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Circuit Theory
of Linear Noisy Networks
TECHNOLOGY PRESS RESEARCH
NoNLINEAR PROBLEMS IN RANDOM THEORY
By Norbert Wiener
Circuit THEORY OF LINEAR Noisy NETWORKS
By Hermann A. Haus and Richard B. Adler
MONOGRAPHS
by sy ued jo
HERMANN A. HAUS
RICHARD B. ADLER
Associate Professors of Electrical Engineering
Massachusetts Institute of Technology
Circuit Theory
of Linear Noisy Networks
MARINE
BIOLOGICAL
LABORATORY
aati
LIBRARY
oe WOODS HOLE, MASS.
The Technology Press of W. H, O, 1.
The Massachusetts Institute of Technology
and
John Wiley & Sons, Inc., New York
Chapman and Hall, Limited, London
Copyright © 1959
by
The Massachusetts Institute of Technology
All Rights Reserved
This book or any part thereof must not
be reproduced in any form without the
written permission of the publisher.
Library of Congress Catalog Card Number: 59-11473
Printed in the United States of America
Foreword
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
hie Aan
Preface
Monographs usually present scholarly summaries of a well-developed
field. In keeping with the philosophy of the new series of Research
Monographs, however, this monograph was written to present a piece of
relatively recent work in a comparatively undeveloped field. Such work
might normally be expected to appear in a series of journal articles, and
indeed originally the authors followed this method of presentation. As
the subject developed, however, a rather general approach to the problem
became apparent which both simplified and unified all the prior research.
Space limitations in the journals made it impossible to publish in that
medium a really suitable picture of the whole development, and this
circumstance led the authors to take advantage of the present Technology
Press Research Monographs.
The principal motivation for this work arose from the obvious desir-
ability of finding a single quantity, a tag so to speak, to describe the
noise performance of a two-terminal-pair amplifier. The possibility of
the existence of such a quantity and even the general functional form
which it might be expected to take were suggested by previous work of
one of the authors on microwave tubes and their noise performance.
This work showed that noise parameters of the electron beam set an
ultimate limit 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
v11
vilt PREFACE
for any linear amplifier could be made without referring again to some
detailed physical mechanism. In order to detach the study of noise
performance from specific physical mechanisms, one had to have recourse
to general circuit theory of active networks. Such a theory had grown
up around the problems associated with transistor amplifiers, and im-
portant parts of it were available to us through the association of one of
us with Professor S. J. Mason. This combination of circumstances led
to the collaboration of the authors.
Two major guiding principles, or clues, could be drawn from the
experience on microwave tubes. One such clue was the general form of
the probable appropriate noise parameter. The other was the recog-
nition that matrix algebra and a proper eigenvalue formulation would be
required in order to achieve a general theory without becoming hope-
lessly involved in algebraic detail.
Essentially by trial and error, guided by some power-gain theorems in
active circuit theory, we first found a few invariants of noisy networks.
Afterward, while we were trying to decide around which quantities we
should build a matrix-eigenvalue formulation leading to these same
invariants, we were aided by the fact that Mr. D. L. Bobroff recognized
a connection between the invariants which we had found and the problem
of the available power of a multiterminal-pair network.
Armed with this additional idea, we consulted extensively with Profes-
sor L. N. Howard of the Massachusetts Institute of Technology, Depart-
ment of Mathematics, in search of the appropriate matrix-eigenvalue
problem. As a result of his suggestions, we were able to reach substan-
tially the final form of the desired formulation.
Once the proper eigenvalue approach was found, additional results
and interpretations followed rapidly. In particular, the idea that the
eigenvalue formulation should be associated with a canonical form
of the noisy network was suggested in a conversation with Professor
Shannon.
One of the principal results of the work is that it furnishes a single
number, or tag, which may be said to characterize the amplifier noise
performance on the basis of the signal-to-noise-ratio criterion. The novel
features of this tag are two in number: First, it clears up questions of
the noise performance of low-gain amplifiers or of the effect upon noise
performance of degenerative feedback; second, it provides for the first
time a systematic treatment of the noise performance of negative-resist-
ance amplifiers. The latter results were not expected in the original
motivation for the study but grew from insistent demands upon the
internal consistency of the theory. It is interesting that the negative-
resistance case will probably turn out to be one of the most important
practical results of our work.
PREFACE ix
Another result worth mentioning here, however, is the canonical form
of linear noisy networks. This form summarizes in a clear, almost visual,
manner the connection between the internal noise of a network at any
particular frequency and its (resistive, positive, or negative) part.
We are hopeful that this second work in the series of Technology Press
Research Monographs will meet the standards and aims envisioned by
Professor Gordon S. Brown, whose personal inspiration and energetic
support brought the present volume into existence.
We wish to express our sincere thanks to Miss Joan Dordoni for the
careful preparation of the manuscript. We also acknowledge gratefully
the thorough and exacting editing work of Miss Constance D. Boyd.
The support in part by the U. S. Army (Signal Corps), the U. S. Air
Force (Office of Scientific Research, Air Research and Development
Command), and the U. S. Navy (Office of Naval Research) is also
acknowledged with gratitude.
Massachusetts Institute of Technology
Cambridge, Massachusetts
January, 1959
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Contents
FOREWORD
PREFACE
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
CHAPTER 5
Introduction
Linear Noisy Networks in the Impedance Representation
2.1 Impedance-Matrix Representation of Linear Networks
2.2 Lossless Transformations
2.3 Network Classification in Terms of Power
Impedance Formulation of the Characteristic-Noise Matrix
3.1 Matrix Formulation of Stationary-Value Problem
3.2 Eigenvalue Formulation of Stationary-Value Problem
3.3. Properties of the Eigenvalues of the Characteristic-
Noise Matrix in Impedance Form
3.4 Lossless Reduction in the Number of Terminal Pairs
Canonical Form of Linear Noisy Networks
4.1 Derivation of the Canonical Form
4.2 Interconnection of Linear Noisy Networks
Linear Noisy Networks in Other Representations
5.1 General Matrix Representations
5.2 Transformation from One Matrix Representation to
Another
x
xt
CHAPTER 6
CHAPTER 7
CHAPTER 8
INDEX
555)
CONTENTS
Power Expression and Its Transformation
5.4 The General Characteristic-Noise Matrix
Noise Measure
6.1 Extended Definitions of Gain and Noise Figure
6.2 Matrix Formulation of Exchangeable Power and Noise
Figure
6.3 Noise Measure
6.4 Allowed Ranges of Values of the Noise Measure
6.5 Arbitrary Passive Interconnection of Amplifiers
Network Realization of Optimum Amplifier Noise Per-
formance
7.1 Classification of Two-Terminal-Pair Amplifiers
7.2 Optimization of Amplifier, Indefinite Case
7.3. The Optimum Noise-Measure Expression for the Con-
ventional Low-Frequency Vacuum Tube
7.4 Optimization of Negative-Resistance Amplifiers,
Definite Case
Conclusions
Introduction
The principal example of a linear noisy network, and the one of greatest
practical importance in electrical engineering, is a linear noisy amplifier.
The noise performance of such amplifiers involves many questions of
interest. One very significant question is the extent to which the amplifier
influences signal-to-noise ratio over a narrow band (essentially at one
frequency ) in the system of which it isa part. We shall address ourselves
exclusively to this feature, without intending to suggest that other fea-
tures of the much larger noise-and-information problem are less important.
The term “‘spot-noise performance” or merely “noise performance” will
be used to refer to the effect of the amplifier upon the single-frequency
signal-to-noise ratio. It is essential to emphasize right at the beginning
the very restricted meaning these terms will have in our discussions.
We undertook the study reported here in the hope of formulating a
rational approach to the characterization of amplifier spot-noise per-
formance, and to its optimization by external circuit operations upon the
terminals. Fortunately, a characterization has resulted which is based
on a single hypothesis about the essential function of an amplifier and
which turns out to avoid pitfalls previously associated with the effect of
feedback upon noise performance. In developing the aforementioned
noise characterization of amplifiers and in pursuing the relevant optimi-
zation problem, we encountered a number of illuminating features relating
power and noise in linear multi-terminal-pair networks. Indeed, it
eventually became clear that the major issues could be presented most
simply by postponing until last the questions we had originally asked
first. The result is a work of broader scope than was originally envisaged,
1
2 INT RODUCTION [Ch. 1
and one for which the title ‘Circuit Theory of Linear Noisy Networks”’
seems appropriate.
Since the introduction by Friis! and Friénz? of the concept of spot-
noise figure F for the description of amplifier noise performance, this
figure has played an essential role in communication practice. The
noise figure is, however, merely a man-made definition, rather than a
quantity deduced from clearly defined postulates or laws of nature. The
possible consequences of this fact were never questioned deeply, although
it has always been known that the (spot-) noise figure F does not con-
stitute a single absolute measure of amplifier noise performance.
In particular, the noise figure is a function of the impedance of the
source connected to the amplifier input. Thus in giving an adequate
conventional description of amplifier noise performance, the source
impedance, as well as the noise figure, must be specified.
Usually, when regarded as a function of source impedance alone, the
noise figure has a minimum value for some particular choice of this
impedance. If with this source impedance the gain of a given amplifier
remains sufficiently high, its noise figure will prescribe the noise figure
of any amplifier cascade in which it is used as the first stage. In this way,
it is possible to build an amplifier cascade with any desired high gain,
and with a noise figure set by the minimum (with respect to source
impedance) of the noise figure of the original amplifier.
If a cascade is to be composed of several individual amplifiers, each of
which alone has a “high enough” gain when driven from the source
impedance that yields its minimum noise figure, the previous argument
shows that the amplifier with the lowest minimum noise figure should be
used as the first stage. Any other choice would result in a higher over-all
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 controlling the
noise performance.
The inadequacy of this viewpoint becomes clear when stage variables
other than source impedance and stage interconnections other than the
1H. T. Friis, ‘(Noise Figure of Radio Receivers,” Proc. [.R.E., 32, 419 (1944).
2K. Franz, “Messung der Empfangerempfindlichkeit bei kurzen elektrischen
Wellen,” Z. Elektr. Elektroak., 59, 105 (1942).
Ch. 1) INT RODUCTION 3
simple cascade become important in amplifier applications. The question
of the quality of noise performance then becomes much more complicated.
For example, when degenerative feedback is applied to an amplifier, its
noise figure can be reduced to as close to unity as desired (for example,
bypassing the entire amplifier with short circuits yields unit noise figure).
But its gain is also reduced in the process. Indeed, if identical stages
with the feedback are cascaded to recover the original single-stage gain
before feedback, the resulting noise figure of the cascade cannot be less
than that of the original amplifier? Moreover, with degenerative feed-
back the gain may easily be so greatly reduced that, as a first stage in a
cascade, this amplifier alone no longer determines the over-all noise
figure of the cascade. The minimum-noise-figure criterion considered
above as a measure of amplifier noise performance breaks down. It
appears that an absolute measure of amplifier noise performance must
include, in addition to the specification of noise figure and source im-
pedance, at least the specification of the gain.
The foregoing reasoning led us to the investigation presented in this
study. Taking our clues from the results previously found by Haus and
Robinson* for microwave amplifiers, and the method of active-network
description presented by Mason,” we searched for a measure of amplifier
noise performance that would not only include the gain explicitly, as
discussed earlier, but could also be minimized by external circuitry in a
nontrivial way. Moreover, we believed that the minimum thus obtained
should be a quantity characteristic of the amplifier itself. It should, for
example, be invariant under lossless feedback, a type of feedback that
does not appear to change the essential ‘“‘noisy” character of the amplifier
because it certainly adds no noise and can always be removed again by
a realizable inverse lossless operation.
The precise form of a suitable noise-performance criterion has actually
been known for many years, although its deeper significance somehow
escaped attention. Indeed, the most glaring example of the correct
criterion arises from the familiar problem of cascading two (or more)
low-gain amplifiers having different noise figures F, and F, and different
available gains G;(>1) and G.(>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
5 A. van der Ziel, Noise, Prentice-Hall, New York (1954).
4H. A. Haus and F. N. H. Robinson, ‘“The Minimum Noise Figure of Microwave
Beam Amplifiers,” Proc. I.R.E., 43, 981 (1955).
5S. J. Mason, “Power Gain in Feedback Amplifiers,” Trans. IRE, Professional
Group on Circuit Theory, CT=-1, No. 2, 20 (1954).
4 INT RODUCTION (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 Fy, and Fo
are the respective noise figures of the cascade when amplifier No. 1
and amplifier No. 2 are placed first, we have
F,—-1
ie =e ve (1.12)
il
F,-1
lrg, 10 ae fa (1.10)
2
The condition that Fj, be less than Fo; is
F,-—1 F,-1
F,-—F —_—— —
to eee!
or
F,-—1 F,-1
F,-—1)- (F, -1 —
(Fy Viaite RS Gp Gy
or
F,-—1 F.e-1
——— <= (1.2)
eed ral
Gy Gy
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
Mu =a (1.3)
iG
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 = G,G2, Eq. 1.1@ becomes
Heat
AM G
HO a i
ane
i Gaal
= M, + au (2—2) (1.4)
Cla, il) INT RODUCTION 5
where AM = M, — M, is the difference between the noise measures of
the second and first amplifiers of the cascade. Equation 1.4 shows that as
long as G, and Gy» are greater than 1 the noise measure of a cascade of
two amplifiers lies between the noise measures of its component amplifiers.
