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WOODS HOLE 
OCEANOGRAPHIC INSTITUTK)N 

LABORATORY 
BOOK COLLECTION 



/ :a I 7 

Purchat© Orier No ....-«^----— 



Circuit Theory 

of Linear Noisy Networks 



TECHNOLOGY PRESS RESEARCH MONOGRAPHS 



Nonlinear Problems in Random Theory 
By Norbert Wiener 

Circuit Theory of Linear Noisy Networks 
By Hermann A. Haus and Richard B. Adler 



HERMANN A. HAUS 
RICHARD B. ADLER 

Associate Professors of Electrical Engineering 
Massachusetts Institute of Technology 



Circuit Theory 

of Linear Noisy Netv^orks 



BIOLOGICAL 
LABORATORY 

lTbrary 



PuhUshd jointly hy [wOODS HOLE, MASS. 

The Technology Press of j W. H. 0. 1. 

The Massachusetts Institute of Technology 

and 
John Wiley & Sons, Inc., New York 



Chapman and Hall, Limited, London 



Copyright © 1959 

by 

The Massachusetts Institute of Technology 



All Rights Reserved 

This book or any part thereof must not 
be reproduced in any form without the 
written permission of the publisher. 



Library of Congress Catalog Card Number: 59-11473 
Printed in the United States of America 



Fore^vord 



There has long been a need in science and engineering for systematic 
publication of research studies larger in scope than a journal article but 
less ambitious than a finished book. Much valuable work of this kind 
is now published only in a semiprivate way, perhaps as a laboratory 
report, and so may not find its proper place in the literature of the field. 
The present contribution is the second of the Technology Press Research 
Monographs, which we hope will make selected timely and important 
research studies readily accessible to libraries and to the independent 
worker. 

J. A. Stratton 



Preface 



Monographs usually present scholarly summaries of a well-developed 
field. In keeping with the philosophy of the new series of Research 
Monographs, however, this monograph was written to present a piece of 
relatively recent work in a comparatively undeveloped field. Such work 
might normally be expected to appear in a series of journal articles, and 
indeed originally the authors followed this method of presentation. As 
the subject developed, however, a rather general approach to the problem 
became apparent which both simplified and unified all the prior research. 
Space limitations in the journals made it impossible to publish in that 
medium a really suitable picture of the whole development, and this 
circumstance led the authors to take advantage of the present Technology 
Press Research Monographs. 

The principal motivation for this work arose from the obvious desir- 
ability of finding a single quantity, a tag so to speak, to describe the 
noise performance of a two-terminal-pair amplifier. The possibility of 
the existence of such a quantity and even the general functional form 
which it might be expected to take were suggested by previous work of 
one of the authors on microwave tubes and their noise performance. 
This work showed that noise parameters of the electron beam set an 
ultimate Hmit to the entire noise performance of the amplifier that 
employed the beam. In the microwave tube case, however, the findings 
were based heavily upon the physical nature of the electron beam, and 
it was not immediately clear that a general theory of noise performance 



viii PREFACE 

for any linear amplifier could be made without referring again to some 
detailed physical mechanism. In order to detach the study of noise 
performance from specific physical mechanisms, one had to have recourse 
to general circuit theory of active networks. Such a theory had grown 
up around the problems associated with transistor amplifiers, and im- 
portant parts of it were available to us through the association of one of 
us with Professor S. J. Mason. This combination of circumstances led 
to the collaboration of the authors. 

Two major guiding principles, or clues, could be drawn from the 
experience on microwave tubes. One such clue was the general form of 
the probable appropriate noise parameter. The other was the recog- 
nition that matrix algebra and a proper eigenvalue formulation would be 
required in order to achieve a general theory without becoming hope- 
lessly involved in algebraic detail. 

Essentially by trial and error, guided by some power-gain theorems in 
active circuit theory, we first found a few invariants of noisy networks. 
Afterward, while we were trying to decide around which quantities we 
should build a matrix-eigenvalue formulation leading to these same 
invariants, we were aided by the fact that Mr. D. L. Bobroff recognized 
a connection between the invariants which we had found and the problem 
of the available power of a multiterminal-pair network. 

Armed with this additional idea, we consulted extensively with Profes- 
sor L. N. Howard of the Massachusetts Institute of Technology, Depart- 
ment of Mathematics, in search of the appropriate matrix-eigenvalue 
problem. As a result of his suggestions, we were able to reach substan- 
tially the final form of the desired formulation. 

Once the proper eigenvalue approach was found, additional results 
and interpretations followed rapidly. In particular, the idea that the 
eigenvalue formulation should be associated with a canonical form 
of the noisy network was suggested in a conversation with Professor 
Shannon. 

One of the principal results of the work is that it furnishes a single 
number, or tag, which may be said to characterize the amplifier noise 
performance on the basis of the signal-to-noise-ratio criterion. The novel 
features of this tag are two in number: First, it clears up questions of 
the noise performance of low-gain amplifiers or of the effect upon noise 
performance of degenerative feedback; second, it provides for the first 
time a systematic treatment of the noise performance of negative-resist- 
ance amplifiers. The latter results were not expected in the original 
motivation for the study but grew from insistent demands upon the 
internal consistency of the theory. It is interesting that the negative- 
resistance case will probably turn out to be one of the most important 
practical results of our work. 



PREFACE ix 

Another result worth mentioning here, however, is the canonical form 
of linear noisy networks. This form summarizes in a clear, almost visual, 
manner the connection between the internal noise of a network at any 
particular frequency and its (resistive, positive, or negative) part. 

We are hopeful that this second work in the series of Technology Press 
Research Monographs will meet the standards and aims envisioned by 
Professor Gordon S. Brown, whose personal inspiration and energetic 
support brought the present volume into existence. 

We wish to express our sincere thanks to Miss Joan Dordoni for the 
careful preparation of the manuscript. We also acknowledge gratefully 
the thorough and exacting editing work of Miss Constance D. Boyd. 

The support in part by the U. S. Army (Signal Corps), the U. S. Air 
Force (Office of Scientific Research, Air Research and Development 
Command), and the U. S. Navy (Office of Naval Research) is also 
acknowledged with gratitude. 

Massachusetts Institute of Technology 
Cambridge, Massachusetts 
January, 1959 



Contents 



FOREWORD 
PREFACE 
CHAPTER 1 
CHAPTER 2 



Introduction 

Linear Noisy Networks in the Impedance Representation 
Impedance-Matrix Representation of Linear Networks 



2.1 
2.2 
2.3 



Lossless Transformations 

Network Classification in Terms of Power 



CHAPTER 3 



Impedance Formulation of the Characteristic-Noise Matrix 

3.1 Matrix Formulation of Stationary- Value Problem 

3.2 Eigenvalue Formulation of Stationary -Value Problem 

3.3 Properties of the Eigenvalues of the Characteristic- 
Noise Matrix in Impedance Form 

3.4 Lossless Reduction in the Number of Terminal Pairs 

CHAPTER 4 Canonical Form of Linear Noisy Networks 

4.1 Derivation of the Canonical Form 

4.2 Interconnection of Linear Noisy Networks 

CHAPTER 5 Linear Noisy Networks in Other Representations 

5.1 General Matrix Representations 

5.2 Transformation from One Matrix Representation to 
Another 



vu 

1 

9 

9 

12 

14 

19 
19 
21 

23 
24 

28 
28 
31 

33 
33 

35 



xt 



act* 



CONTENTS 



5.3 Power Expression and Its Transformation 

5.4 The General Characteristic-Noise Matrix 



CHAPTER 6 Noise Measure 



CHAPTER 7 



6.1 Extended Definitions of Gain and Noise Figure 

6.2 Matrix Formulation of Exchangeable Power and Noise 
Figure 

Noise Measure 

Allowed Ranges of Values of the Noise Measure 
Arbitrary Passive Interconnection of Amplifiers 

Network Realization of Optimum Amplifier Noise Per- 
formance 

7 . 1 Classification of Two-Terminal-Pair Amplifiers 
Optimization of Amplifier, Indefinite Case 
The Optimum Noise-Measure Expression for the Con- 
ventional Low-Frequency Vacuum Tube 
Optimization of Negative-Resistance Amplifiers, 
Definite Case 



6.3 
6.4 
6.5 



7.2 
7.3 

7.4 



CHAPTER 8 Conclusions 



INDEX 



37 
38 

42 
43 

44 
48 
49 

54 

58 
59 
61 

66 

68 
73 

77 




Introduction 



The principal example of a linear noisy network, and the one of greatest 
practical importance in electrical engineering, is a linear noisy amplifier. 
The noise performance of such amplifiers involves many questions of 
interest. One very significant question is the extent to which the amplifier 
influences signal-to-noise ratio over a narrow band (essentially at one 
frequency) in the system of which it is a part. We shall address ourselves 
exclusively to this feature, without intending to suggest that other fea- 
tures of the much larger noise-and-inf ormation problem are less important. 
The term "spot-noise performance" or merely "noise performance" will 
be used to refer to the effect of the amphfier upon the single-frequency 
signal-to-noise ratio. It is essential to emphasize right at the beginning 
the very restricted meaning these terms will have in our discussions. 

We undertook the study reported here in the hope of formulating a 
rational approach to the characterization of amplifier spot-noise per- 
formance, and to its optimization by external circuit operations upon the 
terminals. Fortunately, a characterization has resulted which is based 
on a single hypothesis about the essential function of an amplifier and 
which turns out to avoid pitfalls previously associated with the effect of 
feedback upon noise performance. In developing the aforementioned 
noise characterization of amplifiers and in pursuing the relevant optimi- 
zation problem, we encountered a number of illuminating features relating 
power and noise in linear multi-terminal-pair networks. Indeed, it 
eventually became clear that the major issues could be presented most 
simply by postponing until last the questions we had originally asked 
first. The result is a work of broader scope than was originally envisaged, 

1 



2 INTRODUCTION [Ch. 1 

and one for which the title "Circuit Theory of Linear Noisy Networks" 
seems appropriate. 

Since the introduction by Friis^ and Franz^ of the concept of spot- 
noise figure F for the description of ampUfier noise performance, this 
figure has played an essential role in communication practice. The 
noise figure is, however, merely a man-made definition, rather than a 
quantity deduced from clearly defined postulates or laws of nature. The 
possible consequences of this fact were never questioned deeply, although 
it has always been known that the (spot-) noise figure F does not con- 
stitute a single absolute measure of amplifier noise performance. 

In particular, the noise figure is a function of the impedance of the 
source connected to the amplifier input. Thus in giving an adequate 
conventional description of amplifier noise performance, the source 
impedance, as well as the noise figure, must be specified. 

Usually, when regarded as a function of source impedance alone, the 
noise figure has a minimum value for some particular choice of this 
impedance. If with this source impedance the gain of a given amplifier 
remains sufficiently high, its noise figure will prescribe the noise figure 
of any amphfier cascade in which it is used as the first stage. In this way, 
it is possible to build an amplifier cascade with any desired high gain, 
and with a noise figure set by the minimum (with respect to source 
impedance) of the noise figure of the original amplifier. 

If a cascade is to be composed of several individual amplifiers, each of 
which alone has a "high enough" gain when driven from the source 
impedance that yields its minimum noise figure, the previous argument 
shows that the amplifier with the lowest minimum noise figure should be 
used as the first stage. Any other choice would result in a higher over-aU 
noise figure for the cascade. 

The foregoing discussion seems to suggest that the minimum value 
(with respect to source impedance) of the noise figure of an amplifier may 
be used as an absolute measure of its noise performance and as a basis for 
comparison with other amplifiers. The validity of the argument, how- 
ever, is based upon the two previously mentioned restrictions : 

1. Each stage has ^^high enough"" gain when driven from the "optimum" 
source that yields the minimum noise figure. 

2. Only the source impedance of each stage is varied in controUing the 
noise performance. 

The inadequacy of this viewpoint becomes clear when stage variables 
other than source impedance and stage interconnections other than the 

1 H. T. Friis, "Noise Figure of Radio Receivers," Proc. I.R.E., 32, 419 (1944). 
^ K. Franz, "Messung der Empfangerempfindlichkeit bei kurzen elektrischen 
WeUen," Z. Elektr. Elektroak., 59, 105 (1942). 



Ch. 1\ INTRODUCTION 3 

simple cascade become important in amplifier applications. The question 
of the quality of noise performance then becomes much more complicated. 
For example, when degenerative feedback is applied to an amplifier, its 
noise figure can be reduced to as close to unity as desired (for example, 
bypassing the entire amplifier with short circuits yields unit noise figure). 
But its gain is also reduced in the process. Indeed, if identical stages 
with the feedback are cascaded to recover the original single-stage gain 
before feedback, the resulting noise figure of the cascade cannot be less 
than that of the original amplifier.^ Moreover, with degenerative feed- 
back the gain may easily be so greatly reduced that, as a first stage in a 
cascade, this amplifier alone no longer determines the over-all noise 
figure of the cascade. The minimum-noise-figure criterion considered 
above as a measure of amplifier noise performance breaks down. It 
appears that an absolute measure of amplifier noise performance must 
include, in addition to the specification of noise figure and source im- 
pedance, at least the specification of the gain. 

The foregoing reasoning led us to the investigation presented in this 
study. Taking our clues from the results previously found by Haus and 
Robinson^ for microwave amplifiers, and the method of active-network 
description presented by Mason,^ we searched for a measure of amplifier 
noise performance that would not only include the gain explicitly, as 
discussed earlier, but could also be minimized by external circuitry in a 
nontrivial way. Moreover, we believed that the minimum thus obtained 
should be a quantity characteristic of the amplifier itself. It should, for 
example, be invariant under lossless feedback, a type of feedback that 
does not appear to change the essential "noisy" character of the amplifier 
because it certainly adds no noise and can always be removed again by 
a realizable inverse lossless operation. 

The precise form of a suitable noise-performance criterion has actually 
been known for many years, although its deeper significance somehow 
escaped attention. Indeed, the most glaring example of the correct 
criterion arises from the familiar problem of cascading two (or more) 
low-gain amplifiers having different noise figures Fi and F2 and different 
available gains Gi ( > 1 ) and G2 ( > 1 ) . 

The question is : If the available gain and noise figure of each amplifier 
do not change when the order of cascading is reversed, which cascade 
order leads to the best noise performance for the pair? Usually, "best 
noise performance" has been taken to mean "lowest noise figure" for the 

' A. van der Ziel, Noise, Prentice-Hall, New York (1954). 

* H. A. Haus and F. N. H. Robinson, "The Minimum Noise Figure of Microwave 
Beam Amplifiers," Proc. I.R.E., 43, 981 (1955). 

^ S. J. Mason, "Power Gain in Feedback Amplifiers," Trans. IRE, Professional 
Group on Circuit Theory, CT-1, No. 2, 20 (1954). 



4 INTRODUCTION [Ch. 1 

pair, though in view of the answer obtained on that basis, the criterion 
should have been viewed with a little suspicion. Thus if F12 and F21 
are the respective noise figures of the cascade when amplifier No. 1 
and ampHfier No. 2 are placed first, we have 



F. - 1 



■t'12 - -I'l -r ^ 
^1 




F F ^^'-^ 

^2 




The condition that F12 be less than F21 is 




^2 


i^2-l 
Gl 


or 

{F, - 1) - {F2 - 1)< ^^ - 


F2- 1 
Gl 


or 




F,-\ ^F2-l 




Gi G2 





(1.1a) 



(1.2) 



That is, amplifier No. 1 should come first if Eq. 1.2 is satisfied. 

Equation 1.2 implies that in a cascaded system of amplifiers, where the 
earliest stages are obviously the most critical in regard to noise perform- 
ance, the "best" amplifier is the one having the lowest value not of F 
but of the quantity 

F - 1 
M = ^--j- (1.3) 

'-G 

It is with M that we shall be most concerned, and we shall call it the Noise 
Measure of an amplifier. 

In terms of M, and the fact that the available gain of a cascaded pair 
of amplifiers is G = G1G2, Eq. 1.1a becomes 



^n = ^. + f^(— 4M=M. + AM(|fi) (1.4) 




Ch. 1] INTRODUCTION 5 

where AM = M2 — Mi is the difference between the noise measures of 
the second and first amplifiers of the cascade. Equation 1.4 shows that as 
long as Gi and G2 are greater than 1 the noise measure of a cascade of 
two amplifiers Kes between the noise measures of its component amplifiers. 
In the particular case when the noise measures of the two amphfiers are 
equal, the resulting noise measure of the cascade is that of either amplifier, 
even if the available gains of the individual amplifiers are different. 

Furthermore, since the available gain G = G1G2 is supposed to remain 
the same for either order of cascading, the result (Eq. 1.2) and the defini- 
tion (Eq. 1.3) show that the lowest noise measure for a cascaded pair of 
amphfiers results from placing at the input the amplifier with the lowest 
individual noise measure. 

