on the narrow beam communication system acquisition problem

IEEE TRANSACTIONS ON MILITARY ELECTRONICS

From the preceding analysis of a specific system, itbecomes apparent that in order to achieve the maximumrange resolution combined with a maximum range capa-bility, the receiver time constant must be a variablefunction of the return signal. The simplest means ofachieving this would entail the use of a time constant inthe range 1-2 ,usec for target acquisition and an opti-mized time constant for subsequent range resolution.For instance, the system might use a short train ofpulses with receiver correlation to detect a target withina range interval corresponding to the 1-2-,usec timeconstant. Having identified both the range interval andthe magnitude of the return signal, the receiver couldthen adjust its time constant in order to obtain maxi-mum resolution with a subsequent train of pulses. Asreturn signal correlation plus range-interval gating

techniques could be applied to the second train ofpulses, the necessary value of s could be substantiallyreduced by comparison with that required for acquisi-tion. As can be seen from Fig. 3, such a technique wouldgreatly improve the performance of a ranging systemwhose pulse rise time is somewhat long. For the figuresused in the example, the maximum range of the systemwhen identifying a 1 square meter target is of the orderof 20 miles. However, this performance would requirer = 2 X 10-6 sec which corresponds to a range resolutionof 1000 feet. If the time constant is invariant, then therange resolution of such a system would have a 1000ft lower limit. However, by the proposed technique, theresolution of a similar target would increase progres-sively as range decreased until at a range of approxi-mately 11 miles the system became counter limited.

On the Narrow Beam Communication SystemAcquisition Problem

JOEL S. GREENBERG

Summary-It is desired to establish a communication link be-tween two separated transmit-receive terminals, each having narrow

beamwidths and specified uncertainty as to relative angular locations.To achieve the communication link the narrow beams must pointdirectly at each other. The purpose of this paper is to determine thesearch or acquisition time required by the two terminals to achievethe desired state of knowledge whereby the transmit-receive beamsof both terminals are pointed in the required directions, signals are

mutually recognized and the communication link is thereby estab-lished.

The acquisition time is a random variable and can thus be de-scribed in terms of an expected value and standard deviation. Markovchain concepts are employed to derive equations for the expectedacquisition time in terms of system parameters such as relativeangular uncertainty, beamwidth, probability of detection of a signalin noise, false alarm probability, etc. Single as well as multiplereceive beam systems are considered. Curves relating the expectedacquisition time to probability of detection and pointing probabilityare presented.

INTRODUCTION

IT IS DESIRED to establish a communication linkbetween two separated transmit-receive terminals,termed A and B, each having narrow beamwidths

and some (but limited) knowledge of relative angularlocations. In order to achieve the communication link

Manuscript received July 26, 1963.The author is with Advanced Military Systems, David Sarnoff

Research Center, Radio Corportion of America, Princeton, N.J.

the narrow beams must point directly at each other.The purpose of this paper is to determine the search oracquisition time required by the two terminals toachieve a state of knowledge whereby the transmit-receive beams of both A and B are pointed in the re-quired directions, signals are mutually recognized andthe communication link is thereby established.As will be shown, the establishment of the communi-

cation link is a function of the number of independentobservation or beam positions of the system as deter-mined by the relative angular position uncertainty ofterminals A and B and the probability of detecting a

signal in the presence of noise. Since the time requiredto establish the communication link depends first uponthe probability of both A and B "looking" in the properdirection at the required time, and second upon theprobability of successful detection of the received signalswhen pointing in the correct directions, the acquisitiontime is a random variable. The time required for acquisi-tion can thus be specified in terms of an expected valueand a standard deviation. This paper deals with thedetermination of the expected value of the acquisitiontime in terms of the relative volume of uncertainty, thebeamwidths employed during the acquisition procedureand the available signal-to-noise ratio.

In this initial paper a model of the problem definedby the set of assumptions listed below is considered.

28 January

Greenberg: Narrow Beam Communication Acquisition

Later papers will deal with increasingly difficult prob-lems, wherein the assumptions listed below are re-moved. Within this context the following assumptionsare made:

1) Both transmitters (A and B) radiate CW signalsduring the acquisition phase and both receivershave knowledge of the frequency to be received.'

2) During the acquisition time, change or relativeangular uncertainty due to any motion of theterminals A or B is negligible; i.e., the magnitudeof the solid angle to be searched is invariant duringthe acquisition time.

3) Propagation time delay between A and B isnegligible compared to the least time spent perbeam position during the acquisition phase.

4) Transmit and receive beamwidths of a communi-cation terminal both point in the same direction(unless otherwise specified).

For the model defined by the above assumptions theexpected acquisition time is determined for three dif-ferent cases and then compared. These cases are as fol-lows:

1) Random search with zero false alarm probabilitybut nonunity probability of detection: In this mode ofoperation the antennas of terminals A and B point inrandomly chosen directions for specified short intervalsof time and both terminals transmit and receive simul-taneously. The limits on the choice of directions aredetermined by the available a priori knowledge of thesolid angle to be searched. The random choice scanningprocedure continues until both terminals point at eachother and either both terminals detect each other'stransmissions or one of the terminals (say A) detects theother's (say B) transmission, but not vice versa (i.e., Bdoes not detect A, although pointed at A). When thislatter event occurs the A antenna stops its random scanprocedure and remains transmitting and receiving inthe direction from which the B signal was received. Thesecond antenna B continues scanning until it eventuallypoints again in the A direction and detects the trans-missions of the A terminal. When both terminals receiveand detect each other's CW signals the acquisitionphase is ended.