In the particular case when the noise measures of the two amplifiers 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 = G,G, 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
amplifiers 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 amplifiers with sufficiently high gain, the final performance
evaluation of a practical multistage amplifier always rests numerically
(if not in principle) upon the familiar noise-figure criterion.
From such reasoning, we evolved a criterion for amplifier noise per-
formance. The criterion is based on the plausible premise that, basically,
amplifiers 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 m different types of amplifiers are compared. An un-
limited number of amplifiers of each type is assumed to be available.
A general lossless (possibly nonreciprocal) interconnection of an arbitrary
number of amplifiers of each type is then visualized, with terminals so
arranged that in each case an over-all two-terminal-pair network is achieved.
For each amplifier type, both the lossless interconnecting network and the
number of amplifiers are varied in all possible ways to produce two
conditions simultaneously:
1. A very high available gain (approaching infinity) for the over-all
two-terminal-pair system when driven from a source having a positive
real internal impedance.
2. An absolute minimum noise figure Fyin for the resulting high-
gain system.
The value of (Fmin — 1) for the resulting high-gain two-terminal-patr net-
work is taken specifically as the “‘measure of quality” of the amplifier type
in each case. The “‘best” amplifier type will be the one yielding the smallest
value of (Fmin — 1) at very high gain.
6 INT RODUCTION (Ch. 1
The proof of this criterion will be developed through the concept of
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 linear networks.
Among these, lossless transformations form a group in the mathematical
sense. The quantities invariant under the group transformations must
have a physical significance. Investigation of these invariants forms a
substantial part of the present study. To be sure, for the special case of
a two-terminal-pair amplifier, the optimum noise performance, through
its related noise measure, turns out to be one of the invariants; but several
other interpretations of the invariants prove equally interesting, and
the development of the entire subject is simplified by presenting them
first.
The simplest formulation and interpretation of the invariants of a
linear noisy network result from its impedance representation. The
following chapter is therefore devoted to a discussion of network trans-
formations, or “‘imbeddings,” in terms of the impedance-matrix repre-
sentation. The concept of exchangeable power as an extension of
available power is then introduced.
In Chapter 3, the ” invariants of a linear noisy 7-terminal-pair network
are found as extrema of its exchangeable power, with respect to var-
iations of a lossless 2-to-one-terminal-pair network transformation. It
is found that an u-terminal-pair network possesses not more than these
n invariants with respect to lossless 7-to-n-terminal-pair transformations.
These ” invariants are then exhibited in a particularly appealing way in
the canonical form of the network, achievable by lossless transformations
and characterized by exactly m 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. 1) INT RODUCTION 7
matrix description leads to a different interpretation of the invariants.
In the case of an active two-terminal-pair network, a particularly
important interpretation of the invariants is brought out by the general-
circuit-parameter-matrix description. This interpretation relates directly
to the optimum “‘noise measure”’ of the network used as an amplifier and,
therefore, to the minimum noise figure of the amplifier at arbitrarily high
gain. Chapter 6 is devoted to this noise-measure concept and to the
range of values that the noise measure may assume for a two-terminal-
pair amplifier subjected to arbitrary passive network transformations.
In particular, the minimum value of the noise measure of the amplifier is
found to be directly proportional to one of the two invariants of the
amplifier.
A study is made of those arbitrary passive interconnections of two-
terminal-pair amplifiers which result in an over-all two-terminal-pair
amplifier. The conclusion is that the noise measure of the composite
amplifier cannot be smaller than the optimum noise measure of the best
component amplifier, namely, the amplifier with the smallest optimum
noise measure.
The general theorems having established the existence of an optimum
value of the noise measure of amplifiers, it remains in Chapter 7 to discuss
in detail the network realization of this optimum for two-terminal-pair
amplifiers. Some practical ways of achieving it are presented. Among
these, the realization of optimum noise performance for a maser may be
of greatest current interest.
With proof of the existence and realizability of a lower limit on the
noise measure, and therefore of the noise figure at high gain, the major
objective of the present work is accomplished. It is demonstrated that
the quality with regard to noise performance of a two-terminal-pair
amplifier can be specified in terms of a single number that includes the
gain and that applies adequately to low-gain amplifiers.
We have previously published various separate discussions of some of
these topics in different contexts.°"!° Each of these discussions has
suffered from unnecessary complications because space limitations forced
6H. A. Haus and R. B. Adler, “Invariants of Linear Networks,” 1956 IRE Con-
vention Record, Part 2, 53 (1956).
7H. A. Haus and R. B. Adler, ‘Limitations des performances de bruit des ampli-
ficateurs linéaires,” L’Onde Electrique, 38, 380 (1958).
8H. A. Haus and R. B. Adler, ‘(Optimum Noise Performance of Linear Amplifiers,”
Proc, I.R.E., 46, 1517 (1958).
®R. B. Adler and H. A. Haus, “Network Realization of Optimum Amplifier Noise
Performance,” IRE Trans. on Circuit Theory, CT-5, No. 3, 156 (1958).
10H. A. Haus and R. B. Adler, ‘Canonical Form of Linear Noisy Networks,”
IRE Trans. on Circuit Theory, CT-5, No. 3, 161 (1958).
8 INT RODUCTION [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
in 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 m-terminal-pair net-
work. Such a network has to be classified with respect to its passive—
active character, which depends upon its ability to deliver or absorb
power. The examination of power delivered or absorbed by a multi-
terminal-pair network raises some interesting questions leading to a
generalization of the available-power concept: the exchangeable power.
In Chap. 3, the exchangeable-power concept in turn leads to the discovery
of the invariants mentioned in Chap. 1.
2.1. Impedance-Matrix Representation of Linear Networks
At any frequency, a linear 1-terminal-pair network containing internal
prescribed signal or noise generators is specified completely with respect
to its terminal pairs by its impedance matrix Z and the complex Fourier
amplitudes of its open-circuit terminal voltages £;, Es, --: , E, (Fig. 2.1).
In matrix form, Z denotes a square n-by-n array
eee © © © © © ee ee 8 ee ee 8 8 8 8 8
10 IMPEDANCE REPRESENTATION [Ch. 2
Fig. 2.1. Equivalent representation of linear network with internal noise sources.
The cases in which the Z matrix does not exist in a formal sense can be
handled by suitable perturbations, for example, slight frequency changes
or the addition of perturbing circuit elements. The complex amplitudes
of the open-circuit terminal voltages are represented conveniently by a
column matrix E:
Ey
Ex
E,,
If the internal generators are random-noise sources, the Fourier ampli-
tudes E, -:- E, are complex random variables, the physical significance
of which usually appears in their self- and cross-power spectral densities
E,E;,*.1 The bar indicates an average over an ensemble of noise processes
1W. B. Davenport and W. O. Root, Random Signals and Noise, McGraw-Hill
Book Company, New York (1958).
Sec. 2.1] IMPEDANCE-MATRIX REPRESENTATION 11
with identical statistical properties. Here, as in the rest of this work, we
retain only positive frequencies. In order to preserve the same multipli-
cative factors in power expressions for both random and nonrandom
variables, we shall depart from convention by using root-mean-square values
for all nonrandom complex amplitudes.
A convenient summary of the power spectral densities is the matrix
coos eee ee ee © ew ee ee we we eee ee ee eh eh eh el wl
BoB tales terse nee cia we tai eoeusa one ate ey (50)
where the superscript dagger indicates the two-step operation composed
of forming the complex conjugate of and transposing the matrix to which
it refers. Briefly, A‘ is called the Hermitian conjugate of any matrix A.?
The matrix EE‘ is its own Hermitian conjugate, because E;E;,* =
E,*E; = (E,E;*)*. Such a matrix is said to be Hermitian.
In addition, we can show that EET is a positive definite or semidefinite
matrix. Construct the real nonnegative quadratic form:
(vy*E, +... +0," E;+...+4n* En) (1 Ai*+...+4,E*+...+2,E,*)>0
where the x; are arbitrary complex numbers. In terms of the column
matrix
and the column matrix E, the foregoing expression becomes
xtE(xtE)t = xtEEtx > 0
?F. B. Hildebrand, Methods of Applied Mathematics, Prentice-Hall, New York
(1952).
12 IMPEDANCE REPRESENTATION [Ch. 2
for all x. If the inequality sign applies for all x 4 0, EE? is positive
definite; if the equality sign applies for some (but not all) x # 0, EEt is
positive semidefinite. The equality sign for all x ¥ 0 would occur only
in the trivial case E = 0.
2.2. Lossless Transformations
If the m-terminal-pair network with the generator column matrix E and
the impedance matrix Z is connected properly to a 2n-terminal-pair
network, a new v-terminal-pair network may be obtained. It will have
a new noise column matrix E’ and a new impedance matrix Z’. This
Va = V
Fig. 2.2. General transformation (imbedding) of an m-terminal-pair network.
operation, shown in Fig. 2.2, we shall call a transformation, or an im-
bedding, of the original network. If the added network is lossless, this
arrangement represents the most general lossless modification (including
the addition of multiple, and not necessarily reciprocal, feedback paths)
that can be performed externally upon the original 2-terminal-pair net-
work so as to obtain a new u-terminal-pair network.
The analytical relation between the voltages and currents applied to
the 2n-terminal-pair, lossless network (the “‘transformation network’’)
SIGs (4eA| LOSSLESS TRANSFORMATIONS 13
of Fig. 2.2 can be written in the form
Va a Laala ate Zavle (2.2)
Vp = Zrala + Zoslo (2.3)
The column vectors V, and V; comprise the terminal voltages applied to
the transformation network on its two sides, and the column vectors I,
and I, comprise the currents I7 flowing into it. The four Z matrices in
Eqs. 2.2 and 2.3 are each square and of mth order. They make up the
square 2uth-order matrix Zr of the lossless transformation network. The
condition of losslessness can be summarized in the following relations,
which express the fact that the total time-average power P into the
transformation network must be zero for all choices of the terminal
currents:
=> tI pt (Zr + Zrt)Ip S 0, for all I; (2.4a)
therefore
Fig os (2.4b)
or
Laa a 1 bat Lab =I = They Lop aa =f bie! @ES))
Equation 2.5 does not require that the transformation network be recipro-
cal.
The original 1-terminal-pair network, with impedance matrix Z and
noise column matrix E, imposes the following relations between the
column matrices V and I of the voltages across, and the currents into,
its terminals:
V=ZI+E (2.6)
The currents I into the m-terminal-pair network are, according to Fig. 2.2,
equal and opposite to the currents I, into one side of the 27-terminal-pair
network. The voltages V are equal to the voltages V,. We thus have
VW = Wee Le ik (2.7)
Introduction of Eqs. 2.7 into Eq. 2.6 and application of the latter to
Eq. 2.2 give
I, Sn (Z =F / beg ed bees kp AF (Z oP bee) 1D)
When this equation is substituted in Eq. 2.3, the final relation between
V, and I, is determined:
V, = ZI, + EF’ (2.8)
where
Zo= —Zra(Z + Lian Lat + Zr (2.9)
14 IMPEDANCE REPRESENTATION [Ch. 2
and
Ey eiZp(Z > Zn (2.10)
Equation 2.8 is the matrix relation for the new n-terminal-pair network
obtained from the original one by imbedding it in a 2n-terminal-pair
network. Here Z’ is the new impedance matrix, and E’ is the column
matrix of the new open-circuit noise voltages. Conditions 2.5 must be
applied 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 P., of a one-terminal-pair source is
defined as:
Piy = the greatest power that can be drawn from the source by
arbitrary variation of its terminal current (or voltage)
Linear network
with prescribed
internal sources
Fig. 2.3. The Thévenin equivalent of a one-terminal-pair linear network.
If the Thévenin representation of the source (Fig. 2.3) has the
complex open-circuit voltage EZ and internal impedance Z, with R =
Re (Z) > 0, this definition leads to
tpES 1) REX
See eS. efor Re > 0) ee
If the source is nonrandom, the bar in Eq. 2.11a may be omitted. No
other changes are necessary because £ is then understood to be a root-
mean-square amplitude, as remarked in Sec. 2.1. In Eq. 2.11a, Pay is
also a stationary value (extremum) of the power output regarded as a
function of the complex terminal current 7. Moreover, the available
power (Eq. 2.11@) 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
(Pe or for R < 0
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 mot 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 P,:
P, = 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. 2.11@ for any nonzero value of R in Fig. 2.3. Specifically,
ee ee ee for R #0 (2.110)
When R is negative, P, in Eq. 2.116 is negative. Its magnitude then
represents the maximum power that can be pushed into the terminals by
suitable choice of the complex terminal current J. This situation may
also be realized by connecting the (nonpassive) conjugate-match im-
pedance Z* to the terminals. This impedance actually functions as a
source, pushing the largest possible power into the network terminals.
