The Computation of Correlation and

Spectral Functions by Orthogonal

Filtering

Note that La«, which must always bepositive, can only decrease as the numberof terms in the series increases.

The cross-spectral density approxima­tion is given by the Fourier transform ofequation 3 as

N

'1t/o*(jw) =L: [an<l>n(jw)+bn<l>n( -jw)]n=l

ELMER G. GILBERTNONMEMBER AlEE

(10)

Application of Parseval's theorem toI m in gives:

Since equation 2 is satisfied, the integralsquare error

is finite for finite an and bn. Minimiza­tion of I in respect to the an and bn givesthe following results:

(14)

(15)hn(t) = f.!~-T)cbn(T)dT

an = g(t)hn(t)

where

the response of a linear filter with input

N

=2L: anRe<l>n(jw) (13)n=l

and

N

=L: ancbn( 11"1) (12)n=l

Interchanging the order of the operationsaveraging and integration over T gives:

To obtain the representation for theautocorrelation function and spectraldensity of f(t) it is only necessary to letg=f. Since t/;//(-T) = t/;//(T) , bn=an andequations 3 and 10 become simply:

N

1P//*(r)=L: an[cbn(r)+cbn( -T)]n=l

1 fO)[min = - I'1t/O(jw) - '1t/0*(iw) 1

2dw271" -0)

The procedure for evaluating the anin terms of f(t) and get) rather than t/;/o(T)is found by substituting equation 1 intoequation 8 to obtain:

an= f:"" !(t-T)g(t)cbn(T)dT

N

'1t//*(jw) = L: an[<I>n(jW)+<I>n( -jw)]n=l

(6)

(2)

(4)

(5)cbn( T) = 0, TSO

Orthonormality, rather than orthog­onality, is stipulated for convenience.The second group of terms bnepn(-T) inequation 3 is required because it is as­sumed that:

a condition which is required if the seriesrepresentation is to hold. Since one ofthe spectral densities must be bounded,all periodic and constant components ofthat spectral must be excluded.

Let the series representation of t/;/Q begiven by

N

1/;/0* =L: [anchn(r )+bncbn( - T)] (3)n=l

where the epn are orthonormal, i.e., theysatisfy the conditions

fO) cbn(T)cbm(T)dr=O, n~m-0) =1, n=m

finite power and that at least one of thespectral densities '1'//(jw) or '1'QQ(jw) isbounded. (Upper-case-letter functions ofjw are the Fourier transforms of corre­sponding lower-case-letter functions.) Asshown in Appendix I, these are sufficientconditions to assure that:

f:"" if!to2(T)dT<

co

THE INCREASED consideration ofrandom time functions in engineering

and science has led to the development ofvarious digital and analog techniques forthe computation of autocorrelation andcross-correlation functions or their fre­quency domain equivalents, spectraldensity and cross-spectral density. Ana­log techniques, which are the subject ofthis paper, have been limited primarily toeither the time domain1 - s (employing theoperations of time delay, multiplication,and averaging) or the frequency domains!(employing the operations of narrow-bandfiltering, multiplication, and averaging).A more general butlittle-knownprocedure,which was originally introduced by Lam­pard," is to represent the autocorrelationor cross-correlation function by a finiteseries of predetermined orthogonal ap­proximating functions and to determinecoefficients in the series by the opera­tions of orthogonal filtering, multiplica­tion, and averaging. Advantages of theseries method include analytic represen­tations for both the correlation and spec­tral functions and the possible elimina­tion of delay lines or narrow-band filters.

