signals, processed by integrated circuits, are corrupted by two different types of 
noise: device electronic noise and ?environmental noise?. Noise limits the minimum 
signal level that a circuit can process with acceptable quality. Today?s analog designers 
constantly deal with the problem of noise because it trades with power dissipation, speed 
and linearity. Doing efficient design requires that noise be taken into account as readily 
as any other circuit parameter like gain, input, output impedance etc. 
 Next we discuss the noise sources in a MOSFET and consider methods of 
representing noise in circuits [19]. 
3.4.4.1 MOSFET Thermal noise 
The most significant source [17] is the noise generated in the channel. The random 
thermal motion of electrons in the channel introduces fluctuations in the voltage 
measured across the channel even if the average current is zero. Thus the spectrum of the 
thermal noise is proportional to absolute temperature T. The dependence of thermal noise 
upon temperature suggests that low temperature operation can decrease the noise in 
analog circuits. The thermal noise can be modeled by a current source connected between 
the drain and source terminals as shown in Fig. 3.7 with spectral density given by [19] 
 24
mn
gkTI ?4
2
=
                                                                                                                     (27) 
where k is the Boltzmanns constant , T is the absolute temperature in Kelvin, g
m
 is the 
device transconductance.? is a constant [5] taken to be 2/3 for above threshold and 1/2 for 
below threshold conduction for long channel transistors. Their values for short channel 
transistors is still a topic of research. 
 
Fig. 3.7 MOSFET noise current source model for thermal noise 
3.4.4.2 Flicker noise 
Flicker noise is present under DC conditions and is the result of electron trapping 
(delayed release) due to silicon imperfections in the transistor. Flicker noise is inversely 
proportional to frequency and is commonly referred to as 1/f noise. The flicker noise can 
be modeled with a current source with spectral density [3] in parallel with the transistor 
as shown in Fig 3.8  
fgC
IKF
I
mox
AF
D
fn
)(
*
2
2
/1,
=
                                                                                      (28) 
KF is a process-dependent flicker noise constant with approximate value of 10
-25 
V
2 
F. 
AF is a constant with its value lying between 0.5 and 2, I
D
 is the drain current (DC), C
ox
  
is the capacitance per unit area, g
m
  is the device transconductance and f is the frequency. 
 25
 
Fig. 3.8 MOSFET noise current model for flicker noise 
 The noise sources are given in terms of per unit bandwidth. There can be an 
interesting possibility of lower frequency cut-off flicker noise being zero [19]. As the 
flicker noise has a logarithmic dependence on frequency this yields an infinite value for 
total noise. Secondly if we observe the circuit for a very long time, the very slow noise 
components can randomly assume a very large power level. At such slow rates, noise 
becomes indistinguishable from thermal drift or aging of devices. This leads to the 
following conclusions: first since the signals encountered in most applications do not 
contain significant low frequency components, our observation window need not be very 
long. Secondly, the logarithmic dependence of flicker noise over f
1
 allows some margin 
of error in selecting f
1
. 
 To quantify the significance of 1/f noise with respect to thermal noise for a given 
device, we plot both spectral densities on the same axes and determine their intersection 
point. Called the ?corner frequency?, the intersection point serves as a measure of what 
part of the band is mostly corrupted by flicker noise. 
3
8
Cm
ox
K
fg
CWL kT
=
                                                                                                           (29) 
 26
 
    Fig. 3.9 Total noise spectrum 
3.4.4.3 Representation of noise in circuits 
The natural approach to quantify the effect of noise in a circuit is to calculate the total 
noise at the output due to the various sources of noise in the circuit. While intuitively 
appealing, the output-referred noise does not provide a useful comparison of the 
performance of different circuits because it depends on the gain. 
 Consider a system with noise at the output [19] equal to V
nout
2
 followed by an 
amplifier at the output stage of gain A2 as shown in Fig 3.10. Then the total output noise 
is equal to the V
nout
2
 multiplied by A2. Considering only the output noise we may 
conclude that if the gain A2 increases, the circuit becomes noisier which is an incorrect 
result because a larger A2 also provides a proportionally higher signal level at the output. 
That is the output signal to noise ratio does not depend on A2. 
 
Fig 3.10 Additional gain stage at the output 
To overcome this quandary we specify the ?input-referred noise? of the circuits [17]. The 
idea is to represent the effect of noise sources in the circuit by a single source V
n,in
2
 at the 
input such that the noise at the output is equal to the output noise if all the noise sources 
 27
in the circuit were to be considered separately. If the voltage gain is A
v
, then we must 
have V
n,out 
= AV
n,in
2
 , i.e the input referred noise in the simple case is the output noise 
voltage divided by the gain. 
 
3.5 Conclusions 
A full analysis of the one-stage conventional OTA structure was presented to determine 
the transfer functions and noise contributions of each device in the OTA to the total noise 
at the output. This is the basis for the model for information capacity. The 
transconductance amplifier rejects common mode signals and generates an output current   
dependent on the input voltage. Cascoding of the output shell to increase gain 
necessitates the requirement of a purely capacitive load. The structure is class A and the 
maximum output current is limited by the magnitude of the bias current in the output 
shell. The mirror gain factor K can be increased to increase the maximum output current 
(slew rate) and gain bandwidth at the cost of an increase in static power, and area, as well 
as a decrease in phase margin and stability.  
 This OTA forms the core of our custom amplifier for recording the electrical 
activity of mammalian cells. It is to be optimized to give a good gain and frequency roll 
off at high frequencies. We treat the OTA as a Gaussian channel with a power constraint 
and determine the optimal region of operation of the circuit for maximum information 
transfer rate. In the subsequent chapters we develop this approach and use different 
metrics for evaluating the performance of the OTA. 
 
