IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 2, JUNE 1984

Mean Estimation of Random Process

with Aid of ON-OFF DetectorsDOV WULICH

Abstract-This paper presents a maximum likelihood method (ML) ofmean estimation of a random process when the measurement of the ex-act value of process is not required. The estimation is accomplishedwith an array of ON-OFF detectors.

It is shown, that there exists an optimal value of standard deviationwhich gives the maximum value of accuracy for the worst case, i.e.,when the measured mean value is exactly between detectors. Compari-son of the proposed method (PM) with a commonly used method (CM)indicates that for a wide range of standard deviation of the consideredprocess the accuracy of both methods is almost equal.

I. INTRODUCTIONIN MANY FIELDS such as signal processing, communication,

measurement, and the chemical industry it is necessary toestimate the average value of a stochastic process. Many meth-ods exist which require the exact measurement of the instanta-neous value of the process [2]. On the other hand there existsituations in which it is impossible, or very expensive, to mea-sure the exact value of the process.For example, in the chemical industry there is a need to esti-

mate the volumes of liquids. But from the technological pro-cess it follows that the measurement must be accomplishedwhen the liquid is blended; furthermore, it is impossible to in-troduce into the liquid a float for the exact measurement ofthe level. Consequently, the measurement is possible onlyfrom the outside, often by using an array of gamma ray detec-tors, which are in fact ON-OFF detectors.Another example is the estimation of target azimuth by

means of optical detector arrays. There exist situations inwhich the detectors have a flat angular response, and thereforeare considered as ON-OFF detectors, and the target azimuth ishardly perturbed by atmospheric turbulence.

In this paper, a specific situation is considered as a first andbasic stage of investigation: the process for which the mean isto be estimated is stationary and Gaussian with a known stan-dard deviation. Later these assumptions must be weakened.Nevertheless, such assumptions are quite realistic. In manycases, the process is Gaussian (or approximately Gaussian)and its standard deviation is known a priori, or is easy to mea-sure, (as in the example from the chemical industry).

II. THE PROBLEM STATEMENTEstimation of the mean (average) value of the Gaussian sta-

tionary and ergodic process x(t) is required. The estimation

Manuscript received January 28, 1982; revised December 21, 1983.The author is with The Department of Electrical Engineering, Ben-

Gurion University of the Negev, Israel; and Tadiran, Israel IndustriesLtd., Communications Products Division.

Y, lt)xt

Array() Samplingx()

of N ON-OFF and ML

detectors vector ~ Processor

terminato

Fig. 1. Schematic diagram of the proposed method.

must be accomplished when the following restrictions are

fulfilled:A) There are N ON-OFF detectors available. The action of

the ON-OFF detector is described by

(=1; X(t) > hi,

i t; x(t)<hi5(1)

where x(t) is the input process and yi(t) is the output of theith detector. The array of N such detectors create in fact aquantizer with quantization steps hi - hi1,.B) The output of each detector yi(t) is observed in n equally

spaced discrete times tI, t2, *. ', tn only.The measurement procedure is performed in two stages.

Firstly, the vector k = (ko, kl, *, kN) is determined whereki (i = 0, 1, * * , N) is the number of times that the process iswithin the levels hi and hi, I. Here it is assumed that ho = -00and hN+l = oo. From definition (1) of the ON-OFF detectoraction and from the definition of the vector k, it follows that

nki= Z [yj+1(t)+)yj(ti)], i= 0 1, * N

j=i(2)

where $ denotes the XOR (exclusive or) operation. From (1)and the assumptions that ho = -oo and hN+1 = °° it follows thatyO(tJ) = I andyN+1(tj) = 0 for every tj(j =1, 2, ., n).

Secondly, from the observation of the vector k the estimatei? of the mean value i7 of the process x(t) is obtained. Theproblem is to find the maximum likelihood (ML) algorithm forestimation of q and to evaluate its performance. The measure-ment procedure given above is illustrated in Fig. 1.

III. MAXIMUM LIKELIHOOD ALGORITHM FORMEAN ESTIMATION

Denote as 71 a ML estimate of the mean value of the processx(t). i2 is known to be a solution of the following likelihoodequation [2]:

aln P(k/r,) =0

ai7 (3)

where the likelihood function P,, (k/l7) is a conditional proba-bility of vector k given 7?.

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 2, JUNE 1984

Our goal is to find this likelihood function. For this we as-sume that the samples of yi(tj)(i = 1, 2, N; 1, 2,3,n) are mutually independent random variables. This is satisfiedif the distance between sampling times is greater than the cor-relation interval of the considered process x((t). Under thisassumption the vector k has a generalized Bernoulli distribu-tion [1]

Pn (k/71) = Pn (ko, k I,* , kn M7)- n ! pk k, k~pN(~ 4

ko !kl ! ..*kN!0 (n) -pi P,N ( 4)

where

spect to q and taking the mathematical expectation gives

E 1nf n(k/ _P) Nf Pi(??) IE 2 = E: E{ki}2l (9)

where

Pi(n)= aq2For further progress the mean value of k1 - E{k1} must be

found. From the definition of the vector k follows, that eachof its components k (i = 0, 1, - -, N) has a binomial distribu-tion with parameter Pi(a), therefore [1 ]

Pi(n) = P(hi < x(t) < hi, 1 )It is clear that IN Pi(7a) = 1 for all r.