The purpose of this paper is to developmore fully the series method for computa­tion of correlation and spectral functions.After reviewing the series representationand its determination, the followingtopics are treated: therelationship of time­domain and frequency-domain techniquesto the series method; orthonormal ap­proximating functions; analog computercircuits for orthonormal filtering; ap­proximation errors due to a finite numberof terms in the approximating series,computer component errors, and finiteaveraging time.

where the bars denote averages over all bn=f_""_ if!tO(T)cbn( -T)dTtime. It is assumed that f and g have --

(1) and

Series Representation and ItsDetermination

The cross-correlation function of f(t)and get) is defined by

1/;/OC T) =f(t)g(t+r) =f(t - T)g(t)

(7)

(8)

(9)

Paper 59-1128, recommended by the AlEE BasicSciences Committee and approved by the AlEETechnical Operations Department for presentationat the AlEE Fall General Meeting, Chicago, Tll.,October 11-16, 1959. Manuscript submittedMarch 16, 1959; made available for printingJuly 23, 1959.

ELMER G. GILBERT is with the University ofMichigan, Ann Arbor, Mich.

The author is grateful to F. J. Beutler and EdwardO. Gilbert for helpful discussions during thepreparation of the paper.

954 Gilbert-Computing Correlation Functions by Orthogonal Filtering JANUARY 1960

g(t) MULTIPLIER

MULTIPLIER

(t)

o

SUMMER

ORTHONORMALFILTERS WITH

IMPULSE RESPONSES¢n(t)

~ 4>1 (t) ..... 0

1

~ 4>2 (t) :- O2OUTPUT T

~ IRECORDER· ·uo(t)

..· · N

· · Lan ¢nh=1· ·

~ ¢N (t) :. ON

on

bn

AVERAGINGDEVICE

AVERAGINGDEVICE

(B)

(A)

ORTHONORMALFILTER hn(t)

WITH IMPULSERESPONSE

n (t)

ORTHONORMALFILTER

WITH IMPULSERESPONSE~n (r )

Ht)

f(t)

g(t)

Fig. 1. Physical syatems for evaluation of an(A) and bn(B) Fig. 2. Method for recording positive ". portion of 1/Ifg*

INPUT

TAPPED DELAY LINE

Orthonormal ApproximatingFunctions

f OO IfoocPn(t)cbm(t)dt =- cf>nejw) X

_ 00 21r _ 00

cf>m( -jw)dw

Hence functions orthogonal in the fre­quency domain are also orthogonal in thetime domain.)

orthogonal by causing their transforms4Jn (j UJ ) to be nonoverlapping in the fre­quency domain, i.e., by making the 4Jn(j UJ)be narrow band filters. (By Parseval'stheorem:

The primary factors governing thechoice of orthonormal functions are func­tional form and physical realization. Thefunctional form should yield series repre­sentations for 1/Ifg and 'Vfg which are ac­curate for reasonably small N and are easyto use; physical realization should besimple, accurate, and inexpensive.

When delay elements are allowed in therealization of the orthonormal functions

DELAY TIMES 0

Fig. 3. Realization for orthonormal function cPn(T)

INTEGRATORITH TRANSFER OUTPUT

FUNCTIONI

...[l;Tjw

Relationship to Time-Domain andFrequency-Domain Techniques

In the usual time-domain computa­tion of a correlation function the cPn aremade orthogonal (but not orthonormal)by causing the 4>n to be nonoverlappingfunctions of T. Thus cPn(t) =uo(t-ndT),the impulse response of a delay line withtaps separated by dT. By equations 8and 9 it is seen that an=Y;fg(nd".) andbn=Wfg(-ndT). Actually, because ofdelay line errors in phase and gain, the4>nCT) have finite width and height andonly approximate uo(t-nd",). Still, witha sufficiently good delay line the 4>n do notoverlap appreciably -. and are essentiallyorthogonal.

A similar situation exists for the fre­quency-domain computation of spectraldensity. The functions 4>n are made

square-error approximations may beemployed to diminish certain undesirablecharacteristics of the integral-square­error approximation.?

f(t) and impulse response 4>n(T). Sinceequation 5 holds, the impulse responsecPn("') must be realizable. Fig. l(A) showsthe physical systems needed for evaluation14. For computation of bn a similardevelopment gives the result in Fig. 1(B) .