Compared with the noise figure alone, which always deteriorates in a 
cascade (Eqs. 1.1) and which does not suffice to determine which amplifier 
should come first, the noise measure alone is evidently a more satisfactory 
and self -consistent single criterion of amplifier noise performance. 
Moreover, since noise measure and noise figure become essentially the 
same for amphfiers with sufficiently high gain, the final performance 
evaluation of a practical multistage amphfier always rests numerically 
(if not in principle) upon the familiar noise-figure criterion. 

From such reasoning, we evolved a criterion for amplifier noise per- 
formance. The criterion is based on the plausible premise that, basically, 
amphfiers are supposed to provide "gain building blocks" without adding 
excessively to system noise. In its final stage of evolution, the criterion 
can be described as follows. 

Suppose that n different types of amplifiers are compared. An un- 
hmited number of amphfiers of each type is assumed to be available. 
A general lossless (possibly nonreciprocal) interconnection of an arbitrary 
number of amphfiers of each type is then visuahzed, with terminals so 
arranged that in each case an over-all two-terminal-pair network is achieved. 
For each amplifier type, both the lossless interconnecting network and the 
number of amplifiers are varied in all possible ways to produce two 
conditions simultaneously: 

1. A very high available gain (approaching infinity) for the over-all 
two-terminal-pair system when driven from a source having a positive 
real internal impedance. 

2. An absolute minimum noise figure Fmin for the resulting high- 
gain system. 

The value of (Fmin — 1) for the resulting high-gain two-terminal-pair net- 
work is taken specifically as the ^^ measure of quality of the amplifier type 
in each case. The "besf amplifier type will be the one yielding the smallest 
value of (Fmin — I) at very high gain. 



6 INTRODUCTION [Ch. 1 

The proof of this criterion will be developed through the concept ot 
noise measure. Inasmuch as the general criterion involves (at least) 
arbitrary lossless interconnections of amplifiers, including feedback, 
input mismatch, and so forth, a rather general approach to the noise 
measure is required. In particular, we must show that the noise measure 
has a real significance of its own which is quite different from and much 
deeper than the one suggested by its appearance in Eq. 1.2. There it 
appears only as an algebraic combination of noise figure and available 
gain that happens to be convenient for describing amplifier cascades. 
Here the properties of M with regard to lossless transformations are be- 
coming involved. 

Consideration of these properties brings us into the entire general 
subject of external network transformations of noisy Hnear networks. 
Among these, lossless transformations form a group in the mathematical 
sense. The quantities invariant under the group transformations must 
have a physical significance. Investigation of these invariants forms a 
substantial part of the present study. To be sure, for the special case of 
a two-terminal-pair amplifier, the optimum noise performance, through 
its related noise measure, turns out to be one of the invariants; but several 
other interpretations of the invariants prove equally interesting, and 
the development of the entire subject is simplified by presenting them 
first. 

The simplest formulation and interpretation of the invariants of a 
linear noisy network result from its impedance representation. The 
following chapter is therefore devoted to a discussion of network trans- 
formations, or "imbeddings," in terms of the impedance-matrix repre- 
sentation. The concept of exchangeable power as an extension of 
available power is then introduced. 

In Chapter 3, the n invariants of a Hnear noisy w-terminal-pair network 
are found as extrema of its exchangeable power, with respect to var- 
iations of a lossless w-to-one-terminal-pair network transformation. It 
is found that an w-terminal-pair network possesses not more than these 
n invariants with respect to lossless w-to-w-terminal-pair transformations. 
These n invariants are then exhibited in a particularly appealing way in 
the canonical form of the network, achievable by lossless transformations 
and characterized by exactly n parameters. This form is introduced in 
Chapter 4. 

Through Chapter 4, the invariants are interpreted only in terms of the 
extrema of the exchangeable power. New interpretations are considered 
next. They are best introduced by using other than the impedance- 
matrix description. Accordingly, in Chapter 5, general matrix representa- 
tions are studied, where it is pointed out that usually a different 



Ch. J] INTRODUCTION 7 

matrix description leads to a different interpretation of the invariants. 

In the case of an active two-terminal-pair network, a particularly- 
important interpretation of the invariants is brought out by the general- 
circuit-parameter-matrix description. This interpretation relates directly 
to the optimum "noise measure" of the network used as an amplifier and, 
therefore, to the minimum noise figure of the amplifier at arbitrarily high 
gain. Chapter 6 is devoted to this noise-measure concept and to the 
range of values that the noise measure may assume for a two-terminal- 
pair ampKfier subjected to arbitrary passive network transformations. 
In particular, the minimum value of the noise measure of the amplifier is 
found to be directly proportional to one of the two invariants of the 
amplifier. 

A study is made of those arbitrary passive interconnections of two- 
terminal-pair amplifiers which result in an over-all two-terminal-pair 
amplifier. The conclusion is that the noise measure of the composite 
amplifier cannot be smaller than the optimum noise measure of the best 
component ampHfier, namely, the amplifier with the smallest optimum 
noise measure. 

The general theorems having established the existence of an optimum 
value of the noise measure of amplifiers, it remains in Chapter 7 to discuss 
in detail the network realization of this optimum for two-terminal-pair 
amplifiers. Some practical ways of achieving it are presented. Among 
these, the realization of optimum noise performance for a maser may be 
of greatest current interest. 

With proof of the existence and realizability of a lower limit on the 
noise measure, and therefore of the noise figure at high gain, the major 
objective of the present work is accomplished. It is demonstrated that 
the quality with regard to noise performance of a two-terminal-pair 
amplifier can be specified in terms of a single number that includes the 
gain and that applies adequately to low-gain amplifiers. 

We have previously published various separate discussions of some of 
these topics in different contexts.®~^° Each of these discussions has 
suffered from unnecessary complications because space limitations forced 

® H. A. Haus and R. B. Adler, "Invariants of Linear Networks," 19S6 IRE Con- 
vention Record, Part 2, 53 (1956). 

^ H. A. Haus and R. B. Adler, "Limitations des performances de bruit des ampli- 
ficateurs lineaires," L'Onde Electrique, 38, 380 (1958). 

^ H. A. Haus and R. B. Adler, "Optimiim Noise Performance of Linear Amplifiers," 
Proc. I.R.E., 46, 1517 (1958). 

^ R. B. Adler and H. A. Haus, "Network Realization of Optimum Amplifier Noise 
Performance," IRE Trans, on Circuit Theory, CT-5, No. 3, 156 (1958). 

^"H. A. Haus and R. B. Adler, "Canonical Form of Linear Noisy Networks," 
IRE Trans, on Circuit Theory, CT-5, No. 3, 161 (1958). 



8 INTRODUCTION [Ch. 1 

them to be divorced from each other. It seemed, therefore, desirable to 
present the entire picture at greater leisure, particularly because the 
mathematical and logical complexity of the whole subject is thereby 
actually reduced. 




Linear Noisy Networks 

m the Impedance Representation 



The discussion of the invariants of a linear noisy network under the 
group of lossless transformations is most simply carried out by using the 
impedance description of the network. We shall start by describing 
the effect of a lossless transformation on a general w-terminal-pair net- 
work. Such a network has to be classified with respect to its passive- 
active character, which depends upon its ability to deliver or absorb 
power. The examination of power delivered or absorbed by a multi- 
terminal-pair network raises some interesting questions leading to a 
generaKzation of the available-power concept: the exchangeable power. 
In Chap. 3, the exchangeable-power concept in turn leads to the discovery 
of the invariants mentioned in Chap. 1. 

2.1. Impedance-Matrix Representation of Linear Networks 

At any frequency, a linear w-terminal-pair network containing internal 
prescribed signal or noise generators is specified completely with respect 
to its terminal pairs by its impedance matrix Z and the complex Fourier 
amplitudes of its open-circuit terminal voltages £i, E2, • • • , En (Fig. 2.1). 
In matrix form, Z denotes a square n-hy-n array 



Z = 



Zn- 


• Zu ■ 


' ■ Zik ■ 


■ Zin 


Zn • 


• Zu • 


• Zik • 


^ in 


Zki • 


■ ■ Z]^i • 


• ■ Zkk • 


• Z-kn 


Znl • 


• • Zni ■ 


■ • Znk • 


■ • 7 



10 



IMPEDANCE REPRESENTATION 



lCh.2 







El 


Z 


^1 : 


\!y 




> ■'i 


Ei 


Vi H 


Ky 




' ■'n 


En 


Vn H 


K^ 




I ^ 









Fig. 2.1. Equivalent representation of linear network with internal noise sources. 

The cases in which the Z matrix does not exist in a formal sense can be 
handled by suitable perturbations, for example, slight frequency changes 
or the addition of perturbing circuit elements. The complex amplitudes 
of the open-circuit terminal voltages are represented conveniently by a 
column matrix E: 

'El 



E = 



E 



E 



En. 

If the internal generators are random-noise sources, the Fourier ampli- 
tudes El • ' • En are complex random variables, the physical significance 
of which usually appears in their self- and cross-power spectral densities 
EiEk*} The bar indicates an average over an ensemble of noise processes 

^ W. B. Davenport and W. O. Root, Random Signals and Noise, McGraw-Hill 
Book Company, New York (1958). 



Sec. 2.1] 



IMPEDANCE-MATRIX REPRESENTATION 



11 



with identical statistical properties. Here, as in the rest of this work, we 
retain only positive frequencies. In order to preserve the same multipli- 
cative factors in power expressions for both random and nonrandom 
variables, we shall depart from convention by using root-mean-square values 
for all nonrandom complex amplitudes. 

A convenient summary of the power spectral densities is the matrix 



EEt = 



_EnEi 



EnEi 



EnEk* 



E,E,* ■ 


• • E,Ei* • 


■ ■ E.Ek"^ • 


• • E.Er,* 




EiEi^ ■ 


• • EiEi* ■ 


• EiEk* • 


Hi i-CLn 




EkE,* • 


• • EkEi"" ■ 


' ■ EkEk • 


■ • EkEn" 



F F * 



(2.1) 



where the superscript dagger indicates the two-step operation composed 

of forming the complex conjugate of and transposing the matrix to which 

it refers. Briefly, A+ is called the Hermitian conjugate of any matrix A.^ 

The matrix EE^ is its own Hermitian conjugate, because EiEk* = 



Ek*Ei = (EkEi*)*. Such a matrix is said to be Hermitian. 

In addition, we can show that EE^ is a positive definite or semidefinite 
matrix. Construct the real nonnegative quadratic form : 



(Xi*Ei-\- . . . +Xi''Ei+ . . . -{-Xn^'En) (a:i£i*-i-. . . + XiEi^'-h . . . +Xr,En*) >0 

where the Xi are arbitrary complex numbers. In terms of the column 
matrix 

'~Xi 



X = 



and the column matrix E, the foregoing expression becomes 



xtE(x+E)+ = x+EE+x > 

^ F. B. Hildebrand, Methods of Applied Mathematics, Prentice-Hall, New York 
(1952). 



12 



IMPEDANCE REPRESENTATION 



[Ch.2 



for all X. If the inequality sign applies for all x 5^ 0, EE^ is positive 
definite; if the equality sign applies for some (but not all) x 5^ 0, EE''" is 
positive semidefinite. The equality sign for all x ?^ would occur only 
in the trivial case E = 0. 



2.2. Lossless Transformations 

If the w-terminal-pair network with the generator column matrix E and 
the impedance matrix Z is connected properly to a 2w-terminal-pair 
network, a new w-terminal-pair network may be obtained. It will have 
a new noise column matrix E' and a new impedance matrix Z'. This 



Zt^ = 



aa\ ^ab 



El 






L. 



■la 



V„ = V 



^Z', E' 



Fig. 2.2. General transformation (imbedding) of an M-terminal-pair network. 



operation, shown in Fig. 2.2, we shall call a transformation, or an im- 
bedding, of the original network. If the added network is lossless, this 
arrangement represents the most general lossless modification (including 
the addition of multiple, and not necessarily reciprocal, feedback paths) 
that can be performed externally upon the original w-terminal-pair net- 
work so as to obtain a new w-terminal-pair network. 

The analytical relation between the voltages and currents applied to 
the 2w-terminal-pair, lossless network (the "transformation network") 



P = ilrUZr + Zrt)Ir ^ 0, 


for all Ir 


(2.4a) 


Zr + Zyt = 




(2.4&) 


'oo ^^ •^ao ) ^06 ~ ^ha '■> 


Zfoft = — Zfefot 


(2.5) 



Sec. 2.2] LOSSLESS TRANSFORMATIONS 13 

of Fig. 2.2 can be written in the form 

Va = Zaalo + Zafclft (2.2) 

V5 = Zjalo + Zbblfo (2.3) 

The column vectors Vo and \b comprise the terminal voltages applied to 
the transformation network on its two sides, and the column vectors !„ 
and Ift comprise the currents It flowing into it. The four Z matrices in 
Eqs. 2.2 and 2.3 are each square and of nth order. They make up the 
square 2wth-order matrix Zr of the lossless transformation network. The 
condition of losslessness can be summarized in the following relations, 
which express the fact that the total time-average power P into the 
transformation network must be zero for all choices of the terminal 
currents : 



therefore 



or 



Equation 2.5 does not require that the transformation network be recipro- 
cal. 

The original w-terminal-pair network, with impedance matrix Z and 
noise column matrix E, imposes the following relations between the 
column matrices V and I of the voltages across, and the currents into, 
its terminals: 

V = ZI + E (2.6) 

The currents I into the w-terminal-pair network are, according to Fig. 2.2, 
equal and opposite to the currents !« into one side of the 2w-terminal-pair 
network. The voltages V are equal to the voltages Vo. We thus have 

V = v.; I = -la (2.7) 

Introduction of Eqs. 2.7 into Eq. 2.6 and application of the latter to 
Eq. 2.2 give 

la= -{Z + ZaaT^Zablb + (Z -f Z„o)-1E 

When this equation is substituted in Eq. 2.3, the final relation between 
Vfo and Ift is determined: 

Vfe = Z'ift -h E' (2.8) 

where 

Z' = -Z6a(Z + ZaaT^Zai + Z^fc (2.9) 



14 



IMPEDANCE REPRESENTATION 



[Ch.2 



and 



E = Zj,a(Z + Zaa) E 



(2.10) 



Equation 2,8 is the matrix relation for the new w-terminal-pair network 
obtained from the original one by imbedding it in a 2w-terminal-pair 
network. Here Z' is the new impedance matrix, and E' is the column 
matrix of the new open-circuit noise voltages. Conditions 2.5 must be 
appHed to Eqs. 2.9 and 2.10 if the transformation network is to be lossless. 

2.3. Network Classification in Terms of Power 

In the course of our general study of noise performance of linear am- 
plifiers, it will be necessary to generalize the definition of available power. 
The need arises from situations involving negative resistance. 

Normally, the available power Pav of a one-terminal-pair source is 
defined as : 

Pg^y = the greatest power that can he drawn from the source by 
arbitrary variation of its terminal current (or voltage) 



I 


Linear network 
with prescribed 
internal sources 







E 



Fig. 2.3. The Th6vemn equivalent of a one-terminal-pair linear network. 

If the Thevenin representation of the source (Fig. 2.3) has the 
complex open-circuit voltage E and internal impedance Z, with R = 
Re (Z) > 0, this definition leads to 



P = - 



\EE^ 



1 EE^ 



4 R 



2Z + Z^ 



> 0, for i? > 



(2.11fl) 



If the source is nonrandom, the bar in Eq. 2.11(1 may be omitted. No 
other changes are necessary because E is then understood to be a root- 
mean-square amplitude, as remarked in Sec. 2.1. In Eq. 2.11a, Pav is 
also a stationary value (extremum) of the power output regarded as a 
function of the complex terminal current /. Moreover, the available 
power (Eq. 2 Ala) can actually be delivered to the (passive) load Z*. 



Sec. 2.3] NETWORK CLASSIFICATION IN TERMS OF POWER 15 

When R is negative, however, the foregoing definition of available 
power leads to 

Pay = °o , for i? < 

since this is indeed the greatest power obtainable from such a source, 
and is achievable by loading it with the (passive) impedance —Z. 
Observe that this result is not either a stationary value or extremum of 
the power output as a function of terminal current, nor is it consistent 
with Eq. 2.11a extended to negative values of R. 

To retain the stationary property (with respect to terminal current) of 
the normal available power concept, and accordingly to preserve the form 
of Eq. 2.11a, we define the concept of exchangeable power Pg." 

Pg = the stationary value {extremum) of the power output from the 
source, obtained by arbitrary variation of the terminal current 
{or voltage) 

It is easy to show that this definition of exchangeable power always leads 
to Eq. 2.11a for any nonzero value of R in Fig. 2.3. Specifically, 



lEE* 1 EE* , „ , , 

^^ = 4-^==2zT^' ^^^^^^ ^'-'"'^ 

When R is negative, Pg in Eq. 2.1 IJ is negative. Its magnitude then 
represents the maximum power that can be pushed into the terminals by 
suitable choice of the complex terminal current /. This situation may 
also be realized by connecting the (nonpassive) conjugate-match im- 
pedance Z* to the terminals. This impedance actually functions as a 
source, pushing the largest possible power into the network terminals. 

