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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: 5911473
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 welldeveloped
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 twoterminalpair 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 powergain 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 matrixeigenvalue 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 multiterminalpair 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 matrixeigenvalue
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 signaltonoiseratio criterion. The novel
features of this tag are two in number: First, it clears up questions of
the noise performance of lowgain 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 negativeresist
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
ImpedanceMatrix Representation of Linear Networks
2.1
2.2
2.3
Lossless Transformations
Network Classification in Terms of Power
CHAPTER 3
Impedance Formulation of the CharacteristicNoise 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 CharacteristicNoise 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 TwoTerminalPair Amplifiers
Optimization of Amplifier, Indefinite Case
The Optimum NoiseMeasure Expression for the Con
ventional LowFrequency Vacuum Tube
Optimization of NegativeResistance 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 signaltonoise 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 noiseandinf ormation problem are less important.
The term "spotnoise performance" or merely "noise performance" will
be used to refer to the effect of the amphfier upon the singlefrequency
signaltonoise 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 spotnoise 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 multiterminalpair 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 manmade 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 overaU
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 singlestage 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 overall noise
figure of the cascade. The minimumnoisefigure 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 activenetwork
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 noiseperformance 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)
lowgain 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, PrenticeHall, 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, CT1, 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^2l
Gl
or
{F,  1)  {F2  1)< ^^ 
F2 1
Gl
or
F,\ ^F2l
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 noisefigure 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 overall twoterminalpair 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 overall
twoterminalpair 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 highgain twoterminalpair 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 twoterminalpair 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 impedancematrix 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 wterminalpair network
are found as extrema of its exchangeable power, with respect to var
iations of a lossless wtooneterminalpair network transformation. It
is found that an wterminalpair network possesses not more than these
n invariants with respect to lossless wtowterminalpair 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 twoterminalpair network, a particularly
important interpretation of the invariants is brought out by the general
circuitparametermatrix 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 noisemeasure concept and to the
range of values that the noise measure may assume for a twoterminal
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
terminalpair amplifiers which result in an overall twoterminalpair
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 twoterminalpair
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 twoterminalpair
amplifier can be specified in terms of a single number that includes the
gain and that applies adequately to lowgain 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, CT5, No. 3, 156 (1958).
^"H. A. Haus and R. B. Adler, "Canonical Form of Linear Noisy Networks,"
IRE Trans, on Circuit Theory, CT5, 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 wterminalpair 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
terminalpair network raises some interesting questions leading to a
generaKzation of the availablepower concept: the exchangeable power.
In Chap. 3, the exchangeablepower concept in turn leads to the discovery
of the invariants mentioned in Chap. 1.
2.1. ImpedanceMatrix Representation of Linear Networks
At any frequency, a linear wterminalpair 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 opencircuit terminal voltages £i, E2, • • • , En (Fig. 2.1).
In matrix form, Z denotes a square nhyn array
Z =
Zn
• Zu ■
' ■ Zik ■
■ Zin
Zn •
• Zu •
• Zik •
^ in
Zki •
■ ■ Z]^i •
• ■ Zkk •
• Zkn
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 opencircuit terminal voltages are represented conveniently by a
column matrix E:
'El
E =
E
E
En.
If the internal generators are randomnoise sources, the Fourier ampli
tudes El • ' • En are complex random variables, the physical significance
of which usually appears in their self and crosspower 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, McGrawHill
Book Company, New York (1958).
Sec. 2.1]
IMPEDANCEMATRIX 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 rootmeansquare 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 iCLn
EkE,* •
• • EkEi"" ■
' ■ EkEk •
■ • EkEn"
F F *
(2.1)
where the superscript dagger indicates the twostep 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, PrenticeHall, 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 wterminalpair network with the generator column matrix E and
the impedance matrix Z is connected properly to a 2wterminalpair
network, a new wterminalpair 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 Mterminalpair 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 wterminalpair net
work so as to obtain a new wterminalpair network.