After the an and bn have been computedthe linear systems with impulse responsescPn("') can be employed as a function gen­erator to machine plot the series represen­tation 1/Ifg*. The scheme is shown in Fig.2. The linear system synthesized has an

N

impulse response L an 4>n(t). Thusn=l

when it is excited by an impulse uo(t), or aset of initial conditions producing the sameeffect as an impulse, an output equal tothe positive T portion of 1/Ifg* is obtainedfor recording. The negative r portion of1/1*10 is recorded in exactly the same wayby substituting the bn for the an.

Generalizations of the foregoing pro­cedures are possible. First of all, the ap­proximating functions for negative". neednot be 4>1(-"'), ... 4>N(-T). Any num­ber of other orthonormal functions, say<t>J(--T), . . .<f>MC-".), may be used by substi­tuting <t>n( - T) for 4>(n-"') in all the aboveresults. Second, the approximating func­tions need not be orthogonal. When the<Pn are not orthogonal the computed resultsJ~ 00 1/Ifg(T)4>n(T)d". and J: 00 1/Ifu(T)cPn(- ".)dT appear in 2N linear algebraicequations in 2N unknowns, an and bn .

While nonorthogonal functions are gener­ally easier to realize as impulse responses,the solution of the linear equations for anand bn is lengthy and tends to aggrevatethe effect of errors in the computedJ~00

1/Ifg("')4>n(".)dT and rz: 1/Ifg(T)4>n(-T)dT.Lastly, constrained or weighted integral-

JANUARY 1960 Gilbert-Computing Correlation Functions by Orthogonal Filtering 955

(19)

and

(jw +s) (jw +Sln-l)(jw -S1) (Jw -Sn-l) X

2vanjClJ (21)(jw)2+2aniw+~n2

The corresponding time functions cPn(t)are not written so easily and thereforeare not given here. When all the Sn arereal and equal, the functions cPn are pro­portional to the Laguerre functionsutilized by Lampard.

A realization of the foregoing functionsis shown in Fig. 4. Since the highernumber functions include the dynamiccharacteristics of the lower number func­tions, the cascade arrangement providesminimum complexity. Operational am­plifier circuits for the sections shown inFig. 4 are given in Fig. 5. Except for thenormalization factors +1/2Van and-1/V2a n these sections may be fitteddirectly into Fig. 4. For the functionsdefined by equations 20 and 21 the nth and

made essentially nonoverlapping, buttheir realization is difficult and the form ofthe approximating functions is not simple.

Of greatest practical value are ortho­normal functions ~n(jw) which are ra­tional injw. They may be easily realizedby lumped element systems (an impor­tant advantage in the low-frequencyregions where delay elements are diffi­cult to construct) and have functionalforms which are simple and analyticallytractable. Fortunately, orthogonaliza­tion of such functions is not difficult eventhough they are overlapping in both thetime and frequency domains. The ortho­gonalization procedure is carried out inthe frequency domain and has been de­scribed elsewhere.7

The set of functions ~l, ~2, ••• ~n con­sidered here have poles at jw = Sl, S2, ...Sn (the Sn must have negative real partssince J~CX)cPn2dt=1<oo). When Sn isreal with Sn= - an:

When Sn and Sn+l are complex conjugateswith real parts - an and magnitudes (3n,

(jW+Sl)' .. (jW+Sn-l)cf>n=(. ) (. )XJW- Sl ... JW- Sn-l

2~n~ (20)(jw)2+2anjw+~n2

OUTPUT N

Fn () w) jw+ Sn---:Fn_l(jw) jw - Sn

to a series representation of both if;/u and'!' /u by the simple addition of an integra­tor to a tapped delay line.

