Grup Reprt 1964-2 The Angular Reslutin f Multiple Target J. R. Sklar F. C. Schweppe January 1964 Prepared. tract AF 19 (628)-500 by Lincln Labratry MASSACHUSETTS INSTITUTE OF TECHNOLOGY Lexingtn, M
The wrk reprted in this dcument was perfrmed at Lincln Labratry, a center fr research perated by Massachusetts Institute f Technlgy, with the supprt f the U.S. Air Frce under Cntract AF 19(628)-500.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 80 LINCOLN LABORATORY THE ANGULAR RESOLUTION OF MULTIPLE TARGETS /. R. SKLAR Grup 44 F. C. SCHWEPPE Grup 28 GROUP REPORT 1964-2 14 JANUARY 1964 LEXINGTON MASSACHUSETTS
TABLE OF CONTENTS Page Abstract iii Intrductin 1 Preliminaries 1 Single Target 6 Tw Targets 7 Detectin 10 Discussin 15 Acknwledgement 19 References 20 ii
THE ANGULAR RESOLUTION OF MULTIPLE TARGETS by J. R. Sklar F. C. Schweppe ABSTRACT The angular accuracy btainable frm an aperture f fixed size is cnsidered, with emphasis n the multiple target case. By means f the Cramer-Ra Inequality, a lwer bund t the measurement variance is cmputed. In additin, the degradatin due t uncertainties in the number f targets present is cnsidered. Lss f accuracy is small until the targets apprach t within ne beamwidth, at which pint the degradatin becmes severe. This technical dcumentary reprt is apprved fr distributin. franklin C. Hudsn) Deputy Chief Air Frce Lincln Labratry Office
1. Intrduc tin Reslutin f multiple targets has always been f sme cncern t the radar designer, but the recent interest in discriminatin has given a new imprtance t be prblem. In a multiple target envirnment, energy frm several reflectrs impinges n the receiving antenna and as a result, the data prcessing invlves bth the extractin f salient features f the received signal and the assignment f these features t their respective targets. Range and range rate reslutin can be effected by the design f the radar signal in the time dimensin and angular reslutin can be achieved thrugh design in the special dimensins (antenna design). Hwever, there is at least an rder f magnitude difference between the angular and range reslutin capabilities f tday's mre sphisticated radars. The pssibility that this discrepancy is at least partly due t a failure t utilize all available angular infrmatin in the return signal is indicated by the success f the mnpulse lbe cmparisn technique fr beam splitting when nly a single target is present, and its abrupt failure when there are mre than ne in the same range reslutin interval. Thus, the questin f the existence f a middle grund in which a mderate degree f beam splitting in presence f multiple targets is pssible arises. This reprt investigates the amunt f angular infrmatin inherent in a received signal by the use f the Cramer-Ra r Infrmatin Inequality. Bunds n the minimum btainable angular accuracies are derived fr bth the ne- and tw-target case. The detectin prblem f deciding the number f targets present is als discussed and limited results given. 2. Preliminaries The fllwing tw-dimensinal radar mdel is used. The three-dimensinal case is a direct generalizatin. Several targets, say Q, are reflecting energy cntinuusly tward a linear aperture f fixed length, X. The incident plane -1-
wave is sampled at an arbitrarily large number f pints n the aperture and nise is added t the samples. Since the transmissin medium is linear, the cntributins t the sample value frm each target can be summed, and therefre the samples are a 2 cmpnent vectr cnsisting f the quadrature cmpnents Q z,(x) = ) A. sin (q> + <D(x,a.)) + w (x) j=l 0 < x ^X Z 2 Q (x) =^ Aj cs (cpj + 0>(x,a.)) + w 2 (x) (1) j-l where A. and cp. are the magnitude and phase f the radiatin frm the j target, a. is its angular psitin, x is the lcatin f the sample pint J within the aperture, and $ is a functin f x and CC. which measures the phase J difference between sample pints due t the gemetry. (See Fig. 1.) If X is the wavelength f incident plane wave, then *(x,a ) = ^ cs a w n (x) and w p (x) are Gaussian randm variables f zer mean representing the nise. Since w (x) and w p (x) are rthgnal cmpnents f the rf nise, they are independent. In additin I [ x^y E [w.(x)w i (y)] =) 2 1-1,2 L CT x = y The amplitudes, A., the phases cp., and the angles a. are cnsidered t be unknwn parameters; i.e., n a priri distributin fr them is assumed knwn. Thus, there are 3Q unknwn parameters and the prblem is f the basic frm Z-L(X) = f^x^,...^^) + w x (x) z 2 (x) = fg(x,e 1,..., 3Q ) + w 2 (x) (2)
3-28-4804 TARGET j Fig. 1. Target Gemetry.