A straightforward extension of the exchangeable-power definition to 
«-terminal-pair networks makes it the stationary value (extremum) of 
the total power output from all the terminal pairs, obtained by arbitrary 
variations of all the terminal currents. With reference to Fig, 2.1, we 
search specifically for the stationary values with respect to I of the total 
average output power P of the network 



P = -i[I^(Z + Zt)I + I'^E + E+I] (2.12) 

In Eq. 2.12, P is a quadratic function of the terminal currents. The 
stationary values of interest depend upon a particular classification of 
the impedance matrix Z. This classification is based upon Eq. 2.12, with 
the internal sources inactive. 
With the internal generators inactive, E becomes zero, and the power 



16 IMPEDANCE REPRESENTATION [Ch. 2 

leaving the network is 



Po= -i[r(Z + Z+)I] (2.13) 

Three possible cases have to be distinguished for Po (aside from the 
previously discussed case of a lossless network). 

1. The first case is that of a passive network. Then the power output 
Po must not be positive for any I, indicating a net (or zero) absorption 
of power inside the network. The matrix (Z + Z^) is positive (semi) 
definite. 

2. In the second case, the matrix (Z + Z^) is negative (semi) definite, 
which means that the power Po flowing out of the network is never nega- 
tive, regardless of the terminal currents I. This indicates a net (or zero) 
generation of power inside the network. 

3. Finally, the matrix (Z + Z^) may be indefinite. The power Po 
flowing out of the network may be either positive or negative, depending 
upon the currents I. 

One may imagine the power Po plotted in the multidimensional space 
of the complex current amplitudes I. The three cases may be distin- 
guished according to the nature of the quadratic surface Po. When 
(Z -f Z^) is either positive or negative semidefinite, the surface is a 
multidimensional paraboloid with a maximum or minimum, respectively, 
at the origin. When (Z -f- Z^) is indefinite, the surface is a hyperboloid 
with a stationary point (saddle point) at the origin. The word "origin" 
is used loosely for simplicity; it omits the semidefinite cases, when 
(Z -|- Z''") is singular. Then difiiculties will arise in connection with the 
inverse of (Z + Z"*^). These difficulties will be circumvented by the 
addition of suitable small loss, in order to remove the singularity. Results 
pertaining to the singular case can be obtained in the limit of vanishing 
added loss. Henceforth we shall make no explicit reference to semi- 
definite cases. 

The power P out of the network in the presence of internal generators 
is obtained from Po (Eq. 2.12) by adding to it a plane through the origin. 
The extremum or saddle point ceases to occur at the origin. The new 
position of the stationary point can be determined conveniently by 
introducing an appropriate shift of coordinates. Setting 

r = i-\- (z-\- zt)-iE 

yields for Eq. 2.12 



^ = -i[(lO^(Z + Z^)I' - Et(Z + Z+)-iE] (2.14) 

The shift of origin has led to a completion of the square. The new origin 



Sec. 2.3] NETWORK CLASSIFICATION IN TERMS OF POWER 17 

is obviously the stationary point of the power expressed in the new 
variables I'. The height of the surface at the stationary point is the 
exchangeable power Fe : 



P, = i [E^Z + Z^-^E] (2.15) 

Because the definite characters of (Z + Z"*")"^ and (Z + Z^) are the 
same: Pe > (regardless of E) when (Z + Z"*") is positive definite; 
Pe < (regardless of E) when (Z + Z"^) is negative definite and Pe < 
(depending upon the particular E involved) when (Z + Z'*') is indefinite. 
In view of the term (lO^(Z + Z+)l' in P, the significance of Pe in Eq. 2.15 
is of the same kind as that of Pg in Eq. 2.11& when (Z + Z^) is either 
positive or negative definite. When (Z + Z"^) is indefinite, however, 
Pe in Eq. 2.15 is simply the stationary -point (saddle-point) value of the 
average output power with respect to variations of the terminal currents, 
and has no analog in the case of a one-terminal-pair network (Eq. 2.1 1&). 

We have defined the exchangeable power for a one-terminal-pair net- 
work as the extremum of power output obtainable by arbitrary variation 
of terminal current. In an obvious generalization, we have extended this 
definition to w-terminal-pair networks by considering the extremum of 
the power output of the network obtained by an arbitrary variation of 
all its terminal currents. In this case, we have encountered the possibility 
of the output power assuming a stationary value rather than an extre- 
mum. One may ask whether the stationary value of the output power 
for the multiterminal-pair case could be achieved in a simpler way. One 
obvious method to try is that shown in Fig. 2.4.^ 

The given network is imbedded in a variable {n -\- 1) -terminal-pair 
lossless network. For each choice of the variable lossless network, we 
consider first the power that can be drawn from the {n ■\- l)th pair for 
various values of the complex current /^+i (that is, for various "loadings" 
of this terminal pair). In particular then, we determine the exchangeable 
power Pe,n+i for this terminal pair according to Eq. 2.116, recognizing 
that it may be either positive or negative. In the respective cases, its 
magnitude represents power delivered by, or to, the original network, 
since the imbedding network is lossless. Specifically, its magnitude 
represents the greatest possible value of the power that can be drawn 
from, or delivered to, the original network, for a given choice of the loss- 
less imbedding network. 

' Recently we have learned that, prior to our study, this particular noisy-network 
power-optimization problem was considered and solved independently for receiving 
antennas by J. Granlund, Topics in the Design oj Antennas jor Scatter, M.I.T. Lincoln 
Laboratory Technical Report 135, Massachusetts Institute of Technology, Cambridge, 
Mass. (1956). 



18 



IMPEDANCE REPRESENTATION 



[Ch. 2 



The exchangeable power determined in this way will, in general, be 
different for different choices of the lossless network because the con- 
tributions of the n terminal pairs of the original network are in each case 
combined with different relative magnitudes and phases to make up the 
output at terminals n -\- \. Therefore, we might expect the available 
power of the original w-terminal-pair network to be the extremum value 
of Pe,n+\ obtainable by considering every possible variation of the lossless 



/ 



Variable lossless 

network 

(possibly 

non reciprocal) 



7 



1 '^i 






En 



Fig. 2.4. Imbedding into an (n + l)-terminal-pair lossless network. 



transformation network. As we shall see shortly, this result proves to be 
correct only in one simple case, namely, that in which the original n- 
terminal-pair network contains only coherent signal generators. In such 
a case Pe,n+i has only one stationary value as the lossless network is 
varied, and this value is precisely the exchangeable power discussed 
previously {Pg in Eq. 2.15). 

In the general situation of an arbitrary noisy network, we shall find that 
Pe,n+\ has n stationary values as the lossless transformation network is 
varied in all possible ways. None of these individually is the exchange- 
able power for the original network. The sum of them, however, does 
prove to be the exchangeable power. The major burden of the discussion 
immediately to follow will be to interpret the stationary values of Pe,n+i 
in terms of some physical properties of the original network. In later 
chapters, the relationship of these results with the noise performance 
will emerge. 




Impedance Formulation 

of the Charactenstic^Noise Matrix 



We shall proceed to a close examination of the stationary-value 
problem posed in connection with Fig. 2.4, at the end of the last section 
and prove the assertions made about it. A matrix formulation of 
the problem will be required, which will reduce the problem to one in 
matrix eigenvalues. The corresponding eigenvalues are those of a new 
matrix, the "characteristic-noise matrix." Some general features of the 
eigenvalues will be studied, including their values for two interesting 
special cases. The effect of lossless imbeddings upon the eigenvalues will 
be discussed to complete the background for the noise-performance 
investigations. 



3.1. Matrix Formulation of Stationary- Value Problem 

The network operation indicated in Fig. 2.4 is conveniently accom- 
pUshed by first imbedding the original w-terminal-pair network Z in a 
lossless 2w-terminal-pair network, as indicated in Fig. 2.2. Open-circuit- 
ing all terminal pairs of the resulting w-terminal-pair network Z', except 
the ith, we achieve the w-to-1 -terminal-pair lossless transformation 
indicated in Fig. 2.4. The exchangeable power from the ith. terminal 
pair of the network Z' can be written in matrix form as 



1 E/E/* 1 rE^E^t| 



19 



20 



CHARACTERISTIC-NOISE MATRIX 



[Ch.3 



where the (real) column matrix | has every element zero except the i\h, 
which is 1 : 



ir 




"0" 


^i 


= 


1 


Jn_ 




_0_ 



^i =0, j r^i 

^i = 1 



(3.2) 



Matrix | can be visualized as a double-pole w-throw selector switch which 
chooses electrically only one of the n terminal pairs according to its sole 
nonzero element. 

The variation of the lossless network in Fig. 2.4 now corresponds to 
variation of the transformation network Zt in Fig. 2.2 through all possible 
lossless forms. We wish to find the stationary values of Pe,i correspond- 
ing to variation of Zy, To render explicit this variation, E' is first 
expressed in terms of the original E and Z, using Eq. 2.10. Accordingly, 



where 



E'E'+ = Z,a(Z + Z„„)-lEEt[(Z + Zaa)-^]^Z,/ = T^EE+T (3.3) 

(3.4) 



T+ = Z6a(Z + TaaY^ 



Second, expressing Z' in terms of Z by means of Eq. 2.9 yields 

Z -f Z ''" = — Z6a(Z + Zoo)'~'^Za6 + T^hh + Z^ft 
-Z„6^[(Z + Z„a)-1]+Z,/ 



(3.5) 



The conditions of Eq. 2.5, guaranteeing that the transformation network 
is lossless, convert the foregoing relation to 

TJ 4- Z't = Z6a[(Z -f Zaa)-^ + (Z+ ^ Za^T'^Zj 

= Z6a(Z + Zaa)-'[{Z^ + ZaJ) + (Z + Zaa)] (Z^ + ZaJ)-'Zj 
= T^(Z + Zt)T (3.6) 



It follows that 



p . = 



1 (iV)EEt(Tl) 
2(|V)(Z + Z+)(t|) 



(3.7) 



in which matrix t (not |) is to be varied through all possible values 
consistent with the lossless requirements upon the transformation 
network. 



Sec. 3.2] 



EIGENVALUE FORMULATION 



21 



The significant point now is that t is actually any square matrix of 
order n because Z^a in Eq. 3.4 is entirely unrestricted! Therefore, a new 
column matrix x may be defined as 



= T| = 



Xi 



(3.8) 



in which the elements take on all possible complex values as the lossless 
transformation network Zr in Fig. 2.2 is varied through all its allowed 
forms. Consequently, the stationary values of Pe,i in Eq. 3.7 may be 
found most conveniently by determining instead the stationary values 
of the (real) expression 

^^''"2xt(Z + Z+)x ^^-^^ 

as the complex column matrix x is varied quite arbitrarily. 

Aside from the uninteresting possibility of a lossless original network 
Z, three cases must be distinguished in Eq. 3.9, corresponding to the 
three different characters of Z discussed previously in connection with 
Eqs. 2.13 and 2.15. Since EE^ is positive (semi) definite, these cases are 
described as follows in terms of the variation of Pe,i as a multidimensional 
function of all the complex components of x: 

{a) Z + Z^ positive definite; Pe,i > for all x. 
(&) Z + Z"*^ negative definite; Pe,i < for all x. 
(c) Z + Z^ indefinite; Pe,i < 0, depending upon x. 



3.2. Eigenvalue Formulation of Stationary-Value Problem 

We now turn to the determination of the extrema and stationary 
values of Pe,i in Eq. 3.9. For reasons that will become clear in regard to 
amplifier noise performance, we shall look for extrema of the quantity 
p = -Pe,i. In terms of A = EE^ and B = -2(Z + Z^), 



P = 



c^EE^ 



cUj 



2x^(Z + Z+)x x^Bx 



(3.10) 



22 CHARACTERISTIC-NOISE MATRIX [Ch. 3 

The stationary conditions and corresponding values for p may be ob- 
tained from the solution of the equivalent problem of determining the 
stationary values of x^Bx, subject to the constraint x^Ax = constant. 
Therefore, introduction of the Lagrange multiplier 1/X and recognition 
that p may be regarded as a function of either the set of Xi or the set of 
Xi* lead to the conditions 



d 
dXi" 



U^Bx - - xUx j = 0, i = \,2,--',n (3.11) 



or simply 



Ax - XBx = (A - XB)x = (3.12) 

The values of X are then fixed by the requirement 

det (A - XB) = = det (B-^A - XI) (3.13) 

where 1 is the unit matrix. This means that the values of X are just the 
eigenvalues of the matrix B~^A = —\{^ + Z''^)~^EE''^. 

The matrix x yielding any stationary point of p must satisfy Eqs. 3.12, 
as well as the constraint x^Ax = constant. Let Xs be one eigenvalue of 
B~^A and x^^^ be the corresponding solution (eigenvector) of Eq. 3.12. 
Then premultiplication of Eq. 3.12 by x^®^"*" yields 

x(s)tAx(s) = x^x('^"^Bx(') 
or 

which is real and equal to the corresponding stationary value of p. 

It follows that the stationary values of the exchangeable power Pe,i are 
the negatives of the (real) eigenvalues of the matrix 

AB-i = -i(Z + Z+)-iEE+. 

We therefore define: 

Characteristic-noise matrix = N = - J(Z + Z+)~^EET (3.15) 

and conclude that: 

The stationary values of the exchangeable power Pe,i are the negatives of 
the {real) eigenvalues of the characteristic-noise matrix N. 



Sec. 3.3] PROPERTIES OF THE EIGENVALUES 23 

3.3. Properties of the Eigenvalues of the Characteristic-Noise 
Matrix in Impedance Form 

We shall now confirm the assertion made earlier (Sec. 2.3) to the 
effect that: 

The exchangeable power of the n-terminal-pair network, 



is equal to the algebraic sum of the stationary values of Pe,i, which is alter- 
natively the negative of the trace of the characteristic-noise matrix N. 

Setting W = f (Z + Z'^)~^, we express the typical ijth. element of the 
matrix i(Z + Z^y^E^ = WEE^ in the form 



(WEEt),y = Z WaEkEj'' 
k=i 

so that its trace (sum of diagonal elements) becomes 



Trace (WEE+) = £ WikEkEi"" = -trace (N) (3.16) 

k,l=l 

But Pe of the w-terminal-pair network equals 



Pe = i[E^(Z + Zt)-i]E = E^WE = E Ei*WikEk (3.17) 

k,l=l 

Comparing Eq. 3.17 with Eq. 3.16, we find 

Pe = -trace N = - £ X^ (3.18) 

s 

since the trace of a matrix is the sum of its eigenvalues. 

We must now determine the ranges of values that can be assumed by 
the eigenvalues Xs of the characteristic-noise matrix N as well as the corre- 
sponding ranges of Pe,i. We first recall that the eigenvalues Xs determine 
the stationary values of p in Eq. 3.10. The numerator of this expression 
is never negative, since A(= EE"*") is positive (semi) definite. Thus, the 
algebraic sign of p is determined by that of the denominator. This 
in turn depends upon the definite character of B, which is equal to 
— 2(Z -|- Z^). As noted previously, three cases have to be distinguished, 
in accordance with the second column of Table 3.1. In the first case, the 
denominator is always negative. Accordingly, the eigenvalues Xs pertain- 
ing to this case must all be negative, as shown in the last column of the 
table. The other cases follow in a similar manner. 



24 CHARACTERISTIC-NOISE MATRIX [Ch. 3 

Table 3.1. Classification of Networks and Eigenvalues 



Case 


z + zt 


Network Class p = —Pe.i 


Eigenvalues 
(X,) of N 


1 
2 


positive definite 
negative definite 


passive < 
active (negative resistance) > 


all< 
aU >0 


3 


indefinite 


active ^ 


fsome > Ol 
\some < 0] 



The permissible values of Xs determine the ranges of values that p can 
assume as a function of x in Eq. 3.10. Let us consider Case 1 first. No 
eigenvalue is positive. Among the eigenvalues, there is one of least 
magnitude (possibly zero) and another of largest magnitude. Since ^ is a 



P = -Pe.i 



P = -Pci 



Largest positive 
eigenvalue "■\. 

"D 

E 



Smallest positive -^ 
eigenvalue 



-Pe.i = 

Smallest negative 
eigenvalue 

Intermediate 
eigenvalues 

Largest negative 
eigenvalue 



P = -Pe.i 



I Intermediate 
I eigenvalues 



kTAf 



kTAf 



-Pe.i^O 



:» 
E * 



Intermediate 
eigenvalue 

Smallest 

positive 

eigenvalue 



■Pe.i = 



Smallest 

negative 

eigenvalue 

Intermediate 
eigenvalue 



Z + zt positive definite 
(a) 



Z + Z''' negative definite 
(b) 



Z + zt indefinite 
(c) 



Fig. 3.1. Schematic diagram of permitted values of p for four-terminal-pair networks. 



continuous function of x, its values lie between these two extreme eigen- 
values, as illustrated in Fig. 3.1a. Analogous reasoning applies to the 
second case, illustrated in Fig. 3.16. 