2) Random search with nonzero false alarm proba-bility and nonunity probability of detection: The scanmode is the same as in case 1) above, in that the an-tennas of terminals A and B point in randomly chosendirections for short intervals of time. When A decidesit has detected a signal' its scan sequence is halted andits antenna is fixed in the direction of the received signalfor an interval of x seconds. If by the end of x seconds nodetection is made by B in the direction of A, and there-fore no acknowledgment is received at A, then A re-

I This knowledge may be obtained by using multiple filters tocompensate for frequency uncertainties.

2 Due to the nonzero probability of false alarm for this case, this'detection" can also be a false alarm.

sumes its original random scan procedure. The same istrue if B performs the initial "detection." If A and Bboth detect signals and acknowledgment is received atboth A and B simultaneously or within x seconds, thecommunication link is established. The acknowledg-ment, as in case 1), may take the form of the receiptof a CW signal.

3) Programmed search with zero false alarm proba-bility: In this mode of operation terminal A "looks"in a given direction, transmits and receives, whileterminal B searches out the entire volume of uncer-tainty in a prescribed nonrandom fashion. If no detec-tion is accomplished the procedure is repeated with Amoving to the adjacent beam position and remainingthere while B again searches its entire volume of uncer-tainty, and so on. When either one of the terminalsaccomplishes a detection at time T,, i.e., determinesthe correct pointing direction, its search pattern isstopped while the search pattern of the other terminalis continued until it also accomplishes detection, whichmay be at time Ti, or later.

CASE 1: RANDOM SEARCH WITH ZERO FALSEALARM PROBABILITY

This search problem may be modeled as a "probabil-ity tree," as shown in Fig. 1, where the Si terms repre-sent various possible outcomes of the random process orstates of knowledge. For example, Si is the initial orstarting state of no knowledge and S4 is the final state ofcomplete knowledge. S2 and S3 are intermediate statesof partial knowledge, and the pij terms are the proba-bility of acquiring sufficient knowledge during a giventrial or observation so that the system moves from astate of knowledge Si to a new state of knowledge Sj.

si

S2 S3 S4 SI

S4 S2 S4 S3 S4 S2 S3 S4 S

S4S4S2 S4 S3S4 S2 S4 S3 S4 S2 S3 S4 SiFig. 1-Random search probability tree (for case 1).

The problem is to determine the number of observationsrequired for the system to move from some state Si toa state Sj. The probability tree model correctly repre-sents the search problem on hand in that the probabilityof entering state j from state j-1 is only a function ofthe transition probability pj-j.i Since a finite Markovchain is a stochastic process which moves through afinite number of states, and for which the probabilityof entering a certain state depends only on the last stateoccupied, Markov chain techniques will be used todetermine the expected number of trials or observationsrequired for the processes under consideration to movefrom the initial to the final states. A trial (sample, ob-

1964 29


servation, or step) is performed when a terminalsearches a beam position and makes a decision as to thepresence or absence of a received signal.

In the following paragraphs a number of mathemati-cal definitions and equations are stated without formalproof. For the interested reader the necessary formalmaterial can be found in Kemeny and Snell.3 Also inthe following paragraphs a special or short-hand versionof the probability transition matrix is considered; thatis, all of the ergodic sets (i.e., set of states which cannotbe left once entered) are grouped together or united,and all of the transient sets (i.e., set of states which canbe left) are united. It is assumed that there are s tran-sient states and r-s ergodic states. The transitionmatrix P may then be written as

r -s

(S

IR

s

O

Q

Q i

r - s

} s

(1)

The s Xs matrix Q describes the process as long as itstays in the transient states. The sX (r -s) matrix Rdescribes the transition from transient to ergodic states.The (r - s) X (r - s) matrix S deals with the process afterit has reached an ergodic set. The region 0 consistsentirely of zeroes.

In any finite Markov chain the process proceedstoward an ergodic state; that is, the probability thatafter n trials the process is in an ergodic state ap-proaches unity as n approaches infinity. This propertyis independent of the starting state.

Several conditions or states of knowledge during theacquisition phase may now be defined. These are as fol-lows:

1) S1 antennas of A and B are not pointing in thedesired directions or A and B are pointing in thedesired directions but signals are not detected. Inother words, Si is the state in which no knowledgeis gained.

2) S2-Antenna of A is pointing in the desired direc-tion having achieved a condition whereby A and Bwere both pointing in the desired direction but thesignal was detected only at A. That is, antenna atB was pointing in the desired direction at the cor-rect time but was unaware of this fact.

3) S3-same as S2 except A and B are interchanged.4) S4-antennas of A and B are both pointing in the

desired direction and both receivers have detectedthe signals. In other words, the communicationlink is established. This process is illustrated inFig. 2 where S, designates the above states and pijis the transition probability of going from state ito state j.

3J. G. Kemeny and J.. L. Snell, "Finite Markov Chains," D. VanNostrand Co., Inc., Princeton, N. J.; 1960.

P22 ) P33

P44

Fig. 2-Transition diagram for random search withzero false alarm probability (case 1).