28
 
 
 
Chapter 4 
Information Capacity in OTA 
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.  
 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 
are an amplifier recording signals from electrically active cells and an imager 
acquiring images from natural scenes. The power of these signals is concentrated at 
low temporal frequencies. However we know from basic noise theory [17] 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. 
  
 
29
4.1 Introduction 
To formulate our model, we briefly introduce the relevant concepts from information 
theory. Consider a general communication system [4, 22] consisting of an input, a 
channel, a noise source and the output as shown in Fig. 4.1 
                                         
Fig. 4.1 A Gaussian channel [22] 
Uncertainty in the system is modeled as an additive noise contribution Z at the 
output of the channel, such that Y = [X]  + [Z] where X and Y are the input and 
output signals respectively. The information rate through the system is limited by the 
channel bandwidth F and the noise power N corrupting the signal. The channel 
capacity is determined by maximizing the mutual information I(X;Y) between the 
input and the output. We model the system as a Gaussian channel transmitting 
information at a rate R. 
 A Gaussian channel is an additive noise channel where the noise is a random 
Gaussian process. For a Gaussian channel [22], the transmission bandwidth and the 
ratio of the signal power to noise power are sufficient to determine the channel?s 
capacity to transmit information. The noise entropy per second is 
2
() logH ZF eN?= , 
where F is the bandwidth of transmission over the channel and N =?
z
2
  is the noise 
power. With no constraints on the input, we are free to choose input levels spaced 
arbitrarily far apart, and there is little trouble in distinguishing the outputs. The 
capacity is unbounded. However, unconstrained inputs are physically implausible. 
We?ll resolve this difficulty by imposing a power limitation P on the input signal, so 
 
30
that 
2
1
1
n
i
i
x P
n
=
?
?
. The Gaussian distribution maximizes the entropy for a fixed 
variance, so the output entropy per second is bounded by the entropy of a Gaussian 
distribution with the same output power
2
() log ( )H YF ePN??+. This leads to the 
famous result given by Shannon for the channel capacity of a Gaussian channel; 
2
*log 1
P
CF
N
??
=+
??
??
 where P and N are the average signal and noise power, 
respectively, and the capacity C is given in units of bits per second. This result applies 
to channels with an average signal power constraint P and additive white Gaussian 
noise of power N. The expression of the channel capacity of the Gaussian channel 
makes intuitive sense: 
- As the bandwidth of the channel increases, it is possible to make faster 
changes in the information signal, thereby increasing the information rate linearly. 
- As S/N increases, one can increase the information rate while still preventing 
errors due to noise. 
- For no noise, S/N = ? and an infinite information rate is possible irrespective 
of bandwidth. 
 If a white thermal noise is passed through a filter whose transfer function is  
Y (f), the resulting noise has a power spectrum N (f) = K |Y(f) |
2
 and is known as 
Gaussian noise. We can calculate the capacity of a channel perturbed by any Gaussian 
noise from the white-noise result. Suppose our total transmitter power is P and it is 
distributed among the various frequencies according to P (f).  
Then,  
0
()
W
P fdf P=
?
                                              (1)    
We can divide the band into a large number of small bands, with approximately N (f) 
constant in each. The total capacity for a given distribution P (f) will then be given by 
 
31
1
0
()
log 1
()
W
Pf
Cdf
Nf
??
=+
??
??
?
                                (2) 
since, for each elementary band, the white-noise result applies. The maximum rate of 
transmission will be found by maximizing C
1
 subject to condition (1). We use 
Lagranges multipliers [1] to do this. 
0
()
log 1 ( )
()
W
Pf
P fdf
Nf
?
??
??
++
????
??
??
?
                  (3)   
The condition for this is, by the calculus of variations, or merely from the convex 
nature of the curve log (1+x)  
1
0
() ()Nf Pf
?+=
+
          (4) 
or  N (f) + P (f) must be constant. The constant is adjusted to make the total signal 
power equal to P. For frequencies where the noise power is low, the signal power 
should be high and vice versa as we would expect. 
 
Fig 4.2 Water filling  
The signal that maximizes the capacity can be found using the water-filling analogy 
[4]. Shown if Fig 4.2 is a system with multiple channels each having a different noise 
distribution. Given the total power P to be distributed among the different channels in 
the system, water filling tells us to allocate power to the channels which has the 
lowest noise first and then progressively to the noisier channels. If P is small, we 
 