A straightforward extension of the exchangeable-power definition to
n-terminal-pair networks makes it the stationary value (extremum) of
the total power output from all the terminal pairs, obtained by arbitrary
variations of all the terminal currents. With reference to Fig. 2.1, we
search specifically for the stationary values with respect to I of the total
average output power P of the network
P= -2N(Z4+Z)14+TE+ ET (212)
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 = —2(0 (24 Z)1) (2.13)
Three possible cases have to be distinguished for P, (aside from the
previously discussed case of a lossless network).
1. The first case is that of a passive network. Then the power output
P, 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 P, 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 P,
flowing out of the network may be either positive or negative, depending
upon the currents I.
One may imagine the power P, 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 P,. When
(Z + Z') is either positive or negative semidefinite, the surface is a
multidimensional paraboloid with a maximum or minimum, respectively,
at the origin. When (Z + 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 difficulties 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 P, (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
Y=14+ (24 Z')'E
yields for Eq. 2.12
P= -4,0)(Z4+ ZI — EZ + Z')E] (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 P,:
P. = 3 (E'(Z+ Z')"E] (2.15)
Because the definite characters of (Z + Z')—! and (Z+ Z') are the
same: P, > 0 (regardless of E) when (Z-+ Z') is positive definite;
P, < 0 (regardless of E) when (Z + Z") is negative definite and P, z 0
(depending upon the particular E involved) when (Z + Z') is indefinite.
In view of the term (1’)'(Z + Z')I’ in P, the significance of P, in Eq. 2.15
is of the same kind as that of P, in Eq. 2.116 when (Z + Z') is either
positive or negative definite. When (Z + Z") is indefinite, however,
P, in Eq. 2.15 is simply the stationary-point (saddle-point) value of the
average output power with respect to variations of the terminal currents,
and has no analog in the case of a one-terminal-pair network (Eq. 2.110).
We have defined the exchangeable power for a one-terminal-pair net-
work as the extremum of power output obtainable by arbitrary variation
of terminal current. In an obvious generalization, we have extended this
definition to 2-terminal-pair networks by considering the extremum of
the power output of the network obtained by an arbitrary variation of
all its terminal currents. In this case, we have encountered the possibility
of the output power assuming a stationary value rather than an extre-
mum. One may ask whether the stationary value of the output power
for the multiterminal-pair case could be achieved in a simpler way. One
obvious method to try is that shown in Fig. 2.4.°
The given network is imbedded in a variable (m + 1)-terminal-pair
lossless network. For each choice of the variable lossless network, we
consider first the power that can be drawn from the (m + 1)th pair for
various values of the complex current 7,4, (that is, for various “loadings”
of this terminal pair). In particular then, we determine the exchangeable
power P,.n11 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.
3 Recently we have learned that, prior to our study, this particular noisy-network
power-optimization problem was considered and solved independently for receiving
antennas by J. Granlund, Topics in the Design of Antennas for 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 ” terminal pairs of the original network are in each case
combined with different relative magnitudes and phases to make up the
output at terminals 7 + 1. Therefore, we might expect the available
power of the original m-terminal-pair network to be the extremum value
of P..n41 obtainable by considering every possible variation of the lossless
Variable lossless
network
(possibly
nonreciprocal)
Fig, 2.4. _Imbedding into an (m + 1)-terminal-pair lossless network.
transformation network. As we shall see shortly, this result proves to be
correct only in one simple case, namely, that in which the original n-
terminal-pair network contains only coherent signal generators. In such
a case P.n4i1 has only one stationary value as the lossless network is
varied, and this value is precisely the exchangeable power discussed
previously (P, in Eq. 2.15).
In the general situation of an arbitrary noisy network, we shall find that
P.41 has m 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 Pejn41
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 Characteristic-Noise Matrix
We shall proceed to a close examination of the stationary-value
problem posed in connection with Fig. 2.4, at the end of the last section
and prove the assertions made about it. A matrix formulation of
the problem will be required, which will reduce the problem to one in
matrix eigenvalues. The corresponding eigenvalues are those of a new
matrix, the “‘characteristic-noise matrix.’”’ Some general features of the
eigenvalues will be studied, including their values for two interesting
special cases. The effect of lossless imbeddings upon the eigenvalues will
be discussed to complete the background for the noise-performance
investigations.
3.1. Matrix Formulation of Stationary-Value Problem
The network operation indicated in Fig. 2.4 is conveniently accom-
plished by first imbedding the original -terminal-pair network Z in a
lossless 2n-terminal-pair network, as indicated in Fig. 2.2. Open-circuit-
ing all terminal pairs of the resulting 1-terminal-pair network Z’, except
the ith, we achieve the -to-1-terminal-pair lossless transformation
indicated in Fig. 2.4. The exchangeable power from the ith terminal
pair of the network Z’ can be written in matrix form as
i eae 1 TR/E
jb old SRE al (3.1)
19
20 CHA RACTERISTIC-NOISE MATRIX [Ch. 3
where the (real) column matrix § has every element zero except the 7th,
which is 1:
& 0
E ; §; an 0, il At
= & |e} lip 3.2
€ fi , oie (3.2)
En 0
Matrix § can be visualized as a double-pole n-throw selector switch which
chooses electrically only one of the 1 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 Z, in Fig. 2.2 through all possible
lossless forms. We wish to find the stationary values of P,,; correspond-
ing to variation of Zr. To render explicit this variation, E’ is first
expressed in terms of the original E and Z, using Eq. 2.10. Accordingly,
B/E™ = Zya(Z + Zaa) EE ((Z + Zo)? )'Zoat = ttEE's (3.3)
where
a! = Loa(Z ote bee) i (3.4)
Second, expressing Z’ in terms of Z by means of Eq. 2.9 yields
Zi + Z't = —Zyq(Z + Zaa) 1 Zas + Zoo + Zoo!
—Z'((Z a / beg eal (3.5)
The conditions of Eq. 2.5, guaranteeing that the transformation network
is lossless, convert the foregoing relation to
Zi + Z" = Zeal (Z + Loa) * + (2! + Zao’) *\Zo0"
ae Loa (Z ain bee Nae CA ain Ze) ate (Z air Lag) (zt oF Zia eZee
=T(Z4+Z")t (3.6)
It follows that
1 (tc')EET(s6)
2 (§'r')(Z + Z") (x8)
in which matrix t (not &) is to be varied through all possible values
consistent with the lossless requirements upon the transformation
network.
P= (3.7)
Sec. 3.2] EIGENVALUE FORMULATION 21
The significant point now is that t is actually any square matrix of
order 1 because Zp, in Eq. 3.4 is entirely unrestricted! Therefore, a new
column matrix x may be defined as
C= TE => 1: (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 P.,; in Eq. 3.7 may be
found most conveniently by determining instead the stationary values
of the (real) expression
x'EE'x
1
2x"(Z + Z')x C)
Pos =
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 P,,; as a multidimensional
function of all the complex components of x:
(a) Z + Z' positive definite; P.,; > 0 for all x.
(b) Z + Z' negative definite; P,; < 0 for all x.
(c) Z + Z! indefinite; P,,; 2 0, depending upon x.
3.2. Eigenvalue Formulation of Stationary-Value Problem
We now turn to the determination of the extrema and stationary
values of P,,; 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,,;. In terms of A = EE‘ and B = —2(Z + Z'),
(3.10)
ie CHA RACTERISTIC-NOISE MATRIX [Ch. 3
The stationary conditions and corresponding values for p may be ob-
tained from the solution of the equivalent problem of determining the
stationary values of x'Bx, subject to the constraint x'Ax = constant.
Therefore, introduction of the Lagrange multiplier 1/X and recognition
that p may be regarded as a function of either the set of x; or the set of
x;* lead to the conditions
1
— («'Bs _ ~ x'Ax) ==3(); ede, O02. 77 (3.11)
or simply
Ax — \Bx = (A — \B)x = 0 (3.12)
The values of \ are then fixed by the requirement
det (A — \B) = 0 = det (BA — Al) (3.13)
where 1 is the unit matrix. This means that the values of \ are just the
eigenvalues of the matrix BA = —43(Z + 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 \, 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
xOTAx(s) = d,x6) Bx
or
x) TAx(s)
As = 50h Ole Pstat (3.14)
which is real and equal to the corresponding stationary value of p.
It follows that the stationary values of the exchangeable power P,,; are
the negatives of the (real) eigenvalues of the matrix
AB? = —1(Z + Z')EE!.
We therefore define:
Characteristic-noise matrix = N= — 4(Z+ Z‘) EET (3.15)
and conclude that:
The stationary values of the exchangeable power P.,; are the negatives of
the (real) eigenvalues of the characteristic-noise matrix N.
eG» 3.3] PROPERTIES OF THE EIGENVALUES 23
3.3. Properties of the Eigenvalues of the Characteristic-Noise
Matrix in Impedance Form
We shall now confirm the assertion made earlier (Sec. 2.3) to the
effect that:
The exchangeable power of the n-terminal-pair network,
P. = 3[E'(Z + Z')7E],
is equal to the algebraic sum of the stationary values of P.,;, which 1s alter-
natively the negative of the trace of the characteristic-noise matrix N.
Setting W = 3(Z + Z')}, we express the typical 7th element of the
matrix }(Z + Z')“1EE' = WEE’ in the form
(WEE');; = 2 W Ex E;*
so that its trace (sum of diagonal elements) becomes
Trace (WEE') = 2 Wi.E.Er* = —trace (N) (3.16)
But P, of the m-terminal-pair network equals
P.-E ZE - EWE - 5 EPWak, (3.17)
Comparing Eq. 3.17 with Eq. 3.16, we find
P, = —trace N = —>),; (3.18)
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 \, of the characteristic-noise matrix N as well as the corre-
sponding ranges of P,,;. We first recall that the eigenvalues \, 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 ), 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 CHA RACTERISTIC-NOISE MATRIX [Ch. 3
Table 3.1. CLASSIFICATION OF NETWORKS AND EIGENVALUES
Eigenvalues
Case Z+2zt Network Class p=-—P.i (As) of N
1 positive definite passive <0 all < 0
2 negative definite active (negative resistance) >0 all > 0
: ‘ : > some > 0
3 indefinite active z0 Rees < of
The permissible values of \, 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 pisa
p=-f; p=-P,; p= Fi
mo]
ege 2
Largest positive = Intermediate
eigenvalue 5 | eigenvalue
Qa
s 3
3 = | | Intermediate Ay Smallest
Ss E | | eigenvalues RT Af {Positive
Be (ax eigenvalue
M1
Smallest positive kT Af 5
eigenvalue zo
2
P= 0 Pi =0 2--Ri=0
Smallest negative Smallest
eigenvalue = negative
2 : 3s eigenvalue
= Intermediate 2 3
5 eigenvalues eS = | Intermediate
f 7 eigenvalue
Largest negative rag
eigenvalue
Z + Z' positive definite Z + Z| negative definite Z + Z! indefinite
(a) (b) (c)
Fig. 3.1. Schematic diagram of permitted values of p for four-terminal-pair networks.
continuous function of x, its values lie between these two extreme eigen-
values, as illustrated in Fig. 3.1a. Analogous reasoning applies to the
second case, illustrated in Fig. 3.10.
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 p 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 # is illustrated in Fig. 3.1c.
One particular property of the eigenvalues of N will be of importance
later. Suppose that the original network with the characteristic-noise
matrix N is imbedded in a 2n-terminal-pair lossless network, as shown in
Fig. 2.2. A new m-terminal-pair network results, with the characteristic-
noise matrix N’. The eigenvalues of N’ are the stationary values of the
exchangeable power obtained in a subsequent imbedding of the type
shown in Fig. 2.4. This second imbedding network is completely variable.
One possible variation removes the first 2-terminal-pair imbedding.
Accordingly, the stationary values of the exchangeable power at the
(nw + 1)th terminal pair in Fig. 2.4 do not change when a 2u-terminal-pair
lossless transformation is interposed between the two networks shown.
It follows that:
The eigenvalues of the characteristic noise matrix N are invariant to a
lossless transformation that preserves the number of terminal pairs.
At this point application of our results to two familiar examples of
linear networks helps to establish further significance for the character-
istic-noise matrix and its eigenvalues.
If the 2-terminal-pair network contains only coherent (signal) gener-
ators rather than noise generators, E;E,* = E;E;* 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 P,,; 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 P.,; = kT Af in a frequency band Af, where & is Boltzmann’s
constant. No matter what form the variable lossless network may take,
26 CHA RACTERISTIC-NOISE MATRIX [Ch. 3
P,.,; must remain constant at the foregoing value. Thus, from Eq. 3.9,
x'[EE' — 2kT Af(Z + Z")]x = 0, for all x
or
EE! = 2kT Af(Z + Z")
a result proved previously by Twiss.' In terms of the characteristic-
noise matrix N, we have
N = —2 EE'(Z + Z')1 = —&T Afl
an equation indicating that a passive dissipative network at equilibrium
temperature T always has a diagonal N matrix, with all the eigenvalues
equal to —kT Af.
