69
PROCEEDINGS OF THE IRE
1960
Representation
W. R. Atkinson
G. M. Branch
W. B. Davenport, Jr.
of Noise in Linear
IRE Subcommittee 7.9 on Noise
H. A. HAUS, Chairman
W. H. Fonger
W. A. Harris
S. W. Harrison
Twoports*
W. WV. McLeod
E. K. Stodola
T. E. Talpey
The compilation of standard methods of test ofteni requires theoretical concepts
that are not widely known nor readily available in the literature. While theoretical expositions are not properly part of a Standard they are necessary for its understanding
and it seems desirable to make them easily accessible by simultaneous publication with
the Standard. In suich cases a technical committee report provides a convenient means
of fulfilling this function.
The Standards Committee has decided to publish this technical committee report
immediately following "Standards on Methods of Measuring Noise in Linear Twoports."
A copy of this paper will be attached to all reprinits of the Standards.
Summary-This is a tutorial paper, written by the Subcommittee
on Noise, IRE 7.9, to provide the theoretical background for some of
the IRE Standards on Methods of Measuring Noise in Electron
Tubes. The general-circuit-parameter representation of a linear twoport with internal sources and the Fourier representations of stationary noise sources are reviewed. The relationship between spectral
densities and mean-square fluctuations is given and the noise factor
of the linear twoport is expressed in terms of the mean-square
fluctuations of the source current and the internal noise sources.
The noise current is then split into two components, one perfectly
correlated and one uncorrelated with the noise voltage. Expressed
in terms of the noise voltage and these components of the noise
current, the noise factor is then shown to be a function of four
parameters which are independent of the circuit external to the
twoport.
I. INTRODUCTION
NE of the basic problems in comnmunication
engineering is the distortion of weak signals by
the ever-present thermal noise and by the noise
of the devices used to process such signals. The transducers performing signal processing such as amplificacation, frequency mixing, frequency shifting, etc., may
usually be classified as twoports. Since weak signals
have amplitudes small compared with, say, the grid bias
voltage of a vacuum tube, or emitter bias current of a
transistor, etc., the amplitudes of the excitations at the
ports can be linearly related. Consequently, the description and measurement of noise that is presented in this
paper, and which forms the basis for the "Methods of
Test" described in this same issue,' can be restricted to
linear noisy twoports. Even if inherently nonlinear
characteristics are involved, as in mixing, etc., linear
relations still exist among the signal input and output
amplitudes, although these may not be associated with
the same frequency. Image-frequency components may
be eliminated by proper filtering, but a generalization
Q
Original manuscript received by the IRE, September 15, 1958;
revised manuscript received March 6, 1959.
1 "sIRE Standards on Methods of Measuring Noise in Linear Twoports, 1959," this issue, p. 60.
*
of our results to cases in which image frequencies are
present is not difficult.
The effect of the noise originatinig in a twoport when
the signal passes through the twoport is characterized
at any particular frequency by the (spot) noise factor
(noise figure). Since this has meaning only if the source
impedance used in obtaining the noise factor is also
specified, the noise contribution of a twoport is often
indicated by a minimum noise factor and the source
impedance with which this minimum noise factor is
achieved. However, the extent to which the noise factor
depends upon the input source impedance, upon the
amount of feedback, etc., can be indicated only by a
more detailed representation of the linear twoport.2 '
As a basis for understanding the represenitation of
networks containing (statistical) noise sources, we shall
consider first the analysis of networks contai ning
Fourier-transformable siginal sources. We shall then describe noise in two ways: a) as a limit of Fourierintegral transforms, and b) as a limit of Fourier-series
transforms. The reasons for the wider use of the latter
description will be presented. With this background we
shall be ready to show that the four noise parameters
used in the standard "Methods of Test" completely
characterize a noisy linear twoport.
II. REPRESENTATIONS OF LINEAR TwOPORTS
The excitation at either port of a linear twoport can
be completely described by a time-dependent voltage
v(t) and the time-dependent current i(t). (For waveguide
twoports, where no voltage or current can be identified
uniquely, an equivalent voltage and current can always
2 H. Rothe and W. Dahlke, "Theory of noisy fourpoles," PRoc.
IRE, vol. 44, pp. 811-818; June, 1956. Also, "Theorie rauschender
Vierpole," Arch. elekt. Ubertragung, vol. 9, pp. 117-121; March, 1955.