32
cannot make P (f) + N (f) constant, since this would require negative power at some 
frequencies. It is easily shown, however, that in this case the best P (f) is obtained by 
making P (f) + N (f) constant whenever possible, and making P (f) zero at other 
frequencies. With low values of P, some of the frequencies will not be used at all. If 
we now vary the noise spectrum N (f), keeping the total noise power constant and 
always adjusting the signal spectrum P (f) to give the maximum transmission, we can 
determine the worst spectrum for the noise. This turns out to be the white-noise case. 
It is not just the worst among the Gaussian noise sources, but also among all possible 
noises with the given power in the band. 
 We wish to maximize the information transfer between the input and output 
for the OTA. The starting point for that is maximizing the information capacity, since 
the capacity of a channel depends only on the channel properties and signal 
constraints, such as bandwidth, noise, and constraints on the signal values; it does not 
depend on the specific details of any particular task for which the channel may be 
used [22]. There have been successful efforts in optimizing the performance of analog 
circuits, considering them as a channel [2,12]. The input referred noise waveform 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 signals 
in spectral regions where the channel noise is minimal. As the input referred noise is 
typically cup-shaped, the water filling technique 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 
 
33
noisier parts of the channel [4]. We apply this algorithm to obtain the ideal frequency 
range of operation of an OTA for maximum information transfer. 
 
4.2. System Topology 
We consider a generic system consisting of n analog blocks A1, A2?An with 
equivalent input noise sources V1, V2 ?Vn as shown in Fig. 4.3. The noise from 
each block is assumed to be composed of a flicker noise source and a white Gaussian 
noise source [15]. We treat the entire system as a colored Gaussian channel and 
estimate the ideal frequency range of operation for maximum information transfer 
rate. 
 
Fig 4.3 System comprising of n blocks A1, A2 ? An 
4.2.1. Low Frequency Analysis 
At low frequencies, the transfer function of each block is perfectly flat. The input 
referred noise of the system is given by Equation (5) and shown in Fig 4.4 (a). We 
assume that there is a brick wall filter at the output of the system that blocks all 
frequencies above frequency Fmax to restrict attention to the low frequency case. 
22
222
1 222
1121
......
......
n
IN
n
VV
VV
AAAA
?
=++                 (5) 
We know that the capacity C of a channel or the maximum information transfer rate is 
given by  
2
2
2
():
1
()
max log (1 )
()
f
Sf P
f
Sf
Cdf
Nf
? <
=+
?
        (6) 
 
34
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 (f
2
-f
1
) defines the bandwidth of interest. 
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 
F
max
 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. 4.4 (b). 
 
Fig 4.4 (a) Input referred noise of the system, (b) Ideal distribution of Power for 
maximum information transfer rate. 
   
This differs from the traditional approach of using the systems at low frequencies near 
the first pole. If we shift the operation of the system to higher frequencies, where 
noise is reduced, we will achieve an increase in the signal to noise (SNR) ratio and 
hence capacity of the system.   
4.2.2. High Frequency Analysis 
Under typical working conditions, the transfer functions of these analog blocks vary 
with frequency. We consider a cascade of low pass blocks, and the transfer function 
A1, A2 ... An of each of the blocks are as shown in Fig.4.5 (a). The noise contribution 
from each block is assumed to be similar to that shown in Fig 4.4(a). The input 
referred noise of the system is calculated using Equation (5) and plotted in Fig 4.5(b). 
(a) (b) 
 
35
For efficient use of this system, we should start filling up the signal from the lowest 
point in the noise spectrum and continue till we reach the total power content of the 
signal. This contrasts sharply with the regular usage of the system at frequencies 
below the most dominant pole (f1). Conventionally the amplifiers are used within 
their 3 dB bandwidth where they have maximum gain and maximum noise. Fig 4.5(c) 
shows the system being used at frequencies below f
1
, and Fig 4.5(d) shows the most 
efficient region of operation of the system.   
 
Fig 4.5 (a) Transfer functions of each of the blocks A1, A2? An  
(b) Input referred noise of the entire system, (c) Regular use of the system, (d) Ideal 
distribution of Power for maximum information transfer rate. f1 is the most dominant 
pole of the system, fmin corresponds to the lowest noise level. 
 
4.2.3 Noise at the Output 
The noise of the stage following the system should also be modeled as either a white 
or a colored Gaussian channel depending on its spectrum. If the noise is an order of 
magnitude smaller than the system noise, then it will 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. 
(a) (b) 
(c) (d) 
 
36
4.2.4 Effect of Feedback on Input Noise  
Let us consider a simple first order feedback system as shown in Fig 4.6(a) [19], 
where the block A1 is an open loop voltage amplifier characterized by only an input 
referred noise V
n
 and the feedback network is noiseless. We have  
(V
in
 -?V
out
+V
n
) A
1
 = V
out
                                                                                             (7)                      
Hence, 
1
1
()
1
out in n
A
VVV
A?
=+
+
                                                                                (8) 
The circuit can be simplified as shown in Fig 4.6(b) revealing that the input referred 
noise of the overall circuit remains equal to V
n
. If the feedback network were to 
introduce noise at the output, then the introduction of feedback reduces system 
capacity and power efficiency.                              
       
                  (a)      (b) 
Fig 4.6  (a) A simple first order feedback system, (b) Equivalent schematic of the 
system 
 
4.3. Practical Amplifier Configurations 
Based on the generic system topology we discuss above, we perform this information 
capacity analysis on a custom amplifier which is an Operational Transconductance 
Amplifier in capacitive feedback configuration. Our analysis considers the feedback 
network consisting of the capacitors to be noiseless. 
 