where the 's represent the unknwn parameters, and the f.'s are knwn J functins f x and the e's. We will define e.(j = 1>«««>3Q) as the estimate J f the e. btained frm the bservatins z,(x), z p (x), 0 < x < X. We cnsider the case where the z.(x) are bserved at N pints f the aperture, equally spaced at intervals 6 = X/N. Cnsider p(z(l),...,z(n) e 1,...e 3Q ) where ji(k) is a tw-dimensinal vectr representing the k sample, and p is the prbability f the bserved sample cnditinal n the true values f the unknwn parameters. If ne desires t btain unbiased estimates f a set f unknwn quantities e., based n bserving a set f randm variables z(k), ne J can place a lwer bund n the variances f the estimatrs. by means f the J Cramer-Ra r Infrmatin Inequality. Tw f many derivatins f this fundamental result are fund in Ref. 2, Chap. 32 and Ref. 3, Chap. 12 (the terms Carmer-Ra r Infrmatin Inequality are nt emplyed; see instead the discussins n efficiency). In the case f multiple parameter estimatin, this inequality can best be stated in matrix ntatin. Let I.. = -E a 2 3^-^ lg p (z(l),...z(n)) e 1,... 3Q J be elements f a matrix [i], and let 2 ' -j - [ E <-% - «i> <*, - «.>1 be the elements f a matrix \ a 2 a ]-H is psitive definite, which implies that 4 ^ [A - 1 L J **. Then the Cramer-Ra Inequality states that kk (3) -k-
This result is derived frm the requirement that the estimates be unbiased and a generalized Schwartz inequality and is subject t the cnditin (usuallypresent in practice) that [i] be psitive definite. Thus the desired lwer bund can be btained by calculating the necessary derivatives and inverting the resulting matrix. In the sequel it will be useful t emply the fllwing alternate interpretatin f [I]". Cnsider Eq. (2). Expand the f.(x,,,...,e_ Q ) in a Taylr series abut sme set f 6's,,,..., _. Then f Of (X,,..., - ) -i ^L A j + w^x) + «±,1-1,2 (k) J-l J where the partial derivative is evaluated at the e 's, $. is the remainder.. and Az i (x) = z^x) - t ± (x,e^,.,.,e ). Ae. = e. - e. j J j If the c.'s are clse enugh t the true values t allw the remainder, ($, t be neglected. Eq. (U) can be cnsidered as a linear regressin prblem with unknwn cefficients, Ae.. A minimum variance, unbiased estimate (r a maximum likelihd estimate) fr this prblem has a cvariance matrix equal t [I] [References 2 and 3 give the linear regressin equatins which, when applied t Eq. (!< ), give [I] fr the Gaussian case.] Therefre, [I] can be cn- sidered as either a bunding cvariance matrix r the result f a linearized r small errr, errr analysis. Thus, fr signal-t-nise ratis which are large enugh t imply small errrs, the inequality f Eq. (3) becmes equality.