Variations of these functions are possi­ble. The adjacent time intervals neednot be equal, nor does the last functioncPN(t) have to be zero t>NdT. An ex­ample of the latter is the function:

4ln (jw)Fo (jw)

Fn- I (jw)

0N(t) =0, t~(N-l)AT=v2a Ea(n-l)~T E-a t, t>(N -l)dT

(18)

Still more varied functions may berealized with delay elements if the non­overlapping requirement is not enforced.But then, of course, orthogonalizationis not so easily carried out.

Attempts to form orthogonal functionswhich are nonoverlapping in the frequencydomain encounter theoretical limitationsbecause of realizability considerations.(The Paley-Weiner theorem shows that~n(jw) is not realizable if ~n(jw) = 0 inany finite frequency range.) Band-passfilters with extremely sharp cutoff can be

(17)

(16)

OUTPUT 2

Fig. 4. Realization for rational orthonormal function cf>n(jW)

OUTPUT I

INPUT

Fa (jw)

E-j",(n-l)~T_E-j",n~T

cf> (J. )-------n w - VdT(jW)

(A)

- _I_ 4l (jw)F (jw)~2an n

Fn - I (jw)

result. Here a stepwise representation ofif;/o(T) is achieved. The frequency-do­main approximation is a harmonic seriesof sines and cosines (of fundamentalperiod w=21r/dT) weighted by l/w.A physical realization of the functions isshown in Fig. 3. In practice the integra­tor might be an R-C low-pass filter withtime constant much greater than dT.Thus it is possible to go from a point bypoint determination of if;'o(at T=ndr)

their formulation is particularly straight­forward because the functions can bemade nonoverlapping in time. For ex­ample, by choosing the functions to beconstant in adjacent time intervals ofwidth d T the expressions

cbn(t)=0, t~(n-l)dT, t>ndT1= _/-'-, (n-l)dT<t<ndT

VdT -

and

Fn +lOw)

Fn_l{jw)

(jwtsnHjw+ Sn+l)

(jw-sn )(jw-sn+1)

tPn(jw)Fo(jw) _ 2~ I3 n

Fn _I (jw) (jw)2+2anjw+13~

(8)

4>0"'1 Ow)~(jw)

Fn_l(jW)

2~jw

(jW)2t 2 anjw +13~Fig. 5. Operational amplifiercircuits for sections shown in

Fig. 4

956 Gilbert-C'omputing Correlation Functions by Orthogonal Filtering JANUARY 1960

g(t)

f (t)ORTHONORMALFILTERS WITH

IMPULSERESPONSES

1>n (t)

AVERAGINGDEVICE

Fig. 6. Diagram showing com­ponent errors

STATISTICAL ERRORS DUE TO FINITE

AVERAGING TIME

and when the averaging device is anintegrator with averaging time T,

where l1an is the statistical error in an, andw(t) is the impulse response of theaveraging device. When the averagingdevice is a first-order low-pass filter withtime constant T,

1 -t/1'wet)=- e ,t>OT

this is advantageous it will be assumedthat the constant components of bothf and g have been extracted. Thenneglecting the second order term, ele2gives:

Aan=e3+emC~+el, hn+e2) (24)

Since the average multiplier error isdifficult to compute, the bound

IAanl~le31+lemlmaxlmum (25)

(27)t~O=0,

The time averaging of g(t)hn(t) whichyields an must necessarily be carried outby a low-pass filter or integrator. Ineither situation a statistical error resultsbecause the average is not carried outover all time.v? To derive the resultsthat follow it is assumed that f and g be­long to stationary and ergodic randomprocesses.

If the average of g(t)hn(t) is measuredat time t the result can be expressed by:

an+Aan =ia> g(t-T)hn(t-T)W(T)dT (26)

is more useful. The same result appliesto I1bn • From the above it can be seenthat multiplier error and integrator driftare the important component errors.

To keep the effect of these errors assmall as possible, the levels of l. g, andh« should be scaled as large as possible,observing saturation limitations, In thisregard servo multipliers are severelylimited, even when their linear-systembandwidth is sufficiently large. Velocityand acceleration limits can be encounteredlong before full-scale amplitudes arereached.