In our case the process is Gaussian with known varianctherefore

(hi_ W)Ua J,

i=0, 1, - ,N

(6)

(5) E{ki} = nPi(r7).Substituting (6) into (10) and (9), (10) into (9) yields

U2var( )̂ >

n

1i+ -i-7

N [ (ha r)(ha)]i- hi+n- I

where O(a) is the Gaussian error function

1 `{e-X212 dX.

Substituting (6) into (4) and (4) into (3), the likelihood equa-

tion is now in the form

k ij11) = 0 (7)

i=0 pin

where

fs(7) = aPi() i=O,1,= , N.

The ML estimation of q implies therefore solution of thelikelihood (7) for given values of ko, k1, , kN. This is a

task of the processor-Fig. 1, i.e., the processor should solve(7) when the functions Pi(.) and Pi(.) are stored in its memory.Obviously this is easily realizable by a microprocessor or micro-computer. The algorithm was implemented on an HP-85 micro-computer and the obtained results will be reported in a futurepublication.

IV. PERFORMANCE OF THE PROPOSEDESTIMATION METHOD

A. General DevelopmentThe variance of the proposed ML estimator has a lower bound

given by Cramer-Rao inequality [2]

var (7i)> (-E [2 In Pn(k/7l)]22 -1(8)

where E [.] is the operator of mathematical expectation. Dif-ferentiation of the log likelihood function (4) twice with re-

iP(a) d4d (a) 1 e 2da 2

Instead of var (71) it is more convenient to use another param-eter, "accuracy" defined as A(q)- 1/var (q). As a furtherconsideration the upper bound of the accuracy Au (?7) is investi-gated. From (1 1) it is equal to

a n N [ or(ha);hJ)]APM=AU(E)= (2 £ hi+ -r) 7

(12)

From this point the case of equally spaced thresholds is con-

sidered. Additionally some relative quantities are introduced:

a) relative mean

=

b) relative standard deviation

IA a=

Ah

c) relative thresholds

hA hi- i1 2h

(1 3a)

(1 3b)

(1 3c)

where Ah is the distance between thresholds. Particularly,from above it follows that ht = i - 1 (i = 1, 2, , N).

Fig. 2 shows plots of accuracy versus relative mean 71' forrelative standard deviation a' as a parameter. The number of

(10)

(1 1)

where

76

Pi(q) = 0hi+ I -,qor

WULICH: RANDOM PROCESS WITH ON-OFF DETECTORS

175

- 140

C-)

a: 1050C-O4

0JN

4

a:0z

Relative rmeon 7)

4caC-)4

0wNJ4a:0z

Relative stmdard deviation 0Y'

Fig. 2. Accuracy as a function of the relative mean for relative standard Fig. 3. Accuracy as a function of relative standard deviation a' for de-deviation as a parameter. viation A ' as a parameter.

detectors N = 11 and the number of observation n = 100. In-spection of Fig. 2 reveals that an increase of the standard de-viation a' decreases the accuracy if the relative mean is equalto values 0, 1, 2, ,10 (which are in fact the levels of thethresholds). For other values of the relative mean the accuracyincreases when the standard deviation a' increase (!), reachesa maximum and then the accuracy decreases with increasingof a'. Such behavior of the accuracy is given in Fig. 3. Thisfigure shows plots of accuracy versus relative standard devia-tion a' for deviation zr??' of the relative mean from the discretevalues 0, 1, 2, , 10.1 Inspection of Fig. 3 reveals that anoptimum value of the relative standard deviation which givesthe maximum value of accuracy for the worst case (A/7' = 0.5)is approximately 0.4. This fact provides for designing an opti-mal array.B. Comparison with a Commonly Used Method

Let us describe briefly another method for mean estimationwhich is commonly used. The process x(t) is sampled at theinstants t,, t2, *, tn. The values {x(tj)}7=q are exactly mea-sured. As it is shown in [2] the following estimation of themean:

(14)I n

flCM = - Z x(tj)n=Iis an efficient ML estimate for a Gaussian process if the samples

,t)n= l are mutually uncorrelated. The accuracy of thismethod is given by [2]

nACM = 2 -aT (15)

At this point it is possible to compare the proposed method(PM) (described in Section III) and the commonly used method(CM):

1) in CM it is necessary to measure the process level exactlywhereas the PM needs only the ON OFF detector outputs,

For the left bound value of n' = 0 the deviation is positive, whereasfor the right bound value of t = 10 the deviation is negative.