3 A. G. T. Becking, H. Groendijk, and K. S. Knol, "The noise
factor of four-terminal networks," Philips Res. Repts., vol. 10, pp.
349-357; October, 1955.
PROCEEDINGS OF THE IRE
70
Janttary
The impedanice representation of a linlear twoport
be used.) Let it be assunmed that the voltage and current
functionis can be transformiied fromn the time domain to with internal sources has been reviewed because of its
the frequenicy domaini, and that V and I stand for the famliliarity. However, it is well knownl that other repreFourier transforms wheni the funictioni is aperiodic anid senitations, each leading to a different separation of the
for the Fourier amplitudes when the fuiiction is periodic. ilnternal sources frolmi the twoport, are possible. A parThe linearity of the twoport without internal sources ticularlv conivenienit one for the study of nioise is the
general-Circuit-parameter representation4
theni allows an impedanice represenitation:
V1- 4 V2 + B12+ E
V1 = ZIlIl + Z12I2
V2 = Z21J1 + Z221.
I, = C1V¾ + DI2 + I
(1)
(3)
The subscripts 1 anid 2 refer to the inlput and output where E and I are again functionis of frequency which
ports, respectively, and the coefficienits Zj1. are, in genieral are the Fourier transformns of the time-dependenit funicfunctions of frequency. The currents are defined to be tions e(t) anid i(t) describing the internal sources.
As shown in Fig. 3, the internial sources are niow reprepositive if the flow is into the network as shown in Fig. 1.
by a source of voltage acting in series with the
sented
If the twoport conitaiins internal sources, then (1) anid
and a source of currenit flowing in parallel
voltage
input
the equivalenit circuit mnust be modified. By a generalicurrent. It will be seeni that this particuthe
input
with
zationi of Theveni i 's theoremn, the twoport mav be
of the iinternal sources leads to four
lar
represenitation
separated into a source-free nietwork and two voltage
can easily be derived from singlethat
nioise
parameters
genierators, one in series with the inlput port and one in
of the twoport nioise factor as
measuremnenits
frequenicy
series with the output port. If the time-dependent functions e1(t) and e2(t) describing these equivalent geniera- a function of input miiismiiatch. It has the further adtors can be transformed to funiction-s of frequency E1 vantage that such properties of the twoport as gain anid
and E2, respectively, the impedance represenitation of ilnput conductance do niot eniter inito the nioise factor expression in termns of these four parameters.
the linear twoport with internal sources becomes
V1 = Z11II + Z12I-+ E1
V2 Z21I1 + Z22I2 + E2
and the equivalent inetwork is that shown in Fig. 2.
(2)
II
E
(2)
(1)
Fig. 3-Separationi of twoport with initerildi soLurces inlto a souircefree twoport anid external inpuit currenit anid voltage sources.
(2)
Fig. I-Signi convenition for impedance represen tation
{I)
of linear twoport.
III. REPRESENTATIONS OF STATIO)NARY NoiSE SOIJTR( 'ES
Wheni a liniear twoport conitainis stationary initernial
nioise sources, the frequenicy funictions E anid I in (3)
cannot be found by coniventional Fourier methods fromii
the time functionis e(t) anid i(t), since these are niow
random functions exteniding over all timie anid have inlfinite energy contenit. Two alternatives are possible. Onie
may consider the stibstitute fuiictionis
e(t, T)
Fig. 2-Separation of twoport with internial sources into
free twoport and externial voltage generators.
=
1T
e(t) for---< t <
a source-
=0
2
T
2
T
2
-
for-K< tl
For the purpose of the anialysis to follow, it should be
T
T
emphasized againi that such equations in general characi(t, T) = i(t) for - -2 < t <-2
terize the behavior of the twoport as a functioni of freT
quency. For practical purposes, it should be poinited out
=0
for -K< jtj
(4)
that in most cases the impedance parameters of a linear
2
a
at
be
measured
can
by
frequency
particular
twoport
applying sinusoidal voltages that produce outputs large
4E. Guillemin, "Communication Networks," John Wiley and
Sons, New York, N. Y., vol. 2, p. 138; 1935.
compared with those caused by the internal sources.