P44 is assumed to be unity. S4 is thus an absorbingstate (i.e., it cannot be left once entered) and Si, S2 andS3 are transient states. It is desired to determine theexpected number of trials, or time intervals, spent in thetransient states and the standard deviation of the num-ber of trials. Several different approaches are possible toevaluate the above. The concepts of Markov chains, asstated previously, are used in the following analysis.

For the problem at hand the transition matrix P isgiven by

SI S2 S3 S4

Si rpii P12 P13 P14'S2 P° p22 0 P24

s3 0 0 P33 P34S4 ( O 0 0 1i

(2)

and may be put in the form (1) as follows:

S4 Sl S2 S3

S4r 1 0 0 0 r -s|~

S1 P14 Pll P12 P13S2 Pp24 0 p22 0 s

1S3 p34 O O p33

r-s

(3)

s

Define n, as the function giving the total number oftimes that the process is in Sj. The mean of the totalnumber of times the process is in a given transient statej when the process started in state i, is

{Miv[nij} -= iv = (I - Q)-1 (4)

where Q is the sXs matrix of (1) and I is a matrix hav-ing ones on the main diagonal and zeros elsewhere.Therefore

1 0 0 pll P12 P131- Q = 0 1 0 - 0 P22 0

0 0 1 0 0 p33

f1- Pi -P12 - P131= Io 1 p22 (

O O 1- p33,

30 January

(5)


and

Si

N = (I - Q)-1 = S2

S3

811

1 - Pii

0

0

S2 S3

P12 P13

(1 - pl)(l - P22) (1 - P11)(1 - P33)1

01 - P22

0

1 P33

Thus if the process starts in state Si it will be in thisstate an average of 1/(1 -pi) times. If the processstarts in state Si it will be expected to remain in stateS2 an average of

P12

(1 -p11)(1 - P22)

times, etc.The variance of the process is given by

Vari [nj] } (7)

where

N2=N(2NdO-I) -NsqNd, = matrix N with all values set to zero except the

main diagonal (nii) termsNsq = matrix N with each termed squared.

Let t be the function giving the number of steps ortrials (including the original position) in which theprocess is in a transient state when the process startedin state i. Then

{MILt]} = N= Nv (8)

and

{ Vari [t] } = T2 = (2N I)Tr -f (9)

where

=column vector having all components equal tounity

T=matrix T with each of the elements squared.Therefore

Since the process is to start in state 1) with a proba-bility of unity, the expected number of steps in thetransient process is given bIy

P24P34 + P12P34 + P13P24

(P12 + P13 + P14)(P24P34)(10)

where

P33 + P34 = 1

P22 + P24 = 1

pll+ P12+ P13+P14 = 1.

(10a)

As given by (10), ri is the expected number of searchsteps, or At4 observation intervals, in going from Si to S4and is, therefore, a measure of the expected acquisitiontime (r,At) of the entire system.

In the derivation of (10) it has been assumed thatonce information is gained it is never lost; i.e., P21 =P31==P32=P43=P41=0. This implies that states 2) and 3)cannot degenerate back to state 1) due to a loss ofinformation such as would occur when the two terminals(A and B) have unknown relative angular motions[assumption 2) ]. It is also implied that all of the prob-abilities in the transition matrix are independent oftime [assumptions 2) and 3) ].The transition matrix terms are now written in terms

of system parameters.3 Let

PA = probability of antenna A pointing in the desireddirection (toward B)

PB= probability of antenna B pointing in the desireddirection (toward A)

(P12 + P13 + P14)P24P34

1

P24

P34

Expected number of steps in transientstate if process started in state 1)Expected number of steps in transientstate if process started in state 2)Expected number of steps in transientstate if process started in state 3).

4 A t, the observation interval, is the length of time required tomake a decision as to the presence or absence of a signal.

5 The antenna gain is taken to be the same for both transmissionand reception.

I[nJ,] (6)

1964 31


Pa = probability of detection of a sine wave in noise,at location A, based upon signal and noise powerlevels and assumed to be zero when the mainbeams of the two systems are not pointing in theproper directions

Pb =probability of detection of a sine wave in noise,at location B, based upon signal and noise powerlevels and assumed to be zero when the mainbeams of the two systems are not pointing in theproper directions.

Since the propagation time delay is small compared tothe sample time (time per beam position), and since theinstantaneous pointing directions prior to detection atA and B are independent, (11) may be substituted in(10). In deriving the expressions for P24 and p34 in (11)it is assumed that after A has performed a detectionsufficient information (pointing direction) is obtainedwhereby its beamwidth may be narrowed down suffi-ciently so that the probability of detection Pb, of B whenpointing in the proper direction rises to unity andP24 = PB.6 The same situation arises if B performs thefirst detection (P34= PA). Therefore

P12 = PAPBPa(I -Pb) = probability that signal is re-ceived and detected at Aand not at B

P13 = PAPBPb(- Pa) = probability that signal is re-ceived and detected at Band not at A

P14= PAPBPaPb = probability that signal isreceived and detected at AandB (11)

P24 = PB = probability that B receivedand detected signal given,that A had previously de-tected signal

P34 = PA = probability that A receivedand detected signal given,that B had previously de-tected signal

pll = 1-(P12 +P13 +P14)P22 = -P24P33 = -P34.

