PROCEEDINGS OF THE IRE

Representation of Noise in Linear Twoports*

W. R. AtkinsonG. M. BranchW. B. Davenport, Jr.

IRE Subcommittee 7.9 on NoiseH. A. HAUS, Chairman

W. H. FongerW. A. HarrisS. W. Harrison

W. WV. McLeodE. K. StodolaT. E. Talpey

The compilation of standard methods of test ofteni requires theoretical conceptsthat are not widely known nor readily available in the literature. While theoretical ex-positions are not properly part of a Standard they are necessary for its understandingand it seems desirable to make them easily accessible by simultaneous publication withthe Standard. In suich cases a technical committee report provides a convenient meansof fulfilling this function.

The Standards Committee has decided to publish this technical committee reportimmediately 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 Subcommitteeon Noise, IRE 7.9, to provide the theoretical background for some ofthe IRE Standards on Methods of Measuring Noise in ElectronTubes. The general-circuit-parameter representation of a linear two-port with internal sources and the Fourier representations of station-ary noise sources are reviewed. The relationship between spectraldensities and mean-square fluctuations is given and the noise factorof the linear twoport is expressed in terms of the mean-squarefluctuations of the source current and the internal noise sources.The noise current is then split into two components, one perfectlycorrelated and one uncorrelated with the noise voltage. Expressedin terms of the noise voltage and these components of the noisecurrent, the noise factor is then shown to be a function of fourparameters which are independent of the circuit external to thetwoport.

I. INTRODUCTIONQ NE of the basic problems in comnmunication

engineering is the distortion of weak signals bythe ever-present thermal noise and by the noise

of the devices used to process such signals. The trans-ducers performing signal processing such as amplifica-cation, frequency mixing, frequency shifting, etc., mayusually be classified as twoports. Since weak signalshave amplitudes small compared with, say, the grid biasvoltage of a vacuum tube, or emitter bias current of atransistor, etc., the amplitudes of the excitations at theports can be linearly related. Consequently, the descrip-tion and measurement of noise that is presented in thispaper, and which forms the basis for the "Methods ofTest" described in this same issue,' can be restricted tolinear noisy twoports. Even if inherently nonlinearcharacteristics are involved, as in mixing, etc., linearrelations still exist among the signal input and outputamplitudes, although these may not be associated withthe same frequency. Image-frequency components maybe eliminated by proper filtering, but a generalization

* 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 Two-ports, 1959," this issue, p. 60.

of our results to cases in which image frequencies arepresent is not difficult.The effect of the noise originatinig in a twoport when

the signal passes through the twoport is characterizedat any particular frequency by the (spot) noise factor(noise figure). Since this has meaning only if the sourceimpedance used in obtaining the noise factor is alsospecified, the noise contribution of a twoport is oftenindicated by a minimum noise factor and the sourceimpedance with which this minimum noise factor isachieved. However, the extent to which the noise factordepends upon the input source impedance, upon theamount of feedback, etc., can be indicated only by amore detailed representation of the linear twoport.2 'As a basis for understanding the represenitation of

networks containing (statistical) noise sources, we shallconsider first the analysis of networks containingFourier-transformable siginal sources. We shall then de-scribe noise in two ways: a) as a limit of Fourier-integral transforms, and b) as a limit of Fourier-seriestransforms. The reasons for the wider use of the latterdescription will be presented. With this background weshall be ready to show that the four noise parametersused in the standard "Methods of Test" completelycharacterize a noisy linear twoport.

II. REPRESENTATIONS OF LINEAR TwOPORTS

The excitation at either port of a linear twoport canbe completely described by a time-dependent voltagev(t) and the time-dependent current i(t). (For waveguidetwoports, where no voltage or current can be identifieduniquely, 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 rauschenderVierpole," Arch. elekt. Ubertragung, vol. 9, pp. 117-121; March, 1955.

3 A. G. T. Becking, H. Groendijk, and K. S. Knol, "The noisefactor of four-terminal networks," Philips Res. Repts., vol. 10, pp.349-357; October, 1955.