-= elk n+e2g +ghn+ele2 +

------em(g+el, hn+e2)

+e3 (23)

If Jand gare both zero, the first two errorterms in equation 23 vanish. Because

The component errors considered inthis section are indicated in Fig. 6. Thequantities el, e2, and e3 are constants repre­senting the effects of offsets in the com­puter circuitry. The multiplier staticerror is em and is a function of the twoinputs to the multiplier. Dynamic errorsin the multiplier and realization of the cPnare usually of lesser importance and areneglected.

Referring to Fig. 6, it is seen that(assuming infinite averaging time):

COMPUTER COMPONENT ERRORS

an +Aan = (g+el)(hn+e2) +em(g+el, hn+e2)

+e3

throughout the complex plane in the fre­quency range of importance. In anycase, N should not differ greatly from theN demanded in the more usual time orfrequency-domain computations of 1/;/g.Once a reasonable representation of 1/;/ghas been computed using a large numberof terms, inspection may lead to a moreefficient choice of the cPn.

An additional complication arises inevaluating the goodness of the approxi­mation since J~oo 1/; f g2d r is not knownand 1min cannot be computed. How­ever, comparative calculations are poss-

Nible, since L: (an2+bn2) can be corn-

n=1

puted and becomes larger as approxima-tion accuracy improves. Furthermore,a check on the approximation error at r=

o is readily made since 1/I/g(0) =f(t)g(t) iseasily computed. This check is valuablewhen the rational approximating func­tions are utilized because the discon­tinuities in many of the cPn(r) at T = 0 oftencause the approximation error to begreatest at r = O. Of course, the accuracyof these checks is limited by computercomponent errors and finite averagingtime.

(n+ l)th sections in Fig. 4 are combinedand the output Fn(jw) does not appear.

The singularities Sn of the approximat­ing functions are defined by the potenti­ometer settings an and (3",. Thus the formof the series approximation may be con­veniently changed by simple computeradjustment without any preliminary cal­culations. This flexibility often permitsgood approximation accuracy with arather small N, an advantage which theLaguerre functions do not have.

(22)

Thus III represents the integral squareerror due to sources 2 and 3. Practicallythere is no point in striving to makeeither 1min«1ll or 111<-a;« Thethree error sources will now be examinedin detail.

Approximation Errors

ERRORS DUE TO FINITE N

The N required for a given approxima­tion accuracy varies widely with thechoice of the approximating functions cPn.Often, several of the rational functionsgiven in equations 19, 20, and 21 will givea very satisfactory representation if theyare chosen with care. However, since1/;ra is initially unknown, careful choice ofthe functions is usually not feasible andlarge number of cPn with varied char­acteristics are needed. The necessaryvariation of characteristics can beachieved with rational functions by dis­tributing the points of singularity Sn

Three distinct sources of approxima­tion error exist: 1. the finite number ofterms in the approximating series, 2. de­viations in the an and bn due to computercomponent errors, and 3. statistical de­viations in the an and b« due to finiteaveraging time. The first error sourceyields an error magnitude measured byI min in equation 7. If the deviations inan and bn caused by the last two errorsources are denoted by l1an and I1bn, thenit can be shown that the total integralsquare approximation error is:'

N

I=Imin+AI=Imin+L: (Aan2+Abn2)n=1

JANUARY 1960 Gilbert-Computing Correlation Functions by Orthogonal Filtering 957

(46)

But from equation 15

E[g( - Tt)hn( - Tt)g( - Tt)hn( - T2)]=E[g( -Tt)hn( -Tt)]E[g( -T2)hn( -T2)] +E[g( -Tl)g( -T2)E[hn( -Tt)hn( -T2)] +E[g( - Tt)hn( -T2)]E[g( - T2)hn( - T1)]= 1/;gh2(0)+1/;gg( T)1/;hh( 7) +1/;hg(7)Y;hg( - T),

7=T1-T2 (41)

Thus, noting equation 29, equation 40becomes:

Summation of the terms in equations 48and 49 gives equation 31.