Case 3 is a little more involved. The denominator of Eq. 3.10 can 
certainly become zero for some values of x. Correspondingly, infinite 
values of p will occur. Among the eigenvalues, there is a smallest 



Sec. 3.3] PROPERTIES OF THE EIGENVALUES 25 

positive one and a negative one of smallest magnitude. (In special cases, 
one or both may be equal to zero.) Again, because of the continuous 
nature of /? as a function of x, p may never take a value between and dis- 
tinct from the foregoing extreme eigenvalues. The gap between the 
ranges of allowed values of p is illustrated in Fig. 3.1c. 

One particular property of the eigenvalues of N will be of importance 
later. Suppose that the original network with the characteristic-noise 
matrix N is imbedded in a 2w-terminal-pair lossless network, as shown in 
Fig. 2.2. A new w-terminal-pair network results, with the characteristic- 
noise matrix N'. The eigenvalues of N' are the stationary values of the 
exchangeable power obtained in a subsequent imbedding of the type 
shown in Fig. 2.4. This second imbedding network is completely variable. 
One possible variation removes the first 2w-terminal-pair imbedding. 
Accordingly, the stationary values of the exchangeable power at the 
(w -f l)th terminal pair in Fig. 2.4 do not change when a 2 w- terminal-pair 
lossless transformation is interposed between the two networks shown. 
It follows that: 

The eigenvalues of the characteristic noise matrix N are invariant to a 
lossless transformation that preserves the number of terminal pairs. 

At this point appHcation of our results to two familiar examples of 
linear networks helps to establish further significance for the character- 
istic-noise matrix and its eigenvalues. 

If the w-terminal-pair network contains only coherent (signal) gener- 
ators rather than noise generators, EiEk* = EiEk* because ensemble 
averaging is unnecessary. The matrix EE"'^ is then of rank one ; that is, a 
determinant formed of any submatrix of order greater than one is zero 
because its rows (or columns) are all proportional (Eq. 2.1), The rank 
of N cannot exceed that of either of its factors, so it too is of rank one 
(zero in a trivial case). Matrix N therefore has only one nonzero eigen- 
value, and this is equal to trace N. From Eq. 3.18, we conclude that 
for such networks, containing only coherent signal generators, the 
operations defined by Fig. 2.4 lead merely to the exchangeable power for 
the whole network (in the sense of Eq. 2.15). It is the single stationary 
value of Pe,i and also the negative of the sole eigenvalue of N. 

Another simple but quite different case arises if the original (non- 
reciprocal) network is a passive one with dissipation, (Z -|- Z^) positive 
definite, in thermal equilibrium at absolute temperature T. Then the 
operations defined by Fig. 2.4 must, on thermodynamic grounds, always 
lead to Pe,i = kT A/ in a frequency band A/, where k is Boltzmann's 
constant. No matter what form the variable lossless network may take. 



26 CHARACTERISTIC-NOISE MATRIX [Ch. 3 J 

Pe,i must remain constant at the foregoing value. Thus, from Eq. 3.9, 



x^[EEt - 2kT A/(Z + Zt)]x = 0, for all x 



or 



EE^ = 2kT A/(Z + Z+) 

a result proved previously by Twiss.^ In terms of the characteristic- 
noise matrix N, we have 

N = - J EE^iZ + Z+)-i = -kT A/1 

an equation indicating that a passive dissipative network at equilibrium 
temperature T always has a diagonal N matrix, with all the eigenvalues 
equal to — ^r A/. 

3.4. Lossless Reduction in the Number of Terminal Pairs 

An w-terminal-pair network has a characteristic noise matrix of nth. 
order, with n eigenvalues. If k of the n terminal pairs of the network are 
left open-circuited and only the remaining {n — k) terminal pairs are 
used, the original network is reduced to an {n — ^) -terminal-pair net- 
work. This operation may be thought of as a special case of a more 
general reduction, achieved by imbedding the original w-terminal-pair 
network in a lossless {2n — ^) -terminal-pair network to produce {n — k) 
available terminal pairs (see Fig. 3.2a). The case for n — k = 1 was 
considered in Fig. 2.4. 

The characteristic-noise matrix of the {n — ^) -terminal-pair network 
has {n — k) eigenvalues, which determine the extrema of the exchange- 
able power Pe,n-k+\ obtained in a subsequent (variable) reduction to one 
terminal pair (Fig. 3.2&). The successive reduction of the w-terminal- 
pair network first to (n — k) terminal pairs, and then to one terminal 
pair, is a special case of a direct reduction of the original network to one 
terminal pair. Comparison of the dotted box in Fig. 3.2b with Fig. 2.4 
shows that the exchangeable power Pe,n-k+i obtained by the two succes- 
sive reductions, with variation of only the second network (Fig. 3.2b), 
must lie within the range obtained by direct reduction with variation of 
the entire transformation network (Fig. 2.4). Hence the stationary 
values of Pe,n-k+i found in Fig. 3.2& must lie within the range prescribed 
for Pe,n+i by Fig. 2.4. It follows that the eigenvalues of N for the 
(n — ^) -terminal-pair network must lie within the allowed range of Pe,i 
for the original w-terminal-pair network, illustrated in Fig. 3.1. 

^ R. Q. Twiss, "Nyquist's and Thevenin's Theorem Generalized for Nonreciprocal 
Linear Networks," /. Appl. Phys. 26, 599 (1955). 



I 



Sec. 3.4] 



LOSSLESS REDUCTION OF TERMINAL PAIRS 



27 



1 


Lossless 

(2r— ^)-terminal-pair 

network 


1 


Z, E 






• 
• 
• 


• 
• 
• • 






• 


• 
• 











Fig. 3.2a. Reduction of w-terminal-pair network to (« — k) terminal pairs. 



n-k+l 



Variable 
lossless 
network 



n—k 



Lossless 

(2n-^)-termlnal-pair 

network 



Z, E 



Fig. 3.2b. Successive reductions of «- terminal-pair network to (n — k) terminal pairs and 

one terminal pair. 



Since the number of terminal pairs is changed in the lossless reduction 
just considered, the theorem on the preservation of the eigenvalues under 
lossless transformations (p. 25) does not apply. While the new eigen- 
values of the reduced network do lie within the range of the original ones, 
their values will usually be different. 




Canonical Form 

of Linear Noisy Netv^orks 



Lossless network transformations performed on a noisy network, in 
such a way that the number of terminal pairs is unchanged, change the 
impedance matrix as well as the noise spectra. However, these lossless 
network transformations do not change the eigenvalues of the character- 
istic noise matrix. Thus we know that each noisy network possesses 
some essential noise characteristics, unalterable by those lossless network 
transformations which preserve the number of terminal pairs. On this 
basis, we expect to be able to find a fundamental form of the network 
which places these characteristics directly in evidence. In this chapter, 
we shall develop such a form of the network. This fundamental or "ca- 
nonical" network form is, of course, attainable through lossless network 
transformations performed on the original network. The existence of a 
canonical form for every linear noisy network greatly clarifies its most 
important noise characteristics and simplifies the study of fundamental 
limits on its noise performance. Since the canonical network contains 
not more than n real parameters for every w-terminal-pair network, its 
existence also shows that an «-terminal-pair, linear noisy network does 
not possess more than n (real) invariants with respect to lossless trans- 
formations. 

4.1. Derivation of the Canonical Form 

In this section we shall prove the following theorem : 

At any particular frequency, every n-terminal-pair network can he re- 
duced by lossless imbedding into a canonical form consisting of n separate 

28 



Sec. 4.1] DERIVATION OF THE CANONICAL FORM 29 

{possibly negative) resistances in series with uncorrelated noise voltage 
generators. 

We note first that a lossless imbedding of the w-terminal-pair network 
transforms the two matrices Z + Z^ and EE^ in identical colinear manner, 
as shown in Eqs. 3.3 and 3.6. We have also noted that the matrix t which 
appears in the transformation is entirely unrestricted by the conditions 
(Eqs. 2.5) of losslessness for the imbedding network. 

It is always possible to diagonalize simultaneously two Hermitian 
matrices, one of which (eW) is positive definite (or, as a limiting case, 
semidefinite), by one and the same colinear transformation. Thus, 
suppose that both Z' + Z'^ and E'E'^ have been diagonalized by a 
proper imbedding of the original network (see Fig. 2.2). This means 
that the impedance matrix Z' of the resulting network is of the form 

Z' = diag (Ru R2,--, Rn) + Zrem (4.1) 

where Zrem fulfills the condition of the impedance matrix of a lossless 

network Zrem = -(Zrem)^- 

Suppose, finally, that a lossless (and therefore noise-free) network with 
the impedance matrix —Zrem is connected in series with our network 
(Z'j E') , as shown in Fig. 4.1. The result is a network with the impedance 
matrix 

Z" = Z' - Zrem = diag (i?i, R2, --■,Rn) (4.2) 

The open-circuit noise voltages remain unaffected when a lossless source- 
free network is connected in series with the original network. Thus, 



E'' = E' or E"E''+ = E'E'"^ (4.3) 

Consequently, the two operations lead to a network with the diagonal 
impedance matrix Z" of Eq. 4.2 and a diagonal noise matrix E"E''^. 
This canonical form of the network consists of n separate resistances in 
series with uncorrelated noise voltage generators, as shown in Fig. 4.2. 

Noting that the series connection of a lossless network is a special case 
of a lossless imbedding, we have proved the theorem stated at the begin- 
ning of this section. 

A lossless imbedding leaves the eigenvalues of the characteristic-noise 
matrix invariant. Thus, the eigenvalues of the characteristic-noise 
matrix N'' of the canonical form of the original network are equal to those 
of N of the original network. But, the eigenvalues X„ of N" are clearly 
its n diagonal elements 

// T? *// 



30 



CANONICAL FORM OF LINEAR NOISY NETWORKS [Ch. 4 



The ^th eigenvalue is the negative of the exchangeable power of the ^th 
source. Thus, we have proved the following theorem: 

The exchangeable powers of the n Thevenin sources of the canonical form 
of any n-terminal-pair network are equal and opposite in sign to the n 
eigenvalues of the characteristic noise matrix N of the original network. 



Ideal transformers 
1 




Fig. 4.1. Series connection of networks (Z', E') and — Zrem- 



Since ideal transformers may be applied to each terminal pair of the 
foregoing canonical form in a manner that reduces either all resistors or 
all \Ek\ to unit magnitude, there are actually only n independent (real) 
parameters contained in the canonical form. This fact proves the 
following theorem: 



Sec. 4.2] INTERCONNECTION OF LINEAR NOISY NETWORKS 



31 



A linear noisy n-terminal-pair network possesses not more than n in- 
variants with respect to lossless transformations, and these are all real 
numbers. 




Fig. 4.2. The canonical network. 



4.2. Interconnection of Linear Noisy Networks 

The canonical form is helpful in simplifying the discussion of the inter- 
connection of noisy networks. Consider an ;^-terminal-pair noisy network 
and an independently noisy w-terminal-pair network. Let them be 
connected through a 2(w + w) -terminal-pair lossless network, resulting 
in an {m -\- w) -terminal-pair network, as shown in Fig. 4.3. We shall now 
determine the eigenvalues of the characteristic-noise matrix ^m+n 
of the resulting (m -f- w) -terminal-pair network. 

To do so, we first reduce each of the component networks to canonical 
form of the type shown in Fig. 4.2. This procedure places in evidence, 
but does not alter, the eigenvalues of their respective characteristic-noise 
matrices. Taken together, the two canonical forms represent the canoni- 
cal form of the {m -f w) -terminal-pair network of Fig. 4.3. Accordingly, 
the m-\-n eigenvalues of that network are merely the eigenvalues of the 
component networks. The proof obviously covers the interconnection 



32 



CANONICAL FORM OF LINEAR NOISY NETWORKS [Ch. 4 



of any number of independently noisy networks of any size, provided the 
total number of terminal pairs is preserved. 

We shall be interested in cases in which the number of terminal pairs 
is reduced upon interconnection. A simple example is the interconnection 



1 


Lossless 
2(m+;i)-terminal-pair 
network 


1 


Z/nj ^m 








• 

• 

• 

I 




• 




• 
• 


• 
• 
• 
m 


i 






1 


^ 






_ 


Z„,E„ 








• 
• 


• 
• 
• 


• 




m + n 


• 
• 
• 
n 















Fig. 4.3. Lossless interconnection of w-terminal-pair and «- terminal-pair networks. 



of several two-terminal-pair amplifiers, with feedback, to form a new 
two -terminal-pair amplifier. In such a case, the effect of the reduction 
can be understood by applying the reasoning of Sec. 3.4 to the available 
terminals of the network in Fig. 4.3. The conclusion is immediate. 

The eigenvalues of the characteristic-noise matrix of an n-terminal-pair 
network constructed by lossless interconnection of independently noisy com- 
ponent networks having m terminal pairs in all, m > n, must lie within the 
range defined by the most extreme eigenvalues of the component networks. 




Linear Noisy Networks 
m Other Representations 



In the foregoing analysis we have found all the invariants of a linear 
noisy network with respect to lossless imbeddings that preserve the 
number of terminal pairs. With the aid of the impedance formalism, 
these invariants have been interpreted in terms of the exchangeable power, 
on the one hand, and in terms of the canonical representation of the 
network, on the other. There are, however, additional interpretations of 
the invariants, which are brought out by different matrix representations 
of the network. For each new representation a characteristic-noise matrix 
can be defined. As we might expect, all such characteristic-noise matrices 
have the same eigenvalues, since, after all, these are the only invariants 
of the network. 



5.1. General Matrix Representations 

The impedance-matrix representation, Eq. 2.6, is conveniently re- 
written in the form^ 



[i i -^] 



= E 



(5.1) 



where 1 is the identity matrix of the same order as Z. Any other matrix 

^ V. Belevitch, "Four-Dimensional Transformations of 4-pole Matrices with 
Applications to the Synthesis of Reactance 4-poles," IRE Trans, on Circuit Theory, 
CT-3, No. 2, 105 (1956). 

33 



34 



OTHER REPRESENTATIONS 



[Ch.5 



representation of a linear noisy network can be expressed as 

V — Tu = 8 



(5.2) 



where v is a column matrix consisting of the amplitudes of the terminal 
"response," u is the corresponding column matrix of the terminal "excita- 
tion," and 5 is a column matrix comprising the amplitudes of the internal 
(noise) sources as seen at the terminals. The square matrix T expresses 
the transformation of the network in the absence of internal sources. 

As an example of such a matrix representation of a 2w-terminal-pair 
network, we consider the mixed voltage-current representation, for which 

h 

h 

(5.3) 



V2m—1 
L-* 2ot— 1 . 

-h 

-h 



u = 



and 8 is the noise column vector 



8 = 



V2m 
~^2m 



EnZ 
IfiZ 



En(2m—1) 
_In(,2m—l) J 



(5.4) 



(5.5) 



The equivalent circuit suggested by the representation, Eq. 5.2, with the 
interpretations, Eqs. 5.3, 5.4, and 5.5, is shown in Fig. 5.1. 



Sec. 5.2] TRANSFORMATION BETWEEN REPRESENTATIONS 



35 



h 


Enl 










+ 


Ky 


1 


Vi 




®^.i 




o 




T 




-h 




+ 






V-i 
















• 








• 








•^2m-l 


En(2m-1) 






+ 


\y 


1 


^2m-l 




(P /n{2/n-l) 












-hm 


+ 






y2m 

















Fig. 5.1. Mixed voltage-current representation of 2OT-terminal-pair network. 

Returning to the general expression (Eq. 5.2), we note that it also can 
be written in a form similar to Eq. 5.1. 



[i ! -t] 



= 8 



(5.6) 



The analysis in Sec. 5.2 shows how transformations are performed from 
one matrix representation to another. Then the transformation of the 
power expression will be studied. Combining the results of these two 
studies, we shall be able to define a characteristic-noise matrix for every 
formalism and to show its relation to the characteristic-noise matrix de- 
fined in Eq. 3.15. 