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be used.) Let it be assunmed that the voltage and currentfunctionis can be transformiied fromn the time domain tothe frequenicy domaini, and that V and I stand for theFourier transforms wheni the funictioni is aperiodic anidfor the Fourier amplitudes when the fuiiction is periodic.The linearity of the twoport without internal sourcestheni allows an impedanice represenitation:

V1 = ZIlIl + Z12I2V2 = Z21J1 + Z221. (1)

The subscripts 1 anid 2 refer to the inlput and outputports, respectively, and the coefficienits Zj1. are, in genieralfunctions of frequency. The currents are defined to bepositive if the flow is into the network as shown in Fig. 1.

If the twoport conitaiins internal sources, then (1) anidthe equivalenit circuit mnust be modified. By a generali-zationi of Theveni i 's theoremn, the twoport mav beseparated into a source-free nietwork and two voltagegenierators, one in series with the inlput port and one inseries with the output port. If the time-dependent func-tions e1(t) and e2(t) describing these equivalent geniera-tors can be transformed to funiction-s of frequency E1and E2, respectively, the impedance represenitation ofthe linear twoport with internal sources becomes

V1 = Z11II + Z12I-+ E1

V2 Z21I1 + Z22I2 + E2 (2)

The impedanice representation of a linlear twoportwith internal sources has been reviewed because of itsfamliliarity. However, it is well knownl that other repre-senitations, each leading to a different separation of theilnternal sources frolmi the twoport, are possible. A par-ticularlv conivenienit one for the study of nioise is thegeneral-Circuit-parameter representation4

V1- 4 V2+ B12+ E

I, = C1V¾ + DI2 + I (3)where E and I are again functionis of frequency whichare the Fourier transformns of the time-dependenit funic-tions e(t) anid i(t) describing the internal sources.As shown in Fig. 3, the internial sources are niow repre-

sented by a source of voltage acting in series with theinput voltage and a source of currenit flowing in parallelwith the input current. It will be seeni that this particu-lar represenitation of the iinternal sources leads to fournioise parameters that can easily be derived from single-frequenicy measuremnenits of the twoport nioise factor asa function of input miiismiiatch. It has the further ad-vantage that such properties of the twoport as gain anidilnput conductance do niot eniter inito the nioise factor ex-pression in termns of these four parameters.

EII

and the equivalent inetwork is that shown in Fig. 2.

{I) (2)

Fig. I-Signi convenition for impedance represen tationof linear twoport.

Fig. 2-Separation of twoport with internial sources into a source-free twoport and externial voltage generators.

For the purpose of the anialysis to follow, it should beemphasized againi that such equations in general charac-terize the behavior of the twoport as a functioni of fre-quency. For practical purposes, it should be poinited outthat in most cases the impedance parameters of a lineartwoport can be measured at a particular frequency byapplying sinusoidal voltages that produce outputs largecompared with those caused by the internal sources.

(1) (2)

Fig. 3-Separationi of twoport with initerildi soLurces inlto a souirce-free twoport anid external inpuit currenit anid voltage sources.

III. REPRESENTATIONS OF STATIO)NARY NoiSE SOIJTR( 'ES

Wheni a liniear twoport conitainis stationary initernialnioise sources, the frequenicy funictions E anid I in (3)cannot be found by coniventional Fourier methods fromiithe time functionis e(t) anid i(t), since these are niowrandom functions exteniding over all timie anid have inl-finite energy contenit. Two alternatives are possible. Oniemay consider the stibstitute fuiictionis

1T Te(t, T) = e(t) for---< t < -

2 2

T=0 for-K< tl

2

i(t, T) = i(t)T T

for - - < t <-2 2

T= 0 for-K< jtj

2 (4)

4E. Guillemin, "Communication Networks," John Wiley andSons, New York, N. Y., vol. 2, p. 138; 1935.