Substitution of (11) into (10) yields

Ti =

1 + PAPa + PBpb - Papb(PA + PB)

PAPB(Pa + Pb - Papb)

and PA and PB are given by

/ A 2PA - l X

\OA/

PB -(! (13)

6 Fig. 4. illustrates the mean acquisition time for the above case

and also for the case where beamwidths are not narrowed down;i.e., P24 = PBPb and P34 = PAPa.

where OA2 and 6B2 are the effective solid-angle beam-widths and qA2 and 4B2 are the effective solid angles tobe searched.When the terminals are symmetric, that is, Pa=Pb

and PA=PB, (12) reduces to7

1 + 2PAPa(l - Pa)Ti =

PA2 (2pa - pa2)(14)

Also, when the terminals are symmetric, 6A =OB=mO andOA = q5B=-=. Thus (14) may be written as

0 \2

1 + 2pa (1 - Pa)Ti =

(i4

)(2pa -pa 2)(15)

Pa and Pb are determined by signal and noise considera-tions and are illustrated in Fig. 3 as a function of signal-to-noise ratio for various false alarm probabilities.'

It is interesting to note that

1) whenpa,-1, --(-- )

/69\4 12) when l-)I and pa is small, ri - 2p

(P ~~~~~~2pa499.99

99.999.8

99

9895

z2 90Uw 80aJ 70aU. 60O 50>- 40

30-JX 204

o 10. 5

2

0.5

0.20.I

0.05

2 4 6 8 10 12 14 16 18

S/N (db)

(Taken from: W. Hall, "Prediction of pulse radar peformance, " PROC.IRE, vol. 44; February, 1956.)

Fig. 3-Probability of detection vs S/N for various falsealarm probabilities (one pulse).

For the nonsymmetric case (12) must be used instead of (14).8 The allowable value of false alarm probability is a function of

the number of resolution elements which must be observed in orderto arrive at each decision. The number of resolution elements ob-served per decision includes multiple doppler filters necessary to takecare of uncertainties in radial velocity and multiple receive beamswhich may be used, as discussed in the following pages, to reduce theexpected acquisition time.

- FALSE IALARM 10

- PROBABILITY

0.01,

32 January

41

I I I I


These derived results agree with intuitively expectedones.The received signal-to-noise ratio, and hence the prob-

ability of detection pa, is related to the system param-eters as follows:

S PtGtGrAT 47rR I

KTOXJ

(16)

where

S/N= signal-to-noise ratioPt= transmitted power, wattsGt= transmitter antenna gainGr = receiver antenna gainR=slant range, metersX = wavelength, metersK = Boltzmann's constant, watts/°K/cps;T= effective receiver temperature, OK;j3=receiver bandwidth, cps.

Gt and Gr are functions of the beamwidth 0 and thus(16) can be rewritten as

S k

N\ 0413

where

15j X 106X2Ptk -!. (17)

R2KT

Since At is the time spent per beam position the expectedacquisition time is -rAt. Let the time spent per beamposition determine the receiver bandwidth so that= 1 /At. Therefore

S kAt_ (18)N\ 04

In order to illustrate the previously derived equa-

tions and obtain a "feel for the numbers" the followingexample is presented. It is desired to determine theexpected acquisition time required to establish a narrow

beam communication system when the frequency em-

ployed during the acquisition mode is the same as thatof the communication link, and the power radiated dur-ing the acquisition mode is limited to that required bythe communication link when established. The com-

munication link is to operate with carrier-to-noise ratioC/N, a bandwidth and a beamwidth 6,. In the prioracquisition mode the same power is radiated but thebandwidth is decreased to : and the beamwidth is 0,where 0>0c. Therefore

S k

N 043

C k

N OCC4fC

and

S C Oc4# C cAtA= N_4 =N (6/6c)4

so

(19)

where the expression inside the { } is the signal-to-noise ratio in the search mode.

Eqs. (15) and (19) may be used to determine acquisi-tion time as a function of search and communicationbeamwidths. As an example, let 6,=0.1', 4=50, C/N=100, 3,=106 cps and At=0.1 sec (3=10 cps). ThenPa =f{ 103/64}. In Fig. 4 ri is shown as a function of 0when Pa is obtained from Fig. 3 for a false alarm prob-ability of 10-4. It can be seen from Fig. 4 that for theexample given there exists a minimum expected acquisi-tion time with a search beamwidth of 0 3.5'. Fig. 5illustrates the expected acquisition time in terms of thepointing probability PA and probability of detection Pa.

It is of interest to determine the expected acquisitiontime when the two terminals, A and B, each use multiplereceive beams but single transmit beams. It is assumedthat sufficient receive beams exist such that the volumeof uncertainty is completely filled. With this conditionand assumption 3) that the propagation time delay issmall compared to the observation or sample time, thereresults

p12 = PBTPARPa(1 PA T) + PB TPARPapA TPBR(1- Pb)

p13 = PATPBRPb(I -PBT)+pATPBRPbPBTPAR(1 -pa)

P14 = PATPBTPARPBRpapba

P24 = PBRpb

P34 = PARPa i

(20)

where the second subscripts, T and R, refer to the trans-mit and receive conditions, respectively, and the terms

PA, PB, Pa, Pb, P12, etc., are as previously defined. Eq. (20)may be clarified by referring to Fig. 6. Since the condi-tion of sufficient multiple receive beams has been as-

sumed, PAR and PBR become equal to unity.As before, it is assumed that after A has detected B's

signal it narrows down its transmitting beam pointing atB so that the probability of detection Pb, essentiallyincreases to unity at B and vice versa (Pa = 1 after Bmakes initial detection). Therefore

P24 = 1

P34 = 1. (20a)

Pa = f(S/N) = f{ (t/oc)4}

1964 33


(D

0

4

0

4

w

Cl)

1.0

0.110 102 103

MEAN NUMBER OF OBSERVATIONS - rT

Fig. 4 Mean number of observations for establishmentof communication link (example in text).

rlw104

100

EXAMPLE IN TEXT 2

z 4

0

Io-10

0

IL

0

I-~~~~~~~~~~~~~~2to4

0

a.