3.4. Lossless Reduction in the Number of Terminal Pairs
An n-terminal-pair network has a characteristic noise matrix of uth
order, with m eigenvalues. If & of the m terminal pairs of the network are
left open-circuited and only the remaining (m — k) terminal pairs are
used, the original network is reduced to an (m — k)-terminal-pair net-
work. This operation may be thought of as a special case of a more
general reduction, achieved by imbedding the original 1-terminal-pair
network in a lossless (2n — k)-terminal-pair network to produce (n — k)
available terminal pairs (see Fig. 3.2a). The case for 7 — k = 1 was
considered in Fig. 2.4.
The characteristic-noise matrix of the (7 — k)-terminal-pair network
has (x — k) eigenvalues, which determine the.extrema of the exchange-
able power P.,,x41 obtained in a subsequent (variable) reduction to one
terminal pair (Fig. 3.26). The successive reduction of the -terminal-
pair network first to (~ — 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. 3.2) with Fig. 2.4
shows that the exchangeable power P,,»_%41 obtained by the two succes-
sive reductions, with variation of only the second network (Fig. 3.20),
must lie within the range obtained by direct reduction with variation of
the entire transformation network (Fig. 2.4). Hence the stationary
values of P,.n—x41 found in Fig. 3.26 must lie within the range prescribed
for Penyi by Fig. 2.4. It follows that the eigenvalues of N for the
(n — k)-terminal-pair network must lie within the allowed range of P,,;
for the original m-terminal-pair network, illustrated in Fig. 3.1.
1R. Q. Twiss, ‘““Nyquist’s and Thévenin’s Theorem Generalized for Nonreciprocal
Linear Networks,” J. Appl. Phys. 26, 599 (1955),
Sec. 3.4] LOSSLESS REDUCTION OF TERMINAL PAIRS ar
Lossless
(2n—k)-terminal-pair
network
Fig. 3.2a. Reduction of m-terminal-pair network to (m — k) terminal pairs.
Lossless
(2n—k)-terminal-pair
network
Variable
lossless
Fig. 3.26. Successive reductions of m-terminal-pair network to (m — 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 Networks
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 u real parameters for every n-terminal-pair network, its
existence also shows that an m-terminal-pair, linear noisy network does
not possess more than (real) invariants with respect to lossless trans-
formations.
4.1. Derivation of the Canonical Form
In this section we shall prove the following theorem:
At any particular frequency, every n-terminal-pair network can be 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 m-terminal-pair network
transforms the two matrices Z + Z! and EE! in identical colinear manner,
as shown in Eqs. 3.3 and 3.6. We have also noted that the matrix t which
appears in the transformation is entirely unrestricted by the conditions
(Eqs. 2.5) of losslessness for the imbedding network.
It is always possible to diagonalize simultaneously two Hermitian
matrices, one of which (EE‘) 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
7 = diag (Ri, Ro, Pines ike) a brwsen (4.1)
where Zyem fulfills the condition of the impedance matrix of a lossless
network Zrem = —(Zrem)!.
Suppose, finally, that a lossless (and therefore noise-free) network with
the impedance matrix —Z;em is connected im series with our network
(Z’, E’), as shown in Fig. 4.1. The result is a network with the impedance
matrix
Tih Teen — dae (Ray Bo, «>: 40a) (4.2)
The open-circuit noise voltages remain unaffected when a lossless source-
free network is connected in series with the original network. Thus,
Bae or RVR = Rt (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 B/’E’’’.
This canonical form of the network consists of 7 separate resistances in
series with uncorrelated noise voltage generators, as shown in Fig. 4.2.
Noting that the series connection of a lossless network is a special case
of a lossless imbedding, we have proved the theorem stated at the begin-
ning of this section.
A lossless imbedding leaves the eigenvalues of the characteristic-noise
matrix invariant. ‘Thus, the eigenvalues of the characteristic-noise
matrix N’’ of the canonical form of the original network are equal to those
of N of the original network. But, the eigenvalues \, of N’’ are clearly
its 2 diagonal elements
J ai
JB 18
ie (4.4)
Me = —
30 CANONICAL FORM OF LINEAR NOISY NETWORKS [Ch. 4
The kth eigenvalue is the negative of the exchangeable power of the kth
source. Thus, we have proved the following theorem:
The exchangeable powers of the n Thévenin sources of the canonical form
of any n-terminal-pair network are equal and opposite in sign to the n
eigenvalues of the characteristic noise matrix N of the original network.
& Ideal transformers
Fig. 4.1. Series connection of networks (Z’, E’) and —Zyem.
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 |E,| to unit magnitude, there are actually only m independent (real)
parameters contained in the canonical form. This fact proves the
following theorem:
Sec. 4.2) INTERCONNECTION OF LINEAR NOISY NETWORKS 31
A linear noisy n-terminal-pair network possesses not more than n in-
variants with respect to lossless transformations, and these are all real
numbers.
|
|
|
|
|
I
|
]
|
|
|
|
|
|
|
I
]
I
|
|
|
|
|
|
|
|
}
!
]
|
|
|
|
|
|
|
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 m-terminal-pair noisy network
and an independently noisy m-terminal-pair network. Let them be
connected through a 2(m + m)-terminal-pair lossless network, resulting
in an (m + n)-terminal-pair network, as shown in Fig. 4.3. We shall now
determine the eigenvalues of the characteristic-noise matrix Nmin
of the resulting (m + 2)-terminal-pair network.
To do so, we first reduce each of the component networks to canonical
form of the type shown in Fig. 4.2. This procedure places in evidence,
but does not alter, the eigenvalues of their respective characteristic-noise
matrices. Taken together, the two canonical forms represent the canoni-
cal form of the (m + n)-terminal-pair network of Fig. 4.3. Accordingly,
the m + n eigenvalues of that network are merely the eigenvalues of the
component networks. The proof obviously covers the interconnection
32 CANONICAL FORM OF LINEAR NOISY NETWORKS [Ch. 4
of any number of independently noisy networks of any size, provided the
total number of terminal pairs is preserved.
We shall be interested in cases in which the number of terminal pairs
is reduced upon interconnection. A simple example is the interconnection
Lossless
2(m+n)-terminal-pair
network
Fig. 4.3. Lossless interconnection of m-terminal-pair and #-terminal-pair networks.
of several two-terminal-pair amplifiers, with feedback, to form a new
two-terminal-pair amplifier. In such a case, the effect of the reduction
can be understood by applying the reasoning of Sec. 3.4 to the available
terminals of the network in Fig. 4.3. The conclusion is immediate.
The eigenvalues of the characteristic-noise matrix of an n-terminal-pair
network constructed by lossless interconnection of independently noisy com-
ponent networks having m terminal pairs in all, m > n, must lie within the
range defined by the most extreme eigenvalues of the component networks.
Linear Noisy Networks
in Other Representations
In the foregoing analysis we have found all the invariants of a linear
noisy network with respect to lossless imbeddings that preserve the
number of terminal pairs. With the aid of the impedance formalism,
these invariants have been interpreted in terms of the exchangeable power,
on the one hand, and in terms of the canonical representation of the
network, on the other. There are, however, additional interpretations of
the invariants, which are brought out by different matrix representations
of the network. For each new representation a characteristic-noise matrix
can be defined. As we might expect, all such characteristic-noise matrices
have the same eigenvalues, since, after all, these are the only invariants
of the network.
5.1. General Matrix Representations
The impedance-matrix representation, Eq. 2.6, is conveniently re-
written in the form!
Vv
[a -Z||--|=E (5.1)
I
where 1 is the identity matrix of the same order as Z. Any other matrix
1V. Belevitch, ‘‘Four-Dimensional Transformations of 4-pole Matrices with
Applications to the Synthesis of Reactance 4-poles,” IRE Trans. on Circuit Theory,
CT-3, No. 2, 105 (1956).
33
34 OTHER REPRESENTATIONS (Ch. 5
representation of a linear noisy network can be expressed as
v— Tu=65 (G2)
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 6 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 2m-terminal-pair
network, we consider the mixed voltage-current representation, for which
v= (5.3)
u = : (5.4)
and 6 is the noise column vector
5 = (5.5)
12; (2m—1)
IE (2m—1)
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
I,
if E ni
~Y)
Vi C)) Int
Vom=1 @ I(2m—1)
Fig. 5.1. Mixed voltage-current representation of 2m-terminal-pair network.
Returning to the general expression (Eq. 5.2), we note that it also can
be written in a form similar to Eq. 5.1.
[a -T | eee (5.6)
The analysis in Sec. 5.2 shows how transformations are performed from
one matrix representation to another. Then the transformation of the
power expression will be studied. Combining the results of these two
studies, we shall be able to define a characteristic-noise matrix for every
formalism and to show its relation to the characteristic-noise matrix de-
fined in Eq. 3.15.
5.2. Transformation from One Matrix Representation to Another
\ Vv
The variables | -- | and | -- | can always be related by a linear trans-
I u
36 OTHER REPRESENTATIONS [Ch. 5
E «
where R is a square matrix of order twice that of either V or I. Fora
2m-terminal-pair network, Ris of order 4m. For example, the transforma-
tion from the impedance representation of a 2-terminal-pair network into
the general-circuit-parameter representation (Eqs. 5.3, 5.4, and 5.5) yields
formation of the form
1070 0
O70; 1 0
R= (5.8)
07-10 0
0). Oil
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):
fi 2] Reo | -¥
or
[1 | -Z|R B =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
Ri i Rie
2 Sl)see 9 == (5.10)
Roi |
where the R;; are of the same order as Z. Carrying out the multiplication
in Eq. 5.9, we obtain
Vv
[Ru = ZRo1 Rio = ZRo» | ; / = K (5:11)
u
The correspondence between Eqs. 5.11 and 5.2 is made complete if we
multiply Eq. 5.11 by
M = [Riz ae ZR21|"! (5.12)
Sec. 5.3] POWER EXPRESSION AND ITS TRANSFORMATION 3%
obtaining
[1: -T] -. =8 (5.6)
where
[i:-T]=m[i:-z]R (5.13)
and
8 = ME (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 znto the network is
Vv
a real quadratic form of the excitation-response vector , / . We have
u
v |i Vv
p- |. on |. 6.15)
where Q7 is a Hermitian matrix of order twice that of either u or v. In
the particular case of the impedance-matrix representation,
TGAE
Comparing Eqs. 5.16 and 5.15, we find that the Q matrix for the imped-
ance representation is
1| 0 He!
Qz = ple i (5.17)
ic; 20
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
Comparison with Eq. 5.15 shows that
Qr = R'Q2R (5.18)
where R is the matrix that transforms the general-excitation vector
Vv Vv
[ / into the voltage and current vector ; / , according to Eq. 5.7.
u I
We are now ready to set up the general characteristic-noise matrix N
for any matrix representation, Eq. 5.6.
5.4. The General Characteristic-Noise Matrix
We have introduced the most general matrix representation of a linear
network in Eq. 5.6. We have defined the associated power matrix Qr in
Eq. 5.15. With these two we may define a general characteristic-noise
matrix Nr corresponding to this matrix representation. The require-
ments are that this matrix Nr:
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, Nr will contain the ” 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
Nr = {{1: -T]Qr7[1' —T |} "se (5.19)
For the impedance-matrix representation, using Eqs. 5.1, 5.6, and
5.17, we obtain
[a -Z|Q7"[1! -z|' eV AD) (5.20)
| |
Introducing Eq. 5.20 into Eq. 5.19, we have
Nz = —3(Z + Zt)EEt (5.21)
Sec. 5.4) THE GENERAL CHARACTERISTIC-NOISE MATRIX 39
But, Eq. 5.21 is identical with the definition Eq. 3.15 for N.
Next, let us relate the general noise matrix Nr of Eq. 5.19 to its partic-
ular form in the impedance representation. For this purpose, we note
that according to Eq. 5.14 :
56? = MEE‘? (5.22)
Then, using Eqs. 5.13, 5.18, and 5.20, we find
[ee
M[1: -Z]RQr,1R'[1: -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*1NzM!? (5.24)
According to Eq. 5.24, the characteristic-noise matrix Nr of the general
matrix representation of a network is related by a similarity transforma-
tion to the characteristic-noise matrix Nz of the impedance-matrix repre-
sentation of the same network. Therefore, Nr and Nz have the same
eigenvalues.