70 Janttary

Representation of Noise in Linear Twoports

where T is some long but finite time interval, and use theFourier transforms

1 r T12E(w, T) =-I e(t, T)e-iwtdt1 -T12

I(co, T) - i(t, T)e-iwtdt. (5)

Alternatively, one may construct periodic functions,

T Te(t, T) =e(t) for--< t <-

2 2

e(t + nT, T) = e(t, T) with n an integer,T T

i(t, T) = i(t) for - 2 < t <22 2

i(t + nT, T) = i(t T) with n an integer, (6)

and expand these functions into Fourier series withamplitudes

1rT/2Em((O, T) 2f= e(t, T)e-iwldtT T/x2

1rT/2In(w, T) - f i(t, T)e-iwtdt (7)

T -TJ2

where wc m2ir/T with m an integer, and T is againfinite. In either case, the substitute functions can bemade to approach the actual functions as closely asdesired by making the interval T larger and larger.

In either the Fourier-integral or the Fourier-series ap-proach, we consider a set of substitute functions ob-tained in principle from a series of measurements a) onan ensemble of systems with identical statistical proper-ties or b) on one and the same system at successivetime intervals sufficiently separated so that no statisti-cal correlation exists.'

Since noise is a statistical process, we are in generalinterested in statistical averages6 rather than the exactdetails of any particular noise function. These averagesare important since they relate to physically measurablestationary quantities. For example, in the Fourier-integral approach the spectral density of the noise-voltage excitation is defined by

E(cw, T) 12We() = llim , (8)

T-°° 2irT

where the bar indicates an arithmetic average of theFourier transforms of an ensemble of functions de-

' The equivalence of the ensemble and time average is known asthe ergodic hypothesis. An interesting discussion of the implicationsof this hypothesis can be found in Born, "Natural Philosophy ofCause and Chance," Oxford University Press, New York, N. Y.,p. 62; 1948. See also W. B. Davenport, Jr. and W. L. Root, "An Intro-duction to the Theory of Random Signals and Noise," McGraw-Hill 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.

scribed by (4). This spectral density is proportional tothe noise power (in a narrow frequency band7 around agiven frequency) in a resistor across which the fluctuat-ing voltage e(t) appears.

Unfortunately, the notation developed in the litera-ture for dealing with spectral densities is unwieldy, sincethe quantity with which the density is associated is rele-gated to a subscript. This is one of the reasons why re-searchers on noise have tended to use the historicallyolder Fourier-series approach. The use of spectral denisi-ties is preferred in rigorous mathematical treatments ofnoise and in questions involving definitions of noiseprocesses, since this assures that all points on the fre-quency axis within a narrow range are equivalent. Forpractical purposes, however, the noise amplitudes thatare associated with discrete frequencies in the Fourier-series method can be as closely distributed as desiredby choosing the time interval T sufficiently large.

IV. RELATIONSHIP OF SPECTRAL DENSITIES ANDFOURIER AMPLITUDES TO MEAN-SQUARE

FLUCTUATIONSIf there is an open-circuit noise voltage v(t) across a

terminal pair, the mean-square value v2(t) is related tothe spectral density W,(w) and to the Fourier ampli-tudes Vm(w) as follows:

-- ~~1 wT/2=2(t) lim - 2(t)dtT-.. TF T/2

- f WW,(w)dw = : Vm(w) 12v 0-vo -40

(9)

where the amplitudes Vm are now those obtained asT- oo.Suppose that the stationary time function v(t) is

passed through an ideal band-pass filter with the nar-row bandwidth Af=Awo/2r centered at a frequency

f='0/2ir. The mean-square value of the voltage ap-pearing at the filter output, usually denoted by thesymbol v2 and called the mean-square fluctuation ofv(t) within the frequency interval Af" is

V= 47r\fWl,(wo) = 21 V, 1'. (10)

This equation relates the mean-square fluctuations, thespectral density, and the Fourier amplitude. If the filteris narrow-band but not ideal, the quantity Af is thenoise bandwidth.' The factor 2 in (10) arises becauseW,(w) and Vm have been defined on the negative as wellas the positive frequency axis.

Eq. (10) shows that spectral densities and mean-square fluctuations are equivalent. Although mathe-matical limits are involved in the definitions, these

7 A frequency interval Af is called narrow if the physical quanti-ties under consideration are independent of frequency throughout theinterval, and if Af<<fo, wherefo is the center frequency.