1-

z

cLJLU0C.)

RELATIVE MEAN ql

Fig. 4. Comparison of PM and CM. Coefficient y as a finction of rela-tive mean for relative standard deviation as a parameter.

2) in PM it is necessary to know a priori the standard devia-tion of the process; in CM it is not necessary,

3) let us compare the accuracy of the two consideredmethods: for this the following performance coefficientis introduced

A APM N

ay= =£ACM i=0(16)

The coefficient y indicates how the PM is inferior to the CM.Fig. 4 gives the coefficient y as the function of the relativemean 7q' for relative standard deviation a' as a parameter. In-spection of Fig. 4 reveals that the performance of the PM rela-tive to the CM depends on the value of the standard deviationa. From this figure it follows that for a' = 1.0, for example,the accuracy of the PM is almost identical to the accuracy ofthe CM - y

- 0.92. It should be emphasized at this point thatPM needs not be an accurate measure of the considered process.

77

?, - 77i + I

GF cr

0 i+ I 77 0 hi i?a a

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 2, JUNE 1984

Therefore, this method is justified when an accurate measure-ment is impossible or very expensive.

C. Design ofan Optimal ArrayThe existence of an optimal standard deviation ofthe process

(as it has been investigated in Section IV-A, aopt = 0.4) gives theopportunity to design an optimal array, i.e., to find the optimalnumber of detectors and to determine the placing of the thresh-olds, i.e., the optimal distance Ah0pt between thresholds.From (13b) the Ahopt is

Aho,t= a _ (17)Uopt 0.4

where a is the given a prior standard deviation of the process.The optimal number of the detectors Nopt is

?p?=max - 7lmin (18)Nopt Ahopt

where 7lmin and 7?max are a priori known bounds of the esti-mated mean. In this case h1 A 71min

V. SUMMARYThe maximum likelihood method of mean estimation of a

random process is presented. The estimation is accomplishedwith an array of ON-OFF detectors, i.e., the measurement ofthe exact instantaneous value of the process is not required.

It is shown that the method is applicable to a Gaussian pro-cess with a given standard deviation. The central result statesthat there exists an optimal value of standard deviation whichgives the maximum accuracy for the worst case, i.e., when themeasured mean value is exactly between detectors. Compari-son of the PM with a CM indicates that, for a wide range ofstandard deviation of the considered process, the accuracy ofboth methods is almost equal. Further efforts should be di-rected towards the relaxation of the assumptions that the pro-cess is Gaussian and its standard deviation is known.

REFERENCES

[11 A. Papoulis, Probability, Random Variables and Stochastic Pro-cesses. New York: McGraw-Hill, 1965.

[2] H. L. Van Trees, Detection, Estimation and Modulation Theory,Part I. New York: Wiley, 1968.

A Coaxial Cable Delay System for a Synthesis RadioTelescope

T. L. LANDECKER

Abstract-A switchable coaxial cable delay system for path compensa-tion in a synthesis radio telescope operates over the frequency band15-45 MHz. The requirements of continuum and spectral line observa-tions determine the system specifications. Diode switches include orexclude cables whose lengths form a binary sequence from 0.625 to320 m. Cables are equalized to compensate for the effects of frequencydependent attenuation and of dispersion. A simple technique for dis-persion equalization is described.

I. INTRODUCTIONTHE BASIC ELEMENT of an aperture synthesis radio tele-

scope is a pair of spaced antennas equipped with receivers,and a correlator which forms the time average of the productof the received signals. This is a radio interferometer, depictedin Fig. 1, whose output is the complex cross correlation or visi-

Manuscript received August 21, 1981; revised October 20, 1983.The author is with the Dominion Radio Astrophysical Observatory,

Herzberg Institute of Astrophysics, National Research Council ofCanada, Penticton, B.C. V2A 6K3, Canada.

bility function. Visibility is measured as a function of spacingand orientation of the interferometer, and is related by theFourier transform to the distribution of radio brightness onthe sky [1].An essential part of the system is a device to keep the two

paths to the correlator equal (Fig. 1). Unless this equality isaccurately maintained, correlation is lost, for the signals arebroad-band noise. The path difference changes continuouslyas the earth rotates, and the compensating delay must changewith it. References [2]-[51 contain descriptions of path-compensating delay systems built for radio telescopes.This paper describes a path-compensating delay system built

for the synthesis telescope at the Dominion Radio AstrophysicalObservatory [6]. The telescope consists of four 8.5-m parab-oloidal antennas, with a maximum spacing of 600 m. Two ofthe antennas can be moved on rails. Simultaneous observationsare made with four interferometer pairs at various spacings.Fig. 2 shows the antenna and correlator configuration, andshows how delay cable is introduced to keep all signal paths

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