The eigenvalues of the characteristic-noise matrix of Eq. 5.21 deter-
mined the stationary values of the real quantity p in Eq. 3.10. Com-
parison of these two expressions with the expression for the general
characteristic-noise matrix in Eq. 5.19 shows that its eigenvalues deter-
mine the stationary values of the associated real quantity pr
+ y 85 Ty
a 1 as | T
a
with respect to variations of the arbitrary column matrix y. It is easily
shown by the method of Sec. 3.2 that this “noise parameter” pr has, in
fact, as extrema the eigenvalues of the matrix Nr defined in Eq. 5.19.
The range of values that pr assumes as a function of y is identical with
the range of values of p in Eq. 3.10 and Fig. 3.1.
The network classification in the three cases illustrated in Fig. 3.1
should now be restated in terms of the T-matrix representation. This is
easily accomplished by considering Eq. 5.23. According to it, the matrices
E ! -T Q,! [1 ! -T |’ and —(Z + Z") are related by a colinear
transformation. A colinear transformation preserves the signature of
a matrix. Consequently, the network classification of Table 3.1 can be
carried out in the T-matrix representation, as shown in Table 5.1. The
same conclusion may be reached from the facts that Nr and Nz have the
same eigenvalues, and both EE’ and 48! are positive definite.
pr (5:25)
40 OTHER REPRESENTATIONS (Ch. 5
Table 5.1. CLASSIFICATION OF NETWORKS AND EIGENVALUES
IN T-MATRIX REPRESENTATION
a 1 ii Network Eigenvalues
Gaseu am [1 -T | Ora E -T | Class pr (A,) of Nr
1 positive definite passive <0 all << 0
2 negative definite active (negative >0 all> 0
resistance)
3 indefinite active Z0 fee 2 |
some < 0
Figure 3.1 gives directly the allowed range of pr and the eigenvalues
of Nv, if the notation of Table 3.1 is replaced on the figure by that
of Table 5.1.
In the specific case of the mixed voltage-current representation of
Eqs. 5.3 to 5.5, Nr and pr can be simplified if we introduce the detailed
expressions for the power matrix Q7. This matrix is found most directly
from the explicit expression for the power P flowing into the network in
Vv
terms of the excitation-response vector | -- |. Comparing the resulting
u
expression with Eq. 5.15 allows identification of Q7 by inspection. The
power matrix Q7 is square and of 4mth order
ree a
Qr if “. (5.262)
0 ,-P
where the P’s are matrices of order 2m of the form
Ok ORE On O07;
0 ! 0 0 | 0 0 ! !
OOh nal eOG: FOnOn !
) OL OO seat Oa
Hem Louonnoy Ov eiieoly 1 Can
oi
ire aaa
00, 6, OeuCs Oe cen tra
00 10: 10 0H One eaten
Sec. 5.44 THE GENERAL CHARACTERISTIC-NOISE MATRIX 41
It is easily checked that P has the properties
po! = p'=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 2mth order
[a -T]Qr7 [a —T|' = 2 - TPT’) (5.28)
Thus, from Eq. 5.19 we have for Nr
Nr = 3(P — TPT*)—88° (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
y '88ty
Pr dy'(P — TPT )y (5.30)
The preceding development shows how each matrix representation T
has associated with it a particular noise parameter pr, of which the
extrema are determined by the eigenvalues of its characteristic-noise
matrix. In the next chapter we shall develop in detail the significance of
pr for two-terminal-pair amplifiers represented in terms of their general
circuit constants.
Noise Measure
In Chap. 3, starting from the impedance-matrix representation, we
defined a noise matrix Nz. The eigenvalues of Nz 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 Nz, per-
taining to the general matrix representation in Eq. 5.2. The eigenvalues
of Nr and Nz are identical. Associated with the eigenvalues of Nr are
the extrema of the generalized scalar parameter pr in Eq. 5.25. Special
forms of Nr and fr for the mixed voltage-current representation (Eqs. 5.3
through 5.6) were given in Eqs. 5.29 and 5.30. In the case of a two-
terminal-pair network, this representation reduces to the “general-circuit-
constant” description. Our interest in the noise performance of linear
amplifiers gives the two-terminal-pair case a special importance. The
remaining part of our work will therefore be confined to the interpre-
tation and study of pr for the two-terminal-pair network in the general-
circuit-constant representation.
Our problem is to find the physical operation that leads to the extrema
of pr, in the same manner as the operation of Fig. 2.4 led to the extrema
of pz. Itis obvious that the operations involved in the extremization will
make use of lossless imbeddings, since only such operations leave the
eigenvalues of Nz unchanged. Variations in these imbeddings will
presumably produce variations in the column vector y in Eq. 5.30, and
lead to the extrema of pr.
The general-circuit-constant representation of a two-terminal-pair
network emphasizes its transfer characteristics. ‘Therefore, we expect
42
Sec. 6.1] EXTENDED DEFINITIONS OF GAIN AND NOISE FIGURE 43
that pr in this representation must be related to the noise performance
of the network regarded as a transfer device. The noise figure has been
for many years the most widely used parameter describing the noise
performance of transfer devices. Consequently, we might well investi-
gate first whether or not pr has any relation with the noise figure.
6.1. Extended Definitions of Gain and Noise Figure
The noise figure is normally defined in terms of available power. We
have seen, however, that the available-power concept leads to difh-
culties in cases involving negative resistances. Since such cases must
arise in any general theory of linear amplifiers, the available-power
concept should be replaced everywhere by the exchangeable power.
Accordingly, the same replacement should be made in the available-gain
definition: The exchangeable power P.g of the input source and the
exchangeable power P.o at the network output replace the corresponding
available powers. Specifically,
Peo
Cl am 6.1
G pe (6.1)
where
Eo?
0 =— 6.16
P.o Ro (6.16)
and
Es?
= — 1
Pes 4Rg (6 2)
We find that
G.>0 if Rs/Ro > 0 (6.2a)
G, <0 if Rs/Ro <0 (6.20)
Observe that when Rg > 0 and Ro > 0, G, becomes the conventional
available gain of the two-terminal-pair network.
The foregoing ideas lead to an extended definition, F., of the noise
figure of a two-terminal-pair network!
Ne
GekTo Af
where JV,; is the exchangeable noise power at the network output when
the source has a given (noiseless) impedance Zs. Thus, N.; contains
B= (6.3)
1H. 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 MEASU RE [Ch. 6
only the effect of zmternal 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 To, provided this
impedance is passive (that is, has a positive real part). In the case of a
source impedance with a negative real part, kT) Af merely represents an
arbitrary but convenient normalization factor.
From Eqs. 6.1, 6.2, and 6.3, it is clear that
Fe-—1>0 if Rs >0 (6.4a)
F,-—1<0 if Rs <0 (6.40)
If Rs > Oand Ro > 0, F, becomes the standard noise figure F.
Zs qi
Es; Vy
Fig. 6.1. Thévenin equivalent of two-terminal source.
6.2. Matrix Formulation of Exchangeable Power and Noise Figure
The parameter pr of Eq. 5.30, whose relation with the noise figure we
are going to investigate, is expressed in matrix notation. Accordingly,
to facilitate comparison, we must express the (extended) noise figure and
the related parameters in the same manner.
Exchangeable Power. A linear two-terminal source of power can be
represented by its Thévenin rms open-circuit voltage Es, in series with
its internal impedance Zs. Let V,; and J, be the terminal voltage and
current of the source, respectively, as shown in Fig. 6.1, so that
Vi +Zsl; = Eg (6.5)
To express Eq. 6.5 in matrix form suitable for cascade applications, we
define column vectors (matrices) v and x as
Eh i) a
x'vy = Eg (6.7)
Sec. 6.2] MATRIX FORMULATION OF EXCHANGEABLE POWER 45
We note that Eq. 6.7 is the one-terminal-pair form of Eq. 5.1, where all
the submatrices have become scalars.
Since multiplication of Eq. 6.7 by a constant ¢ does not alter it, we can,
purely as a matter of form, always make a new column vector
MO is oa) (ee ae a a
ile ea
and a new scalar
y = ckEs
so that the source equation (Eq. 6.7) becomes
yv=yY (6.8)
with
2 ge (6.9)
ail
where Zz 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 P, of the source may be written in
matrix form
Dy = |Es|? ee |Es|? os ly|? (6.10)
where the square “permutation” matrix P is the two-terminal-pair form
of Eq. 5.260.
il
p=[? 4] an
It has the properties P' = P and P~! = P as indicated in Eqs. 5.27.
The usefulness of the last expression in Eq. 6.10 lies in the fact that it
can be written by inspection for any two-terminal source with a source
equation in the form of Eq. 6.8.
Exchangeable-Power Gain. Consider a linear source-free two-
terminal-pair network, described by its general-circuit constants A, B,
C, D, as shown in Fig. 6.2. If v is the ‘“‘input”’ column vector defined by
Eq. 6.6, and we let u be the “output” column vector,
= el (6.12)
and T is the general-circuit matrix,
T = E Al (6.128)
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 P.o from terminal pair 2 can be obtained at
Fig. 6.2. The general-circuit-matrix representation of a source-free two-terminal-pair
network.
once from a source equation in the form of Eq. 6.8 written with u as
the voltage-current column variable. Multiplication of Eq. 6.13 by y’,
with the use of Eq. 6.8, yields this new relation,
(y'T)u = ¥ (6.14)
Accordingly, by applying the steps of Eqs. 6.8, 6.10, and 6.11 to Eq. 6.14,
we find that
= Gals
se 2y' TPTty 61>)
and with Eq. 6.10, and Eq. 6.1a¢ for the exchangeable gain, we have
ti
y Py
=o 6.16
. y'TPT'y Ket)
Extended Noise Figure. To apply the matrix formulation of ex-
changeable power to the calculation of the (extended) noise figure of a
two-terminal-pair noisy network, we still describe the network by its
general-circuit constants, but we also allow for noise voltages or currents
at the terminals in the absence of external sources. With reference to
Fig. 6.3, the network equations for the dotted box”? would be
v= Tu+6 (6.17)
2 A. G. Th. Becking, H. Groendijk, and K. S. Knol, ‘““The Noise Factor of 4 Termi-
nal Networks,” Philips Research Repts. 10, 349-357 (1955).
3H. Rothe and W. Dahlke, “Theory of Noisy Four Poles,” Proc. I.R.E., 44,
811-817 (1956).
Sec. 6.2} MATRIX FORMULATION OF EXCHANGEABLE POWER 47
with 6 a ‘‘noise column vector,”
§ = | (6.18)
Now Equation 6.17 can be rewritten as two relations
v = Tu (6.192)
=v +6 (6.195)
/
If we visualize v’ = BI as referring simultaneously to the input
terminals of a noise-free network T and the output terminals of a pure-
noise network 6, the cascade division of the system is represented in
Fig. 6.3. The noiseless part T does not affect the noise figure of the
system. Thus, for noise-figure calculations, we need consider only the
noise network § driven by a source of internal impedance Zs.”
Noise-free
network T
Noisy amplifier T, 6
Fig. 6.3. The general-circuit-matrix representation of a linear two-terminal-pair network
with internal sources.
The source equation appropriate to the right-hand terminals in Fig. 6.4
can be obtained from Eqs. 6.196 and 6.8 with Es =
ylv’ = —y'8 (6.20)
Therefore, the output exchangeable power J;, produced by the internal
noise only, is given by
ipat.
y 58'y
et — 6.21
and, since G, = 1 for this network, we have
| ‘BaF
aan Nes y'68'y
~ Ey Af ~ yFPy(QkTo Af) se
48 NOISE MEASU RE [Ch. 6
Noise network 6
Fig. 6.4. Noise network of a linear noisy two-terminal-pair network in general-circuit-matrix
representation.
6.3. Noise Measure
The ‘‘excess-noise figure,” Eq. 6.22, has the same numerator as the
noise parameter pr, Eq. 5.30, that we are trying to identify. The two
equations differ only in the subtractive term y'TPT'y in the denominator,
aside from the multiplicative constant kT) Af. In order to facilitate the
identification, we rewrite Eq. 5.30 in the form
y'88ly
vr)
ire lc (6.23)
2(y'Py) (1 y'Py
(Diag
From a comparison of Eq. 6.23 with Eqs. 6.22 and 6.16, it is obvious that
pr NBs A
kTo Af
i (6.24)
G.
The expression on the right-hand side of Eq. 6.24 may now be identi-
fied as the noise parameter with the extremal properties corresponding to
the network invariants. In the cases in which the extended definition of
noise figure F, coincides with the conventional noise figure # and the
exchangeable gain G, 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 F, and G, 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 M.:
F.-1 657
Mo S&S a eto a eda aia ae (6.25)
,_-2 y(P— TPT )y 2kTo Af)
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 M,
of Eq. 6.25 is achieved.
In the discussion of the noise performance of amplifiers, it turns out to
be important to bear in mind the algebraic sign that M@, assumes under
various physical conditions. These are summarized in Table 6.1.
Table 6.1. ALGEBRAIC SIGNS OF EXCHANGEABLE GAIN
AND DERIVED QUANTITIES
Rs Ro Ge Een an|Gal M.