Representation of Noise in Linear Twoports
1960
where T is some long but finite time interval, and use the
Fourier transforms
1 r T12
E(w, T) =-I1 -T12 e(t, T)e-iwtdt
I(co, T)
i(t, T)e-iwtdt.
-
(5)
Alternatively, one may construct periodic functions,
T
T
e(t, T) =e(t) for--< t <2
2
e(t + nT, T)
=
e(t, T)
i(t, T)
=
i(t) for - 2 < t <2
with n an integer,
T
2
i(t + nT, T) = i(t T)
T
2
with n an integer,
(6)
and expand these functions into Fourier series with
amplitudes
Em((O, T) 2f=1rT/2 e(t, T)e-iwldt
T
T/x2
1rT/2
In(w, T) -T f-TJ2i(t, T)e-iwtdt
(7)
where cw m2ir/T with m an integer, and T is again
finite. In either case, the substitute functions can be
made to approach the actual functions as closely as
desired by making the interval T larger and larger.
In either the Fourier-integral or the Fourier-series approach, we consider a set of substitute functions obtained in principle from a series of measurements a) on
an ensemble of systems with identical statistical properties or b) on one and the same system at successive
time intervals sufficiently separated so that no statistical correlation exists.'
Since noise is a statistical process, we are in general
interested in statistical averages6 rather than the exact
details of any particular noise function. These averages
are important since they relate to physically measurable
stationary quantities. For example, in the Fourierintegral approach the spectral density of the noisevoltage excitation is defined by
We() =
llim
T-°°
E(cw,, T) 12
2irT
71
scribed by (4). This spectral density is proportional to
the noise power (in a narrow frequency band7 around a
given frequency) in a resistor across which the fluctuat-
ing voltage e(t) appears.
Unfortunately, the notation developed in the literature for dealing with spectral densities is unwieldy, since
the quantity with which the density is associated is relegated to a subscript. This is one of the reasons why researchers on noise have tended to use the historically
older Fourier-series approach. The use of spectral denisities is preferred in rigorous mathematical treatments of
noise and in questions involving definitions of noise
processes, since this assures that all points on the frequency axis within a narrow range are equivalent. For
practical purposes, however, the noise amplitudes that
are associated with discrete frequencies in the Fourierseries method can be as closely distributed as desired
by choosing the time interval T sufficiently large.
IV. RELATIONSHIP OF SPECTRAL DENSITIES AND
FOURIER AMPLITUDES TO MEAN-SQUARE
FLUCTUATIONS
If there is an open-circuit noise voltage v(t) across a
terminal pair, the mean-square value v2(t) is related to
the spectral density W,(w) and to the Fourier amplitudes Vm(w) as follows:
--
=2(t) lim
~~1 wT/2
T-.. FT
-f
-
T/2
WW,(w)dw
2(t)dt
=
v-vo0
:
-40
Vm(w) 12
(9)
where the amplitudes Vm are now those obtained as
T- oo.
Suppose that the stationary time function v(t) is
passed through an ideal band-pass filter with the narrow bandwidth Af=Awo/2r centered at a frequency
f='0/2ir. The mean-square value of the voltage appearing at the filter output, usually denoted by the
symbol v2 and called the mean-square fluctuation of
v(t) within the frequency interval Af" is
V= 47r\fWl,(wo) =
21 V, 1'.
(10)
This equation relates the mean-square fluctuations, the
(8) spectral density, and the Fourier amplitude. If the filter
where the bar indicates an arithmetic average of the
Fourier transforms of an ensemble of functions de' The equivalence of the ensemble and time average is known as
the ergodic hypothesis. An interesting discussion of the implications
of this hypothesis can be found in Born, "Natural Philosophy of
Cause and Chance," Oxford University Press, New York, N. Y.,
p. 62; 1948. See also W. B. Davenport, Jr. and W. L. Root, "An Introduction to the Theory of Random Signals and Noise," McGrawHill Book Company, Inc., New York, N. Y., p. 66 ff.; 1958.
6 W. R. Bennett, "Methods of solving noise problems," PROC.
IRE, vol. 44, pp. 609-637; May, 1956.
is narrow-band but not ideal, the quantity Af is the
noise bandwidth.' The factor 2 in (10) arises because
W,(w) and Vm have been defined on the negative as well
as the positive frequency axis.