The analytical relation between the voltages and currents applied to
the 2wterminalpair, 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 2wthorder 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 timeaverage 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 wterminalpair 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 wterminalpair network are, according to Fig. 2.2,
equal and opposite to the currents !« into one side of the 2wterminalpair
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 wterminalpair network
obtained from the original one by imbedding it in a 2wterminalpair
network. Here Z' is the new impedance matrix, and E' is the column
matrix of the new opencircuit 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 oneterminalpair 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 oneterminalpair linear network.
If the Thevenin representation of the source (Fig. 2.3) has the
complex opencircuit 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
meansquare 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) conjugatematch 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 exchangeablepower definition to
«terminalpair 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 (saddlepoint) value of the
average output power with respect to variations of the terminal currents,
and has no analog in the case of a oneterminalpair network (Eq. 2.1 1&).
We have defined the exchangeable power for a oneterminalpair 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 wterminalpair 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 multiterminalpair 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) terminalpair
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 noisynetwork
poweroptimization 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 wterminalpair 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)terminalpair 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
terminalpair 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 stationaryvalue
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 "characteristicnoise 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 noiseperformance
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 wterminalpair network Z in a
lossless 2wterminalpair network, as indicated in Fig. 2.2. Opencircuit
ing all terminal pairs of the resulting wterminalpair network Z', except
the ith, we achieve the wto1 terminalpair 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
CHARACTERISTICNOISE 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 doublepole wthrow 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 StationaryValue 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 CHARACTERISTICNOISE 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
ABi = i(Z + Z+)iEE+.
We therefore define:
Characteristicnoise 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 characteristicnoise matrix N.
Sec. 3.3] PROPERTIES OF THE EIGENVALUES 23
3.3. Properties of the Eigenvalues of the CharacteristicNoise
Matrix in Impedance Form
We shall now confirm the assertion made earlier (Sec. 2.3) to the
effect that:
The exchangeable power of the nterminalpair 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 characteristicnoise 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 wterminalpair 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 characteristicnoise 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 CHARACTERISTICNOISE 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 fourterminalpair 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 characteristicnoise
matrix N is imbedded in a 2wterminalpair lossless network, as shown in
Fig. 2.2. A new wterminalpair 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 2wterminalpair 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 terminalpair
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
isticnoise matrix and its eigenvalues.
If the wterminalpair 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 CHARACTERISTICNOISE 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 wterminalpair network has a characteristic noise matrix of nth.
order, with n eigenvalues. If k of the n terminal pairs of the network are
left opencircuited and only the remaining {n — k) terminal pairs are
used, the original network is reduced to an {n — ^) terminalpair net
work. This operation may be thought of as a special case of a more
general reduction, achieved by imbedding the original wterminalpair
network in a lossless {2n — ^) terminalpair 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 characteristicnoise matrix of the {n — ^) terminalpair network
has {n — k) eigenvalues, which determine the extrema of the exchange
able power Pe,nk+\ obtained in a subsequent (variable) reduction to one
terminal pair (Fig. 3.2&). The successive reduction of the wterminal
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,nk+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,nk+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 — ^) terminalpair network must lie within the allowed range of Pe,i
for the original wterminalpair 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— ^)terminalpair
network
1
Z, E
•
•
•
•
•
• •
•
•
•
Fig. 3.2a. Reduction of wterminalpair network to (« — k) terminal pairs.
nk+l
Variable
lossless
network
n—k
Lossless
(2n^)termlnalpair
network
Z, E
Fig. 3.2b. Successive reductions of « terminalpair 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 wterminalpair network, its
existence also shows that an «terminalpair, 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 nterminalpair 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 wterminalpair 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 noisefree) 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 opencircuit 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 characteristicnoise
matrix invariant. Thus, the eigenvalues of the characteristicnoise
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 nterminalpair 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 nterminalpair 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 ;^terminalpair noisy network
and an independently noisy wterminalpair network. Let them be
connected through a 2(w + w) terminalpair lossless network, resulting
in an {m \ w) terminalpair network, as shown in Fig. 4.3. We shall now
determine the eigenvalues of the characteristicnoise matrix ^m+n
of the resulting (m f w) terminalpair 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 characteristicnoise
matrices. Taken together, the two canonical forms represent the canoni
cal form of the {m f w) terminalpair 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)terminalpair
network
1
Z/nj ^m
•
•
•
I
•
•
•
•
•
•
m
i
1
^
_
Z„,E„
•
•
•
•
•
•
m + n
•
•
•
n
Fig. 4.3. Lossless interconnection of wterminalpair and « terminalpair networks.