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quantities can be measured to anly desired degree of ac-curacy. For example, if one measures the power flowinginto a termination connected to the terminals withwhich 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 imped-ance so that the voltage across the terminals remnainsessentially the open-circuit voltage v(t), or the internalimpedance associated with the two terminlals is alsoknown so that any change in voltage can be computed.The noise voltage conitributed by the termination musteither be negligible, or its statistical properties mlust bekniown so that its effect can be taken iilto account.

2) The termination is fed through a filter with a passband sufficiently narrow so that the spectral denisity andinternal impedance are essentially constant throughoutthe band; or the terminatioin is itself a resonant circuitwith a high Q corresponding to a sufficieintly narrowbandwidth.

3) The power measurement is made over a period oftime that is long compared to the reciprocal bandwidth,1/Af, of the filter. In this way the power-measuring de-vice takes an average over many long tilme intervalsthat is equivalent to an ensemble average.

In the description of noise transformations by lineartwoports that follows, the mean-square fluctuations willbe used. Mean-square current fluctuations can be re-lated to the Fourier amplitudes Im(w) by a proceduresimilar to the one just described. Since fluctuations ofthe cross products of statistical functions will also beused it should be noted that

1 T/2 X0

lim j (t, T)i(t, T)dt = 2 VmIm* (11)T eo T -T12 M_-oo

where i(t) is a noise current and Vm and Im are again theamplitudes obtained as T - o .We may interpret the real part of the complex quan-

tity 2 VmIm* as the contribution to the average of vi fromthe frequency increment Af= I/T at the angular fre-quency w -mAw. However, the imaginary part of VmIm*also contains phase information that has to be used innoise computations. We shall, therefore, use the ex-pression

=i*= 2 VmIm form > O,w > 0

as the complex cross-product fluctuations of v(t) and i(t)in the frequency increment Af. They are related to thecross-spectral density by

vi* = 4irAfWu(co) (13)

where

There are several ways of specifyinig the fluctuations(or the spectral denisities) that characterize the internalnoise sources. A mean-square voltage fluctuationi can begivenl directly in units of volt2 second. It is oftenconvenient, however, to express this quantity ill re-sistance units by using the Nyquist formiiula, whichgives the mnean-square fluctuation of the open-circuitnoise voltage of a resistor R at temiiperature 1' as

e2 = 4kTRAJ.

where k is Boltzmann's conistaint. For any m-ieain-squarevoltage 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 useof R. has the advantage that a direct comparison canibe made between the noise due to internal sources andthe noise of resistances generally present in the circuit.Note that R. is not the resistance of a physical resistorin the network in which e is a physical noise voltage andtherefore does not appear as a resistance in the equiv-alent circuit of the network.

In a similar manner, a mean-square current fluctua-tion can be represented in terms of an equivalent noiseconductance G. which is defined by

G k=4kTolf

(16)

V. NOISE TRANSFORMATIONS BY LINEAR TWOPORTSThe statistical averages discussed in Sections III and

IV will now be used to describe the internal inoise sourcesof a linear twoport. We may consider functions of thetype given by (6) and represent the nioise sources E andI in (3) by the Fourier amplitudes E,,(w, T) and Im(w, T).Thus, if the circuit shown in Fig. 3 represents a separa-tion of inoise sources from a linear twoport, the noise-free circuit is preceded by a noise network. Since anoise-free network connected to a terminal pair does Inotchange the signal-to-noise ratio (noise factor evaluatedat that terminal pair) the noise factor of the over-allnetwork 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 internaladmittance Y8 and a currenit source again representedby a Fourier amplitude hs(w, T). The network to be usedfor the noise-factor computation is then as shown inFig. 4. By definition,8 the spot noise factor (figure) of anetwork at a specified frequency is given by the ratio

Wi18(w) = lim I*(w, T)V(co, T)T-_oo 27rT

(14) 8 "IRE Standards on Electron Tubes: Definitions of Terms,1957," PROC. IRE, vol. 45, pp. 983-1010; July, 1957.