 
 
 
 
37
4.3.1. Operational Transconductance Amplifier 
 
We perform noise analysis on a simple cascoded OTA [9], and then apply the water 
filling approach to the noise waveform.  
 
Fig. 4.7 (a) Operational Transconductance Amplifier with cascodes 
 
We model the noise sources by flicker and thermal noise [19]. The input referred 
noise for a MOS transistor is determined by dividing the noise current source between 
the drain and the source by the g
m 
of the device. 
2
2
*41
() *
*
AF
D
n
moxm
KF IkT
V
gCfg
? ????
=+
????
????
                (9) 
where kT is thermal energy, g
m
 is the transconductance of the MOSFET, ? 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 V
2
F, I
D
 is the current level and C
ox
 
is the gate oxide capacitance per unit area. The first term in Equation (9) corresponds 
to the thermal noise and the second term to the flicker noise. 
12345768
;;
mmmm m mm m
g gg g g gg g=====
 
38
 Based on our understanding of the noise sources in the OTA, we perform the 
small signal analysis to determine the transfer function and the total noise at the 
output.  At low frequencies, assuming perfectly matched devices, the input referred 
noise for the circuit is given by Equation (10): 
2222
22 2 2 2 236
136
1111
24 2
mmmNmP
inOTA n n n nN nP
gggg
VV V V V V
?? ?? ?? ??
=+ + + +
?? ?? ?? ??
?? ?? ?? ??
                                    (10) 
In order to minimize the noise level we must maximize g
m1
 and minimize g
m3
, g
m5
, 
g
mN
 and g
mP
. However we cannot arbitrarily reduce the sizes of M
3
 and M
5
 because of 
stability considerations. Reduction in the bias current will help reduce the flicker 
noise, but at the cost of a reduced slew rate. The 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.  
 At high frequencies, the parasitic capacitances associated with the devices can 
no longer be ignored. The gate-source capacitance of each of the MOSFETs and the 
Miller effect [13] become important factors in the noise analysis. Assuming that the 
output resistances of the transistors M
7
, M
8
, M
N
 and M
P
 are all equal to r
o
, the transfer 
function for the OTA is calculated as shown in Equation (11). 
10
01 0
601 0
3
67
3
6
11
2
**
12
2(1
1
2
1
m
mP
min L
gd
m
m
gr
Vgr
grVrsC
C
CC
s
g
s
g
?
??+
=
??
+??? ?
??
++
+
??? ?
+??
??
??
 
10
02 0
302 0
37
3
1
2
*
1
2(1 )
2
1
m
mN
min L
gd
m
gr
Vgr
grVrsC
sC C
g
??+
=
??
+??
??
++
??
??
+
                                           (11) 
00102
VVV=+
 
 
39
C
n
 and C
gdn
 denote the gate to source and the gate to drain capacitance of each 
transistor.   
 The noise contribution of each of the device to the total noise is calculated by 
tracing the path from the input noise source for that device to the output. The noise is 
shaped by the transfer function for each device.  
 
Fig 4.7(b)  Noise analysis of OTA. Each device has a noise current source between 
drain and source. 
 
The region of operation for maximum information transfer rate is simulated using 
MATLAB [16] .We observe that the spectral region corresponding to the maximum 
capacity shifts to the higher frequency with increase in the signal power and also the 
bias current. An increase in bandwidth implies faster changes in the information 
transfer and hence higher capacity. The capacity or the maximum information that 
can be transferred by this system increases with the increase in the input power level 
and also the bias current. This result is in agreement with out theoretical 
understanding of the inverse dependence of noise on the bias current and the direct 
 
40
dependence of capacity on the input signal level.  
 
 
Fig 4.8 Noise contributions of each of the devices in the OTA 
M1-green, M2-red, M3-blue, M4-cyan, M5-magenta, M6-yellow, M7-black, 
 M8-red(:), MN-blue(:), MP-g(:), Total-Blue 
 
 We explore different assumptions on the system bandwidth allocation. In one 
case we fix f1 at 10 Hz, and vary f2 to find the optimal regions of operation for 
different values f2.  As the distribution of the signal shifts to a higher frequency, the 
signal plus noise level (represented by blue horizontal lines) goes down. For a given 
signal power constraint the noise is lesser, so we see an increase in the capacity. As f
2
 
varies from 1 kHz to 10 MHz, we find an increase in information rate by a factor of 
104 dB! The optimal region and the variation in the information transfer rates are 
plotted in Fig4.9. These optimal signal allocations result in the system capacity shown 
in Fig4.10 reading the frequency axis from left to right. With f1 fixed the capacity 
increases as we increase f2, and with f2 fixed the capacity increases as we decrease 
f1.  The signal level variation with variation in f1 is not as high as that of that 
variation with f2. We find the optimal region of operation of this circuit in terms of 
 
41
information capacity to be around the second pole, which is quite contrary to its 
regular usage below the first pole!  
 
Fig 4.9 Ideal distribution of signal power for constant f1 and varying f2 
Horizontal lines represent the signal+noise level and the vertical lines represent brick 
wall filters. 
 
 
Fig 4.10 Information transfer rates as a function of frequency 
Fig 4.11 is a 3-dimensional plot of the maximum transfer rates as a function of bias 
current and total signal power. Noise is inversely proportional to square root of the 
 
42
bias current and hence reduces with an increase in the bias current. An increase in 
signal power causes an increase in the signal-to-noise ratio. Thus capacity increases 
with an increase in signal power and bias currents. 
 