of several twoterminalpair amplifiers, with feedback, to form a new
two terminalpair 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 characteristicnoise matrix of an nterminalpair
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 characteristicnoise matrix
can be defined. As we might expect, all such characteristicnoise matrices
have the same eigenvalues, since, after all, these are the only invariants
of the network.
5.1. General Matrix Representations
The impedancematrix 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, "FourDimensional Transformations of 4pole Matrices with
Applications to the Synthesis of Reactance 4poles," IRE Trans, on Circuit Theory,
CT3, 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 2wterminalpair
network, we consider the mixed voltagecurrent 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
+
Vi
•
•
•^2ml
En(2m1)
+
\y
1
^2ml
(P /n{2/nl)
hm
+
y2m
Fig. 5.1. Mixed voltagecurrent representation of 2OTterminalpair 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 characteristicnoise matrix for every
formalism and to show its relation to the characteristicnoise 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
2wterminalpair network, R is of order 4w. For example, the transforma
tion from the impedance representation of a 2 terminalpair network into
the generalcircuitparameter 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 excitationresponse 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 impedancematrix 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 generalexcitation vector
into the voltage and current vector
, according to Eq. 5.7.
We are now ready to set up the general characteristicnoise matrix N
for any matrix representation, Eq. 5.6.
5.4. The General CharacteristicNoise 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 characteristicnoise
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 impedancematrix 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 CHARACTERISTICNOISE 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 characteristicnoise matrix Ny of the general
matrix representation of a network is related by a similarity transforma
tion to the characteristicnoise matrix N^ of the impedancematrix repre
sentation of the same network. Therefore, Ny and N^ have the same
eigenvalues.
The eigenvalues of the characteristicnoise 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
characteristicnoise 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 Tmatrix 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 Tmatrix 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 TMATRix 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 voltagecurrent 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 excitationresponse 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 CHARACTERISTICNOISE MATRIX 41
It is easily checked that P has the properties
pi = 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 characteristicnoise
matrix. In the next chapter we shall develop in detail the significance of
pT for twoterminalpair amplifiers represented in terms of their general
circuit constants.
Noise Measure
In Chap. 3, starting from the impedancematrix 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 voltagecurrent representation (Eqs. 5.3
through 5.6) were given in Eqs. 5.29 and 5.30. In the case of a two
terminalpair network, this representation reduces to the "generalcircuit
constant" description. Our interest in the noise performance of linear
amplifiers gives the twoterminalpair case a special importance. The
remaining part of our work will therefore be confined to the interpre
tation and study of ^r for the twoterminalpair network in the general
circuitconstant 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 generalcircuitconstant representation of a twoterminalpair
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 availablepower concept leads to diffi
culties in cases involving negative resistances. Since such cases must
arise in any general theory of linear amplifiers, the availablepower
concept should be replaced everywhere by the exchangeable power.
Accordingly, the same replacement should be made in the availablegain
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 twoterminalpair network.
The foregoing ideas lead to an extended definition, Fg, of the noise
figure of a twoterminalpair 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
Fe1>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 twoterminal source of power can be
represented by its Thevenin rms opencircuit 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 oneterminalpair 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 twoterminalpair 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 twoterminal source with a source
equation in the form of Eq. 6.8.
ExchangeablePower Gain. Consider a linear sourcefree two
terminalpair network, described by its generalcircuit 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 generalcircuit 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 generalcircuitmatrix representation of a sourcefree twoterminalpair
network.
once from a source equation in the form of Eq. 6.8 written with u as
the voltagecurrent 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
twoterminalpair noisy network, we still describe the network by its
generalcircuit 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, 349357 (1955).