72 Janufary

Representation of Noise in Linear Twoports

of 1) the total output noise power per unit bandwidthavailable at the output port to 2) that portion of 1) en-gendered by the input termination at the standardtemperature To.

E

Fig. 4-Truncated nietwork for noise-factor computation.

Now the total short-circuit noise current at the outputof the network shown in Fig. 4 can be represented interms of Fourier amplitudes by

I,(co, T) + I (co, T) + YsEm(w, T).

Let us assume that the internal noise of the twoport andthe noise from the source are uncorrelated. If we thensquare the total short-circuit noise-current Fourieramplitudes, take ensemble averages and use equationssimilar to (10) and (12) to introduce mean-square fluc-tuations, we obtain a mean-square current fluctuation

j,2+ i+Ye12 = s2 +2+ IYe12e2 +Y*ie*+ Y8i*e (17)

linear twoport by a voltage geinerator and a currentgenerator lumps the effect of all internal noise sourcesinto the two generators. A complete specification ofthese generators is thus equivalent to a complete de-scriptioin of the internal sources, as far as their contribu-tion to terminal voltages and currents is concerned.Since the sources under considerationi are noise sources,their description is confined to the methods applicableto noise. The extent and detail of the description de-pends on the amount of detail envisioned in the analysis.Since, in the case of the noise factor, only the mean-square fluctuations of output currents are sought, thespecification of self and cross-product fluctuations of thegenerator voltages and currents is adequate.The expression for the noise factor giveni in (18) can

be simplified if the noise current is split into two com-poinents, one perfectly correlated and one uncorrelatedwith the inoise voltage. The uncorrelated noise current,designated by i,, is defined at each frequency by therelations

ei?,* = 0 (20)

(i-ijiu) = 0. (21)

The correlated noise current, i-i., can be written asY,e, where the complex constant Y,= G,+jB, has thedimensions of an admittance and is called the correla-tion admittance. The cross-product fluctuation ei* maythen be written

to which the total output noise power is proportional.It should be noted here that when e and i are complex,the symbols e9 and i2 denote, by convention, el 2 andIi 2. Since the noise power due to the source alone isproportional to ij2A the noise factor becomes

F =1 + 1 i (18)j,S2

In the denominator, the mean-square source noise cur-rent is related to the source conductance G8 by the Ny-quist formula

i,2= 4kToG8Af. (19)

ei* = e(i - = Y*e2. (22)

The noise-voltage fluctuation can be expressed interms of an equivalent noise resistance Rn as

e2- 4kToR,Af (23)

and the uncorrelated noise-current fluctuationi in termsof an equivalent noise conductance G, as

i-= 4kToGu.Af. (24)

The fluctuations of the total noise current are then

j2 = i- uI12++ti2= 4kTo [ Y 122R?, + Guj]Af. (25)

In the numierator of (18), the four real variables in-volved in i2, e9 and ie*, where e* is the complex conjugateof e, describe the internal noise sources. If the Fouriertransforms, (5), are used to characterize the noise, theself-spectral densities of noise current and voltage andthe cross-spectral density of these quantities would haveto be specified.

Before proceeding, let us review in general terms themethods employed and the conclusions reached. Therepresentation of the internal noise sources of a noisy

From (18)-(24), the formula for the noise factor be-comes

F= 1+-l [j2 + Y + y 12-2]F=I+4kToG,Af e (26)

Gu Rn= 1 +-+-[(G, + G-)2+ (B. + B)2] . (27)Gs G,

Thus, the noise factor is a function of the four param-eters G,, Rn, G, and B. These depend, in general, upon

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the operatin-g point aind operating frequency of thetwoport, but nlot upoIn the external circuitry. In a vacu-im tube triode the correlation susceptance Be is niegligiblysnmall at frequeincies such that transit timnes are sjmiallcompared to the period. Also, GC, anid G, are vanlish-ingly smiall wheni there is little grid loading. Thus,tube noise at low frequenicies is adequately characterizedby the single nionizero constant R,. Tube noise at highfrequencies anid tranisistor nioise at all frequencies havenio such simple represenitationi.