E( d an2)= l"'l"'(I(Tl-T2)W(Tl)X

w( T2)d71dT2 (42)

Since g and hn have Gaussian distributions,the fourth tnoment in equation 40 may beexpressed in terms of the second moments i'?

where

where k=cPww(O) is the upper bound ofcPww( 7 ). Since the averaging times areusually very long compared to values ofI'T I where Y;II.. 'T), 1/I1Ig( 'T), Y;Ig('T), andconsequently fJ( T) have appreciable magni­tude, E [Aan2] approaches closely theupper bound given in equation 45. Thus:

fJ( 7) = fJ( - 7) = Y;gg( 7 )l/Ihh( 7) +l/Ihg( T)1/;hg( - T) (43)

E[Aan2]'::::.~fCO ['I!gg(jW)'I!hh( -jw)+21r -co

'1'hg2(jW)]dw (47)

By defining

d>/DW( T)=l'"w( Tl)W( T+n)dTl (44)

Application of Parseval's theorem toequation 46 gives:

Similarly,

equation 42 can be written as:

E[ d an2] =f-"'ex> (I(T)fJtDfD(T)dT~k f:ex> (I(.,.)dT

(45)

'1'hg(jW) ~ '1'Jg(jw)<I»n( -jw)

Thus

'1'hh( - jw ) =Whh(jW) ='1'II(jw ) I <l»n(jW) 12

and

(35)

(36)

(37)

(38)I'I!Ig 12 ::;2'1'11'1'gg

Addition of equations 36 and 37 gives:

(Re'l! Ig)25:'1'II'I!gg

If co Mfco- '1'gg'~ Itdw 5:- '1'ggdw1(' -co 1(' -co

Appendix II

The variance of Aan can be written as

E[Aan2] =E[(an+Aan)2] -an2

Substituting an+Aan from equation 26and for convenience letting t=O gives:

E[Mnl ] = lex>lex> E[g( -Tl)hn( -Tl)X

g( - T2)hn( - T2)]W( 71)W( 72)d71d72 -an2 (40)

Now if one of the spectral densities isbounded (say 'l!11~M), then:

(lm'1' Ig)2::;,!, JI'I!gg

Similar consideration of the spectradensity of Xf+jg yields the result:

Appendix I

or

This is the desired result.

4(Re'1' Ig)2-4'1'II'I!gg::;O

Let X be an arbitrary real parameter.Then the spectral density of the timefunction Xf+g must be nonnegative. Thus:

o::;'1'(~I+g)(~/+g) =X2'1'II+X'1' Ig+X'1'gl+'1'gg

But '1'gl=conj'1'lg, so:

and

X2'1f/1+2ARe'1'Ig+'1' gg~O

Since the result must hold for all values ofX the discriminant must be negative orzero and

All- of these results can be applied toautocorrelation function measurements byreplacing the g subscripts by f subscripts.

they can be measured using series repre­sentations. Then the right member ofequation 33 is approximately:

2kfco N-; _ex> "igg*'ID/* ~11<l>n12dW

N

= k 2: agnalmCnm (34)n,m=1

where agn and a1m are coefficients in theseries '1'Dg* and '1'11* and

8fco NCnm=- ReipnReipm2:I<l»Jl 2dw

1(' -co 1=1

(33)

(32)

(28)

(29)

1:=- for equation 28

2T

Thus the average increment of integral­square-approximation error is inverselyproportional to the averaging time.