"H. Rothe and W. Dahlke, "Theory of Noisy Four Poles," Proc. I.R.E., 44,
811817 (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 noisefree 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 noisefigure 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
Noisefree
network T
"H
.J
Noisy amplifier T, S
Fig. 6.3. The generalcircuitmatrix representation of a linear twoterminalpair network
with internal sources.
The source equation appropriate to the righthand 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 twoterminalpair network in generalcircuitmatrix
representation.
6.3. Noise Measure
The "excessnoise 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 righthand 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
^ _ Fel _ 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 Fe1 \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 twoterminalpair 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 twoterminalpair 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 fourtermi
nalpair, 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'"^
= (mVMi)88^(mVmi)'^
= CWC (6.27)
and
(P  T'PT't) = M'iZ' + Z'+)M'^ = MV(Z + Z+)tM'+
= (mVm^)(p  tptO(m t"^Mi)+
= Ct(P  TPTt)C (6.28)
where
C^ = MVm1 (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 twoterminalpair amplifier.
the noisemeasure 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
twoterminalpair 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
negativeresistance 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 twoterminalpair networks.
When C in Eq. 6.30 is varied through all possible values, M/ reaches two
extrema, which are the two eigenvalues of the characteristicnoise 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 twoterminal
pair amplifier into a new twoterminalpair 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 characteristicnoise matrix of the original amplifier.
The fact that Me' > Xi/ikTo A/) also puts a lower limit upon the
excessnoise 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 excessnoise 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
excessnoise figure, and this Hmit depends upon the exchangeablepower
gain achieved in the particular connection.
If Ge > 0, that is, the output impedance of the amplifier has a positive
real part, the excessnoise figure can be less than \i/(kTo A/) only to the
extent of the gaindependent factor (1 — l/Gg).
If Ge < 0, that is, the output impedance has a negative real part, the
lower limit to the excessnoise 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
excessnoise figure of the latter cannot be less than that of the former.
Equation 6.31 has established a gaindependent lower bound for the
excessnoise figure achievable with lossless imbeddings of a given amplifier.
At large values of Ge, the excessnoise 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 excessnoise 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 excessnoise 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 twoterminalpair
amplifier with an excessnoise 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 2wterminalpair network results. By open
circuiting all but two of the resulting terminal pairs, we obtain the most
general twoterminalpair 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 characteristicnoise matrix of the network lie between the
most extreme eigenvalues of the characteristicnoise 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

4nterminalpair
lossless imbedding
network
h
Tl,5i
+
V4n
Anl
h
+
V4n1
V2
•
•
hi1
Ti, 5;
+
m
•
•
hi
+
V2i
•
•
^2rtl *
■^2n+2
Tn, 5„
+ *
1^2/1+2
V2n1
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;zterminalpair 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 4wterminalpair 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
4wterminalpair network with a characteristicnoise matrix, the eigen
ES
1
0— — 1
2
, 1
• .
• '
n
1
2
n
•
• '
1
'
2n
1
2n
L
Passive dissipative 4reterminalpair
imbedding network
Fig. 6.8. Imbedding of n twoterminalpair amplifiers in lossy 4«terminalpair network.
values of which are all negative (Sec. ?>.?)). The n amplifiers can be
grouped correspondingly into a 2wterminalpair network with a charac
teristicnoise matrix that has (in general, positive and negative) eigen
values equal to those of the characteristicnoise 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 twoterminalpair
amplifiers that results in a twoterminalpair network. When the resulting
twoterminalpair 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 characteristicnoise matrix.
Statement 3 of Section 6.4 is an immediate corollary of the previous
theorem; that is,
Passive dissipative imbedding of a given twoterminal 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 excessnoise 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 characteristicnoise
matrix. We also showed that an arbitrary lossless or passive interconnec
tion of twoterminalpair amplifiers, which leads to a new twoterminal
pair amplifier, yields an excessnoise 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 twoterminalpair 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
terminalpair 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 impedancetransformation
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 excessnoise figure of the highgain 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 excessnoise 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 TwoTerminalPair Amplifiers
The noiseperformance optimization problem is solved conveniently by
referring to a detailed classification of nonpassive twoterminalpair 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 impedancematrix 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 vacuumtube 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, CT1, 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) =7T^  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 "negativeresistance" amplifier.