Fig 4.11 Capacity as a function of bias current and signal power 
 (1?A < ?
b
 <10?A         1e-7 V
2
 < P
0
 < 1e-6V
2
)
 
 
We assume the crossover point between the sub-threshold and above threshold 
regions to be current levels of 0.1?A for a square device. Modifying our model to 
incorporate the sub-threshold behavior by changing the ? in the noise model and the 
g
m 
calculation for the device, we compute the capacity of the OTA as a function of 
signal power and bias current. Fig 4.12 shows that the trend of increasing capacity 
with increasing bias current and signal power remains unchanged in the sub-threshold 
region as well. 
 Our next experiment is to reduce the signal power levels by an order of 
magnitude from 10e-7 to 10e-8 while keeping the bias current levels same as shown 
in Fig 4.13. 
 
 
43
 
Fig 4.12 Capacity as a function of bias current and signal power 
(100nA < ?
b
 <1?A         1e-7 V
2
 < P
0
 < 1e-6V
2
 
 
Fig 4.13 Capacity as a function of bias current and signal power 
   (1?A < ?
b
 <20?A   1e-8 V
2
 < P
0
 < 1e-7V
2)
 
We verify that capacity increases with increase in signal power and bias 
current across a wide spectrum of power levels and current levels. In practical systems 
power levels less than -50dBm are difficult to discern simply because the 
environmental noise is higher by almost an order of magnitude.  The interesting 
 
44
regions are current levels between 1nA and 20 ?A and power levels between 100 ?V 
and a few mV.  
4.3.2 Custom Amplifier 
 
Fig 4.14 Bio-amplifier 
We have designed a custom VLSI amplifier based on the OTA discussed in chapter3.  
This amplifier has been fabricated in a commercially available 0.5?m technology 
through MOSIS. The amplifier is an OTA in capacitive feedback configuration. This 
configuration requires a large feedback resistance, which can be achieved using back 
to back diode connected PMOS devices [9] or using a MOSFET controlled by a bias 
voltage [10]. For the bio-amplifier design, considering the feedback network 
consisting of capacitors C
1
 and C
2
 to be noiseless [9], the input referred noise of the 
circuit is given as  
2212
1
*
in
inAmp inOTA
CCC
VV
C
??++
=
??
??
                                                                                    (12) 
For a typical midband gain of 100, the input referred noise is almost the same as that 
of the OTA. Therefore, the optimal region of operation of this circuit is close to that 
of the OTA. 
 
 
45
4.4 Experimental Results 
The experimental data were collected from the custom amplifier described in the 
previous section.  We made a test fixture with the IC mounted on a vector board. The 
output of the amplifier feeds into a voltage follower amplifier made from discrete 
components (Texas Instruments TL074). To prevent environmental noise from 
affecting the measurements, the vector board was placed in a die cast aluminum box. 
 Transfer function and noise measurements on the OTA based amplifier were 
done using the HP 4395A (10Hz ? 500 MHz) spectrum/network analyzer. The 
arrangement is shown in the Fig 4.15. The spectrum /network analyzer was operated 
in the spectrum analyzer mode and the input to the amplifier was grounded for the 
noise measurements. The bias current was sourced through a Keithley 236. Our 
fabricated IC had amplifiers with gains of 20, 40, and 100. We performed the noise 
measurements for all the amplifiers with bias currents ranging from 10 ?A to 20 ?A.  
 
 
 
 
Fig 4.15 (a) Block diagram of measurement set up 
 
46
 
Fig 4.15 (b) Actual set up for measurements 
We ground the inputs and measure the noise at the output. Fig 4.16 is the measured 
noise at the output for the amplifier of gain 100 and different current levels. Notice 
that at low frequencies the noise is lowest for lowest current (10?A). Next we perform 
the transfer function measurements. The spectrum analyzer was used in the network 
analyzer mode and the input was sourced through it. The measured transfer function 
for gain of 100 and different current levels is shown in Fig 4.17. 
 The input referred noise is determined by dividing the noise at the output by 
the transfer function of the circuit. Using numerical methods, we fit the data to the 
closest curve. The measured and curve fitted data are shown in Fig 4.18 
 
 
 
47
 
Fig 4.16 Measured noise at the output. 
Blue- 10?A, Red - 15?A, Green-20?A 
 
 
Fig 4.17 Measured transfer functions 
Blue- 10?A, Red - 15?A, Green-20?A 
 
The next step is to apply the water filling model to this data. Our circuit?s bandwidth 
is only around 2 kHz for a midband gain of 100. The noise minima is in kHz range. 
We keep the lower cut off frequency fixed at 100 Hz and shift the higher cut off 
frequency from 750Hz to 100 kHz. We fix 100 kHz as the upper limit since the gain 
becomes negligible beyond this frequency. 
 