10-2 lfio-'POINTING PROBABILITY (PA)

Fig. 5-Curves of constant number of samples (Ti) as a function ofprobability of detection and pointing probability [case 1-(14)].

p'2 PBT PAR PH

0S. THAT

2OINTS TO BRECEIVE --

(I'PAT) + PST PAR PO

PROB. THAT

A DOES NOT POINT

TO ON TRANSMIT

PROS. THATA DETECTS SIGNALWHEN RECEIVED

Fig. 6-Clarification of (2

I PAT PBR (IPb9)

PROS. THAT P0OB. THATA POINTS TO 8 B MAKES NOON TRANSMIT DETECTION

LTHOUGH SIGNALIS RECEIVED

PR08. THAT9 POINTS TO AON RECEIV E-=I

?0.

When the symmetric condition between A and B isagain imposed for simplicity and (20) and (20a) are

substituted into (10), the expected acquisition time isgiven by (21)

1 + 2PA Tpa(l - PaPA T)

paPA T(2 PaPA T)

(21)

In the limit as Pa 41 and if PAT is small, r-l>1/(2PAT).For the trivial case where both Pa and PAT approach

unity, rl-*1 as expected. As before, PAT = (061)2 and P.is determined from signal and noise considerations.The subscript T is maintained above to remind the

reader that (21) applies to the multiple receive beamcase. However, it should also be noted that PAT-PA.

In Table I the single and multiple beam (receivingsystem) cases are compared. When PA is small there areadvantages to employing a system composed of multiplebeams.

TABLE ICOMPARISON* OF SINGLE AND MULTIPLE BEAM CASES

System Expected Number of Samples (r,)

Pa,l, PA small PA-l, Pa small

Single receive beam &4/04 1/(2pa)Multiple receive beams 42/202 1/(2p.)

* Difference in false alarm rates due to differences in number ofobservable resoliition elements per unit time have been neglected.

CASE 2: RANDOM SEARCH WITH FINITE FALSEALARM PROBABILITY

The introduction of false alarms into the system doesnot basically alter the method of analysis; it just com-plicates the analysis. The mode of operation examinedhere is the sanme as that discussed previously, i.e., theantennas at terminals A and B point in randomlychosen directions for short intervals of time. When Adetects a signal9 its search sequence is halted and itsantenna is fixed in the direction of the received signal fora time interval of x seconds, the time necessary for nobservation or decision periods. Thus a decision is madeat the end of each of the n periods. If by the end of xseconds no detection is made by B in the direction of A(no acknowledgment received at A), A resumes itsrandom search procedure. The same is true if B per-forms the initial detection. If A and B both detect sig-nals (and acknowledgment received at A and B) simul-taneously or within the n observations, the communica-tion link is established.The complication due to false alarms arises from the

fact that when A detects a signal it does not knowwhether this was due to a false alarm (noise) or due toA and B pointing in the proper directions but B notdetecting the transmission of A.'0 If B was indeedpointing in the proper direction (at A) during the Jthobservation it will not necessarily be pointing in thesame direction during the J+1 observation, since Bchooses its pointing directions at random. Therefore theproblem arises as to how long A should remain pointingin the same direction (the "detection" direction fromwhich a signal supposedly was received) before a deci-

I The term "detects a signal" is used to describe the processwherein a decision is made that a signal is present because some thresh-old has been exceeded, although in fact no signal may be presentand the decision is due to a false alarm caused by noise.

10 The same statements are true for B detecting, except all theA's and B's are interchanged.

34 January

_P24 PBt_P34=PAI

z L %~~~~~P24" PBPb

I~~~~~P34=PA Po

_c 0. DEG.

# -5DEG.- CN=0-100

-e loto6cps.-At=O.ISEC

FALSE ALARM PROB. 10O4_ Ll 11111 1~ 111l1 1 1 IIIl1i 1111

10 -

I

PROB.THAT PROSIGNAL IS A PCRECEIVED AND ON PDETECTED ATA AND NOT III

PROB.THAT9 POINTS TO A.ON TRANSMIT


sion is made that the threshold crossing must have beendue to a false alarm and the random search sequenceresumed again.The random search (with finite false alarm probabil-

ity) transition diagram and transition matrix are shownin Figs. 7 and 8, respectively, for the condition thatnP12' and nP13' are small (P12' and P13' are the transitionprobabilities due to false alarms). As in case 1), severalconditions or states of knowledge may be defined duringthe acquisition phase. These are

1) SI-antennas of A and B are not pointing in thedesired directions or A and B are pointing in thedesired directions but signals are not detected.

2) S2-antenna of A is pointing in the desired direc-tion having achieved a condition whereby A andB were both pointing in the desired direction butthe signal was only detected at A.

3) S2a, S2b, , S2, antenna of A is pointing inthe same direction as in state 2). In other words,these are intermediate states where A is "markingtime" before making a decision. If at the end ofn observations at A, B has not detected thetransmission of A and no acknowledgement isreceived at A, the scan process of A is againresumed and state S2n reverts back to Si.

