C) TliE RESÜLVABILITY OF PODTT SOURCES'

Peter Swer'J iru; Sn>^lneerinß Division The RAOT Corporation

P-l '2

Deceaber ö, 19">9

COPV

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To be presented -i* the o/uiposiu:;: or: Decibio;. T-.eorj at Ko-,e Air Development Cer:ter, May 1)0.

r h r P A N D C ü r p o ' a • i o n • 5 n n I a M a " i t a • C o I ■ f o f n i ti

Fho views e.'picssed in this paper are not npressrinly ttioso of the Corporation

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P-l ■ 2 li

SUMMARY

The probler, invcGtigated is the resolvability, in the presence of noise,

of point so'drces vhich are Gepturated In aziruuth by less than the width of

the main lobe of the gain pattern.

The probability of correctly resolving two sources is derived for a

particular class of decision methods as a function of the strengths of the

sources, their angular separation and noise level. Numerical results are

presented for a case which corresponds :;iore accurately to optical or infra-

red aevices than to redar.

1

I. INTRODUCTION

The question of resolvability of point targeto, or point sources, arises

in radar, infrared, and optical applications.

First let us distinguish the problem of resolution from that of accuracy.

The accuracy probleir. is concerned with the errors in determining the position

(l 2) of an isolated source. It is, for example, well known ' that the error

in determining angular position of a radar target iray be much less than the

width of the main lobe of the antenna beam.

The word "resolution" covers several related concepts. In photography

or radar ground mapping, resolution has to do with the quality in some sense

of the picture, or with the ability to pick out certain features of the

picture. On the other hand, the type of resolution with which we shall

here be concerned is the following: suppose the received signal is known

to be due to a relatively small number of point sources; resolution consists

in deciding exactly how many sources are actually there. We will also limit

the discussion to angular resolution; resolution in range or range rate will

not be considered=

A widely used rule of thumb is that point sources are resolvable If

they are separated by (roughly) at least the width of the main lobe of the

gain pattern. (By gain pattern is meant, in radar applications, the antenna

gain pattern; in infrared or optical applications, the intensity pattern of

the image of a point source. ) While this is a useful rule of thumb, it

falls to answer many important questions. This can be illustrated by the

fact that, in the absence of all noise, and if the gain pattern were per-

fectly known, it would in many cases be possible to make error-free decisions

as to the number of sources regardless of their relative positions or strengths,

P-lc.2 2

A re solvability criterion should therefore involve the probabilities

of making correct decisions as to the number of sources present, and the

manner in which these probabilities depend on the noise level, on the strengths

and positions of the sources, and on the gain pattern.

Ideally, the criterion should reflect the intrinsic dependence of re-

solvability on these factors, where by "intrinsic" is meant the (in some

sense) irreducible dependence, apart from the particular decision method

employed. However, due to difficulties in arriving at a satisfactory treat-

ment of "intrinsic" resolvability, the approach to be used here Is based on

analysis of the performance of a particular class of decision methods.

It is clear that the resolution problem as defined above is a problem

of statistical hypothesir testing. However, the problem of finding an

optimum test cannot be answered by the Neyman-Pearson likelihood ratio test,

since, if the problem is not to be simplified out of existence, the positions

and strengths of the sources must be regarded aa not known a priori. Thus,

the problem is one of testing composite hypotheses against composite alter-

natives, leading to all the well-known difficulties of defining optimum

tests in-situations of this type.

*. ~ -L . u

:5

II. A CLASS OF RESOLUTION DECISION METHODS

It will be a.-.sumed. that we are dealing with a detection device scanning

in azimuth. The received cignal is assumed to be due to at most two point

sources. It will be assumed that the output of the detection device is

sampled at a series of angular positions x., i » 1, •••, K, resulting in the

presentation to a decision-making device of a series of magnitudes v., 1=1,

•••, K. (If we eure dealing with a pulsed radar, this sampling occurs auto-

matically; otherwise, such a sampling may always be defined to be part of

the decision procedures which are to be considered.)

It will be supposed that, when two sources are present, one can repre-

sent v by

v, i c^ FU. -■ \) + ti2 F(x. " "^ ^ V (1)

Here QL and Q are equivalent to the strength of sources located at angular

positions T and x ; F(x ) represents the gain pattern; and n is the noise

component of v.. The single-source case may be obtained by putting eithor

O, " 0, a = 0 or i, a t . (Not all physical situations can be represented

by Eq. 1; more will be said about this below. ) a

The following assumptions will be made regarding F and n.:

(a) F is a known function. Thus, attention is focusaed on tre limitations

on resolvability due to noise; that due to imperfect knowledge of the gain

pattern is ignored.

(b ) The gain pattern F is assumed to be a non-negative, even function.