Eq. (10) shows that spectral densities and meansquare fluctuations are equivalent. Although mathematical limits are involved in the definitions, these
7 A frequency interval Af is called narrow if the physical quantities under consideration are independent of frequency throughout the
interval, and if Af<<fo, wherefo is the center frequency.
PROCEEDINGS OF TIHE IRE
72
quantities can be measured to anly desired degree of accuracy. For example, if one measures the power flowing
into a termination connected to the terminals with
which v(t) is associated, one measures essentially Wv(coo),
provided that the following requiremiients are mnet:
1) The termination is known anid has a high impedance so that the voltage across the terminals remnains
essentially the open-circuit voltage v(t), or the internal
impedance associated with the two terminlals is also
known so that any change in voltage can be computed.
The noise voltage conitributed by the termination must
either be negligible, or its statistical properties mlust be
kniown so that its effect can be taken iilto account.
2) The termination is fed through a filter with a pass
band sufficiently narrow so that the spectral denisity and
internal impedance are essentially constant throughout
the band; or the terminatioin is itself a resonant circuit
with a high Q corresponding to a sufficieintly narrow
bandwidth.
3) The power measurement is made over a period of
time that is long compared to the reciprocal bandwidth,
1/Af, of the filter. In this way the power-measuring device takes an average over many long tilme intervals
that is equivalent to an ensemble average.
In the description of noise transformations by linear
twoports that follows, the mean-square fluctuations will
be used. Mean-square current fluctuations can be related to the Fourier amplitudes Im(w) by a procedure
similar to the one just described. Since fluctuations of
the cross products of statistical functions will also be
used it should be noted that
X0
T/2
1
lim j (t, T)i(t, T)dt = M_-oo
2 VmIm*
T eo T -T12
(11)
where i(t) is a noise current and Vm and Im are again the
amplitudes obtained as T o .
We may interpret the real part of the complex quantity 2 VmIm* as the contribution to the average of vi from
the frequency increment Af= I/T at the angular frequency w -mAw. However, the imaginary part of VmIm*
also contains phase information that has to be used in
noise computations. We shall, therefore, use the expression
-
form > O,w > 0
=i* 2 VmIm
as the complex cross-product fluctuations of v(t) and i(t)
in the frequency increment Af. They are related to the
cross-spectral density by
=
vi*
=
4irAfWu(co)
(13)
where
T)
Wi18(w) = lim I*(w, T)V(co,
27rT
T-_oo
(14)
Janufary
There are several ways of specifyinig the fluctuations
(or the spectral denisities) that characterize the internal
noise sources. A mean-square voltage fluctuationi can be
givenl directly in units of volt2 second. It is often
convenient, however, to express this quantity ill resistance units by using the Nyquist formiiula, which
gives the mnean-square fluctuation of the open-circuit
noise voltage of a resistor R at temiiperature 1' as
e2
=
4kTRAJ.
where k is Boltzmann's conistaint. For any m-ieain-square
voltage fluctuation e2 within the frequency interval Af,
one defines the equivalent noise resistance R,, as
e2
4kToAf
(15)
where To is the standard temperature, 2900K. The use
of R. has the advantage that a direct comparison cani
be made between the noise due to internal sources and
the noise of resistances generally present in the circuit.
Note that R. is not the resistance of a physical resistor
in the network in which e is a physical noise voltage and
therefore does not appear as a resistance in the equivalent circuit of the network.
In a similar manner, a mean-square current fluctuation can be represented in terms of an equivalent noise
conductance G. which is defined by
Gk
=
4k Tolf
(16)
V. NOISE TRANSFORMATIONS BY LINEAR TWOPORTS
The statistical averages discussed in Sections III and
IV will now be used to describe the internal inoise sources
of a linear twoport. We may consider functions of the
type given by (6) and represent the nioise sources E and
I in (3) by the Fourier amplitudes E,,(w, T) and Im(w, T).
Thus, if the circuit shown in Fig. 3 represents a separation of inoise sources from a linear twoport, the noisefree circuit is preceded by a noise network. Since a
noise-free network connected to a terminal pair does Inot
change the signal-to-noise ratio (noise factor evaluated
at that terminal pair) the noise factor of the over-all
network is equal to that of the noise network.