>0 >0 >0 >0 al >0
>0 >0 >0 >0 <i <0
>0 <0 <0 >0 zi >0
<0 >0 <0 <0 z1 <0
<0 <0 >0 <0 >1 <0
<0 <0 >0 <0 <1 >0
With reference to Table 6.1, it should be pointed out that for Rs > 0,
(F, — 1) > 0. When Rg > 0, conventional available gain greater than
1 occurs in only two ways:
(a) G. > 1
(6) G. <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 6 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 M, 1s found to be greater than zero.
In the succeeding portions of our work, we shall restrict our considera-
tion of amplifier performance to cases in which the source has an internal
impedance with a positive real part. This is the only case of practical
interest. Indeed, any amplifying system, however complicated, is
essentially a two-terminal-pair network driven by a signal transducer
with a positive real part to its output impedance (for example, antenna,
microphone, and so forth).
6.4. Allowed Ranges of Values of the Noise Measure
Let us consider a noisy two-terminal-pair network with the noise
column matrix 6 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 Zg. 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
22 = Zs" (6.9)
41
Next, we suppose that the original network is imbedded in a four-termi-
nal-pair, lossless network (Fig. 6.5) before we connect it to the source.
A new network results, with the noise column vector 8’ 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:
M fee a eVeON Maye = eng es (6 26)
oy (P= Py Dil GAG
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
FE? = stEE's (3.3)
and
Z+ ZZ =e(Z+ Z")r (3.6)
However, from Eqs. 5.22, 5.23, and 5.28 we have
87877 = M’ E’E™M"t = M’s'EE'eM"t
= (M’t'M~!)88" (M’stM—?)t
= Clssic (6.27)
and
(P — T’PT") = M’(Z’ 4+ Z’)M"t = —M’s1(Z 4+ Z!)emM"t
= (M’s'M"!)(P — TPT") (M’s'M?)?
= Cl(P — TPT")C (6.28)
where
ci = M’7'mM"! (6.29)
The matrix C involved in the colinear transformations of 65’ 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
Imbedding
network
Amplifier
T, 6
New
amplifier
i é'
14 Vo
qT, Ug Ty Va
u= e e e = ° 5 YS = e e °
V3 uy V4 Vp
I, I,
Fig. 6.5. Imbedding of two-terminal-pair amplifier.
the noise-measure expression, Eq. 6.26, we have
Cy) *88'(C 1
oti EUR (6.30)
(Cy)'(P — TPT") (Cy) 2kT Af
Through variations of the imbedding network, the column vector Cy
can be varied arbitrarily. In this manner M,’ can be varied over its
entire allowed ranges, which are limited by the two eigenvalues \, and A»
52 NOISE MEASU RE [Ch. 6
of Nr. By making use of Table 5.1 and Eq. 5.28 in the special case of a
two-terminal-pair network, we see that the following three cases have to
be distinguished:
1. P — TPT" is negative definite; T is the general circuit matrix of a
passive network, M,’ < 0.
2. P — TPT" is positive definite; T is the general circuit matrix of a
negative-resistance network, M,’ > 0.
3. P — TPT" is indefinite; T is the general circuit matrix of a network
capable of absorption, as well as delivery of power, M.’ z 0.
M. Me M.
ic
= g
a} E
5 3
uw a
c AA
3 NN ———_
aS) S) 2 kTp Af
5 3 |kT) df
S P=
iS
ae E
kT, Af =
5
M.=0 M = ot
a Al ac
£ [RT Af S kT Af
€ 3
amet 2 3
NTN S E=|
(0) ue =
f= pa
(3) CT)
no} a.
aS
2a
eS
P — TPT negative definite P — TPT? positive definite P — TPT? indefinite
(a) (b) (c)
Fig. 6.6. Schematic diagram of permitted values of M,’ for two-terminal-pair networks.
When C in Eq. 6.30 is varied through all possible values, M .’ reaches two
extrema, which are the two eigenvalues of the characteristic-noise matrix
Nr divided by kT) Af. 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 Rs > 0 in all cases except Ro > 0,0 < G, <1.
This case does not correspond to an amplifier. Hence we shall be inter-
ested in achieving only positive values of M,.’, which occur in Cases b and
c of Fig. 6.6. These cases both have an available gain, G, in the con-
Sec. 6.4] ALLOWED RANGES OF NOISE MEASU RE 53
ventional sense, greater than unity (G=G,,ifG,>1;G= ~,if
EG <e
We observe from Eq. 6.30 that the numerator is never negative.
Therefore, changes in sign of M,’ 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 \,/ (kT Af) and Az/(kTo Af).
The two cases of interest, Cases 6 and ¢ of Fig. 6.6, have a least positive
eigenvalue of N which we call \; > 0. We have therefore proved the
following theorem:
Consider the set of lossless transformations that carry a two-terminal-
pair amplifier into a new two-terminal-pair amplifier with a conventional
available gain G greater than 1. When driven from a source that has an
internal impedance with positive real part, the noise measure of the trans-
formed amplifier cannot be less than \1/(kTo Af), where dy is the smallest
positive eigenvalue of the characteristic-noise matrix of the original amplifier.
The fact that M.’ > 1/(kTo Af) also puts a lower limit upon the
excess-noise figure. Suppose that the amplifier is imbedded in a lossless
network and then connected to a source with an internal impedance
having a positive real part. Let the resulting exchangeable power gain
be G,. Then, the excess-noise figure of the resulting amplifier has to
fulfill the inequality
ips is is (1 = =) (6.31)
Consequently, an amplifier has a definite lower limit imposed on its
excess-noise figure, and this limit depends upon the exchangeable-power
gain achieved in the particular connection.
If G, > 0, that is, the output impedance of the amplifier has a positive
real part, the excess-noise figure can be less than \;/ (kT Af) only to the
extent of the gain-dependent factor (1 — 1/G,).
If G, < 0, that is, the output impedance has a negative real part, the
lower limit to the excess-noise figure is higher than A,/(kTo Af) by
(1 + |1/G,).
Thus, if two amplifiers with the same eigenvalue d, of their characteristic-
noise matrix are driven with a positive source impedance, one of which has a
positive output impedance, the other a negative one, then the minimum
excess-noise figure of the latter cannot be less than that of the former.
Equation 6.31 has established a gain-dependent lower bound for the
excess-noise figure achievable with lossless imbeddings of a given amplifier.
At large values of |G,|, the excess-noise figure is evidently equal to the
54 NOISE MEASU RE [Ch. 6
noise measure. Large values of |G,| must correspond to large values of
conventional available gain. Therefore, the excess-noise figure at large
conventional available gain is limited to values greater than, or equal to,
\i/kTo Af 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,
M1
kT, Af
Meopt = (6.32)
is a significant noise parameter of the amplifier. The significance of
M.,opt Will be further enhanced by the proofs, given in the remaining
sections, of the following statements:
1. The lower bound M,,.p¢ 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 M,opt as the excess-noise figure at arbitrarily high gain.
2. An arbitrary passive interconnection of independently noisy ampli-
fiers with different values of M,,.»_ cannot yield a new two-terminal-pair
amplifier with an excess-noise figure at large conventional available gain
lower than M, opt of the best component amplifier (the one with the
smallest value of M, 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 M, 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 m independently noisy amplifiers, as
shown in Fig. 6.7. A 2n-terminal-pair network results. By open-
circuiting all but two of the resulting terminal pairs, we obtain the most
general two-terminal-pair amplifier obtainable from the original ones by
lossless interconnection. In Sec. 4.2 we have developed the general
theory of such an imbedding and reduction of terminal pairs. Indeed,
Fig. 4.3 includes the situation of Fig. 6.7. We know that the eigen-
values of the characteristic-noise matrix of the network lie between the
most extreme eigenvalues of the characteristic-noise matrices of the
original 7 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 M, opt of the re-
duced network cannot be less than that of the best of the original
amplifiers.
4n-terminal-pair
lossless imbedding
network
Fig. 6.7. Lossless interconnection of several amplifiers.
We shall now extend our proof to cover passive interconnections of
amplifiers. We again consider m amplifiers but shall now imbed them
into a dissipative 4n-terminal-pair network in the configuration of Fig. 6.7.
Suppose, for a moment, that we disconnect the amplifiers and that we
throw the imbedding network into its canonical form, as discussed in
Sec. 4.1. This is accomplished by a lossless network transformation, and
results in an arrangement like Fig. 4.2 but with 4 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 MEASU RE [Ch. 6
that produced the canonical form, we obtain a 4n-terminal-pair network
which has the same terminal behavior as the original dissipative im-
bedding network. The resulting network is shown in Fig. 6.8. It has
2n available terminal pairs. The 4m resistances can be comprised in a
4n-terminal-pair network with a characteristic-noise matrix, the eigen-
8n-terminal-pair
lossless inverse
canonical
transformation
network
Output terminal pairs
n amplifiers
Passive dissipative 4n-terminal-pair
imbedding network
Fig. 6.8. Imbedding of » two-terminal-pair amplifiers in lossy 4n-terminal-pair network.
values of which are all negative (Sec. 3.3). The m amplifiers can be
grouped correspondingly into a 2m-terminal-pair network with a charac-
teristic-noise matrix that has (in general, positive and negative) eigen-
values equal to those of the characteristic-noise matrices of the original
amplifiers. The network operation in Fig. 6.8 is obtained from that
corresponding to Fig. 4.3 by a subsequent reduction from 6x to 2 output
terminal pairs. Again, the least positive eigenvalue of the characteristic-
noise matrix is not less than that of the best amplifier. Thus, we may
state the following theorem:
Consider a general lossless or passive dissipative interconnection of an
arbitrary number of different and independently noisy two-terminal-pair
amplifiers that results in a two-terminal-pair network. When the resulting
two-terminal-pair network is driven from a source that has an internal
Sec. 6.5] PASSIVE INTERCONNECTION OF AMPLIFIERS 57
impedance with positive real part, its optimum noise measure M, op cannot
be less than that of the best amplifier, that is, the amplifier with the least
positive eigenvalue of tts characteristic-noise matrix.
Statement 3 of Section 6.4 is an immediate corollary of the previous
theorem; that is,
Passive dissipative wmbedding of a given two-terminal-pair amplifier
cannot reduce its noise measure to a positive value below M,. opt, provided
that the source impedance has a positive real part.
Network Realization
of Optimum Amplifier
Noise Performance
In Chap. 6 we have shown that the excess-noise figure of an amplifier
with a high gain cannot be less than M, opt,
Ay
kT Af
where , is the smallest positive eigenvalue of the characteristic-noise
matrix. We also showed that an arbitrary lossless or passive interconnec-
tion of two-terminal-pair amplifiers, which leads to a new two-terminal-
pair amplifier, yields an excess-noise figure at high gain that is higher than,
or at best equal to, the M.,opt of the best amplifier used in the inter-
connection. These proofs established the quantity M.opt as a lower
bound on the noise performance of a two-terminal-pair amplifier.
In this chapter we shall show that the lower bound M,,op+ on the excess-
noise figure at high gain can always be realized. Specifically, we shall
show that the minimum positive noise measure M,op4 of any two-
terminal-pair amplifier can be achieved by suitable external network
operations, which, however, do not usually result in an amplifier with a
high gain. Nevertheless, the source impedance, and the amplifier output
impedance with the source connected, will always have positive real parts
in the realizations of M.op+ presented. It follows that an amplifier with
an arbitrarily high gain can be constructed by cascading identical,
optimized amplifiers that have appropriate impedance-transformation
networks between the stages. By an adjustment of the transformation
networks, the optimum noise measure of the cascade can be made equal
to the M.op+ of the individual amplifiers in the cascade, as explained in
58
Meopt = (6.32)
Sec. 7.1] CLASSIFICATION OF AMPLIFIERS 59
Chap. 1. The excess-noise figure of the high-gain cascade is equal to the
optimum noise measure of the cascade, and thus in turn equal to Mz opt
of the individual amplifiers. This arrangement therefore accomplishes
the realization of the lower limit of the excess-noise figure at high gain.
Since M.opt of any given amplifier determines the lowest (excess-)
noise figure that can be achieved at high gain with the amplifier, either
singly or in interconnection with other amplifiers of the same type, we
may conclude from the criterion chosen in Chap. 1 that Me opt is an
absolute measure of the quality of noise performance of a given amplifier.
7.1. Classification of Two-Terminal-Pair Amplifiers
The noise-performance optimization problem is solved conveniently by
referring to a detailed classification of nonpassive two-terminal-pair net-
works (that is, amplifiers). Mason’ has shown that every such network
can be reduced by lossless reciprocal imbedding to one of the three basic
types shown in Fig. 7.1. His classification is based primarily upon the
range of values of the unilateral gain U
[Zo1 = Zis/?
U =o
A(R Roo ae Ry2Ro1)
(7.1)
where the R’s are the real parts of the impedance-matrix elements. Since
the numerical value of U is invariant to lossless reciprocal transforma-
tions,’ none of the three types can be carried into any other type by such
transformations.
The first type (Fig. 7.1¢), with U > 1, is by far the most common.