(c) The range of observations v., i = 1. ••■, K, the gain pattern F, the

possible source positions T and the density of observations are such that,

to goci approximation.

P-lb£2

K K S F(xj - T) - y~^ F(x,) - 3, all T (c)

1 = 1 1*1

K y" F(x, - T] F(xi - T2) - P(T1 - T2) = p(-l-2 - T1) (5)

(d) The noise components n^ have knovn means n. (hpnce, the means are inde-

pendent of the unknown parametere CL , a0, T , T0). Thus, it is no loss of

generality to aesume that

n = 0; all i (U)

It will not, however, be asEumed that the higher order n-ionente of n, are

necessarily independent of a , a , -r^ , T0. 1. ti L c

(e) The noises n. Bind n, are statistically independent for i ^ J.

These asauaptionü, incidentally^ simplify many of the mathematical ex-

presBions but are not really eßsential to the basic approach.

Now, define

K

v - - y v (5) 1 B 4-T- i KJ,

i=l

K

>w ^r Vl'

Observe that, if n. = 0, all i, then

0(T - T )"] V/ - V,, « 2a.ap 1 i ^- (7)

^ 2 L p(0) j

p-ie62 5

A reasonable decision nethod would therefore be to announce two sources

or one source according to whether V is greater or less than a pre-assigned

threshold. Observe that, if n » 0, all i, then V is zero if, and only if,

either O. ■ 0, a ■ 0, or T « T .

It iums out, however, that this decision niethod must be modified in

most cases of interest, for the following reason: if n ^0, then (denoting

expected values by bars )

K

1 1 2 V - V » 2a a vl 2 12

1 ± i_

p(0) J i 2

B p (0)J i-1 n. (8)

2'

' 2 In some interesting cases, the quantities n. depend on OL and a0.

/vlso, the higher moments of V as defined by Ej.. (?) niay depend on OL and a

even if the statistics of n, do not.

The class of decision methods to be analyzed can be defined as follows:

announcement of two sources or one source according to whether V is cheater

to or less than a pre-assigned threshold, where V is some function of V and

V0. The specific form of V will depend on the manner in which the moments

of n. depend on OL , a , T,, and T„, in the particular case under considerati

It is clear that all necessary information about the performance of

this class of decision procedures can be deduced from the Joint probability

density of V1 and V^.

on,

P-1562

III. THE JOIM1 PROBABILITY DENSITY OF V AKD V

It will now be assumed that the nui.-.ber K of observations is sufficiently

great so that both V. and V are norrÄl randorr. variables (within the range

of probabilities which we wish to compute). Thus, the Joint density of

V and V will be completely specified if one can compute V , V , V , V ,

and V V , or the equivalent. Computation of chese quantities ia a straight-

forward process. The results are,

V ■» a + Q 1 1 2 (9)

-lO

LV1J

K

B . i«l / n. (10)

2 2a

1Q2 P^i " T2^

K 1 \ ' —;

V\ - a/" + a.;" + —^—— — + — I , n^ 1

)(0) p(0) i=l (11)

and, vritir.g for convenience

^ - o^ ^(x^^ - T1) + a^ F(xi - Tg), (12)

K

(0) i»l ^ : ' i i 2/^« ', ■■-' i i

D£'(0) i-1 (13)

1

r (o) i-i n.

viv2

v'iv2

J

BTTO]

K \ • p

2 / n. S. c ' i i i-1

K

i»l j (IM

This, of course, does not imply that V is norraal.

P-lK'S

7

IV. RESULTS FOR A PARTICULAR CASE

In order to proceed further it is necessary to make definite assumptions

ais to the moments of n , which depend on the particular detection device

under consideration.

As an illustrative case, it vill be assumed that

ni

2 ni

3 ni

T ni

(15)

This (with suitable nonaalization ) would be the case if the n. were Gaussian

with means and standard deviations independent of a. , Q0. T., and T_. This

might describe some infrared or optical detection devices; it is not so easy

to +link of radar cases which would be adequately described in this manner.

The first and second order moments of V and V for this case can be

obtained by inserting (l^) into (9) - (l^)j the results need not be written

out in detail. It turns out that

2,^ K[B - p(0)] , 1 2 B p(0) ^ c

Pt^ - T2) 1 RöT" (16)

An appropriate resolution test is defined by V > threshold, where

V . i (V 2 v KjB^Mll . (17) Vl I 1 2 B2 p(0) J

Note that in the absence of noise, one would have

p(T1 - T ) 1 - ~ ~

P(O)

I-X::i2 8

(16)

Even with all the special assumptions which have thus far been made,

it would still be possible to generate numerical results ad inTinltura.