To derive the noise factor, let us connect the nioise
network to a statistical source comnprising an internal
admittance Y8 and a currenit source again represented
by a Fourier amplitude hs(w, T). The network to be used
for the noise-factor computation is then as shown in
Fig. 4. By definition,8 the spot noise factor (figure) of a
network at a specified frequency is given by the ratio
8 "IRE Standards on Electron Tubes: Definitions of
1957," PROC. IRE, vol. 45, pp. 983-1010; July, 1957.
Terms,
1960
Representation of Noise in Linear Twoports
73
of 1) the total output noise power per unit bandwidth linear twoport by a voltage geinerator and a current
available at the output port to 2) that portion of 1) en- generator lumps the effect of all internal noise sources
gendered by the input termination at the standard into the two generators. A complete specification of
these generators is thus equivalent to a complete detemperature To.
scriptioin of the internal sources, as far as their contribution to terminal voltages and currents is concerned.
E
Since the sources under considerationi are noise sources,
their description is confined to the methods applicable
to noise. The extent and detail of the description depends on the amount of detail envisioned in the analysis.
Since, in the case of the noise factor, only the meansquare fluctuations of output currents are sought, the
specification of self and cross-product fluctuations of the
Fig. 4-Truncated nietwork for noise-factor computation.
generator voltages and currents is adequate.
expression for the noise factor giveni in (18) can
Now the total short-circuit noise current at the output be The
if the noise current is split into two comsimplified
of the network shown in Fig. 4 can be represented in poinents, one perfectly
correlated and one uncorrelated
terms of Fourier amplitudes by
with the inoise voltage. The uncorrelated noise current,
designated by i,, is defined at each frequency by the
I,(co, T) + I (co, T) + YsEm(w, T).
relations
Let us assume that the internal noise of the twoport and
ei?,* = 0
(20)
the noise from the source are uncorrelated. If we then
square the total short-circuit noise-current Fourier
(21)
(i-ijiu) = 0.
amplitudes, take ensemble averages and use equations
similar to (10) and (12) to introduce mean-square fluctuations, we obtain a mean-square current fluctuation
The correlated noise current, i-i., can be written as
Y,e, where the complex constant Y, = G,+jB, has the
j,2+ i+ Ye12 = s2 +2+ IYe12e2 +Y*ie*
dimensions of an admittance and is called the correla+ Y8i*e
(17) tion admittance. The cross-product fluctuation ei* may
then be written
to which the total output noise power is proportional.
= Y*e2.
ei* = e(i (22)
It should be noted here that when e and i are complex,
the symbols e9 and i2 denote, by convention, el 2 and
The noise-voltage fluctuation can be expressed in
Ii 2. Since the noise power due to the source alone is terms of an equivalent noise resistance Rn as
proportional to ij2A the noise factor becomes
e2- 4kToR,Af
(23)
and the uncorrelated noise-current fluctuationi in terms
(18) of an equivalent noise conductance G, as
F =1 + 1 i
j,S2
i-= 4kToGu.Af.
(24)
In the denominator, the mean-square source noise current is related to the source conductance G8 by the Ny- The fluctuations of the total noise current are then
quist formula
j2 =
i uI12++ti2
i,2= 4kToG8Af.
(19)
= 4kTo [ Y 122R?, + Guj]Af.
(25)
In the numierator of (18), the four real variables in- From (18)-(24), the formula for the noise factor bevolved in i2, e9 and ie*, where e* is the complex conjugate comes
of e, describe the internal noise sources. If the Fourier
[j2 + Y + y 12-2]
transforms, (5), are used to characterize the noise, the
F= 1 +-l
e
(26)
self-spectral densities of noise current and voltage and
the cross-spectral density of these quantities would have
Gu Rn
= 1 +-+-[(G, + G-)2+ (B. + B)2] . (27)
to be specified.