Recalling that the optimum noise measure is not changed by any loss
less transformation, we conclude that noiseperformance 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 characteristicnoise 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
2wP  1
1 \T 2
En\Inl +
In\
2 kr  1
(7.8)
(6.25)
2w2  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 impedancematching 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 voltagecurrent formalism. We assume that transmission lines of
1ohm 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{V2h)
These transformations are conveniently summarized in matrix form. We
define the matrix
R=5
^[1 1] (^)
We then have
The generalcircuitparameter 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' = RTRi (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+ = jRi (7.17)
Using Eqs. 7.16 and 7.17, we can write the noise matrix 88^ as
88^ = R^8^Ri = 4R8^R^ (7.18)
Furthermore, we have from Eqs, 7.15 and 7.17
P  TPT^ = P  RiT'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 ^ 2lyiy2\
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 negativeconjugate 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
ldTW
\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.=^
wr1
"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 /
«P1
? '72 I Ts'T2 I
\0l +02 +
,WP1
, ^ , ^\2
Isj / 2 J Is / 2 I
lOl I +02 I ) '
wr1
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 generalcircuitmatrix 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 excessnoise
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 (lSnin/nir ~ lEnl^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 conjugateimage 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 conjugateimage
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 vacuumtube
and transistor amplifiers meet these conditions over a significant fre
quency range.
7.3. The Optimum NoiseMeasure Expression for the Conventional
LowFrequency Vacuum Tube
We shall now derive from Eq. 7.29 the expression for the minimum
noise measure of a conventional lowfrequency 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 noisevoltage 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 opencircuit noise voltages of the overall network are
El
E'
'Ri
E2
L_ " 'P_l
(7.34)
Finally, in generalcircuitparameter 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 noisemeasure
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 excessnoise 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 NegativeResistance Amplifiers,
Definite Case
There remains the problem of achieving the optimum noise measure of
negativeresistance 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 negativeresistance
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 twoterminalpair 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 characteristicnoise matrix N are directly related:
J^nl ^ _x^; ^ ^ _X2 (7.38)
4i?i ^' 4i?2
2 A. van der Ziel, Noise, PrenticeHall, New York (1954)
Sec. 7.4]
OPTIMIZATION, DEFINITE CASE
69
In the special case of a negativeresistance 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 twoterminalpair 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 1ohm characteristic impedance are connected
to all four terminal pairs of the circulator.) Terminal pair (4) of the
circulator is matched to a 1ohm 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 twoterminalpair network can easily
be derived using the scatteringmatrix representation. We find that the
r
1
1
1
70 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch. 7
Input (1) o
:i?n=i '
rMV4WW
(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 1ohm source is
= (ItI)'
The excessnoise 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 oneterminalpair negative resistance Ri in series with a noise
voltage generator Eni To make a twoterminalpair 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 twoterminalpair
network. Lossless imbedding of these two resistances therefore leads to a
twoterminalpair 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 reemphasized that the as
sumed presence of a positive resistance R2 in the circuit is an artifice that
enables the use of the theory of twoterminalpair networks for the noise
study of the oneterminalpair maser. The assumed temperature of the
resistance is immaterial because it determines only the negative eigen
value of the characteristicnoise 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. Negativeresistance 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 (1ohm) balancing resistor,
as shown in Fig. 7.3. The resulting unilateral twoterminal pair network,
driven from a 1ohm 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 twoterminalpair linear amplifiers. It
was necessary at the outset to elect a criterion of noise performance, which
we chose to be the signaltonoise ratio achievable at high gain. This
criterion is not clear for systems without gain, nor for multiterminalpair
networks. For multiterminalpair networks, the noise parameter pT
expressed in terms of the general circuit constants has been set down as
an extension of the twoterminalpair noisemeasure definition but has
not been given any physical interpretation in this work. One reason for
this omission is the fact that a generalcircuitconstant (or wavematrix)
description of multiterminalpair systems has been of little use in the past.