5.2. Transformation from One Matrix Representation to Another 



The variables - - 



rv 




V 


-- 


and 


-- 


[i J 




u_ 



can always be related by a linear trans- 



36 



OTHER REPRESENTATIONS 



[Ch.5 



formation of the form 





V 




"V" 


R 


-- 


= 


-- 




_ u_ 




_I_ 



(5.7) 



where R is a square matrix of order twice that of either V or I, For a 
2w-terminal-pair network, R is of order 4w. For example, the transforma- 
tion from the impedance representation of a 2 -terminal-pair network into 
the general-circuit-parameter representation (Eqs. 5.3, 5.4, and 5.5) yields 



R = 



'1 








0" 








1 








1 

















-1_ 



(5.8) 



The relation between the matrices Z and T is derived in the following way. 
We start from Eq. 5.1 and introduce the transformation (Eq. 5.7): 



b i -^] 



RR- 



V 

LiJ 



= E 



or 



[l; -z]r 



= E 



(5.9) 



In order to relate Eq. 5.9 to Eq. 5.2, we note that the order of R is twice 
that of Z. The matrix R is therefore conveniently split into submatrices 
as 

Rl2 



R = 



Rii 
R21 



R 



22 J 



(5.10) 



where the Rij are of the same order as Z. Carrying out the multiplication 
in Eq. 5.9, we obtain 



[R, 



ZR 



21 



R12 ~ ZR 



22 



u 



= E 



(5.11) 



The correspondence between Eqs. 5.11 and 5.2 is made complete if we 
multiply Eq. 5.11 by 

M = [Rn - ZRai]-^ (5.12) 



Sec. 5.3] POWER EXPRESSION AND ITS TRANSFORMATION 
obtaining 



37 



where 



and 



[1 ; -T] 



u 



= 8 



[i! -t] = m[i: -z]r 



8 = ME 



(5.6) 



(5.13) 



(5.14) 



Equations 5.12 to 5.14 summarize the transformation from one matrix 
representation to another. 



5.3. Power Expression and Its Transformation 

In any matrix representation, the power P flowing into the network is 

V 

a real quadratic form of the excitation-response vector - - . We have 

_u_ 

V ' V 

P = --- Qr -- (5.15) 

_u J _u_ 

where Qt is a Hermitian matrix of order twice that of either u or v. In 
the particular case of the impedance-matrix representation, 

P = \ [V+I + I+V] 



"V" 


t/0 1 1\ 


"V" 


_I_ 


\i i 0/ 


_I_ 



(5.16) 



Comparing Eqs. 5.16 and 5.15, we find that the Q matrix for the imped- 
ance representation is 

1 



Qz = 



1 







(5.17) 



A transformation from one representation into another transforms the 
Q matrix. Let us study how Q changes when we transform from the 
impedance representation into the general representation of Eq. 5.6. 



38 



OTHER REPRESENTATIONS 



[Ch.5 



We have 





"V" 


t 


"V 




p = 


_I_ 


Qz 


_I_ 






V 


t 


V 


= 


-- 


R+QzR 


-- 




u_ 






_u_ 



Comparison with Eq. 5.15 shows that 

Qt = R^QzR 



(5.18) 



where R is the matrix that transforms the general-excitation vector 



into the voltage and current vector 



, according to Eq. 5.7. 



We are now ready to set up the general characteristic-noise matrix N 
for any matrix representation, Eq. 5.6. 

5.4. The General Characteristic-Noise Matrix 

We have introduced the most general matrix representation of a linear 
network in Eq. 5.6. We have defined the associated power matrix Qt in 
Eq. 5.15. With these two we may define a general characteristic-noise 
matrix Nr corresponding to this matrix representation. The require- 
ments are that this matrix Ny : 

1. Should reduce exactly to the form of Eq. 3.15 when the network is 
described on the impedance basis. 

2. Should be related to Eq. 3.15 by a similarity transformation when 
the same network is described on other than impedance basis (for example, 
admittance, scattering, and so forth). 

Under these conditions, Ny will contain the n network invariants as its 
eigenvalues. 

Here we shall follow the simple expedient of defining Nr and then 
proving its relationship to the matrix defined by Eq. 3.15. Thus, let 

Ny = {[l I -t] Qt-' [l I -t]}~'88T (5.19) 

For the impedance-matrix representation, using Eqs. 5.1, 5.6, and 
5.17, we obtain 

[l ; -Z] Qz-' [l ; -Zj = -2(Z + Z+) 
Introducing Eq. 5.20 into Eq. 5.19, we have 

Nz = -J(z + z+)-^eeT 



(5.20) 
(5.21) 



Sec. 5.4] THE GENERAL CHARACTERISTIC-NOISE MATRIX 39 

But, Eq. 5.21 is identical with the definition Eq. 3.15 for N. 

Next, let us relate the general noise matrix Nr of Eq. 5.19 to its partic- 
ular form in the impedance representation. For this purpose, we note 
that according to Eq. 5.14 

88^ = Mii^M^ (5.22) 

Then, using Eqs. 5.13, 5.18, and 5.20, we find 

[i! -t]q.-[i| -t]^ 

= m[i I -z]RQr~^R^[l | -z]^M^ 
= -2M(Z + Z+)Mt (5.23) 

Combining Eqs. 5.21, 5.22, and 5.23 with 5.19, we have finally 

Nr = M'^-^NzM'^ (5.24) 

According to Eq. 5.24, the characteristic-noise matrix Ny of the general 
matrix representation of a network is related by a similarity transforma- 
tion to the characteristic-noise matrix N^ of the impedance-matrix repre- 
sentation of the same network. Therefore, Ny and N^ have the same 
eigenvalues. 

The eigenvalues of the characteristic-noise matrix of Eq. 5.21 deter- 
mined the stationary values of the real quantity p in Eq. 3.10. Com- 
parison of these two expressions with the expression for the general 
characteristic-noise matrix in Eq. 5.19 shows that its eigenvalues deter- 
mine the stationary values of the associated real quantity pT 

yt[l , -t]Qj^-i[i ; -T] y 

with respect to variations of the arbitrary column matrix y. It is easily 
shown by the method of Sec. 3.2 that this "noise parameter" pr has, in 
fact, as extrema the eigenvalues of the matrix Nr defined in Eq. 5.19. 
The range of values that pr assumes as a function of y is identical with 
the range of values of p in Eq. 3.10 and Fig. 3.1. 

The network classification in the three cases illustrated in Fig. 3.1 
should now be restated in terms of the T-matrix representation. This is 
easily accomplished by considering Eq. 5.23. According to it, the matrices 

[l I -t] Qt~^ [l 1 -t]^ and -(Z 4- Z^) are related by a colinear 

transformation, A colinear transformation preserves the signature of 
a matrix. Consequently, the network classification of Table 3.1 can be 
carried out in the T-matrix representation, as shown in Table 5.1. The 
same conclusion may be reached from the facts that Nr and N^ have the 
same eigenvalues, and both EE^ and 85^ are positive definite. 



40 



Case 
1 

2 
3 



OTHER REPRESENTATIONS 

Table 5.1. Classification of Networks and Eigenvalues 
IN T-MATRix Representation 



[Ch.5 



[i;-t]q.-[i!-tJ 
positive definite 
negative definite 

indefinite 



Network 
Class 
passive 

active (negative 
resistance) 

active 



pT 
<0 

>0 



^0 



Eigenvalues 
(X.) of Nr 
alK 

all> 



jsome > 
Isome < 



Figure 3.1 gives directly the allowed range of pr and the eigenvalues 
of Nr, if the notation of Table 3.1 is replaced on the figure by that 
of Table 5.1. 

In the specific case of the mixed voltage-current representation of 
Eqs. 5.3 to 5.5, Nr and pr can be simplified if we introduce the detailed 
expressions for the power matrix Qy. This matrix is found most directly 
from the explicit expression for the power P flowing into the network in 



terms of the excitation-response vector 



Comparing the resulting 



expression with Eq. 5.15 allows identification of Qr by inspection. The 
power matrix Qr is square and of 4wth order 



Qt = 



1 



P 






-P 



where the P's are matrices of order 2m of the form 



P = 



"0 1 


















0" 


1 



























1 


















1 






























1 

















1 





-- 


-- 




















-- 


-- 


1 


_0 
















_ 


1 0. 



(5.26a) 



(5.266) 



Sec. 5.4] THE GENERAL CHARACTERISTIC-NOISE MATRIX 41 

It is easily checked that P has the properties 

p-i = pt = p (5.27) 

Substituting the particular form of Qr from Eq. 5.26a into the matrix in 
column 2 of Table 5.1, we obtain the matrix of 2wth order 

[l I -t] Qt-' [l ; -t]^ = 2(P - TPT^) (5.28) 

Thus, from Eq. 5.19 we have for Nr 

Nr = i(P - TPT+)-^85^ (5.29) 

With the introduction of the specific expression Eq. 5.28 into Eq. 5.25 
for the noise parameter pr, we find that it reduces to 

^^ = 2y + (P - TPT+)y ^^-^"^ 

The preceding development shows how each matrix representation T 
has associated with it a particular noise parameter pr, of which the 
extrema are determined by the eigenvalues of its characteristic-noise 
matrix. In the next chapter we shall develop in detail the significance of 
pT for two-terminal-pair amplifiers represented in terms of their general 
circuit constants. 




Noise Measure 



In Chap. 3, starting from the impedance-matrix representation, we 
defined a noise matrix N^. The eigenvalues of N^ gave the extrema of 
a scalar, pz, which was found to be the exchangeable power derived from 
the polyterminal network under consideration by the arrangement of 
Fig. 2.4. In Chap. 5, we defined a generalized noise matrix Ny, per- 
taining to the general matrix representation in Eq. 5.2. The eigenvalues 
of Nr and N^ are identical. Associated with the eigenvalues of Nr are 
the extrema of the generalized scalar parameter pT in Eq. 5.25. Special 
forms of Nr and px for the mixed voltage-current representation (Eqs. 5.3 
through 5.6) were given in Eqs. 5.29 and 5.30. In the case of a two- 
terminal-pair network, this representation reduces to the "general-circuit- 
constant" description. Our interest in the noise performance of linear 
amplifiers gives the two-terminal-pair case a special importance. The 
remaining part of our work will therefore be confined to the interpre- 
tation and study of ^r for the two-terminal-pair network in the general- 
circuit-constant representation. 

Our problem is to find the physical operation that leads to the extrema 
of pT, in the same manner as the operation of Fig, 2.4 led to the extrema 
of pz. It is obvious that the operations involved in the extremization will 
make use of lossless imbeddings, since only such operations leave the 
eigenvalues of Ny unchanged. Variations in these imbeddings will 
presumably produce variations in the column vector y in Eq. 5.30, and 
lead to the extrema oi pT- 

The general-circuit-constant representation of a two-terminal-pair 
network emphasizes its transfer characteristics. Therefore, we expect 

42 



Sec. 6.1] EXTENDED DEFINITIONS OF GAIN AND NOISE FIGURE 43 

that pT in this representation must be related to the noise performance 
of the network regarded as a transfer device. The noise figure has been 
for many years the most widely used parameter describing the noise 
performance of transfer devices. Consequently, we might well investi- 
gate first whether or not pr has any relation with the noise figure. 

6.1. Extended Definitions of Gain and Noise Figure 

The noise figure is normally defined in terms of available power. We 
have seen, however, that the available-power concept leads to diffi- 
culties in cases involving negative resistances. Since such cases must 
arise in any general theory of linear amplifiers, the available-power 
concept should be replaced everywhere by the exchangeable power. 
Accordingly, the same replacement should be made in the available-gain 
definition: The exchangeable power Pes of the input source and the 
exchangeable power Peo at the network output replace the corresponding 
available powers. Specifically, 



where 



and 



We find that 





G. = ^- 




P ^°' 




P ^'" 


G,>0 


if Rs/Ro > 


G. <0 


if Rs/Ro < 



(6,1a) 
(6.16) 

(6.k) 

(6.2a) 

(6.26) 

Observe that when Rs > and Ro > 0, Gg becomes the conventional 
available gain of the two-terminal-pair network. 

The foregoing ideas lead to an extended definition, Fg, of the noise 
figure of a two-terminal-pair network^ 

GekTo Af 

where Net is the exchangeable noise power at the network output when 
the source has a given (noiseless) impedance Zs- Thus, Nei contains 

^ H. A. Haus and R. B. Adler, "An Extension of the Noise Figure Definition," 
Letter to the Editor, Proc. I.R.E., 45, 690 (1957). 



44 



NOISE MEASURE 



[Ch.6 



only the effect of internal network noise sources. The term kTo Af in 
Eq. 6.3 represents the available noise power from the input source im- 
pedance held at equilibrium at standard temperature Tq, provided this 
impedance is passive (that is, has a positive real part). In the case of a 
source impedance with a negative real part, kTo Af merely represents an 
arbitrary but convenient normalization factor. 
From Eqs. 6.1, 6.2, and 6.3, it is clear that 



Fe-1>0 
Fe- KO 



if Rs>0 
if Rs <0 



(6.4a) 
(6.4&) 



li Rs > and Ro > 0, Fe becomes the standard noise figure F. 



E, 



Zs 

■AAAAAr 



-o + 



o 



Fig. 6.1. Thevenin equivalent of two- terminal source. 



6.2. Matrix Formulation of Exchangeable Power and Noise Figure 

The parameter pr of Eq. 5.30, whose relation with the noise figure we 
are going to investigate, is expressed in matrix notation. Accordingly, 
to facilitate comparison, we must express the (extended) noise figure and 
the related parameters in the same manner. 

Exchangeable Power. A linear two-terminal source of power can be 
represented by its Thevenin rms open-circuit voltage Es, in series with 
its internal impedance Zs. Let Vi and /i be the terminal voltage and 
current of the source, respectively, as shown in Fig. 6.1, so that 



Fi + Zsh = E, 



(6.5) 



To express Eq. 6.5 in matrix form suitable for cascade applications, we 
define column vectors (matrices) v and x as 



Then, for Eq. 6.5, we have 



xV = Ei 



(6.6) 
(6.7) 



Sec. 6.2] MATRIX FORMULATION OF EXCHANGEABLE POWER 45 

We note that Eq. 6.7 is the one-terminal-pair form of Eq. 5.1, where all 
the submatrices have become scalars. 

Since multiplication of Eq. 6.7 by a constant c does not alter it, we can, 
purely as a matter of form, always make a new column vector 



B:] = - = UA 



y = 

and a new scalar 

7 = cEs 

so that the source equation (Eq. 6.7) becomes 

y\ = 7 (6.8) 



with 



^ = Zs* (6.9) 

yi 



where Zs is still the internal impedance of the source. This formal 
multiplication feature of the source equation is helpful in interpreting the 
following analyses. 

Now the exchangeable power Pg of the source may be written in 
matrix form 

£|2 \T? \2 I-, 1 2 

„ = . S\ ^ 1-^51 ^ |7| (. .fxx 

^^ 2(Zs + Zs*) 2(xtPx) 2(y+Py) ^ ' ""^ 

where the square "permutation" matrix P is the two-terminal-pair form 
of Eq. 5.266. 



P = 



c a ^«"' 



It has the properties P^ = P and P~^ = P as indicated in Eqs. 5.27. 
The usefulness of the last expression in Eq. 6.10 lies in the fact that it 
can be written by inspection for any two-terminal source with a source 
equation in the form of Eq. 6.8. 

Exchangeable-Power Gain. Consider a linear source-free two- 
terminal-pair network, described by its general-circuit constants A,B, 
C, D, as shown in Fig. 6.2. If v is the "input" column vector defined by 
Eq. 6.6, and we let u be the "output" column vector, 

u = p;] (6.12.) 

and T is the general-circuit matrix, 



46 



NOISE MEASURE 



[Ch. 6 



the network equations are expressed in matrix form as 

V = Tu (6.13) 

When the network of Fig. 6.2 is driven by the source of Fig. 6.1, the 
exchangeable output power Peo from terminal pair 2 can be obtained at 



+ 




h 

> o 

+ 


Vi 


T = 


^2 







o 



Fig. 6.2. The general-circuit-matrix representation of a source-free two-terminal-pair 

network. 



once from a source equation in the form of Eq. 6.8 written with u as 
the voltage-current column variable. Multiplication of Eq. 6.13 by y^, 
with the use of Eq. 6.8, yields this new relation, 



(y^T)u = 7 



(6.14) 



Accordingly, by applying the steps of Eqs. 6.8, 6.10, and 6.11 to Eq. 6.14, 
we_find that 

|2 



PeO = 



T 



2ytxpTV 

and with Eq. 6.10, and Eq. 6.1a for the exchangeable gain, we have 

y^Py 



(6.15) 



G. = 



(6.16) 



ytXPXV 

Extended Noise Figure. To apply the matrix formulation of ex- 
changeable power to the calculation of the (extended) noise figure of a 
two-terminal-pair noisy network, we still describe the network by its 
general-circuit constants, but we also allow for noise voltages or currents 
at the terminals in the absence of external sources. With reference to 
Fig. 6.3, the network equations for the dotted box^'^ would be 

V = Tu -f 8 (6.17) 

2 A. G. Th. Becking, H. Groendijk, and K. S. Knol, "The Noise Factor of 4 Termi- 
nal Networks," Philips Research Repts. 10, 349-357 (1955). 