48
         
Fig 4.18 Input Referred Noise (blue- actual data, red- curve fitted data) for bias 
current of 20uA and gain of 40dB 
 
For a given signal power level of 300?V and bias currents of 10?A-20?A , the 
capacity increases as we shift to the higher frequency ranges as shown in Fig 4.19. 
This is in compliance with the prediction of our model. We observe from Fig 4.17 that 
for lower frequencies the noise at the output is lowest for 10?A. The transfer 
functions are almost the same for all the three currents levels measured as shown in 
Fig 4.18. As we move to the higher frequencies, the noise at the output is the same for 
all the currents (Fig 4.17), but the transfer function is highest for current of 20?A (Fig 
4.18). This explains why the capacity for 10?A is highest at lower frequencies and for 
20 ?A is highest at higher frequencies. The cross over point is around 10 kHz.  
 This is in accordance with the fact that higher currents give more gain, more 
noise at the output, less input referred noise and hence more capacity but at the cost of 
higher static power dissipation. 
 
49
  
 Fig 4.19 Variation of capacity with f2 shifting to higher frequencies. 
Blue- 10?A, Red - 15?A, Green-20?A 
 
 
 
 
 
 
                
 
 
 
Clockwise from top 
10?A , 15?A and 20?A 
Model 
Data 
Fig 4.20 Comparison of experimental data with model for capacity variation with 
changing f2. 
 
50
4.5 Discussion 
We compare the noise predicted by our model to the noise of the OTA device. While 
there are obvious quantitative differences between the model and the data, there are 
striking qualitative similarities as shown in Fig 4.20. 
 The channel capacity given by Equation (2) is an upper bound on the rate of 
information transmission, assuming that the signal is limited only in average power 
and the noise is normally distributed. Bandwidth limitations alone do not limit 
information transmission; a bandwidth limitation which affects signal and noise 
equally does not affect the channel capacity. Capacity can be increased arbitrarily by 
increasing the signal power; we must specify how the signal power is constrained 
such that Equation (2) is meaningful.  
 
4.6 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. 
 Noise limits information transmission in practical analog systems and sets 
lower bound on power. Water-filling algorithm gives efficient distribution of signal 
power in a noisy channel. Given noise and transfer characteristics we compute 
optimal information rates for the linearized OTA. This also specifies the most 
efficient way to use the channel. We conclude that operating the OTA at higher 
frequency where the noise is less increases the capacity of the system considerably. 
 51
 
 
 
Chapter 5 
Bit-Energy  
The minimum cost incurred by the transmission of one bit of information through the 
channel is a fundamental figure of merit characterizing the most economical way to 
communicate reliably. Its reciprocal, the capacity per unit cost, is defined similar to the 
conventional capacity, except that the ratio of the logarithm of the number of the 
codewords to their block length (rate) is replaced by the ratio of the logarithm of the 
number of codewords to the their cost (rate per unit cost). 
As shown by Verdu [24], the capacity per unit cost can be computed from the capacity ?
cost function by finding
0
sup ( ) /C
?
? ?
>
, or alternatively, as 
 
(;)
sup
[[ ]]
X
IXY
C
E bX
=                                                      (1) 
 For any system with a power constraint the capacity is a measure of the maximum 
information transfer rate between the input and output [2]. This holds if we consider our 
system as a zero cost system i.e. we say that there is no cost involved in achieving these 
rates. But in the real world there is a price for everything. We introduce this cost in our 
analysis. The bit-energy gives the minimum power required to transmit a single bit of 
information through the system; it provides an appropriate standard for comparing the 
efficiency of communication among different technologies. Besides the power constraint 
 52
on the input of a system, there is an implicit cost i.e. the static power dissipation required 
by the system to operate. For the OTA it is a function of the bias currents and voltage 
requirements of the circuit.  
 We compute the capacity per unit power of input signal (
sig
C
P
 ) and the capacity 
per unit total power budget of the system i.e. the sum of power constraint on the input 
signal and the static power dissipation (
sig STATIC
C
PP+
). It is important to note here that our 
capacity analysis has the both these power constraints built in. The dependence of the 
input referred noise on the bias current level introduces the static power dissipation and 
the signal to noise ratio has the input power constraint. 
 
5.1 Capacity per unit signal power 
Capacity is computed as log 1 *
sig
P
CF
N
??
=+
??
??
   where P
sig
 is the signal power constraint, N 
is the noise spectral density and F is the bandwidth. 
sig
C
P
 is the capacity per unit  signal power . It is the actual cost in terms of signal 
power for maximum information transfer; 
(;)
max
sig sig
CIXY
PP
=
. This sets the limits on the 
signal power for achieving the maximum capacity. The maximum information transfer 
rate is determined by the water filling approach and is a function of P
sig
 and the I
bias
. So 
our actual cost metric is
(, )
sig bias
sig
CP I
P
. As the noise power reduces with the increase in bias 
current, the capacity per unit power goes up asymptotically as a logarithmic function of 
 53
bias current. As the signal power is increased, the capacity per unit signal power goes 
down as a logarithmic function of signal power. This is shown in Fig 5.1. 
 