Sinice the noise factor is anl explicit funictioni of thesource coniductanice anid susceptance, it cani readily beshown that the nioise factor has an optimumii (minimumii)value at somiie optilImum1 source admiiittance Y,OG,o+jBowhlere

G'-) [GU ] (28)Rill

Bo= -Be (29)

and the value of this minimiuinum noise factor is

Fo- 1 + 2Ro,(G7 + Go). (30)

In terms of Go, Bo anid Fo, the noise factor for any arbi-trary source impedance then becomiies

F = Fo,R-[(s - G.)2 + (B, o-

)2I (31)

Eq. (31) shows that the four real parameters Fol Go,Bo and R, give the noise factor of a twoport for everyinput terminationi of the twoport.9 As showni in themlethods of test that are published in this issue, ac meas-urement of the minimum nioise factor and of the sourceadmittance YV with which Fo is achieved gives the firstthree parameters. The paramneter R. can be comiputedfrom an additionial measuremiietnt of the lnoise factor fora source admittanice Y. other than YO.From the given values, F,,G,, Bo and R, one canl

compute, if desired, the nioise fluctuationis j2, e2, and ei*(or the corresponding spectral denisities). To do this,one uses (28) to (30) to find Ye anid GC. From these onieevaluates e2, i2, and ei* from (23), (25), and (22). TheflulCtuations of any terminal voltage or currenit of thetwoport produced with a given source anld load cani theni

9 Reasoning similar to that leading to (31) can be carried olit in alual represenitation, where imiipedanices anid admittances are initer-changed. An equation similar to (31) results, again involving fournoise parameters, some of which are different from the ones used here.

be evaluated from the kulnown coefficients, (A, B, C, D)of (3), and the knowni nioise fluctuationis. Thus, the noisein a twoport is completely characterized (with regard tothe fluctuations or the spectral (leisities -at the ter-inals) by the noise fluctuationis e-, tand ei*, or alter-

natelv by the four nloise plaralmeters F,, ,B,,B ladR,.

VI. CONCLUSION

Th-e preceding discussioni showed that, with limitedobjectives, the nioise in a liniear twoport cani be chclr-acterized adequately by a limited nlumber of parameters.Thus, if one seeks oinly information concerning themiieani-squiare fluctuationis or the spectral denisities ofcurrenits or voltages inlto or across the ports of a lineartwoport at a particular frequenicy anid for arbitrary cir-cuit conniiections of the twoport, it is sufficient to specifytwo miieani-square flucttiationis anid one product fluctua-tionl or two spectral denisities and onie cross-spectraldensitv. This inivolves the specification of four realniumlbers. If a band of frequencies is beinig conisidered,these quaintities have to be given as funictionis of ire-quenicy uniless they are approximately constant over thebauid.

Different separations of initernial nioise sources willlead, in general, to different frequenicy dependences ofthe restilting fluctuations or spectral densities. Trhus, aparticular separationi of the nioise sources mnay be pref-erable to (another separation if it is founid that fluctua-tionls (spectral densities) are less frequency-depenidentin the band. In lparticular, if available informationabout the physics of the nioise inl a particular (levicesuggests introducing, inside the twoport, appropr-iatenioise genierators of fluctuationls (spectral (lensities)with nio frequency dependence, or with al sim-iple fre-quency dependence, it imiay be more advantageous tospecify the device in termiis of these physically sugges-tive genierators.

I-lowever, usually one resorts to the characterizatiolnsof lnoise presented in this paper, sinice they have the aid-vanitage that they do not require any knowledge of thephysics of the initernial n-oise. Furthermiiore, noise-factormleasuremiienits performed oni the twoport yield (muore orless) directly the noise paramneters of the genieral circuitre)resentation, niamely the fluctuationis (spectral (lensi-t-ies) of the voltage anid currenit generattors attached tothe input of the nioise-free equivalenit of the twoport.rhis fact recommienids the use of the nioise parametersniatuiral to the genleral-circuiit-paramiieter representationi.

74 JaInuaryV