"H. Rothe and W. Dahlke, "Theory of Noisy Four Poles," Proc. I.R.E., 44, 
811-817 (1956). 



Sec. 6.2] MATRIX FORMULATION OF EXCHANGEABLE POWER 47 



with 8 a "noise column vector," 



8 = 



fc] 



(6.18) 

(6.19a) 
(6.19&) 



Now Equation 6.17 can be rewritten as two relations: 

v' = Tu 
V = V + 8 

If we visualize v' = K. ^, as referring simultaneously to the input 

terminals of a noise-free network T and the output terminals of a pure- 
noise network 8, the cascade division of the system is represented in 
Fig. 6.3. The noiseless part T does not affect the noise figure of the 
system. Thus, for noise-figure calculations, we need consider only the 
noise network 8 driven by a source of internal impedance Z^.^'^ 



+ 
^1 






h' 




v/ 



T = 



A B 
C D 



Noise network S 



Noise-free 
network T 



"H 



.J 



Noisy amplifier T, S 

Fig. 6.3. The general-circuit-matrix representation of a linear two-terminal-pair network 

with internal sources. 



The source equation appropriate to the right-hand terminals in Fig. 6.4 
can be obtained from Eqs. 6.19& and 6.8 with Es = 0. 

yW = -y+8 (6.20) 

Therefore, the output exchangeable power Nei, produced by the internal 
noise only, is given by 

y'*'88^y 



N,i = 



2y^Py 



and, since Ge = 1 for this network, we have 

Nei yWy 



F«- 1 = 



kTo A/ y^Py(2^ro A/) 



(6.21) 



(6.22) 



48 



NOISE MEASURE 



[Ch.6 



Enl 



.0 



// 



Noise network 5 

Fig. 6.4. Noise network of a linear noisy two-terminal-pair network in general-circuit-matrix 

representation. 



6.3. Noise Measure 

The "excess-noise figure," Eq. 6.22, has the same numerator as the 
noise parameter pr, Eq. 5.30, that we are trying to identify. The two 
equations differ only in the subtractive term y^TPT^y in the denominator, 
aside from the multiplicative constant kTo A/. In order to facilitate the 
identification, we rewrite Eq. 5.30 in the form 



pT = 



rW^ 



(6.23) 



From a comparison of Eq, 6.23 with Eqs. 6.22 and 6.16, it is obvious that 



pT 



kToAf 



(6.24) 



1 - 



The expression on the right-hand side of Eq. 6.24 may now be identi- 
fied as the noise parameter with the extremal properties corresponding to 
the network invariants. In the cases in which the extended definition of 
noise figure Fe coincides with the conventional noise figure F and the 
exchangeable gain Ge is equal to the available gain G, the quantity in 
Eq. 6.24 is identical with the noise measure, Eq. 1.3. We shall now adopt 
this same name in the general case when Fe and Gg differ from the con- 
ventional F and G, and denote this extended definition of noise measure 



Sec. 6.4] ALLOWED RANGES OF NOISE MEASURE 49 

by Mel 

^ _ Fe-l _ yWy 1__ 

'~ ,±~ jHP - TPTt)y {2kTo A/) ^^-^^^ 

Ge 

According to Eq. 6.9, the column vector y is determined, within a constant 
multiplier, by the source impedance Zs at which the noise measure Me 
of Eq. 6.25 is achieved. 

In the discussion of the noise performance of ampKfiers, it turns out to 
be important to bear in mind the algebraic sign that Me assumes under 
various physical conditions. These are summarized in Table 6.1. 

Table 6.1. Algebraic Signs or Exchangeable Gain 
AND Derived Quantities 

RS RO Ge Fe-1 \Ge\ Me 



>0 


>0 


>0 


>0 


>1 


>0 


>0 


>0 


>0 


>0 


<1 


<0 


>0 


<0 


<0 


>0 


^1 


>o 


<0 


>0 


<0 


<0 


^1 


<0 


<0 


<0 


>0 


<0 


>1 


<0 


<0 


<0 


>0 


<0 


<1 


>o 



With reference to Table 6.1, it should be pointed out that for Rs > 0, 
(Fe — 1) > 0. When Rs > 0, conventional available gain greater than 
1 occurs in only two ways : 

(a) Ge > 1 

(b) Ge<0 

Case a holds whenever the amplifier has an output impedance with 
positive real part and an available (or exchangeable) gain greater than 1. 
Case b corresponds to an amplifier with an output impedance having a 
negative real part. Such an amplifier has an infinite available gain in 
the conventional sense. In both cases Me is found to be greater than zero. 
In the succeeding portions of our work, we shall restrict our considera- 
tion of amplifier performance to cases in which the source has an internal 
impedance with a positive real part. This is the only case of practical 
interest. Indeed, any amplifying system, however complicated, is 
essentially a two-terminal-pair network driven by a signal transducer 
with a positive real part to its output impedance (for example, antenna, 
microphone, and so forth). 

6.4. Allowed Ranges of Values of the Noise Measure 

Let us consider a noisy two-terminal-pair network with the noise 
column matrix 8 and the matrix of general circuit constants T. We 



50 NOISE MEASURE [Ch. 6 

suppose that the input terminal pair of this network is connected to a 
source with the internal impedance Zs. The noise measure of the net- 
work as measured at its output terminal pairs is then given by Eq. 6.25, 
where the column vector y satisfies the relation 

^ = Zs* (6.9) 

Next, we suppose that the original network is imbedded in a four-termi- 
nal-pair, lossless network (Fig. 6.5) before we connect it to the source. 
A new network results, with the noise column vector 5' and the matrix T'. 
If one of the terminal pairs of the resulting network is connected to the 
same source, a new noise measure M/ is observed at the other terminal 
pair: 

, y+8'8'V 1 

We shall now determine how the primed matrices in Eq. 6.26 are related 
to the unprimed matrices of the original network. First, from Eqs. 3.3 
and 3.6 we know that after a lossless transformation 

E^E^ = T^EE^T (3.3) 

and 

Z' + Z'+ = T^(Z -h Z+)t (3.6) 

However, from Eqs. 5.22, 5.23, and 5.28 we have 

W^ = M' E^E^M't = M VEE^tM'"^ 
= (mVM-i)88^(mVm-i)'^ 
= CWC (6.27) 

and 

(P - T'PT't) = M'iZ' + Z'+)M'^ = -MV(Z + Z+)tM'+ 
= (mVm-^)(p - tptO(m t"^M-i)+ 
= Ct(P - TPTt)C (6.28) 

where 

C^ = MVm-1 (6.29) 

The matrix C involved in the colinear transformations of 88^ and 
P — TPT^ can be adjusted arbitrarily by arbitrary changes in the im- 
bedding network, on account of the matrix t of the lossless transformation 
that appears in C. 

Introducing the explicit transformations, Eqs. 6.27 through 6.29, into 



Sec. 6.4] 



ALLOWED RANGES OF NOISE MEASURE 



51 





/4 


Imbedding 
network 

Side Side 
(b) (a) 


/i 




Amplifier 
T, 5 


+ 






+ 
^1 




^3 


h 




+ 




+ 


° 






" 





h 


New 

amplifier 

T', 5' 


"l! 






I' 











— — 










— 


~ 










^1 










V2 








^1 




"a 








h 






Va 




. . . 


= 


• • 


• 


; V = 




' ' 


= 


• 


• 


• 


^3 




"6 








Va 






Vfc 




^3 










h 



















_ 













Fig. 6.5. Imbedding of two-terminal-pair amplifier. 



the noise-measure expression, Eq. 6.26, we have 

(Cy)W(Cy) 



MJ = 



(Cy)+(P - TPT^)(Cy) 2kTo A/ 



(6.30) 



Through variations of the imbedding network, the column vector Cy 
can be varied arbitrarily. In this manner Mj can be varied over its 
entire allowed ranges, which are limited by the two eigenvalues Xi and X2 



52 



NOISE MEASURE 



[Ch. 6 



of Nr. By making use of Table 5.1 and Eq. 5.28 in the special case of a 
two-terminal-pair network, we see that the following three cases have to 
be distinguished: 

LP — TPT"*" is negative definite; T is the general circuit matrix of a 
passive network, Mj < 0. 

2. P — TPT^ is positive definite; T is the general circuit matrix of a 
negative-resistance network, Mj > 0. 

3. P — TPT""" is indefinite; T is the general circuit matrix of a network 
capable of absorption, as well as delivery of power, Mj < 0. 



M^ 



Mp 



-m: = 



kTQ^f 

X2 



kTQ^f 



Mi 



kToAf 



kToAf 



-m; = o 



kTo^f 



■MJ = 



kToAf 



P - TPTt negative definite P - TPTt positive definite 
(a) (b) 



P - TPTT indefinite 
(c) 



Fig. 6.6. Schematic diagram of permitted values of Me' for two-terminal-pair networks. 

When C in Eq. 6.30 is varied through all possible values, M/ reaches two 
extrema, which are the two eigenvalues of the characteristic-noise matrix 
Nr divided by kTo A/. Now, we have pointed out that in practical 
situations amplifiers are driven from sources having an internal im- 
pedance with positive real part. According to Table 6.1, the noise meas- 
ure is positive when i?^ > in all cases except Rq > 0,0 < Ge < 1. 
This case does not correspond to an amplifier. Hence we shall be inter-" 
ested in achieving only positive values of Mj , which occur in Cases h and 
c of Fig. 6.6. These cases both have an available gain, G, in the con- 



Sec, 6.4] ALLOWED RANGES OF NOISE MEASURE 53 

ventional sense, greater than unity (G = Gg, if Gg > 1 ; G = <» , if 
Ge < 0). 

We observe from Eq. 6.30 that the numerator is never negative. 
Therefore, changes in sign of Mj occur with those of the denominator. 
In Case c, which includes most conventional amplifiers, M/ changes 
sign only at a zero of the denominator. Thus, the values of M/ cannot 
lie between \i/{kTo A/) and \2/{kTo A/). 

The two cases of interest. Cases b and c of Fig. 6.6, have a least positive 
eigenvalue of N which we call Xi > 0. We have therefore proved the 
following theorem : 

Consider the set of lossless transformations that carry a two-terminal- 
pair amplifier into a new two-terminal-pair amplifier with a conventional 
available gain G greater than 1 . When driven from a source that has an 
internal impedance with positive real part, the noise measure of the trans- 
formed amplifier cannot be less than \i/{kTo A/), where Xi is the smallest 
positive eigenvalue of the characteristic-noise matrix of the original amplifier. 

The fact that Me' > Xi/ikTo A/) also puts a lower limit upon the 
excess-noise figure. Suppose that the amplifier is imbedded in a lossless 
network and then connected to a source with an internal impedance 
having a positive real part. Let the resulting exchangeable power gain 
be Ge. Then, the excess-noise figure of the resulting ampHfier has to 
fulfill the inequality 

kTo^fV "Ge) ^^'^^^ 



F..-l> 



Consequently, an amplifier has a definite lower Hmit imposed on its 
excess-noise figure, and this Hmit depends upon the exchangeable-power 
gain achieved in the particular connection. 

If Ge > 0, that is, the output impedance of the amplifier has a positive 
real part, the excess-noise figure can be less than \i/(kTo A/) only to the 
extent of the gain-dependent factor (1 — l/Gg). 

If Ge < 0, that is, the output impedance has a negative real part, the 
lower limit to the excess-noise figure is higher than \i/{kTo A/) by 

(1 + WGe\). 

Thus, if two amplifiers with the same eigenvalue Xi of their characteristic- 
noise matrix are driven with a positive source impedance, one of which has a 
positive output impedance, the other a negative one, then the minimum 
excess-noise figure of the latter cannot be less than that of the former. 

Equation 6.31 has established a gain-dependent lower bound for the 
excess-noise figure achievable with lossless imbeddings of a given amplifier. 
At large values of |Ge|, the excess-noise figure is evidently equal to the 



54 NOISE MEASURE [Ch. 6 

noise measure. Large values of \Ge\ must correspond to large values of 
conventional available gain. Therefore, the excess-noise figure at large 
conventional available gain is limited to values greater than, or equal to, 
\i/kTo A/ under the most general lossless external network operations on 
the amplifier. These include, for example, lossless feedback, input mis- 
match, and so forth. On the supposition that the noise figure at large 
conventional available gain is a meaningful measure of the quality of 
amplifier noise performance, the minimum positive value of the noise 
measure, 

is a significant noise parameter of the amplifier. The significance of 
■^e.opt will be further enhanced by the proofs, given in the remaining 
sections, of the following statements : 

1. The lower bound Me, opt on the noise measure of an amplifier can 
actually be achieved by appropriate imbedding. Moreover, this is ac- 
complished in such a way that subsequent cascading of identical units 
realizes Me,opt as the excess-noise figure at arbitrarily high gain. 

2. An arbitrary passive interconnection of independently noisy ampU- 
fiers with different values of Me, opt cannot yield a new two-terminal-pair 
amplifier with an excess-noise figure at large conventional available gain 
lower than Me, opt of the best component amplifier (the one with the 
smallest value of Me, opt) • 

3. The use of passive dissipative imbedding networks for a given 
amplifier driven with a positive source impedance cannot achieve a posi- 
tive noise measure less than its Me, opt- 

We shall take up statements 2 and 3 first. 

6.5. Arbitrary Passive Interconnection of Amplifiers 

To prove statements 2 and 3 of Sec. 6.4, we begin by considering a 
general lossless interconnection of n independently noisy amplifiers, as 
shown in Fig. 6.7. A 2w-terminal-pair network results. By open- 
circuiting all but two of the resulting terminal pairs, we obtain the most 
general two-terminal-pair amplifier obtainable from the original ones by 
lossless interconnection. In Sec. 4.2 we have developed the general 
theory of such an imbedding and reduction of terminal pairs. Indeed, 
Fig. 4.3 includes the situation of Fig. 6.7. We know that the eigen- 
values of the characteristic-noise matrix of the network lie between the 
most extreme eigenvalues of the characteristic-noise matrices of the 
original n amplifiers. Therefore the lowest positive eigenvalue of the 



Sec. 6.5] PASSIVE INTERCONNECTION OF AMPLIFIERS 



55 



reduced network cannot be less than the lowest positive eigenvalue in the 
original lot. Accordingly, the optimum noise measure Me,opt of the re- 
duced network cannot be less than that of the best of the original 
amplifiers. 



l^n 


- 

4n-terminal-pair 

lossless imbedding 

network 


h 


Tl,5i 


+ 
V4n 




An-l 


h 


+ 
V4n-1 


V2 




• 
• 

hi-1 








Ti, 5; 




+ 


m 
• 
• 


hi 




+ 

V2i 




• 
• 

^2rt-l * 


■^2n+2 




Tn, 5„ 


+ * 

1^2/1+2 


V2n-1 


hn+1 


hn 


V2n + l 


+ 
V2n 







Fig. 6.7. Lossless interconnection of several amplifiers. 



We shall now extend our proof to cover passive interconnections of 
amplifiers. We again consider n amplifiers but shall now imbed them 
into a dissipative 4;z-terminal-pair network in the configuration of Fig. 6.7. 
Suppose, for a moment, that we disconnect the amplifiers and that we 
throw the imbedding network into its canonical form, as discussed in 
Sec. 4.1. This is accomplished by a lossless network transformation, and 
results in an arrangement Hke Fig. 4.2 but with 4:n separate resistances 
and uncorrelated noise generators. These resistances are arranged 
in the way shown at the bottom of Fig. 6.8. If we now apply to this 
canonical form the lossless network transformation inverse to the one 



56 



NOISE MEASURE 



[Ch.6 



that produced the canonical form, we obtain a 4w-terminal-pair network 
which has the same terminal behavior as the original dissipative im- 
bedding network. The resulting network is shown in Fig. 6.8. It has 
2n available terminal pairs. The 4w resistances can be comprised in a 
4w-terminal-pair network with a characteristic-noise matrix, the eigen- 



ES 





1 


0— — 1 




2 








, 1 






• . 






• ' 


n 


-1 
-2 








n 








• 






• ' 




-1 


' 


2n 


1 




2n 





L 




Passive dissipative 4re-terminal-pair 
imbedding network 



Fig. 6.8. Imbedding of n two-terminal-pair amplifiers in lossy 4«-terminal-pair network. 



values of which are all negative (Sec. ?>.?)). The n amplifiers can be 
grouped correspondingly into a 2w-terminal-pair network with a charac- 
teristic-noise matrix that has (in general, positive and negative) eigen- 
values equal to those of the characteristic-noise matrices of the original 
amplifiers. The network operation in Fig. 6.8 is obtained from that 
corresponding to Fig. 4.3 by a subsequent reduction from 6n to 2« output 
terminal pairs. Again, the least positive eigenvalue of the characteristic- 
noise matrix is not less than that of the best amplifier. Thus, we may 
state the following theorem: 

Consider a general lossless or passive dissipative interconnection of an 
arbitrary number of different and independently noisy two-terminal-pair 
amplifiers that results in a two-terminal-pair network. When the resulting 
two-terminal-pair network is driven from a source that has an internal 



Sec. 6.5] PASSIVE INTERCONNECTION OF AMPLIFIERS 57 

impedance with positive real part, its optimum noise measure Me, opt cannot 
be less than that of the best amplifier, that is, the amplifier with the least 
positive eigenvalue of its characteristic-noise matrix. 