Since equation 31 is somewhat toocomplicated to use for simple estimatesof Ill, the following upper bound on E [Ill]is derived in Appendix III:

2kfco NE[M]~-; _ex> 'iJfgg"l l1~ lef>nl 2dw

k=lex> w(t)2dt

1=Tfor equation 27

( ' 1w tJ=r' O<t5:T

=0, t5:0, .c-rThe proper gain of the averaging deviceis achieved by requiring

l'" w(t)dt=l

E[an+Aanl =an+E[Aanl

= l'" E[g(t-T)hn(t-T)]W(T)dT

=E[ghn]l'" w(T)dT

=E[ghnl =g(t)hn(t) =an (30)

where

N

Once 2: ftPnf2 has been evaluated for an==l

given set of approximating functions, theabove integral can be quickly estimatedto determine a suitable averaging time.If '1'DD and '1'11 are not known initially,

Because get) and/(t) are stationary andergodic, the expected value of an+Ilanis

Thus E [Ilan] = 0 and the estimate of anis unbiased. In an analogous fashion thesame result can be developed for Ilbn •

In order to derive reasonably simpleexpressions for the variance of Ilan it isnecessary to further assume that g andI are Gaussian. For processes which arenot Gaussian the results presented herewill at least serve as an approximateguide. In Appendix II it is shown that:

958 Gilbert-Computing Correlation Functions by Orthogonal Filtering JANUARY 1960

A Small Automatic TeletypewriterSwitching System

R. A. VANDERLIPPENONMEMBER AlEE

Appendix III

Equation 47 can be written as:

k fcoE[~an2] =2; -co ['Irgg(jW)Vhh(jW)+

ReVhg2(jW)+jImVhg2(jW)] dw

Since EI~an2] is real the integral of Imvhg2must vanish. Hence:

Application of inequality 36 with f replacedby h gives:

E. R. SHIMMINNONMEMBER AlEE

Two teletypewriter selective callingline systems, the TADS (Teletype­

writer Automatic Dispatch System)system! and the 83B1 system.s have beendescribed in recent papers. These sys­tems are for single multistation lines;those in service at the present time havean average of about ten stations on a linealthough, theoretically, if the trafficoffered per station is low, there may be asmany as 40 stations on a line. As soonas the required number of stations ex­ceeds that which can be efficientlyhandled on one line, other lines are re­quired, with the need for transferring mes­sages from one line to another.

For simple networks of not more thanthree lines, the transferring can be doneefficiently and economically by an exist­ing automatic relay arrangement usingreperforator-transmitters with interme­diate punched tape as the switchingdevice. For more than three lines such aswitching device becomes increasinglymore complicated and prohibitively ex­pensive.

To fill the need for interconnecting fromfour to nine selective calling lines a new

Similarly, the same upper bound appliesto E[~bn2].

The summationN

L (E[~an2] +E[~bn2])n=-l

therefore gives equation 33.All of the foregoing results can be applied

to autocorrelation function measurementsby replacing the g subscripts by f subscripts.

ReFerences

1. A DEVICE FOR COMPUTING CORRELATIONFUNCTIONS, A. E. Hastings, J. E. Meade. Reviewof Scientific Instruments, New York, N. Y., vol.23, 1952, p. 347.

2. A FIVE CHANNEL ELECTRONICANALOG CORRE­LATOR, M. J. Levin, J. F. Reintjes. Proceedings,National Electronics Conference, Chicago, 111.,vol. 8, 1952, p. 647.

3. A HIGH-SPEED CORRELATOR, H. Bell, V. C.Rideout. Transactions, Professional Group onElectronic Computers, Institute of Radio Engineers,

A. L. WHITMANMEMBER AlEE

automatic switching unit, known as the84A1 teletypewriter switching unit, wasdeveloped. In the interests of simplicityand economy, the requirements for thesystem were set with certain recognizedlimitations. As a result, the capacity ofthe switching unit to handle crossofficetraffic efficiently is limited. There is nofixed upper limit of capacity; whathappens if the system becomes overloadedis that the messages are subject to longerdelays in delivery. The plan for the 84A1system was laid out on the basic assump­tion that it would be used in situationswhere the principal communities ofinterest would be along the individualselective calling lines, and that only arelatively small amount of traffic wouldbe exchanged between lines. Quantita­tively, about 60% of the total traffic on aparticular line should be intraline, withonly 40% interline.