4) S3-applies to B in the same manner as S2 appliesto A.

5) S3a SMb, , S3,-applies to B in the same man-ner as S2a, S2b, * * * , S2n applies to A.

6) S2' terminal A has made a detection based uponfalse information due to noise (false alarm).

7) S2a', S2b', , S2fl'-antenna of A is now point-ing in incorrect direction as determined by stateS2'. In other words, these are intermediate stateswhere A is "marking time" before making a deci-sion, which in this case must eventually be thatthe detection was a false alarm.

8) S3'-terminal B obtains false information due tonoise.

9) S3a', S3b', , S3n' applies to B in the samemanner as S2a', S2b', * * S2J' applies to A.

10) S4-antennas of A and B are both pointing in thedesired direction and both receivers have de-tected the signals.

The problem now, as in case 1, is to find the r matrix,specifically the term Ti which is the expected number ofsteps that the system remains in the transient state ifthe process started in state 1). In order to accomplishthis the N matrix or inverse of the (I-Q) matrix mustbe determined. Since the (I-Q) matrix is an sXs

Fig. 7-Transition diagram for random search (finite false alarmprobability) on the condition that np52' and np13' are small.

Note: Empty spaces signify zeros.

Fig. 8 Transition matrix.

matrix with the value of s unknown, solving for(I_ Q) -1 becomes difficult. However the inverse matrixcan be determined relatively easily when n is known andis small, and from this ri for general n can be determinedas shown below. The equations for the expected numberof trials or steps in the transient state when the processstarts in state 1) is given in (22) and (23) when n, thenumber of "marking time" states, is 2 and 3, respec-tively. Eqs. (22) and (23) are obtained in the same man-ner as (10) except that the transition matrix of (3) is

11 Because the number of "marking time" states, n, is not knowna priori, s is not known.

1964 35


replaced by that given in Fig. 8. Therefore

rl (2) =

T1 (3) =

1 + P12(l + q24 + q242) + P13(l + q34 + q342) + 3pl2' + 3p13'P14 + P12(l - q243) + P13(l - q341)

1 + P12(l + q24 + q242 + q241) + P13(l + q34 + q342 + q343) + 4pl2' + 4pl3'P14 + P12(l - q244) + P13(1 - q344)

where q24= 1 -P24 and q34= 1 -p34.12 The probability oftransition from S2aS2b, or S2b-S2c, * *, or S2n-*SI isq24. Similarly, q34 iS the probability of transition fromS3a +S3b, or S3bS3c * * * or S3n )SlBy deduction, the general equation for rI(n) may be

obtained. It is

1+ P12( t q24i) + P13( E q34i) + (n + 1)P12' + (n + 1)p13'

P14 + P12[1 - q24('+1)] + P13[1 - q34('+')]

When n is large and q24 and q34 are less than unity an

approximate formula for rl(n) is obtained and is given by

1+P12 + P13 + p12' + npl3

1-q24 1-q34 ( ;

TI(n)P12(l - q24 ) + p13(1 - q34n) + P14

When the symmetric conditions, i.e., P12 = P13, q24 = q34

and PJ2/= P13, are imposed, then Tl(n) in (25) reduces to

1 + - + 2npl2'

T1rl(1 - q24) (26)

2P12(I q24n) + pi

As in case 1)

P12 =PA2pa( Pa)P241= -q24= PA

P14 = PA2pa2P12'=false alarm probability.

Therefore, the expected number of trials may be writ-ten as

1 + 2PAPa(l - Pa) + 2np12 2

PA Pa[2 - pa - 2(t1 Pa)(1(p)I (7 )

and it can be seen that r1(ff) depends upon the choice ofn. There is a value of n for given values of PA, Pa and

P12' which minimizes 7T(n). When the parameters of theexample given in case 1 are used the minimum value ofT1(n) and the value of n which minimizes Trl() are shown

in Table II.

12 The second subscripts in the parentheseis for r refer to the

value of n.

When the values given in Table II are plotted (rl(n)vs 0), the points fall extremely close to the curve givenin Fig. 4 for the case of zero probability of a false alarm.Therefore it may be concluded that as long as the falsealarm probability is small little or no error is introducedin determination of the expected acquisition time, if theapproximation is made that the false alarm probabilityis zero and (15) is used instead of (27) (this agrees withintuition). For sizeable false alarm probability (27)must be used, and optimum n to minimize -r1(n) foundby some convergent trial and error process.

TABLE II

EXPECTED NUMBER OF TRIALS

(Degrees)

0.51.02.03.04.05.0

PA

10-21/254/259/2516/251.0

Pa

111o.80.120.01

TI(n)

10,00062539912.351

n

small*small*small*

10small*small*

Finite false alarm probability, P12'= 10-4* From (27), which is an approximation accurate for large n, n

is determined. Thus when the minimum value of Tl(,), as obtainedfrom (27), occurs for n on the order of say, less than 5, nothingfurther can be said about n except that it is small [unless of course

equations such as (22) or (23) are used].