3. Single Target The first case t be cnsidered is a single target at unknwn angle a with unknwn amplitude A and phase <!>. The target is knwn t exist. When the nise is gaussian, with variance CT and zer mean, the bservatin f N samples equally X spaced at 6 alng the aperture leads t a cnditinal prbability fr the samples f p(z(6),z(28),... a,a,0) ( 2TT ) O I ±ml N 2TTiB z (ib) -A sin(0 + ±2- cs a) -L A N V z 2 (i6) -A cs( + 2TTiB cs i-i / y frm Eq. (l). After taking the prper derivatives and averages we btain, fr N large, N A 2 /2nxf sin 2 a CT N A /2rtc\, _ 2 S{ j Sina m = N "2" c N A 2 T ( sin a m in which the indices 1, 2 and 3 crrespnd t a, A and $ respectively. Frm this we calculate -6-
12 4 ( ) 2 sin a NA /2T1X\ [II N NA. fgrgj^) sin a NA Cnsequently, the variance in the angle measurement is lwer bunded by V 12 R P NA /2T«Y 4 2 ~ "~?VT7 sina (5) The fractinal beamwidth errr is therefre bunded by '»Wi where R = p- is the signal-t-nise rati f the same aperture if all array element utputs are added and BW = \/X. is the beamwidth fr the aperture f2\ 2 measured t the 1! relative pwer level. We see that this is cmparable t \TT. that btained by mnpulse lbe cmparisn techniques in Ref. 1. k. Tw Targets The same prcedure can be fllwed in the tw-target case when exactly tw targets are knwn t exist. The cnditinal prbability expressin is
P(z(6),z(26),... a 1,a 2,A 1^2,» 1,» 2 ). (^)"? exp A -A 2 sin(<j> 2 + =2(i6) cs a 2 ) zaib) -A, sin(0. v + =^16) cs a.) r 1 1 X 1 N "I 2 CT i-1 z 2 ( 16) -k 1 cs($ 1 + ^(i) cs ^) -A 2 sin($ 2 + ~(i&) cs ^jl Taking derivatives and cmputing averages the result is i «[D] [rj [D] (6) where [r] is a 6 x 6 matrix given in Fig. 2 and [D] is the 6x6 diagnal matrix,?^l A. sin a, X 1 1?f A 2 sin a 2 *V 2
"21 in 1/2 a 6l a 21 1/3 -%1 l 6l 1/2 [n = 0 a Ul -%! 0 1 a Ul l U3 l 63 ^63 1/2 a 6l a 63 %3 a 6l 1/2 ^63 l *3 a. 21 5- cs(ax + b) + r-^y sin(ax + b) ^ sin(ax + b) + (ax) (ax)" 2 «sin b (ax)" a i,i B U1 rztt cs(ax + b) *- sin(ax + b) + 1 * sin b W (ax) 2 (ax) 2 a Ai 3 7T5T sin (ax + b) + 5- cs(ax + b) = cs b 61 {ax) (ax) 2 (ax) 2 a U3 * f5xt sin(ax + b) "(ixt sinb a 63 = r^xt cs(ax + b) TixT csb 2n.. a = (cs a cs OL) b - cp 1 - cp 2 Figure 2 28-4805 [n Indices 1, 2, 3, ^> 5, and 6 crrespnd t parameters a, a p, A.., Ag, <p x, and q> 2> -9-
This factrizatin f the fll matrix is attractive since it places in evidence the dependence f the elements f [I]" n the varius parameters. (Recall the inverse f a diagnal matrix is a diagnal matrix with the elements inverted and als [Il~ = [D]~ [r]~ [D]~.) Thus we may nte that the element f [I] crrespnding t the variance f amplitude is independent f the amplitude. This fact will be imprtant in the detectin discussin f Sectin 5«Cmputing the inverse f the [I] matrix is difficult, "but when we are nly cncerned with the angular accuracy, nly part f the inverted matrix is required. This can be btained with a smewhat less tedius calculatin by partitining the [r] matrix. Althugh it is then pssible t btain the variance f the angular measurement in clsed frm, the expressin is t lengthy t be very useful. Therefre, the angular variance is pltted as a functin f the target separatin in Fig. 3 with the phase difference b = 0^ <t»_ as a parameter. Frm this nrmalized curve results fr varius values f a signal-t-nise rati and angular psitin can be btained. In additin, the rati f the variance in the tw-target case t the variance in the ne-target case is als pltted. One can bserve that the results are essentially identical fr target separatins greater than ne beamwidth (BW = ), but that fr smaller separatins the measurement errr in the twtarget case becmes large very rapidly as the separatin decreases. The phase difference b causes a wide variatin in the frm f the curves. Hwever, fr all values f this parameter, the variance is radically larger if the separatin is less than abut ne beamwidth. Cnsequently, any attempt at beam splitting fr separatins much smaller than ne beamwidth will prve futile. 5. Detectin The preceding tw sectins have cnsidered estimating accuracy fr multiple targets under the assumptin that the exact number f targets present is knwn. -10-