This will indicate why the system hasbeen called a "small" system. It is in nosense competitive, so far as traffic­handling capacity is concerned, with thelarger and more elaborate automaticswitching systems of the 81 type. 3 How-

New York, N. Y., vol. PGEC-3, no. 2, June 1954,p.70.

4. AFOSR TN 58-338, A. Huberstrich, F. R.Hama, USAF Office of Scientific Research,Washington, D. C., 1958.

5. TECHNIQUE FOR MEASUREMENT OF CROSS­SPECTRAL DENSITY OF Two RANDOM FUNCTIONS,M. S. Uberoi, E. G. Gilbert. Review of ScientificInstruments, vol. 30, Mar. 1959, p. 176.

6. A METHOD OF DETERMINING CORRELATIONFUNCTIONS OF STATIONARY TIME SERIES, D. G.Lampard, Proceedings, Institution of ElectricalEngineers, London, England, pt. C, no. 1,· Mar.1955, p. 35.

7. LINEAR SYSTEM ApPROXIMATION BY DIFFEREN­TIAL ANALYZER SIMULATION OF ORTHONORMALApPROXIMATING FUNCTIONS, E. G. Gilbert. Trans­actions, Professional Group on Electronic Com­puters, Institute of Radio Engineers, vol. PGEC-8,no. 2, June 1959.

8. STATISTICAL ERRORS IN MEASUREMENTS ONRANDOM TIME FUNCTIONS, W. B. Davenport, Jr.,R. A. Johnson, D. Middleton. Journal of AppliedPhysics, New York, N. Y., vol. 23, Apr. 1952,p.377.9. THE MEASUREMENT OF POWER SPECTRA FROMTHE POINT OF VIEW OF COMMUNICATIONS ENGI­NEERING, R. B. Blackman, J. W. Tukey. BellSystem Technical Journal, New York, N. Y., vol.37, 1958, PART I, p. 185; PART II, p. 485.

10. RANDOM PROCESSES IN AUTOMATIC CONTROL(book), J. H. Laning, r-, R. H. Battin. McGraw­Hill Book Company, Inc., New York, N. Y., 1956.

ever, it is believed that it will fill a signif­icant need for businesses which do notneed or cannot afford the larger sys­tems. In addition, where the customerhas a need to terminate some half-duplexmultistation circuits, these circuits couldterminate on an 84A1 switching unit.The transfer of traffic between the twosystems could be effected manually bythe tom tape method or by an auto­matic interconnection now under COD­

sideration.A significant difference between the 84

and 81 systems, is that the lines of the 84system operate half duplex (HDX) , whilethe lines of the 81 system are full duplex(FDX). Half-duplex operation permitsfree interchange of traffic between stationson one line without the intervention of theswitching center. If a. substantial pro­portion of the traffic is of this kind, HDXoperation relieves the switching center ofa large part of its load.

The multiple-address capacity of theswitching unit is limited only by thetotal number of stations on the linesswitched by the unit. Since the directorprocesses each address separately, itwill continue to set up crossoffice paths

Paper 59-1105, recommended by the AlEE Tele­graph Systems Committee and approved by theAlEE Technical Operations Department forpresentation at the AlEE Fall General Meeting,Chicago, Ill., October 11-16, 1959. Manuscriptsubmitted April 22, 1959; made available forprinting July 15, 1959.

E. R. SHIMMIN is with the Pacific Telephone andTelegraph Company, San Francisco, Calif. ;R. A. VANDERLIPPE and A. L. WHITMAN are withBell Telephone Laboratories, Inc., New York, N. Y.

JANUARY 1960 Shimmin, Vanderlippe, Whitman-Auto. Teletypewriter Switching System 959