CASE 3: PROGRAMMED SEARCH WITH ZERO

FALSE ALARM PROBABILITY

The mode of operation is such that terminal A looks

in a given direction, transmits and receives until ter-

minal B searches out its entire volume of uncertainty. If

no detection is performed the procedure is continued

with A moving to and remaining at an adjacent beam

position each time B searches its entire volume of un-

(22)

(23)

(24)

36 January

'r(19) :


certainty. When one of the terminals performs a

detection, i.e., determines the correct pointing direction,its search pattern is stopped and the search pattern ofthe other terminal continued until it also performs a

detection. It is assumed that when A detects B, theprobability of B (the next time B looks at A) detectingA is unity; the same is true if B does the initial detection(A and B reversed). This is due to the beam-narrowingaction taken by A after detecting B (and vice versa)previously discussed.The subcase where A and B each have only two pos-

sible beam positions (PA =PB= 2) is considered initially.This is then expanded upon to yield an equation for theexpected acquisition time for any number of beampositions (O <PA = PB < 1). Since two beam positions are

to be considered initially, terminal A will sample each ofits beam positions twice before moving on to an adjacentbeam position, and terminal B will sample each of itsbeam positions once before it moves on to an adjacentbeam position. Therefore at any instant of time theterminals may be in any one of the eight conditionsillustrated in Fig. 9 where the Ai, and Bij represent thebeam positions of terminals A and B, respectively; i

indicates the beam position and j the sample number(i.e., All represents terminal A beam position one on thefirst sample in that beam position, A12 represents ter-minal A beam position one on the second sample, etc).The conditions shown in Fig. 9 are also the possibleinitial search conditions and are all equally likely (theprobability of starting the search procedure in any

given initial condition is w). As will be seen, the timerequired to establish the communication link is a func-tion of the initial or starting condition. Therefore allpossible initial conditions or starting states must beconsidered. The transition diagrams for each of theseinitial conditions or starting states is shown in Fig. 10.As in the previous cases, several conditions or states ofknowledge may be defined during the acquisition phase.These are

S1, S2, S3, S4-as previously defined for case 1.S1,-intermediate states where antennas of A and B

are not pointing towards each other, and neitherterminal has detected the other's presence.

S2T-intermediate states resulting from A performinga detection when A and B were pointing at eachother and B not performing a detection. Thusthe S2i states result from B not pointing in thecorrect direction due to the programmed natureof B's search pattern.

S3X-intermediate states resulting from B performinga detection when A and B were pointing at eachother, and A not performing a detection. Thusthe S3i states result from A not pointing in thecorrect direction due to the programmed natureof A's search pattern.

,0TION5

AtD-FAA _ j1 ELFl El EL FT-] FI-]ETIWAa - EDf ~fl- F - FT-.] FT3 .1- FT-] FT-.

(b) (C) () (e) () (g) 'h)

Fig. 9-Possible relative beam positions of terminals A and B.

St~~~~~~~~~~~~~~~~.

SIb SiID tDsib

_aJc~~~ ~~~41 < - 3b t lC , ,

NITIAL CONDITION ) INITIAL CONDITION. INITrAL CoNwrtON

,bjs

$ I

4 f,~~~~~~S Ic

Sib

I i,

INITIAL CONDITION (D INITAL CONDITION 6

Fig. 10-Transition diagram for programmed search (case 3).

Fig. 11-Search sequence (programmed search) initial condition e.

PA= PB= 2-

1964 37


The above definitions may be clarified by referringto Fig. 11 which illustrates the programmed search se-quence when the search sequence starts from initialcondition e (Fig. 9). S12 is therefore the initial or startingstate (when considering initial condition e).

It should be noted (Fig. 10) that once state Si isentered there exist only two different transition dia-grams, initial conditions a and b. The expected numberof trials required to reach S4 when starting from initialcondition d is thus 3 plus that required for initial condi-tion a. The expected number of trials to arrive in stateS4 is the summation of the expected number of trialswhen starting in each initial condition multiplied bythe probability of starting in that initial condition. Thus

T1 =

Tla

'Tlb

1 + rlb

3 + Tia

2 + rla

2 + Tlb

3 + Trlb

1 + Tla

PaPbPcPd

Pe

Pf

-Ph-

(PA = PB = 2) (28)

investigated given (for the symmetric case) by

11 = (2p Pa2) [2m' + (m2 - 4m - 1)pa2

+ (-2m2 + 4m + 2)pa] (34)where

1

PAA comparison of the expected acquisition time of the

random and programmed cases is shown below for sev-eral different limiting conditions. The subscripts R andP refer to the random and programmed cases, respec-tively.

lim -) 1m- large TIPPa-Small

(TIR\lim -) 2

m-*large Trp/

lim (-2i *1m-4Small TIPPa-SIMall

where rla and rlb are the expected number of trials whenstarting in initial conditions a and b, respectively, andpi is the probability of starting in initial condition i(for the case under consideration, p,=). Solution of(8) when using the transition matrix of initial condi-tions a and b yields

Tla = 1 [1 + 3pl,ia + 2pi2 + P13] (29)1 Pl,la

rlb = 1 [1 + 3pla+ 2pl2 + 3pl3]. (30)1 - l11

Substitution of the transition probabilities

P12 = Pa(l - Pb)p13 = Pb(' - Pa)

Pl,la = (I -Pa)(l - pb) (31)

and (29) and (30) in (28) (with pi = 1) results in

1 F8 + Pa + Pbb5PaPbl2 [L1 (I pa) (l1 Pb)i (PA =pB=)* (32)

As before, for the symmetric case (Pa Pb),1 r8 + 2pa - 5p4u

T = -

2 - 2pa - Pa(PA = PB = 2 )

Pa-1 TIP

2m2

m2+ 1(35)

Thus, again, it may be generalized that the expectedacquisition time of the particular programmed scanconsidered may be approximated, with a fair degree ofaccuracy (within a factor of 2), by the equations derivedfor the expected acquisition time for the random searchprocedure (case 1). This is illustrated in Fig. 12 wherethe ratio of the expected acquisition time for the randomand programmed cases (TIrR/TIp) is shown in terms ofprobability of detection (pa) and the number of beampositions to be investigated (m).