The majority of vacuum-tube and transistor amplifiers belong to this
class. The second type (Fig. 7.10), with U < 0,isless common. It does,
however, share one important property with the first; namely, both have
det (P — TPT") <0 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. 7.10 has a unilateral gain
U = —|ul? <0.
By connecting the network in series with a lossless gyrator with the
1S. J. Mason, “Power Gain in Feedback Amplifiers,” Trans. IRE, Professional
Group on Circuit Theory, CT-1, No. 2, 20 (1954).
60 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch.7
q;
—_——_—_—_>
+
Vi 1 ohm
U>1; jul >1
Det(P — TPT?) = ae — 1< 0; indefinite
(a)
q;
—__S
+
V; 1 ohm
u<0; IulS1
1
Det(P = TPT) =— (az o 1) < 0; indefinite
(b)
O<U<1; |ul<)
Det(P — TPTT) =p = 1>0; positive definite
(c)
Fig. 7.1. Classes of amplifiers.
ec. 7.2] OPTIMIZATION, INDEFINITE CASE 61
impedance matrix
Th ee 4 (3)
where + is real, we obtain, for the unilateral gain (Eq. 7.1) of the com-
bined network,
_ [Re @) ae oF se toe)
oa —1+7? + 2r Re (uw) ne)
In Eq. 7.4, U can always be made positive and greater than unity by
choosing
r>+vi[Re (vu)? + 1 — Re (uw)
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’) > 0 (7.5)
and cannot, under any terminal condition, absorb power. Obviously it
cannot be reduced to any of the other forms by any Jossless transforma-
tion whatsoever. It is a “negative-resistance’”’ amplifier.
Recalling that the optimum noise measure is not changed by any loss-
less transformation, we conclude that noise-performance optimization of
amplifiers need only be carried out on networks of the specific forms
Figs. 7.1@ 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') <0 is
initially in, or is reduced to, the form of Fig. 7.1¢. We shall show that
M..,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
ahd °
and the noise column matrix (elements not shown in Fig. 7.12) is
ae 7
62 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch.7
The characteristic-noise matrix of this amplifier can be computed to be
N = 4(P — TPT')—88"
Enilnt ?
5 ; 2|u | er | a7 8p Ina i 2|u\? 1 ap Tn
| 2iel2 2 ay at :
Bee i Gemma a ng i
Bal Z |
D} nl £nl \ ni
Ent 2|u\? 4 Ents ate 2 ||? a
The amplifier noise measure in matrix form is given by Eq. 6.25:
y'dd'y 1
ene y'(P — TPT"')y 2kTy Af
(6.25)
where y is a column vector fulfilling the condition:
in which Z¢g 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 Zg. Hence, the noise measure can be optimized by
a lossless impedance-matching network at the input of the amplifier if
the actual source impedance with a positive real part can be transformed
into the value Zs‘ prescribed by the eigenvector y“,
(1)\ *
Zs) = (25) (7.9)
Thus, if the noise measure is to be optimized by a lossless mismatching
network, it is necessary and sufficient that
yo)?
Re (25) = Re (Zs?) >0 (7.10)
1
The proof that the inequality (Eq. 7.10) is fulfilled for || > 1 will now
be carried out.
The proof is greatly facilitated if we use a wave formalism rather than
the voltage-current formalism. We assume that transmission lines of
1-ohm characteristic impedance are connected to the amplifiers. The
incident waves a, and az and the reflected waves 0; and bs on these
Sec. 7.2] OPTIMIZATION, INDEFINITE CASE 63
transmission lines are related to the terminal voltages and currents by
a =3(Vi4+1); by = 3(V2 + Ie)
by =3(Vi = Jk)? a2 = 3(V2 — Ie)
These transformations are conveniently summarized in matrix form. We
define the matrix
1/1 1
R=5 f | (7.11)
We then have
Hseeg LAN i Vi as
iy = ea =R ie = Rv (7.12)
ate bo fs Vo ne
a i =R ila Ru (7.13)
The general-circuit-parameter representation of the network is
v— Tu =58 G22)
The matrix equation for the new choice of variables v’ and u’ has the
general form
fata? ao (7.14)
The relations between the matrices T’ and 8’, on the one hand, and T
and 6, on the other hand, are easily derived by using Eqs. 7.12 and 7.13
in Eq. 5.2 above. We obtain
T’ = RTR” : (7.15)
1/6; +6 6,’
ipl we pO | Le |) OR
eA a6
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' =2R7! (7s)
Using Eqs. 7.16 and 7.17, we can write the noise matrix 85° as
65)’ = R78/5’tR = 4R8’8'RI (7.18)
Furthermore, we have from Eqs. 7.15 and 7.17
P — TPT’ = P — R“'T’RPR'T'(R")t
= 2R[2RPR' — T’(2RPR')T”]R*
= 2R(P’ — T’P’T")R'
64 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch.7
where
Pp’ = 2RPRt = RPR™ = E ne (7.19)
Combining Eqs. 7.18 and 7.19, we obtain for the noise measure, Eq. 6.25,
w'88 Ww 1
MrT ITE POS (7.20)
with
1 | Et at = ete
= = R'y =- eo
i Ee TD Ma a V2 ( )
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't)“18'577 (7.22)
The requirement that Re (y2!?/y;“) > 0 imposed on the eigenvector
y‘) imposes a corresponding limitation on w‘?. From Eq. 7.21,
—=— 1
We Va a1)
== | = | _ -(=—— 7.23
Wy ya 44 os ( )
¥1
Thus, w/w, is the negative-conjugate reflection coefficient corresponding
to the source impedance Zs. Therefore, we must have
We
aul (7.24)
Wi
From Eqs. 7.15, 7.16, 7.19, and 7.22, it is easily found that, for the net-
work of Fig. 7.14,
|u|?15,’]? |u|76,/52'*
|u|? — 1 |u|? — 1
N’ = (7.25)
5% _- TP
Sec. 7.2] OPTIMIZATION, INDEFINITE CASE 65
The positive eigenvalue of N’ is
nae 1 |x|? 2 Js 72
=| Er
oh [6,’P?— vP) ine aL 5 (an? [S2’|?—|61'62*|? |
(7.26)
From the definition of the eigenvector, the matrix equation results:
Nw® = \yw
of which the second component is
. /xS 7 7 1
—6, #05 w nae 9 2wo")? = Ai We )
and thus
wo)?
ae
: ~ 2575
U 2 7 U 2 7
rE (tee wre) —4 na a
(7.27)
But |u| > 1, and the eigenvalue , is real. Therefore, from Eq. 7.26
seen 2 wee pe
4|51’52/*/? < 4 ee 1 [51'62'*|? < (FS jz — ; (6? ar eP)
which, in Eq. 7.27 yields
eee (7.28)
The eigenvector w‘!) for which the noise measure is optimized thus
corresponds to a reflection coefficient less than unity, that is, to a passive
source impedance. The condition in Eq. 7.28 is equivalent to the condi-
tion, Eq. 7.10, in the general-circuit-matrix notation. We have thus
proved that the noise measure of a unilateral amplifier can be optimized
with a lossless mismatching network between the source (having an
impedance with a positive real part) and the amplifier. The output im-
pedance of this amplifier always has a positive real part. It follows that
any number of optimized amplifiers, with det (P — T PT") <0, can be
cascaded with appropriate lossless mismatching networks between suc-
cessive stages so as to achieve an arbitrarily high gain. The excess-noise
66 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch 7
figure of the cascade is then equal to the minimum noise measure of each
stage, which, with the aid of Eq. 7.8 for N, is
[Em|? + [Zn)? Bi wus pee
M1 1 ni 2417 Sale pe ele *2 Re ESTs
ae See esea ay | age i ee
qian FS = 2Re Eat)
(Peo ae (|Ena|?|Zn1|? = fae)
(7.29)
The proof of the inequality in Eq. 7.28 can easily be extended to cover
the case of nonunilateralized amplifiers of the class of Fig. 7.1a, provided
they have passive conjugate-image impedances. We start with the net-
work in the form that has the scattering matrix
with
[Soi] > 1
and
[Sie < il
The only differences occur as minor modifications in Eqs. 7.25 ff., where
|So1|? replaces |e? and (1 — |S,9|") appears in other terms. Thus,
unilateralization is not a necessary step to achieve optimum noise measure
with input mismatch. Amplifiers that have passive conjugate-image
impedances can be optimized for noise measure by an input mismatch
alone. However, the output impedance under optimized conditions is
guaranteed to have a positive real part only if the amplifier is also stable
under arbitrary passive input and output loading. Most vacuum-tube
and transistor amplifiers meet these conditions over a significant fre-
quency range.
7.3. The Optimum Noise-Measure Expression for the Conventional
Low-Frequency Vacuum Tube
We shall now derive from Eq. 7.29 the expression for the minimum
noise measure of a conventional low-frequency vacuum tube. The noise
in the tube is characterized by a grid noise resistance R,,, the input imped-
ance is R,, and the plate resistance is r,. The noise-voltage column
Sec. 7.3) OPTIMUM PERFORMANCE OF CONVENTIONAL TUBE 67
matrix is
| 28a
E = | | (7.30)
with
|Z]? = 4kTy) Af (Rn + Ri) pe? (7.31a)
and
E, E.* = —AkTo Af uR, (7.31c)
The impedance matrix is
IR oi
ees ! .| (7.32)
a wR, lp
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' = ie Ff (7.33)
NIE
and the new open-circuit noise voltages of the over-all network are
Ey
! V Ry
E = (7.34)
Ey
Vp
Finally, in general-circuit-parameter form, the parameters of the triode
become
10 Me aoe |
r-->|; |
pea [= (7.35)
Dp
where
Also
68 NETWORK REALIZATION OF OPTIMUM PERFORMANCE ([Ch.7
With these specific values we obtain for the optimum noise-measure
expression of the triode
lp : Toe tn
Kn cts Al hn RRs a ee 2 eee
ll i lt
Ar
Re ay
1 ue
M opt =2 (7.37)
For large values of », the minimum noise measure is effectively equal
to the minimum excess-noise figure, and all terms in Eq. 7.37 divided by
uw can be disregarded. We obtain in the limit up >
D}
Tee =) if =P R. Gee = V Re -- R,R,)
1
This result is well known.”
7.4. Optimization of Negative-Resistance Amplifiers,
Definite Case
There remains the problem of achieving the optimum noise measure of
negative-resistance amplifiers, that is, the class illustrated in Fig. 7.1c.
This problem we now wish to solve, employing a positive source imped-
ance and guaranteeing that a positive output impedance results.
While it is actually possible to accomplish our purpose by performing
a consecutive series of lossless reciprocal imbeddings, starting from the
specific amplifier form given in Fig. 7.1c, the particular method we found
for doing it was rather involved. It was also of little interest beyond its
application to the present proof. )
Fortunately, there exists another method of optimizing the noise per-
formance of any nonpassive network, including negative-resistance
networks. This method is not only simple analytically but has a practical
bearing upon the noise optimization of the new maser amplifier. We
shall present this solution and its relation to the maser.
We have shown in Chap. 4 that every two-terminal-pair network can
be reduced by lossless nonreciprocal imbedding to the canonical form of
Fig. 7.2, comprising two isolated (positive or negative) resistances in
series with uncorrelated noise voltage generators. Moreover, the open-
circuit noise voltages E,; and Ey», and the two eigenvalues A; and dg of
the characteristic-noise matrix N are directly related:
oe JDiipe
Te 43 = —h2 (7.38)
2 A. van der Ziel, Noise, Prentice-Hall, New York (1954).
Sec. 7.4] OPTIMIZATION, DEFINITE CASE 69
In the special case of a negative-resistance amplifier, the eigenvalues A,
and A» are both positive. Accordingly, resistances R,; and Ro of the
canonical form are both negative. We suppose now that the eigenvalue
\, has the smaller magnitude. According to the theory of Chap. 6, this
|
| Rs
|
a
EnEng =0
Fig. 7.2. Canonical form of two-terminal-pair amplifier.
eigenvalue determines the lowest achievable value of the noise measure
M,. We shall now prove that this lowest value, \1/(&T Af), can indeed
be achieved using only that terminal pair of Fig. 7.2 which contains the
negative resistance R, and noise generator Eni.
As shown in Fig. 7.3, the terminal pair (R, En») of the canonical form
is connected to terminal pair 2 of an ideal lossless circulator with the
scattering matrix
So ©
O(iP oo
- © Oo ©
Sronon
(Transmission lines with 1-ohm characteristic impedance are connected
to all four terminal pairs of the circulator.) Terminal pair (4) of the
circulator is matched to a 1-ohm load at a temperature To, terminal pair
(1) is used as the input, and terminal pair (3) is used as the output
(Fig. 7.3).