Fig 5.1 Capacity per unit Signal Power 
 
5.2 Capacity per unit total power 
If we want to quantify the total power constraint for the system, we should also take into 
account the power needed to bias the system i.e. the static power dissipation. The static 
power dissipation is determined by the total power needed to operate the system and the 
biasing level. For the OTA it is given as **(2 )
static dd bias
PVI K= +   where, K is the current 
mirror gain factor.  
 Our metric now is 
(, )
sig bias
sig static
CP I
PP+
 where P
sig 
+ P
static
  is the total power budget of the 
system. As we increase the signal power keeping the current level fixed, we notice an 
increase in the capacity per unit total power. Alternatively, as we increase the bias current 
keeping the signal power constant, the capacity per unit power decreases. 
 54
 
Fig 5.2 Capacity per unit Total Power 
 
5.3 Conclusions 
 
I
bias
 , P
sig 
 
Capacity 
 
Capacity/P
sig
 
 
Capacity/P
sig
+P
static 
I
bias 
increases,  
P
sig
 constant 
Increases Increases Decreases 
P
sig
 increases, 
I
bias
 constant 
Increases Decreases Increases  
 
Simulations 
 
 
 
 
 
Fig 5.3 Variation of Capacity Metrics with Power constraints 
 
The analysis for maximum capacity has been extended to cover the cost incurred in 
achieving that capacity. An increase in bias current reduces the noise and hence increases 
 55
the capacity but at the cost of higher static power dissipation. An increase in the power 
content of the signal increases the capacity. 
 The given power budget is the deciding factor. If there is a constraint on the input 
signal power level, increase the bias current to increase capacity. Alternatively, if there is 
a constraint on the maximum power available to the circuit for operation then signal 
power has to be boosted for improving the capacity. 
 56
 
 
 
Chapter 6 
Noise Efficiency Factor 
Noise is a limiting factor in improving the capacity of a system. Steyaert et al [23] 
proposed a metric for comparing the noise performance of analog amplifiers called the 
Noise Efficiency Factor (NEF). The noise performance of the circuit is compared to the 
noise performance of a simple bipolar transistor considering only thermal noise and no 
base resistance [13]. In line with the definition of NEF, we determine the capacity of a 
bipolar transistor (C
NEF
). Next, we compute the capacity of OTA (C
OTA
) considering only 
thermal noise [9] since it is the dominant noise source in the circuit and compare it with 
that of the bipolar transistor. 
 
6.1 Noise Efficiency Factor for a bipolar transistor 
The NEF of the system is defined as the ratio of total input referred noise of the circuit to 
the total input referred noise of the bipolar transistor. The NEF describes how many times 
the noise of a given system with the same drain current and bandwidth is higher 
compared to the ideal case of a simple bipolar transistor. The total equivalent input noise 
of an ideal bipolar transistor (only thermal noise and no base resistance) is given by [13] 
 57
,
4* *4* *
** **
22
T
rms in
C
kT UkT
VBW BW
gm I
??
==                                     (1) 
where, BW for the bipolar transistor this is the unity gain bandwidth f
t
.  
The definition of NEF simplifies to, 
,
2*
**4*
tot
rms in
T
I
NEF V
UkTBW?
=                                        (2) 
where, I
tot
 is the total drain current in the system and V
rms,in
 is the total equivalent input 
noise of the circuit. 
The total thermal noise for the OTA circuit is given as [9]  
2 37
,
111
16
12
3
mm
thermal in
mmm
ggkT
Vf
ggg
??
??
=++?
????
??
??
                         (3) 
Substituting (3) in (2) and with g
m3
, g
m7
 << g
m1, 
we find 
1
11
4 16
33
tot D
Tm T m
I I
NEF
Ug U g
??
==
??
??
                                            (4) 
 where, I
D1
 is the drain current through M
1
  which is ? of the total amplifier supply 
current.  
 
6.2 Capacity of BJT and OTA 
The capacity for a continuous band limited system is given by [22] 
 
2
*lo g 1
sig
in
P
CBW
V
??
=+
??
??
                                        (5) 
For a bipolar transistor BW = 
2( )
m
t
g
f
CC
??
?
=
+
                                                                (6) 
Substituting V
rms,in 
 from (1)  and BW from (6) in (5) we get 
 58
()
log 1
2( ) 2
sig
bias
T
PC C
I
C
UC C kT
??
??
?
+??
=+
??
+
??
                                            (7) 
We compute the capacity as a function of signal power and bias currents as shown in Fig 
6.1 and the capacity of the OTA with only thermal noise and without applying the water 
filling optimization as shown in Fig 6.2. 
 
Fig 6.1 Capacity of a simple Bipolar Transistor 
 
 
Fig 6.2 Capacity of OTA with Thermal noise 
 59
6.3 Conclusion 
The capacity of the BJT and the OTA under similar constraints of bandwidth and total 
current, are plotted on the same scale in Fig 6.3. Since the noise performance of the BJT 
is better than the OTA, the capacity of the BJT is higher than the capacity of the OTA. 
This is very evident from Fig 6.3.  
 
Fig 6.3 Capacity of BJT and Capacity of OTA as a function of signal power and bias 
current. The upper surface is the BJT and the lower surface is the OTA. 
 
 
Fig 6.4 C
OTA
/C 
BJT 
 as a function of bias current and signal power 
 60
Fig 6.4 shows how the ratio of the capacities of the OTA and BJT vary with changing 
signal power and current levels. As we increase the signal power, the capacity of the 
OTA increases faster than the capacity of the BJT. Increasing the bias current shows the 
reverse trend; the capacity of the OTA reduces slower than the capacity of the BJT.  
 We can conclude that at lower current levels, the capacity of the OTA will be 
closer to the capacity of the BJT. In other words, to achieve the noise performance of the 
BJT, we need to reduce the bias current and shift the operation in the sub-threshold 
domain. This is in agreement with the NEF comparison of OTA with BJT in [9]. 
Operating some of the devices in OTA in the sub?threshold region is one way of 
improving the noise performance of the system. 
 