Gs G,
Before proceeding, let us review in general terms the
methods employed and the conclusions reached. The Thus, the noise factor is a function of the four paramrepresentation of the internal noise sources of a noisy eters G,, Rn, G, and B. These depend, in general, upon
-
F=I+4kToG,Af
PROCEEDING(S OF lFHE IRE
74
JaInuaryV
the operatin-g point aind operating frequency of the be evaluated from the kulnown coefficients, (A, B, C, D)
twoport, but nlot upoIn the external circuitry. In a vacu- of (3), and the knowni nioise fluctuationis. Thus, the noise
im tube triode the correlation susceptance Be is niegligibly in a twoport is completely characterized (with regard to
snmall at frequeincies such that transit timnes are sjmiall the fluctuations or the spectral (leisities -at the terinals) by the noise fluctuationis e-, tand ei*, or altercompared to the period. Also, GC, anid G, are vanlishingly smiall wheni there is little grid loading. Thus, natelv by the four nloise plaralmeters F,, ,B,,
l
B adR,.
tube noise at low frequenicies is adequately characterized
VI. CONCLUSION
by the single nionizero constant R,. Tube noise at high
frequencies anid tranisistor nioise at all frequencies have
Th-e preceding discussioni showed that, with limited
nio such simple represenitationi.
objectives, the nioise in a liniear twoport cani be chclrSinice the noise factor is anl explicit funictioni of the acterized adequately by a limited nlumber of parameters.
source coniductanice anid susceptance, it cani readily be T hus, if one seeks oinly information concerning
the
shown that the nioise factor has an optimumii (minimumii) miieani-squiare fluctuationis or
the
spectral
denisities
of
value at somiie optilImum1 source admiiittance Y,OG,o+jBo currenits or voltages inlto or across
of
a linear
the
ports
whlere
twoport at a particular frequenicy anid for arbitrary circuit conniiections of the twoport, it is sufficient to specify
]
(28) two miieani-square flucttiationis anid one product fluctuaG'-) [GU
tionl or two spectral denisities and onie cross-spectral
(29) densitv. T his inivolves the specification of four real
Bo= -Be
niumlbers. If a band of frequencies is beinig conisidered,
and the value of this minimiuinum noise factor is
these quaintities have to be given as funictionis of ireFo- 1 + 2Ro,(G7 + Go).
(30) quenicy uniless they are approximately constant over the
bauid.
In terms of Go, Bo anid Fo, the noise factor for any arbiDifferent separations of initernial nioise sources will
trary source impedance then becomiies
lead, in general, to different frequenicy dependences of
the restilting fluctuations or spectral densities. Trhus, a
F = Fo,R
(31) particular separationi of the nioise sources mnay be pref-[(s - G.)2 + (B, o )2I
erable to (another separation if it is founid that fluctuaEq. (31) shows that the four real parameters Fol Go, tionls (spectral densities) are less frequency-depenident
Bo and R, give the noise factor of a twoport for every in the band. In lparticular, if available information
input terminationi of the twoport.9 As showni in the about the physics of the nioise inl a particular (levice
mlethods of test that are published in this issue, ac meas- suggests introducing, inside the twoport, appropr-iate
urement of the minimum nioise factor and of the source nioise genierators of fluctuationls (spectral (lensities)
admittance YV with which Fo is achieved gives the first with nio frequency dependence, or with al sim-iple frethree parameters. The paramneter R. can be comiputed quency dependence, it imiay be more advantageous to
from an additionial measuremiietnt of the lnoise factor for specify the device in termiis of these physically suggestive genierators.
a source admittanice Y. other than YO.
I-lowever, usually one resorts to the characterizatiolns
From the given values, F,,G,, Bo and R, one canl
of
lnoise presented in this paper, sinice they have the aidcompute, if desired, the nioise fluctuationis j2, e2, and ei*
that they do not require any knowledge of the
vanitage
(or the corresponding spectral denisities). To do this,
of
physics
the initernial n-oise. Furthermiiore, noise-factor
one uses (28) to (30) to find Ye anid GC. From these onie
mleasuremiienits
performed oni the twoport yield (muore or
evaluates e2, i2, and ei* from (23), (25), and (22). The
less)
directly
the
noise paramneters of the genieral circuit
flulCtuations of any terminal voltage or currenit of the
re)resentation,
the fluctuationis (spectral (lensiniamely
twoport produced with a given source anld load cani theni
t-ies) of the voltage anid currenit generattors attached to
9 Reasoning similar to that leading to (31) can be carried olit in a
the input of the nioise-free equivalenit of the twoport.
lual represenitation, where imiipedanices anid admittances are initer- rhis fact recommienids the use of the nioise parameters
changed. An equation similar to (31) results, again involving four
noise parameters, some of which are different from the ones used here. niatuiral to the genleral-circuiit-paramiieter representationi.
Rill
-