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Detectin is the prblem f deciding the number f bjects present. Certain general aspects f multiple target detectin are briefly reviewed, and a measure prvided n the relative detectin capability in the ne- and twtarget case. A standard apprach t the detectin prblem emplys the thery f hypthesis testing.* In the case where at mst ne target may exist, it is hypthesized that n target exists (IL. is the hypthesis). The alternate hypthesis is that a target des exist (H. is the alternate hypthesis). If A is the amplitude f the signal, these hyptheses are equivalent t V H,: A = A^O If up t tw targets are pssible, it is then necessary t check the pssibilities f zer, ne and tw targets. Such a multi-level decisin can be built ut f tw-level tests such as used fr the ne-target prblem. Fr example, a first test might hypthesize that n targets are present vs the alternate hypthesis that at least ne target is present. That is H Q : A x = 0, A 2 = 0 H,: At least ne A / 0 j = 1,2 (7) where the A. are the amplitudes f the signals. If the hypthesis that n targets are present is rejected, ne culd then hypthesize the existence f exactly ne target vs the alternate that exactly tw targets exist. That is HQ: A X ± 0, A 2 = 0 E ± : A ± + 0, A 2 I 0 (8) *The fllwing discussin is a very special applicatin f the general statistical thery. References 2 and 3 are tw f many which cntain far mre extensive discussins. -12-
There are, f curse, many ther pssible sequences, and the abve is nly an example. The basic prblem is thus the testing f sme hypthesis IL. vs sme alternate H,. The test is based n sme test statistics,, where is sme functin f the bserved data. There are many pssible test statistics, tw f which have particularly nice prperties. The first f these is based directly n the estimated amplitudes (A.) f the signals. Fr example, in J the ne-target case, the hypthesis that n target exists (H-) is rejected if the magnitude f the estimated amplitude exceeds sme chsen value. In the multiple-target case, similar tests are emplyed n the vectr f the estimated amplitudes. The secnd test statistic is the likelihd rati; i.e., the rati f the maximum likelihd attainable under H, t the maximum likelihd attainable under H-. (See, fr example, Ref. 3>) The amplitude statistic is mre pwerful, but the likelihd rati test is independent f the signal-t-nise rati. We shall cnfine discussins t the amplitude statistic. T be explicit, assume the signal-t-nise rati is high enugh s that the variance given by the Cramer-Ra Inequality can actually be realized r, equivalently, that we are dealing with thesystem f linear equatins given by Eq. (!< ).* The vectr estimate [A,,A? ] is then a tw-dimensinal Gaussian randm variable with cvariance matrix given by the crrespnding elements f the matrices f [i]". Let [l]7. dente this 2x2 sub-matrix f [l]~. If it is desired t test the hypthesis f Eq. (7), the test statistic is, in matrix ntatin, e- r -1 A A A- I _ 1 2J AA A A l A A 2 _ 1 *This apprach is similar t the investigatin f the asympttic behavir f test statistics as emplyed in the statistical literature. -13-
2 The equatin, = cnstant, defines an ellipse f cnstant prbability in A A 2 2 A., A? space. Thus the hypthesis test, which rejects K- when i > n, means A A the hypthesis is rejected when IA.,, Allies utside the cnstant prbability 2 /\ ellipse specified by 0. Let cfes dente the variance f A_ as evaluated, i using the crrespnding diagnal term f [l]. A «Fr the case f Eq. (8), the test statistic is,/v 2 2 " V «fr / which is the ne-dimensinal versin f the preceding case. The questin f prime interest in this paper is hw well detectin can be perfrmed. Unfrtunately, a cmplete analysis requires the chice f explicit sequences f tests, and this chice depends n the particular prblem f interest. In additin, the calculatin f the resulting prbabilities f the varius types f errr is smewhat labrius. Thus we shall merely indicate the relative degradatin in detectin capability which results frm the presence f a secnd target. Cnsider the case f Eq. (8), where the existence f ne target is definitely knwn (A, / 0), and the pssible exis- A p tence f a secnd is t be tested. The test statistic,, is (Ap/ ). Thus A 2 A 2 where y\ is a Gaussian, zer-mean randm variable f unity variance. Nw, cnsider the prblem f deciding whether there are ne r n targets. Exactly the same test is used, except that A is replaced by the a* btained frm the crrespnding element f [i] as evaluated in Sec. 3, Eq. (5). Thus, fr a fixed false detectin prbability, the prbability f detecting a secnd target f amplitude A p in the presence f anther knwn t be present, equals the prbability f detecting a single target f amplitude A when -llli-