1.0, .,

0.8

0.6

0.4

0.2 F

(33)

Determination of Ti for three and four beam positions(i.e., PA is ' and 1, respectively) leads, by deduction,to a general equation for the expected acquisition timein terms of the number of beam positions (m) to be

0.5 1.0 1.5 2.0

rlRrIP

Fig. 12-Ratio of random-to-programmed acquisition time as afunction of probability of detection and number of beam posi-tions.

/ yII

1

'

m5 *m=5o- I yfmzl

j tm-100

I1

l l ll/-- IX I I I

38 January

0m

0

Urkowitz: Maximum Likelihood Angle Estimates

CONCLUSIONS

The concepts of Markov chains have been applied tothe narrow-beam communication system acquisitionproblem. Application of Markov techniques allows theexpected value (8) and standard deviation (9) of theacquisition time to be determined.The expected acquisition time for a communication

link consisting of terminals having limited knowledge ofrelative angular locations has been determined. It hasbeen shown that the expected acquisition time is de-pendent upon the probability of detecting a signal innoise (pa) and the probability of looking in the correctdirection (PA). It has been shown that the expectedacquisition time is essentially independent of the systemfalse alarm probability provided that the false alarmprobability is not large."3 Curves of the expected num-ber of samples (ri) as a function of probability of detec-tion and pointing probability are shown in Fig. 5, inwhich the example given in the text is plotted. It isnoted (for the example considered) that a very sharpminimum exists in the expected number of samples re-quired to establish the communication link. It shouldalso be noted (Fig. 5) that the pointing probability must

13 This is uisually the case since small changes made in power andthreshold levels yield large changes, or reductions, in false alarmprobability (see Fig. 3).

be quite large if short acquisition times are to beachieved. This implies searching with as broad a beamas possible (making the search beamwidth approach theangular uncertainty). The effect upon pa of searchingwith a broad beam cannot be neglected since pa is afunction of search beamwidth [see (19)].

It has been shown that the expected acquisition timeof the random search and the programmed (raster scan)search are approximately equal (1<TIR/TrP< 2). Thusindications are that the particular form of scanningprocedure chosen will probably depend strongly uponthe standard deviation of the acquisition time andhardware considerations, in addition to the expectedacquisition time. Mote optimized forms of search which,for example, make use of the probability density func-tion of the uncertainty volume and/or use sequentialdetection techniques, etc., deserve further considerationfor reducing acquisition time. The concepts of Markovchains as outlined in this paper may be used to greateradvantage when investigating these more complexmodels of the acquisition procedure.

ACKNOWLEDGMENT

The author wishes to express his appreciation to Dr.H. Mills and Dr. M. Handelsman of Advanced MilitarySystems, RCA, for their many helpful suggestions.

The Accuracy of Maximum Likelihood Angle Estimatesin Radar and Sonar

HARRY URKOWITZ, SENIOR MEMBER, IEEE

Summary-By extending the results of Kelly, Reed, and Root,'formulas are derived for the variances of maximum likelihood esti-mates of azimuth, and azimuth and elevation, jointly, by dense,discrete and discrete-continuous apertures for the strong signalcase. The accuracy of angle measurements depends upon

1) total signal energy captured by the aperture,2) the mean-square aperture size,3) carrier frequency,4) the mean-square signal bandwidth.

Mean-square quantities are the second moments about the centroids.The actual signal form and aperture form do not matter, except asthey affect the mean-square quantities.

When joint estimates of azimuth and elevation are made, theerrors are generally coupled. Minimum variances are obtained whenthe errors are uncoupled. This condition is obtained, in the narrow-band case, when the two-dimensional illumination function is factor-able into the product of one-dimensional functions.

Manuscript received October 16, 1963.The author is with the Philco Scientific Laboratory, Blue Bell, Pa.' E. J. Kelly, I. S. Reed and W. L. Root, "The detection of radar

echoes in noise, II," J. Soc. Ind. and Appl. Math., vol. 8, pp. 481-510; September, 1960.

The formulas for the dense and discrete apertures are identical inform, the various factors being discrete or continuous analogs of oneanother in which integrations are replaced by summations. Theformulas for the discrete-continuous array differs in form by thepresence of terms which reflect the anisotropy of the beam patterns.

INTRODUCTION

N GAUSSIAN NOISE, the signal ambiguity func-tion plays an important part in obtaining a maxi-mum likelihood estimate of signal parameters. In

fact, in white noise, the estimate is obtained by con-structing the ambiguity function as a function of theparameters to be estimated and the values which maxi-mize the function are taken as the desired estimate.Woodward2 has shown how this is done for delay andDoppler and has derived formulas for the variances of

2 P. M. Woodward, "Probability and Information Theory withApplications to Radar," Pergamon Press, New York, N. Y., ch. 6;1953.

1964 39

on the narrow beam communication system acquisition problem

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