 
 61
 
 
 
Chapter 7 
Conclusion 
The focus of this work is to investigate the information efficiency of an OTA based 
amplification system. We apply principles of information theory to analog circuits in an 
effort to optimize their performance. The specific application is to design efficient sensor 
systems for neural recordings that are able to transfer maximum information between 
input and output. We start with treating the amplifier system as a channel for transmitting 
information. We model the system as a Gaussian channel with power constrained input 
signal and corruption due to noise and determine the capacity of the system.  
 Chapter 1 describes the experiments to record the electrical activity of living cells. 
These experiments give us the actual power content of the electrical signals that we get 
from the cells. We aim to design amplifiers according to the measured spike spectrum 
and power levels. In Chapter 3 we describe the OTA circuit and the various 
characterization parameters. We introduce the noise sources and discuss their 
representation in a circuit. Chapter 4 presents a detailed description of the analytical 
model for determining the information capacity of the OTA.  We employ the water filling 
algorithm to determine the spectral region where the capacity can be maximized. 
Capacity increases with increasing bias current and signal power levels. As we shift the 
 62
bandwidth allocation to the higher frequencies where noise is minimal, there is an 
increase in capacity. We measure the noise at the output and the transfer function from 
the fabricated device. We apply our model to the actual data and perform comparisons 
between the results from the data and the model. We observe that the measurements 
follow the trend the model predicts; at higher frequencies where noise is minimal, 
capacity of the system is increased. In chapter 5, we introduce power efficiency in our 
analysis. The cost for achieving the desired capacity is two-fold; the power constraint on 
the signal and the static power dissipation in the circuit. We contrast capacity per unit 
signal power with the capacity per unit total power. Noise efficiency factor has been used 
to compare the noise performance of amplifiers. A simple bipolar transistor has the 
lowest NEF. We compute the capacity of a bipolar transistor under the same power 
constraints and similar biasing requirements as our circuit. Operating the circuits at lower 
current levels reduces the gap between the noise performance of the OTA and the bipolar 
transistor. 
 This work is the first step towards maximizing the information transfer in 
amplifier systems for neural recordings. Applying principles of information theory to 
characterize the system as a channel, we determine the channel capacity of the OTA. This 
approach can be developed into an engineering design tool for designing efficient 
systems. This analysis provides us information about the frequency bandwidth for signal 
allocation, the cost in terms of signal power constraint as well as the static power and bias 
current levels for reducing the noise in the system.  
 
 
 63
7.1 Future Work 
7.1.1 Encoding/Decoding Scheme 
The findings of this analysis are quite contrary to the regular designs of amplification and 
filtering circuits. These amplifiers are usually operated at frequencies near their first pole. 
But to get maximum capacity, they should be operated at much higher frequencies. These 
frequencies might be outside the useful bandwidth of these amplifiers.  This brings up 
another challenging area of research. Is it feasible to operate these amplifiers at these 
higher frequencies? If we could shift the spectral content of the signal to the higher 
frequency without affecting the noise spectral density distribution then we could achieve 
higher signal to noise ratios. In the case where the spectral content of the signal cannot be 
varied then even a shift to the frequency range where noise level is a minimum gives rise 
to an increase in SNR leading to an increase in the information transfer rate. The cost 
involved in this process is the reduction in the gain of the system. Fig 7.1 is an illustration 
of this observation.  
 
  (a)                                     (b)                                        (c)                              
Fig 7.1 (a) A system with input signal whose frequency spectrum cannot be altered  
(b) System used for maximum gain (c) System used for maximum capacity 
 
 64
This puts forth the need for an encoding/decoding scheme to shift the input signal to a 
higher spectrum and reconstruct the original signal at the output. The over-head involved 
in introducing these intermediate stages in terms of power requirement and the noise 
performance has to be carefully analyzed. 
 
7.1.2 Range of the Analytical Model 
In chapter 4 we present results for the capacity computation for the OTA for the above 
threshold and below threshold operation. There is a discontinuity in this crossover as seen 
is Fig 7.2, since our model uses different parameters in the above threshold and sub 
threshold case. To better understand and determine the effect of subthreshold and above 
threshold operation, we need to have a continuous  model applicable to both regions. 
 
Fig7.2 Crossover between sub-threshold and above threshold. The first 5 points are 
subthreshold and the next 10 points are above threshold 
 
 65
My study of efficient communication has been primarily limited to fundamental noise 
sources. In different contexts, however, what we mean by ?noise? and ?signal? changes, 
and in many applications the dominant noise sources are practical rather than 
fundamental. These practical noise sources must be characterized empirically, and 
included in the analysis.  
 The focus of this work has been the quantification of information and 
information?power efficiency for Operational Transconductance Amplifier. I chose 
channel capacity and bit-energy as performance measures for this research. While these  
measures are fundamental and physically-motivated, they are not representative of all 
design constraints and tradeoffs, and further work is necessary to transition this approach 
into a practical design tool. 
 
 66
 
 
 
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