Statement 3 of Section 6.4 is an immediate corollary of the previous 
theorem; that is, 

Passive dissipative imbedding of a given two-terminal- pair amplifier 
cannot reduce its noise measure to a positive value below Me,opt, provided 
that the source impedance has a positive real part. 




Net\vork Realization 
of Optimum Amplifier 
Noise Performance 



In Chap. 6 we have shown that the excess-noise figure of an amplifier 
with a high gain cannot be less than Me,opt, 

^.,ope = ^^ (6.32) 

where Xi is the smallest positive eigenvalue of the characteristic-noise 
matrix. We also showed that an arbitrary lossless or passive interconnec- 
tion of two-terminal-pair amplifiers, which leads to a new two-terminal- 
pair amplifier, yields an excess-noise figure at high gain that is higher than, 
or at best equal to, the ikfg, opt of the best ampHfier used in the inter- 
connection. These proofs established the quantity Me,opt as a lower 
bound on the noise performance of a two-terminal-pair amplifier. 

In this chapter we shall show that the lower bound Me, opt on the excess- 
noise figure at high gain can always be realized. Specifically, we shall 
show that the minimum positive noise measure Me,opt of any two- 
terminal-pair amplifier can be achieved by suitable external network 
operations, which, however, do not usually result in an amplifier with a 
high gain. Nevertheless, the source impedance, and the ampHfier output 
impedance with the source connected, will always have positive real parts 
in the realizations of Me,opt presented. It follows that an amplifier with 
an arbitrarily high gain can be constructed by cascading identical, 
optimized amplifiers that have appropriate impedance-transformation 
networks between the stages. By an adjustment of the transformation 
networks, the optimum noise measure of the cascade can be made equal 
to the Me,opt of the individual amplifiers in the cascade, as explained in 

58 



Sec. 7.1] CLASSIFICATION OF AMPLIFIERS 59 

Chap. 1. The excess-noise figure of the high-gain cascade is equal to the 
optimum noise measure of the cascade, and thus in turn equal to Me, opt 
of the individual amplifiers. This arrangement therefore accomplishes 
the realization of the lower limit of the excess-noise figure at high gain. 

Since Me, opt of any given amplifier determines the lowest (excess-) 
noise figure that can be achieved at high gain with the amplifier, either 
singly or in interconnection with other amplifiers of the same type, we 
may conclude from the criterion chosen in Chap. 1 that Me, opt is an 
absolute measure of the quality of noise performance of a given amplifier. 

7.1. Classification of Two-Terminal-Pair Amplifiers 

The noise-performance optimization problem is solved conveniently by 
referring to a detailed classification of nonpassive two-terminal-pair net- 
works (that is, amplifiers). Mason^ has shown that every such network 
can be reduced by lossless reciprocal imbedding to one of the three basic 
t)^es shown in Fig. 7.1. His classification is based primarily upon the 
range of values of the unilateral gain U 

U = ^^ ^ (7.1) 

4(i?ni?22 — ^12^21) 

where the i?'s are the real parts of the impedance-matrix elements. Since 
the numerical value of U is invariant to lossless reciprocal transforma- 
tions,^ none of the three types can be carried into any other type by such 
transformations. 

The first type (Fig. 7.1a), with C/ > 1, is by far the most common. 
The majority of vacuum-tube and transistor amplifiers belong to this 
class. The second type (Fig. 7.1&), with C/ < 0, is less common. It does, 
however, share one important property with the first ; namely, both have 

det (P - TPT+) < (7.2) 

which means that they can absorb as well as deliver power. It is perhaps 
not surprising, therefore, that with lossless nonreciprocal transformations 
amplifiers of the type of Fig. 7,16 can be carried into the form shown in 
Fig. 7.1a. This we now show. 
The network of Fig. 7.16 has a unilateral gain 

U = -\u\^ <0. 

By connecting the network in series with a lossless gyrator with the 

^ S. J. Mason, "Power Gain in Feedback Amplifiers," Trans. IRE, Professional 
Group on Circuit Theory, CT-1, No. 2, 20 (1954). 



60 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch. 7 



1 ohm 






1 ohm 



+ 

V2 



C/>1; lul>l 
Det(P - TPTt) = j-^ - l< 0; indefinite 



2uli 



1 ohm 



-1 ohm 



U-<0; lul^l 
Det(P - TPTt) = - (|-^ + l) < 0; 
(b) 



+ 
V2 



indefinite 



V^ -1 ohm 



2w7i 



-1 ohm 



+ 

V2 



0<U<1; |u|<l 
Det(P - TPTt) =7-T^ - 1 > 0; positive definite 

(c) 

Fig. 7.1. Classes of amplifiers. 



Sec. 7.2] OPTIMIZATION, INDEFINITE CASE 61 

impedance matrix 



Z = 



I (^-3) 



where r is real, we obtain, for the unilateral gain (Eq. 7.1) of the com- 
bined network, 

[Re(^)+rf + Im^(z^) . 

-1 +r2 + 2rRe(w) ^^^ 

In Eq. 7.4, U can always be made positive and greater than unity by 
choosing 



r > + \/[Re (w)]2 + 1 - Re (w) 

Thus, the network of Fig. 7.16 can always be given a unilateral gain that 
is positive and greater than unity. Then, according to Mason's work,^ 
it can be unilateralized and brought into the form of Fig. 7.1a. 
The amplifier in Fig. 7.1c, however, has 

det (P - TPT+) > (7.5) 

and cannot, under any terminal condition, absorb power. Obviously it 
cannot be reduced to any of the other forms by any lossless transforma- 
tion whatsoever. It is a "negative-resistance" amplifier. 

Recalling that the optimum noise measure is not changed by any loss- 
less transformation, we conclude that noise-performance optimization of 
amplifiers need only be carried out on networks of the specific forms 
Figs. 7.1a and 7.1c, and such a procedure will be sufficient to include all 
nonpassive cases. 



7.2. Optimization of Amplifier, Indefinite Case 

We imagine that the given amplifier with det (P — TPT^) < is 
initially in, or is reduced to, the form of Fig. 7.1a. We shall show that 
-^e.opt can be realized for this circuit by suitable input mismatch, retain- 
ing positive source impedance. 

The general circuit matrix of the amplifier in Fig, 7.1a is 

and the noise column matrix (elements not shown in Fig. 7.1a) is 

6 = [t:] (") 



62 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch. 7 
The characteristic-noise matrix of this amplifier can be computed to be 

N = J(P - TPT+)-W 



2 w 



1 



4(W^ - 1) 



-Eni 



2 w^ - 1 



+ En\ In\ 



\Enl\ + 



-Snl Inl 



2|wP - 1 



-1- \T 2 



En\Inl + 



In\ 



2 kr - 1 



(7.8) 



(6.25) 



2|w|2 - 1 
The amplifier noise measure in matrix form is given by Eq. 6.25 

yWy 1 

' y+(P - TVT^)y IUTq ^f 

where y is a column vector fulfilling the condition: 

yi 



in which Zs is the source impedance. 

The noise measure of Eq. 6.25 reaches its least positive value when 
the vector y is equal to that eigenvector y^^^ of N which pertains to the 
positive eigenvalue of N. The vector y is adjusted by an adjustment of 
the source impedance Zs. Hence, the noise measure can be optimized by 
a lossless impedance-matching network at the input of the amplifier if 
the actual source impedance with a positive real part can be transformed 
into the value Zs'^^ prescribed by the eigenvector y^^\ 



«»-©■ 



(7.9) 



Thus, if the noise measure is to be optimized by a lossless mismatching 
network, it is necessary and sufficient that 



Re 






Re {Zs^^n > 



(7.10) 



The proof that the inequality (Eq. 7.10) is fulfilled for \u\ > 1 will now 
be carried out. 

The proof is greatly facilitated if we use a wave formalism rather than 
the voltage-current formalism. We assume that transmission lines of 
1-ohm characteristic impedance are connected to the amplifiers. The 
incident waves ax and a^ and the reflected waves hi and 62 on these 



Sec. 7.2] OPTIMIZATION, INDEFINITE CASE 63 

transmission lines are related to the terminal voltages and currents by 

ai =i(Fi+7i); b2 = HV^ -\- h) 
b, =i(^i --^i); cL2 = i{V2-h) 

These transformations are conveniently summarized in matrix form. We 
define the matrix 



R=5 



^[1 -1] (-^) 



We then have 



The general-circuit-parameter representation of the network is 

V - Tu = 8 (5.2) 

The matrix equation for the new choice of variables v' and u' has the 
general form 

v' - T'u' = 8' (7.14) 

The relations between the matrices T' and 8', on the one hand, and T 
and 8, on the other hand, are easily derived by using Eqs. 7.12 and 7.13 
in Eq. 5.2 above. We obtain 

T' = RTR-i (7.15) 



5' = R8 = i 

2 



'8i + 62 
_5i — §2 



] = B:'] (^-^^^ 



Let us now rewrite the noise measure, Eq. 6.25, in terms of the matrices 
T' and 8'. For this purpose we note that, from the definition of Eq. 7.11 
for R, we have 

R = R+ = jR-i (7.17) 

Using Eqs. 7.16 and 7.17, we can write the noise matrix 88^ as 

88^ = R-^8^R-i = 4R8^R^ (7.18) 

Furthermore, we have from Eqs, 7.15 and 7.17 

P - TPT^ = P - R-iT'RPR+T'^(R-^)+ 

= 2R[2RPR+ - T'(2RPR^)T't]R^ 
= 2R(P' - T'P'T'+)R+ 



64 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch. 7 
where 



P' = 2RPR+ = RPR-^ = 



■ ■ [: -?] 



(7.19) 



Combining Eqs. 7,18 and 7.19, we obtain for the noise measure, Eq. 6.25, 

w^SV^w 1 



M. = 



w\F' - T'F'T'^)w kTo Af 



(7.20) 



with 



Lw2j ^ 2lyi-y2\ 



The noise measure expressed in terms of the voltage and current variables 
is optimized when y is an eigenvector of the noise matrix N. Correspond- 
ingly, the noise measure is also optimized in terms of the wave formulation 
if w is an eigenvector of the noise matrix 



N' = (P' - T'P'T'"^)-i8'5 



/T>/rr/t\-iFFT 



(7.22) 



The requirement that Re (>'2^^V3'/^^) > imposed on the eigenvector 
y^^^ imposes a corresponding limitation on w^^\ From Eq. 7.21, 



^2 




( Zs"" - 1 \ 
U^* + 1/ 



(7.23) 



Thus, W2/W1 is the negative-conjugate reflection coefficient corresponding 
to the source impedance Zs. Therefore, we must have 



^2 



< 1 



(7.24) 



From Eqs. 7.15, 7.16, 7.19, and 7.22, it is easily found that, for the net- 
work of Fig. 7.1a, 



N' = 



5/ 



2s /S /* 



\u? - 1 



l-dTW 



\u\^di 82 

IwP - 1 



-IT^ 



2 1 J 



(7.25) 



Sec. 7.2] 



OPTIMIZATION, INDEFINITE CASE 



65 



The positive eigenvalue of N' is 



x.=^ 



wr-1 



"1 "2 



I // Is /|2 Is /|2\ 1/1 r*" /I? /|2h; /|2 |s /? /* 2\ 



(7.26) 



From the definition of the eigenvector, the matrix equation results: 
of which the second component is 



-b^'H^W^ - WVW2^'^ = \,W2^'^ 



and thus 






U) 



-25/*52 



/*x / 



«P-1 



? '72 I Ts'T2 I 

\0l +02 + 



,WP-1 



, ^ , ^\2 

Isj / 2 J Is / 2 I 
lOl I +|02 I ) -' 



wr-1 



|5iV*r 

(7.27) 



But |w| > 1, and the eigenvalue Xi is real. Therefore, from Eq. 7.26 



|w| - 
which, in Eq. 7.27 yields 



Oi 02 



< 



(112 \ 

1^1 Is /|2 I Is '121 

.12 _ 1 I5l I + 1^2 1 J 



^2 



(1) 



Wl 



(1) 



< 1 



(7.28) 



The eigenvector w^^^ for which the noise measure is optimized thus 
corresponds to a reflection coefficient less than unity, that is, to a passive 
source impedance. The condition in Eq. 7.28 is equivalent to the condi- 
tion, Eq. 7.10, in the general-circuit-matrix notation. We have thus 
proved that the noise measure of a unilateral amplifier can be optimized 
with a lossless mismatching network between the source (having an 
impedance with a positive real part) and the amplifier. The output im- 
pedance of this amplifier always has a positive real part. It follows that 
any number of optimized amplifiers, with det (P — TPT'^) < 0, can be 
cascaded with appropriate lossless mismatching networks between suc- 
cessive stages so as to achieve an arbitrarily high gain. The excess-noise 



66 NETWORK REALIZATION OF OPTIMUM PERFORMANCE \Ch 7 

figure of the cascade is then equal to the minimum noise measure of each 
stage, which, with the aid of Eq. 7.8 for N, is 



M, 



Xi 



1 



e.opt 



kTo A/ 8kTo A/ 



^ + ^ ^2h^^^^____ 



- 1 




' +^4^ ^ 2 Re (£„i/.i*) 



u' - 1 



"^ 1^ I 12 _ 1 (l-Snin/nir ~ l-Enl-^nl*r) 

(7.29) 

The proof of the inequality in Eq. 7.28 can easily be extended to cover 
the case of nonunilateralized amplifiers of the class of Fig. 7.1a, provided 
they have passive conjugate-image impedances. We start with the net- 
work in the form that has the scattering matrix 



r 5i2"| 
[-§•21 J 



with 
and 



\S2l\ > 1 
1^121 < 1 



The only differences occur as minor modifications in Eqs, 7.25 ff., where 
|52il^ replaces \u\^ and (1 — |5'i2|^) appears in other terms. Thus, 
unilateralization is not a necessary step to achieve optimum noise measure 
with input mismatch. Amplifiers that have passive conjugate-image 
impedances can be optimized for noise measure by an input mismatch 
alone. However, the output impedance under optimized conditions is 
guaranteed to have a positive real part only if the amplifier is also stable 
under arbitrary passive input and output loading. Most vacuum-tube 
and transistor amplifiers meet these conditions over a significant fre- 
quency range. 



7.3. The Optimum Noise-Measure Expression for the Conventional 
Low-Frequency Vacuum Tube 

We shall now derive from Eq. 7.29 the expression for the minimum 
noise measure of a conventional low-frequency vacuum tube. The noise 
in the tube is characterized by a grid noise resistance Rn, the input imped- 
ance is Ri, and the plate resistance is rp. The noise-voltage column 



Sec. 7.3] OPTIMUM PERFORMANCE OF CONVENTIONAL TUBE 67 
matrix is 



E = 



B:] 



with 
and 



fE^P = UTo A/ {Rr, + Ri),,' 



£i 2 = ^kToAfRi 



E1E2* = -4:kToAffxRi 



The impedance matrix is 



Ri 
-tJ^Ri 



(7.30) 

(7.31a) 

(7.31&) 
(7.3k) 

(7.32) 



In order to make the impedance matrix of the triode represent the 
normalized form (Fig. 7.1a), suitable ideal transformers have to be 
connected to the input and output. The new impedance matrix then 
becomes 

1 0' 



Z' = 



-/? « 



(7.33) 



and the new open-circuit noise voltages of the over-all network are 

El 



E' 



'Ri 

E2 



L_ " 'P_l 



(7.34) 



Finally, in general-circuit-parameter form, the parameters of the triode 
become 



where 



Also 



T = - 



w = t/i 



2u\\ ij 



8 = 



2w 



Eo 2uEi 
+ 



VRi 



Eo 



Enl 



(7.35) 



(7.36) 



68 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch. 7 

With these specific values we obtain for the optimum noise-measure 
expression of the triode 



Mont 



—^ + ylRn^ + RlRn H T — ^ ~l 






(7.37) 



For large values of m, the minimum noise measure is effectively equal 
to the minimum excess-noise figure, and all terms in Eq. 7.37 divided by 
II can be disregarded. We obtain in the limit ju — ^ °° 



/^min = 1 + — (i?n + Vi?^^ + R^R^) 

This result is well known.^ 

7.4. Optimization of Negative-Resistance Amplifiers, 
Definite Case 

There remains the problem of achieving the optimum noise measure of 
negative-resistance amplifiers, that is, the class illustrated in Fig. 7.1c. 
This problem we now wish to solve, employing a positive source imped- 
ance and guaranteeing that a positive output impedance results. 