There have not been any systems incorporating gain whose noise per
formance on a multiterminalpair basis was of interest. It is true that
in the past some special problems involving frequency conversion have
called for proper interpretations, and that twoterminalpair networks
processing sidebands may be analyzed theoretically as multiterminalpair
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 timevarying, 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
multiterminalpair 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
multiterminalpair networks have been neglected completely. Indeed, in
the impedancematrix formulation we have given extensive attention to
the exchangeablepower interpretations of the network invariants in the
multiterminalpair 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
multiterminalpair network. The merit of this form is that it leads to a
simplification in thinking about singlefrequency 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 spotnoise
discussions that we have conducted. Certainly, in a twoterminalpair
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 networksynthesis 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 signaltonoise ratio at
high gain, at each frequency within the band." It is by no means obvious
that, with the overall 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 overall 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
noiseperformance 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, negativeresistance, 61, 68, 72
nonunilateral, 66, 71
parametric, 73
twoterminalpair, 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
Characteristicnoise matrix, eigenvalues of, 23
general formulation of, 38
impedance formulation of, 22
mixed voltagecurrent formulation of, 41
trace of, 23
Circulator, 69, 72
Classification of networks and eigenvalues, 24
in Tmatrix representation, 40
Classification of twoterminalpair ampli
fiers, 59
Crosspower spectral densities, 10
Dahlke, W., 46, 47
Eigenvalues, 22
classification of, 24
least positive, 53
Excessnoise figure, 48
lower limit imposed on, 53
Exchangeable power, 15, 44
matrix form for, 19
matrix formulation of, 44
nterminalpair networks, 15
stationary values of, 22
Exchangeablepower 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
exchangeablepower, 43, 46
77
78
INDEX
Gain, extended definitions of, 43
unilateral U, 59
Generalcircuitparameter representation, 36,
61
Granlund, J., 17, 74
Gridnoise 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, characteristicnoise, 22, 38, 41
generalcircuit, 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 exchangeablepower gain, 45
of extended noise figure, 46
Matrix formulation, of stationaryvalue
problem, 19
Mismatch, input, 6, 54, 61, 71
»towterminalpair network transforma
tions, 6, 25
»tooneterminalpair network transforma
tion, 6, 18
Negative definite matrix, 16, 52
Network transformations, ntowterminal
pair, 6, 25
wtooneterminalpair, 6, 18
Networks, classification of, 14, 24, 40
in thermal equihbrium, 25
purenoise, 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 lowfrequency vacuum
tube, 66
for triode, 66
optimum (Afe.opt), 54, 55
Noise parameter pT, 39
in mixed voltagecurrent representation, 41
Noise performance, criterion for amplifier, 5
optimization of, 61, 68
singlefrequency, 1, 73
spot, 1
Nonreciprocal transformations, 59
Nonunilateral amplifier, 66, 71
Optimization, of amplifier, indefinite case, 61
of maser amplifiers, 7 1
of negativeresistance amplifiers, definite
case, 68
Optimum noise measure (ilfe.opt), 54, 55
for lowfrequency 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
Scatteringmatrix representation, 69
Selfpower spectral densities, 10
SemideJBnite matrix, 11, 16
Signaltonoise ratio, 1, 73
Singlefrequency noise performance, 1, 73
Spectral densities, crosspower, 10
power, 11
selfpower, 10
Spotnoise figure, 2
Stationaryvalue problem, eigenvalue formu
lation of, 21
matrix formulation of, 19
Stationary values of exchangeable power,
22
Tmatrix representation, 34, 39, 40
Terminalvoltage and current vector, 38
Thevenin representation, 14
Trace of characteristicnoise matrix, 23
Transformation, lossless, 9, 12, 19, 61
wtowterminalpair network, 6, 25
»tooneterminalpair 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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