The equations for the resulting two-terminal-pair network can easily
be derived using the scattering-matrix representation. We find that the
70 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch.7
Input (1) (3) Output
Fig. 7.3. Realization of optimum amplifier noise performance from canonical form of the
amplifier.
amplifier is unilateral and is described by the equations:
Ik AP a
a= R af 1 23 + 6,’ (7.39)
by = ony, (7.40)
where
ong = kT Af (7.412)
yet eal 7.4
01 @= Re (7.416)
5162’* = 0 (7.41c)
The available gain G of the amplifier with a 1-ohm source is
R, — 1\?
¢= (R21) a.
The excess-noise figure of the resulting amplifier is
5,’
Be a? (7.43)
Sec. 7.4] OPTIMIZATION, DEFINITE CASE 71
Thus, according to the foregoing results and Eq. 7.38,
MM dele lo: pei OE aide RBS
I tne ioe Re Ai era Ay Cie)
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 Jossy network (ideal lossless circulator
plus Ro) has been employed with the original amplifier to optimize its
noise performance.
The optimization carried out in connection with Fig. 7.3 has a useful
corollary concerning circuit connections of maser amplifiers for optimum
noise performance. One of the forms of the maser has for an equivalent
circuit a one-terminal-pair negative resistance R, in series with a noise
voltage generator E,;. To make a two-terminal-pair network, we may
consider as an artifice not only the noisy negative resistance R, of the
maser but also another positive resistance R at a temperature Tj. The
two resistances can be treated as the canonical form of a two-terminal-pair
network. Lossless imbedding of these two resistances therefore leads to a
two-terminal-pair amplifier with the eigenvalues \, = — |En:|?/(4R1) > 0
and Ay = —kT, Af <0. The best noise measure that can be expected
from the resulting amplifier is Mopt = A1/(RTo Af). The circulator
arrangement has been shown to achieve this noise measure. ‘Thus, it
provides one of the optimum network connections of the maser with
regard to noise performance. It should be re-emphasized that the as-
sumed presence of a positive resistance Rg in the circuit is an artifice that
enables the use of the theory of two-terminal-pair networks for the noise
study of the one-terminal-pair maser. The assumed temperature of the
resistance is immaterial because it determines only the negative eigen-
value of the characteristic-noise matrix, which has no relation to the
optimum noise measure achieved with gain.
The results of this chapter lead to the following theorem:
1. Any unilateral amplifier with U > 1 may be optimized with input
mismatch alone.
2. A nonunilateral amplifier with U > 1, which is also stable for all
passive source and load impedances, may be optimized with input mismatch
alone.
3. Any amplifier with U > 1 may be optimized by first making 1t umt-
lateral, using lossless reciprocal networks, and subsequenily employing input
mismatch.
72 NETWORK REALIZATION OF OPTIMUM PERFORMANCE ([Ch.7
4. Amplifiers of the class U < 0 can be optimized by first transforming
them into the class U > 1 by lossless nonreciprocal imbedding. The optimi-
zation methods 1 to 3 can then be applied to this class.
5. Negative-resistance amplifiers (0 < U < 1) can be optimized by first
transforming them into the canonical form. The terminal pair of the canon-
ical form that possesses the exchangeable power of smaller magnitude is
connected into a lossless circulator with a positive (1-ohm) balancing resistor,
as shown tn Fig. 7.3. The resulting unilateral two-terminal-pair network,
driven from a 1-ohm source, achieves the optimum noise measure.
Conclusions
The developments that we have undertaken have been rather lengthy.
Therefore, it is worth while to reassess and summarize our principal re-
sults as well as our omissions.
As pointed out in the introduction, the original motivation for the
present work was the desire to describe in a systematic manner the single-
frequency noise performance of two-terminal-pair linear amplifiers. It
was necessary at the outset to elect a criterion of noise performance, which
we chose to be the signal-to-noise ratio achievable at high gain. This
criterion is not clear for systems without gain, nor for multiterminal-pair
networks. For multiterminal-pair networks, the noise parameter pr
expressed in terms of the general circuit constants has been set down as
an extension of the two-terminal-pair noise-measure definition but has
not been given any physical interpretation in this work. One reason for
this omission is the fact that a general-circuit-constant (or wave-matrix)
description of multiterminal-pair systems has been of little use in the past.
There have not been any systems incorporating gain whose noise per-
formance on a multiterminal-pair basis was of interest. It is true that
in the past some special problems involving frequency conversion have
called for proper interpretations, and that two-terminal-pair networks
processing sidebands may be analyzed theoretically as multiterminal-pair
networks. But a sophisticated theoretical approach to noise problems of
this nature was never necessary. Problems of this type were easily dis-
posed of by inspection.
Recently, parametric amplifiers (nonlinear-, or time-varying-, reactance
amplifiers) have received a great deal of attention because of their low-
73
74 CONCLUSIONS [Ch. 8
noise characteristics. In parametric amplifiers correlation between signal
sidebands often occurs and must be taken into account in the mathe-
matical analysis. For such an analysis a systematic theory of noise in
multiterminal-pair networks involving correlation between signal and/or
noise sidebands is required. It is also probable that, because of these very
same signal correlations between sidebands, the appropriate theory for
the parametric case may not be merely the theory of the generalized noise
parameter pr introduced in this work. We have not had the opportunity
to pursue this interesting question in much detail. But we have studied
the question sufficiently to be convinced that the general matrix methods
of dealing with power and power ratios employed in the present study
will help greatly in the analysis and understanding of these somewhat
more difficult problems.
The remarks of the previous paragraphs do not, of course, imply that
multiterminal-pair networks have been neglected completely. Indeed, in
the impedance-matrix formulation we have given extensive attention to
the exchangeable-power interpretations of the network invariants in the
multiterminal-pair case. Asa practical application of these ideas, we may
refer again to the work of Granlund! regarding the problem of combining
a multiplicity of antenna outputs into a single receiver, when the inputs
to the antennas are statistically related. Furthermore, from the imped-
ance formulation we have been able to develop a canonical form for the
multiterminal-pair network. The merit of this form is that it leads to a
simplification in thinking about single-frequency noise and gain char-
acteristics of linear networks.
The problem of considering noise performance over a broad band,
rather than at a single frequency, appears to be covered by the spot-noise
discussions that we have conducted. Certainly, in a two-terminal-pair
amplifier one could adopt the position of optimizing the noise measure at
each frequency in the band. Although such a procedure might involve
complicated feedback variations with frequency and/or intricate match-
ing systems, these are principally network-synthesis problems that pre-
sumably could be solved on the basis of suitable approximations, if it
appeared desirable to do so. There is no doubt that such a solution would
give the ‘optimum noise performance” of the amplifier. By this we mean
that the optimum is to be interpreted as the “best signal-to-noise ratio at
high gain, at each frequency within the band.” It is by no means obvious
that, with the over-all system in mind, such a solution is always the
best. There are many other considerations besides noise performance
which enter into the design of wideband amplifiers, such as the behavior
1J. Granlund, Topics in the Design of Antennas for Scatter, M.I.T. Lincoln Labora-
tory Technical Report 135, Massachusetts Institute of Technology, Cambridge, Mass.
(1956).
Ch. 8] CONCLUSIONS 75
of the phase characteristic, transient response, and the uniformity of the
gain—to say nothing of over-all circuit complexity. Therefore, it seems
clear that no general theory of noise performance of such wideband
systems should be undertaken without attention to other system require-
ments.
The last point brings us to the question of the usefulness of an optimum
noise-performance criterion of the type we have presented in this study.
It is probable that such a criterion will serve primarily as an indication of
the extent to which a given design, which has met a variety of other
practical conditions, fails to achieve its best noise performance. In other
words, one may very well not attempt to realize the optimum noise
measure directly but use it instead as a guide to detect the onset of
diminishing returns in further efforts to improve noise performance.
Index
Amplifier, negative-resistance, 61, 68, 72
nonunilateral, 66, 71
parametric, 73
two-terminal-pair, classification of, 59
U <0,72
unilateral, 71
unilateral U > 1,71
wideband, 74
Amplifier cascade, 2
Amplifier noise performance, criterion for, 5
Amplifiers, interconnection of, 55
Available power, 14
Becking, A. G. Th., 46, 47
Belevitch, V., 33
Canonical form, 31
derivation of, 28
Cascading, problem of, 3
Characteristic-noise matrix, eigenvalues of, 23
general formulation of, 38
impedance formulation of, 22
mixed voltage-current formulation of, 41
trace of, 23
Circulator, 69, 72
Classification of networks and eigenvalues, 24
in T-matrix representation, 40
Classification of two-terminal-pair ampli-
fiers, 59
Cross-power spectral densities, 10
Dahlke, W., 46, 47
Eigenvalues, 22
classification of, 24
least positive, 53
Excess-noise figure, 48
lower limit imposed on, 53
Exchangeable power, 15, 44
matrix form for, 19
matrix formulation of, 44
n-terminal-pair networks, 15
stationary values of, 22
Exchangeable-power gain, 43, 46
algebraic signs of, 49
Extended noise figure, 43, 46
Feedback, 3
lossless, 54
Franz, K., 2
Friis, H. T., 2
Gain, available, 43
exchangeable-power, 43, 46
78 INDEX
Gain, extended definitions of, 43
unilateral U, 59
General-circuit-parameter representation, 36,
61
Granlund, J., 17, 74
Grid-noise resistance, 66
Groendijk, H., 46
Gyrator, lossless, 59
Hermitian conjugate, 11
Hermitian matrix, 11
Imbedding, 12
lossless, 20
passive dissipative, 57
reciprocal, 59
Impedance representation, 6, 9, 38
Indefinite matrix, 16, 52
Interconnection of amplifiers, lossless, 6, 54
passive dissipative, 55
Invariants, 25, 31
Knol, K. S., 46
Lossless circulator, 69
Lossless feedback, 54
Lossless gyrator, 59
Lossless imbedding, 20
Lossless interconnection, 54
Lossless transformation, 9, 12, 19, 61
Losslessness, condition of, 13
Maser, 7, 68
Mason, S. J., 3, 59, 61
Matrix, characteristic-noise, 22, 38, 41
general-circuit, 61
general representation, 34
Hermitian, 11
impedance, 9
indefinite, 16, 52
negative definite, 16, 52
noise column, 47, 61
permutation, 45
positive definite, 11, 16, 52
scattering, 69
semidefinite, 11, 16
T, 34, 39, 40
Matrix formulation, of exchangeable power,
44
of exchangeable-power gain, 45
of extended noise figure, 46
Matrix formulation, of
problem, 19
Mismatch, input, 6, 54, 61, 71
stationary-value
n-to-n-terminal-pair network transforma-
tions, 6, 25
n-to-one-terminal-pair network transforma-
tion, 6, 18
Negative definite matrix, 16, 52
Network transformations, -to-n-terminal-
pair, 6, 25
n-to-one-terminal-pair, 6, 18
Networks, classification of, 14, 24, 40
in thermal equilibrium, 25
pure-noise, 47
with coherent sources, 25
Noise column matrix, 47, 61
Noise figure, excess, 48
extended, 43, 46
matrix formulation of, 46
Noise measure, 4, 42, 48
allowed ranges of, 49
for conventional low-frequency vacuum
tube, 66
for triode, 66
optimum (Me opt), 54, 55
Noise parameter pr, 39
in mixed voltage-current representation, 41
Noise performance, criterion for amplifier, 5
optimization of, 61, 68
single-frequency, 1, 73
spot, 1
Nonreciprocal transformations, 59
Nonunilateral amplifier, 66, 71
Optimization, of amplifier, indefinite case, 61
of maser amplifiers, 71
of negative-resistance amplifiers, definite
case, 68
Optimum noise measure (M¢,opt), 54, 55
for low-frequency vacuum tube, 66
Parametric amplifier, 73
Passive dissipative imbedding, 56
Passive dissipative interconnection of ampli-
fiers, 56
Passive network at equilibrium, 26
Permutation matrix, 45
Positive definite matrix, 11, 16, 52
Power, available, 14
exchangeable, 15, 44
INDEX 79
Power spectral densities, 11
Reciprocal imbedding, 59
Reduction of number of terminal pairs, 32,
56
Robinson, F. N. H., 3
Rothe, H., 46, 47
Scattering-matrix representation, 69
Self-power spectral densities, 10
Semidefinite matrix, 11, 16
Signal-to-noise ratio, 1, 73
Single-frequency noise performance, 1, 73
Spectral densities, cross-power, 10
power, 11
self-power, 10
Spot-noise figure, 2
Stationary-value problem, eigenvalue formu-
lation of, 21
matrix formulation of, 19
Stationary values of exchangeable power,
22
T-matrix representation, 34, 39, 40
Terminal-voltage and current vector, 38
Thévenin representation, 14
Trace of characteristic-noise matrix, 23
Transformation, lossless, 9, 12, 19, 61
n-to-n-terminal-pair network, 6, 25
n-to-one-terminal-pair network, 6, 18
Transformation from one matrix representa-
tion to another, 35
Transformation network, 12
Transistor amplifiers, 59
Twiss, R. Q., 26
Unilateral amplifier, 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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