Therefore, numerical results will be shown only for the following special

case:

Let 2

r(x) - e"X (19)

1 1+1 1 20

(20)

Then, approximately.

B - 20 xHT (21)

p(T1 - T2) - 20 j~|~ exp L | (^ - T2)2j (22^

The range of observation will be chosen from x - -2 to x « +2. With

this choice of observation range, K ^ 50. The choice of observation range

ought really to be the solution of an optimization problem. However, the

choice is here somewhat restricted by the requirement that the sums in Eq's.

(2) and (3) should satisfy Eq's. (2) and (o) to good approximation, for all

values of T and x under consideration. We will be interested in values

of 1^ - T up to about 1.5, and the range x ■ -2 to x • +2 was chosen as,

roughly, the smallest range for which (2) and (i) would hold for all values

of T^ - T up to 1.5.

The results obtained foi this case will clearly also apply to any case

of the form:

P{x) = exi (' 1~' ) ' coservation ran^e x n -2;- to x » +2D; x, -i " ;,: = ~r'

and all i /aluec multiplied by l:.

All ;;arar„eter vulje,. will be chosen :;c t-hat, the prcbability o:' fall'are

tu detect any source at all !s i.efjl ;,■;. i'io.

Figureo L-10 bhow probability d-str; but Ion -/unctions for V (as defined

by (I?)) for a variety of 7alueü of u , u,,, and T - T.,. In these figurea,

Pm - ProV- (V T) (2.0

One ;;..;.;ht dej.re LL proben' this ;;.:'..rrr.ation in alternative :\:;:.;. An

illustration of thiü .c r.-'-e-'. in Fns. 11-1.'.

Define

P [twc I Ot . u , T - T,,] (21. )

» probability of correctly anncmci:^

two sources, .: t).e;-e are in realit;«

two sources ci'.arHCterized ny tne

ara-T.eters .1. . ;_, 1, , T . i c i 2

Also, let

P[t.wo 1-^, 0] (25)

= probability cf falsely anncuncing two

sources, if in reeuity only one soui'ce is

prebent character''zed by the parameter ct .

Frei:. Fig. 1, we ■ ee trat ■ r. tlie 1 resent ca^c, P[ two I '^ ; C] is nearly independent

of ■« n . provided t:.',.s probabil^t', :s relativelv j::.nll. L

Figures li, IT, ar.d !.• show P[two I /. , ü,., i, - T0] subject to the con- 1 C 1 it

iition P[two 1 30, G; = .10.

10

It ran also he verified .'n the particolar raue ander consideration that

•esolution perTorrr-ance, when rresented in the i'or::. of curveB like those of

"if'u, -.1-1', rer:.a;n5 roiu-'hiy ti.e aai.-.e if x . - x., Q and rj.„ a-e all

::ulti]lLed iij the jo.,.e 1'actor (other ijararneterü renyiining unaltered).

11

V. ADDITIONAL TOPTCS FOR INVE3TIGAT:ON

Come additional topics which woula Le i-sel'.il tc investigate are >-^

f'ollovs:

i. Investigation of cuces 1;J w;., •;. f.-.e '.or.' .A.^ cf n. (oti.er than the

l'irüt) depend D.I x , Ci,,, T , and T0. TIILü wo'ild include some radar caües;

for exa;-iple :

(a) Noise-like taryetü (or noise 'aj.jr.ln^) detected b^ a radar.

In this caae, the output of a square lav envelope detector, properly norral

ized, would have probability density function

exp 1 + 07 F(x. - Tp + a F(x^ - 7,jj ,(v ) . h i 1 i : : 1- (:()

1 + a. F(x. - •:. )>.'... r(x. - TJ 1. A L tic

(b ) Non-fluci.uating targets, the relative phase of one target

return with respect to the other varying randorily i'rot;. pulse to pulse.

One would expect that in these cases the resolvabillty of targets

separated by less than a bearawidth would be seriously degraded aa compared

with the case considered in the previous section.

2. Dropping the requirement that F b§ an even function and always

positive. This might be of interest in analyzing a scanning monopulse radar,

Here it might be necest-ary to use functions of the data ether than V and V ,

Dropping the requirement that n.n » 0 for i / .i.

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REFKKENCtS

?-ldO 25

Swe:-l r.. ,

, "Maxi.:.'. - A;.(.;,J;u- Ace ,-'";,,• •'•>••, no. ', Gerjt.e;:.^sr i '■ ' .

a FulG'.'d Searci". Radar," Proc

Pernsr.cn, R., "A:i Anaiysiu or An-A a: Acc.;rp.cy i;: liearch Ruir.r/' l/^:,- L.R.E. Cor.vent^on Record, K'.rt ;, :• . ■'- ' .