While it is actually possible to accomplish our purpose by performing 
a consecutive series of lossless reciprocal imbeddings, starting from the 
specific amplifier form given in Fig. 7.1c, the particular method we found 
for doing it was rather involved. It was also of little interest beyond its 
application to the present proof. 

Fortunately, there exists another method of optimizing the noise per- 
formance of any nonpassive network, including negative-resistance 
networks. This method is not only simple analytically but has a practical 
bearing upon the noise optimization of the new maser amplifier. We 
shall present this solution and its relation to the maser. 

We have shown in Chap. 4 that every two-terminal-pair network can 
be reduced by lossless nonreciprocal imbedding to the canonical form of 
Fig. 7.2, comprising two isolated (positive or negative) resistances in 
series with uncorrelated noise voltage generators. Moreover, the open- 
circuit noise voltages Eni and En2 and the two eigenvalues Xi and X2 of 
the characteristic-noise matrix N are directly related: 



J^nl ^ _x^; ^ ^ _X2 (7.38) 



4i?i ^' 4i?2 

2 A. van der Ziel, Noise, Prentice-Hall, New York (1954) 



Sec. 7.4] 



OPTIMIZATION, DEFINITE CASE 



69 



In the special case of a negative-resistance amplifier, the eigenvalues Xi 
and X2 are both positive. Accordingly, resistances R\ and R2 of the 
canonical form are both negative. We suppose now that the eigenvalue 
Xi has the smaller magnitude. According to the theory of Chap. 6, this 






^ 



i?] 



R2 



EnlEn2 = 



J 



Fig. 7.2. Canonical form of two-terminal-pair amplifier. 



eigenvalue determines the lowest achievable value of the noise measure 
Me. We shall now prove that this lowest value, Xi/{kTo A/), can indeed 
be achieved using only that terminal pair of Fig. 7.2 which contains the 
negative resistance Ri and noise generator Eni- 

As shown in Fig. 7.3, the terminal pair {Ri, En\) of the canonical form 
is connected to terminal pair 2 of an ideal lossless circulator with the 
scattering matrix 



S = 



(Transmission lines with 1-ohm characteristic impedance are connected 
to all four terminal pairs of the circulator.) Terminal pair (4) of the 
circulator is matched to a 1-ohm load at a temperature Tq, terminal pair 
(1) is used as the input, and terminal pair (3) is used as the output 
(Fig. 7.3). 

The equations for the resulting two-terminal-pair network can easily 
be derived using the scattering-matrix representation. We find that the 












r 


1 














1 














1 






70 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch. 7 



Input (1) o- 



:i?n=i ' 



rM-V4WW 




(2)'? 



-o (3) Output 



i?i 



Fig. 7.3. Realization of optimum amplifier noise performance from canonical form of the 

amplifier. 



amplifier is unilateral and is described by the equations : 






&3 + 5/ 



where 



WV = kTo A/ 



h 



'2 _ 



^nl 



(1 - RiY 



'1 02 



The available gain G of the amplifier with a 1-ohm source is 

= -(ItI)' 

The excess-noise figure of the resulting amplifier is 

F - 1 



5/ 



kToAf 



(7.39) 
(7.40) 

(7.41a) 
(7.41&) 
(7.41c) 

(7.42) 
(7.43) 



Sec. 7.4] OPTIMIZATION, DEFINITE CASE 71 

Thus, according to the foregoing results and Eq. 7.38, 



M = M. = ^ = - ,/"^ ,, = -^, (7.44) 



_ l^ ^RikTo A/ kTo A/ 

G 

We have therefore proved that the circulator arrangement indeed 
achieves the lowest possible noise measure. Since it also leads to a 
unilateral amplifier with positive real input and output impedances, an 
arbitrary gain can be achieved through cascading of such identical ampli- 
fiers. We observe, however, that a lossy network (ideal lossless circulator 
plus Rq) has been employed with the original amplifier to optimize its 
noise performance. 

The optimization carried out in connection with Fig. 7.3 has a useful 
corollary concerning circuit connections of maser amplifiers for optimum 
noise performance. One of the forms of the maser has for an equivalent 
circuit a one-terminal-pair negative resistance Ri in series with a noise 
voltage generator Eni- To make a two-terminal-pair network, we may 
consider as an artifice not only the noisy negative resistance Ri of the 
maser but also another positive resistance R2 at a temperature T2. The 
two resistances can be treated as the canonical form of a two-terminal-pair 
network. Lossless imbedding of these two resistances therefore leads to a 
two-terminal-pair amplifier with the eigenvalues Xi = — \Eni\/{4:Ri) > 
and X2 = —kT2 A/ < 0. The best noise measure that can be expected 
from the resulting amplifier is If opt = ^i/{kTo A/). The circulator 
arrangement has been shown to achieve this noise measure. Thus, it 
provides one of the optimum network connections of the maser with 
regard to noise performance. It should be re-emphasized that the as- 
sumed presence of a positive resistance R2 in the circuit is an artifice that 
enables the use of the theory of two-terminal-pair networks for the noise 
study of the one-terminal-pair maser. The assumed temperature of the 
resistance is immaterial because it determines only the negative eigen- 
value of the characteristic-noise matrix, which has no relation to the 
optimum noise measure achieved with gain. 

The results of this chapter lead to the following theorem : 

1. Any unilateral amplifier with U > 1 may be optimized with input 
mismatch alone. 

2. A nonunilateral amplifier with U > 1, which is also stable for all 
passive source and load impedances, may be optimized with input mismatch 
alone. 

3. Any amplifier with U > 1 may be optimized by first making it uni- 
lateral, using lossless reciprocal networks, and subsequently employing input 
mismatch. 



72 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch. 7 

4. Amplifiers of the class U < can be optimized by first transforming 
them into the class U > 1 by lossless nonreciprocal imbedding. The optimi- 
zation methods \ to Z can then be applied to this class. 

5. Negative-resistance amplifiers (0 < Z7 < 1) can be optimized by first 
transforming them into the canonical form. The terminal pair of the canon- 
ical form that possesses the exchangeable power of smaller magnitude is 
connected into a lossless circulator with a positive (1-ohm) balancing resistor, 
as shown in Fig. 7.3. The resulting unilateral two-terminal- pair network, 
driven from a 1-ohm source, achieves the optimum noise measure. 




Conclusions 



The developments that we have undertaken have been rather lengthy. 
Therefore, it is worth while to reassess and summarize our principal re- 
sults as well as our omissions. 

As pointed out in the introduction, the original motivation for the 
present work was the desire to describe in a systematic manner the single- 
frequency noise performance of two-terminal-pair linear amplifiers. It 
was necessary at the outset to elect a criterion of noise performance, which 
we chose to be the signal-to-noise ratio achievable at high gain. This 
criterion is not clear for systems without gain, nor for multiterminal-pair 
networks. For multiterminal-pair networks, the noise parameter pT 
expressed in terms of the general circuit constants has been set down as 
an extension of the two-terminal-pair noise-measure definition but has 
not been given any physical interpretation in this work. One reason for 
this omission is the fact that a general-circuit-constant (or wave-matrix) 
description of multiterminal-pair systems has been of little use in the past. 
There have not been any systems incorporating gain whose noise per- 
formance on a multiterminal-pair basis was of interest. It is true that 
in the past some special problems involving frequency conversion have 
called for proper interpretations, and that two-terminal-pair networks 
processing sidebands may be analyzed theoretically as multiterminal-pair 
networks. But a sophisticated theoretical approach to noise problems of 
this nature was never necessary. Problems of this type were easily dis- 
posed of by inspection. 

Recently, parametric amplifiers (nonlinear-, or time-varying-, reactance 
amplifiers) have received a great deal of attention because of their low- 

73 



74 CONCLUSIONS [Ch. 8 

noise characteristics. In parametric amplifiers correlation between signal 
sidebands often occurs and must be taken into account in the mathe- 
matical analysis. For such an analysis a systematic theory of noise in 
multiterminal-pair networks involving correlation between signal and/or 
noise sidebands is required. It is also probable that, because of these very 
same signal correlations between sidebands, the appropriate theory for 
the parametric case may not be merely the theory of the generalized noise 
parameter pT introduced in this work. We have not had the opportunity 
to pursue this interesting question in much detail. But we have studied 
the question sufficiently to be convinced that the general matrix methods 
of dealing with power and power ratios employed in the present study 
will help greatly in the analysis and understanding of these somewhat 
more difficult problems. 

The remarks of the previous paragraphs do not, of course, imply that 
multiterminal-pair networks have been neglected completely. Indeed, in 
the impedance-matrix formulation we have given extensive attention to 
the exchangeable-power interpretations of the network invariants in the 
multiterminal-pair case. As a practical application of these ideas, we may 
refer again to the work of Granlund^ regarding the problem of combining 
a multiplicity of antenna outputs into a single receiver, when the inputs 
to the antennas are statistically related. Furthermore, from the imped- 
ance formulation we have been able to develop a canonical form for the 
multiterminal-pair network. The merit of this form is that it leads to a 
simplification in thinking about single-frequency noise and gain char- 
acteristics of linear networks. 

The problem of considering noise performance over a broad band, 
rather than at a single frequency, appears to be covered by the spot-noise 
discussions that we have conducted. Certainly, in a two-terminal-pair 
amplifier one could adopt the position of optimizing the noise measure at 
each frequency in the band. Although such a procedure might involve 
complicated feedback variations with frequency and/or intricate match- 
ing systems, these are principally network-synthesis problems that pre- 
sumably could be solved on the basis of suitable approximations, if it 
appeared desirable to do so. There is no doubt that such a solution would 
give the "optimum noise performance" of the amplifier. By this we mean 
that the optimum is to be interpreted as the "best signal-to-noise ratio at 
high gain, at each frequency within the band." It is by no means obvious 
that, with the over-all system in mind, such a solution is always the 
best. There are many other considerations besides noise performance 
which enter into the design of wideband amplifiers, such as the behavior 

^ J. Granlund, Topics in the Design of Antennas for Scatter, M.I.T. Lincoln Labora- 
tory Technical Report 135, Massachusetts Institute of Technology, Cambridge, Mass. 
(1956). 



Ch. 8] CONCLUSIONS 75 

of the phase characteristic, transient response, and the uniformity of the 
gain — to say nothing of over-all circuit complexity. Therefore, it seems 
clear that no general theory of noise performance of such wideband 
systems should be undertaken without attention to other system require- 
ments. 

The last point brings us to the question of the usefulness of an optimum 
noise-performance criterion of the type we have presented in this study. 
It is probable that such a criterion will serve primarily as an indication of 
the extent to which a given design, which has met a variety of other 
practical conditions, fails to achieve its best noise performance. In other 
words, one may very well not attempt to realize the optimum noise 
measure directly but use it instead as a guide to detect the onset of 
diminishing returns in further efforts to improve noise performance. 



Ind 



ex 



Amplifier, negative-resistance, 61, 68, 72 

nonunilateral, 66, 71 

parametric, 73 

two-terminal-pair, classification of, 59 

Z7 < 0, 72 

unilateral, 71 

unilateral U > 1, 71 

wideband, 74 
Amplifier cascade, 2 

Amplifier noise performance, criterion for, 5 
Amplifiers, interconnection of, 55 
Available power, 14 

Becking, A. G. Th., 46, 47 
Belevitch, V., 33 

Canonical form, 31 

derivation of, 28 
Cascading, problem of, 3 
Characteristic-noise matrix, eigenvalues of, 23 

general formulation of, 38 

impedance formulation of, 22 

mixed voltage-current formulation of, 41 

trace of, 23 
Circulator, 69, 72 
Classification of networks and eigenvalues, 24 

in T-matrix representation, 40 



Classification of two-terminal-pair ampli- 
fiers, 59 
Cross-power spectral densities, 10 

Dahlke, W., 46, 47 

Eigenvalues, 22 

classification of, 24 

least positive, 53 
Excess-noise figure, 48 

lower limit imposed on, 53 
Exchangeable power, 15, 44 

matrix form for, 19 

matrix formulation of, 44 

n-terminal-pair networks, 15 

stationary values of, 22 
Exchangeable-power gain, 43, 46 

algebraic signs of, 49 
Extended noise figure, 43, 46 

Feedback, 3 
lossless, 54 
Franz, K., 2 
Friis, H. T., 2 

Gain, available, 43 
exchangeable-power, 43, 46 



77 



78 



INDEX 



Gain, extended definitions of, 43 

unilateral U, 59 
General-circuit-parameter representation, 36, 

61 
Granlund, J., 17, 74 
Grid-noise resistance, 66 
Groendijk, H., 46 
Gyrator, lossless, 59 

Hermitian conjugate, 11 
Hermitian matrix, 11 

Imbedding, 12 

lossless, 20 

passive dissipative, 57 

reciprocal, 59 
Impedance representation, 6, 9, 38 
Indefinite matrix, 16, 52 
Interconnection of amplifiers, lossless, 6, 54 

passive dissipative, 55 
Invariants, 25, 31 

Knol, K. S., 46 

Lossless circulator, 69 

Lossless feedback, 54 

Lossless gyrator, 59 

Lossless imbedding, 20 

Lossless interconnection, 54 

Lossless transformation, 9, 12, 19, 61 

Losslessness, condition of, 13 

Maser, 7, 68 

Mason, S. J., 3, 59, 61 

Matrix, characteristic-noise, 22, 38, 41 

general-circuit, 61 

general representation, 34 

Hermitian, 11 

impedance, 9 

indefinite, 16, 52 

negative definite, 16, 52 

noise column, 47, 61 

permutation, 45 

positive definite, 11, 16, 52 

scattering, 69 

semidefinite, 11, 16 

T, 34, 39, 40 
Matrix formulation, of exchangeable power, 
44 

of exchangeable-power gain, 45 

of extended noise figure, 46 



Matrix formulation, of stationary-value 

problem, 19 
Mismatch, input, 6, 54, 61, 71 

»-to-w-terminal-pair network transforma- 
tions, 6, 25 
»-to-one-terminal-pair network transforma- 
tion, 6, 18 
Negative definite matrix, 16, 52 
Network transformations, n-to-w-terminal- 
pair, 6, 25 

w-to-one-terminal-pair, 6, 18 
Networks, classification of, 14, 24, 40 

in thermal equihbrium, 25 

pure-noise, 47 

with coherent sources, 25 
Noise column matrix, 47, 61 
Noise figure, excess, 48 

extended, 43, 46 

matrix formulation of, 46 
Noise measure, 4, 42, 48 

allowed ranges of, 49 

for conventional low-frequency vacuum 
tube, 66 

for triode, 66 

optimum (Afe.opt), 54, 55 
Noise parameter pT, 39 

in mixed voltage-current representation, 41 
Noise performance, criterion for amplifier, 5 

optimization of, 61, 68 

single-frequency, 1, 73 

spot, 1 
Nonreciprocal transformations, 59 
Nonunilateral amplifier, 66, 71 

Optimization, of amplifier, indefinite case, 61 

of maser amplifiers, 7 1 

of negative-resistance amplifiers, definite 
case, 68 
Optimum noise measure (ilfe.opt), 54, 55 

for low-frequency vacuum tube, 66 

Parametric ampUfier, 73 
Passive dissipative imbedding, 56 
Passive dissipative interconnection of ampli- 
fiers, 56 
Passive network at equilibrium, 26 
Permutation matrix, 45 
Positive definite matrix, 11, 16, 52 
Power, available, 14 
exchangeable, 15, 44 



INDEX 



79 



Power spectral densities, 11 

Reciprocal imbedding, 59 

Reduction of number of terminal pairs, 32, 

56 
Robinson, F. N. H., 3 
Rothe, H., 46, 47 

Scattering-matrix representation, 69 
Self-power spectral densities, 10 
SemideJBnite matrix, 11, 16 
Signal-to-noise ratio, 1, 73 
Single-frequency noise performance, 1, 73 
Spectral densities, cross-power, 10 

power, 11 

self-power, 10 
Spot-noise figure, 2 

Stationary-value problem, eigenvalue formu- 
lation of, 21 

matrix formulation of, 19 
Stationary values of exchangeable power, 
22 



T-matrix representation, 34, 39, 40 
Terminal-voltage and current vector, 38 
Thevenin representation, 14 
Trace of characteristic-noise matrix, 23 
Transformation, lossless, 9, 12, 19, 61 
w-to-w-terminal-pair network, 6, 25 
»-to-one-terminal-pair network, 6, 18 
Transformation from one matrix representa- 
tion to another, 35 
Transformation network, 12 
Transistor amplifiers, 59 
Twiss, R. Q., 26 

Unilateral ampUfier, 71 

Unilateral amplifier with U > 1, 71 

Unilateral gain U, 59 

Vacuum tube, 59, 66 
van der Ziel, A., 2, 68 
Vector, noise column, 47, 66 

Wideband amplifiers, 74 



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