\ a The rati, qft /a/s, therefre measures the degradatin f detectin perfrmance caused by the presence f a secnd target. As with the case f the angular accuracies f Sec. k, the clsed frm 2 2 slutins are t cmplex t be very useful. Thus q^ /W is pltted in Fig. h as a functin f target separatin with the phase difference b as a parameter. 6. Discussin Figures 3 and k cnstitute the majr results f this study and shw that, as target separatins increase beynd ne beamwidth, the infrmatin lss caused by multiple targets rapidly disappears. Since these curves resulted frm a limited analysis n a simplified mdel, we must discuss their limita- tins befre drawing actual cnclusins. We cnsidered nly errrs due t additive nise. Thus the antenna pattern is assumed t be knwn exactly. This assumptin results in the angular accuracy f ne target being independent f the crss sectins (amplitude) f ther targets. With an exactly knwn antenna pattern, this is nt unreasnable, since the effect f ther targets can be "subtracted ut." In practice, hwever, uncertainties in antenna pattern can cause a large crss sectin target t sizeably degrade estimates frm a nearby smaller target. In additin, an Inequality was used in place f an exact errr analysis. We can hpe t apprach the resulting bund nly with a sufficiently high signal-t-nise rati. Thus Figs. 3 and h indicate actual capabilities nly fr S/N ratis high enugh t apprach the bund but lw enugh t keep antenna uncertainties frm predminating. The range f such S/N ratis depends n the particular prblem under investigatin. Hwever, it is nt unreasnable t expect sme -15-
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systems fr which it is nnempty. Fr example, the same basic assumptins hld in bth the ne-target and tw-target cases and the single target bund is ften apprached in practice. We applied the basic thery t a very simple mdel: a ne-dimensinal aperture with tw statinary targets and white nise. The tw-dimensinal aperture and mre than tw targets merely intrduces mre parameters int the prblem. This is als true fr mving targets as the unknwn phase $ J f the return signal frm the j target becmes a parameterized functin f time such as t accunt fr range and range rate and the angular psitin, a, becmes a J similar parameterized functin f time t accunt fr the angular mtin. A mving target als requires specificatin f the carrier mdulatin. The intrductin f these parameters int the mdel is straightfrward and wuld nt change the basic methd f analysis. Hwever, it wuld greatly enhance the cmputatinal difficulties assciated with explicit results. The simple mdel is valuable as it indicates the target separatins fr which the angular reslutin prblem can be ignred and either single target analysis used r range and range rate multiple target reslutin studied by itself. Such a dichtmy prvides a simplified but very useful picture f the ver-all multiple target prblem. The incrpratin f n rder Markvian bservatin nise changes the equatin fr infrmatin t a nnlinear differential equatin, which can be numerically integrated n a digital cmputer. Hwever, ur basic prblem is nt well enugh defined t justify a nnwhite nise mdel. The mst imprtant limitatin in the scpe f the study is the fact that we have nt cnsidered an actual data prcessing system. If cnditins are such that the bund can be apprached, the classical maximum likelihd -17-
estimatin technique prvides the desired estimates. If we are nt wrried abut cmputatin time and cst, such estimates can be btained using a large digital cmputer. Hwever, fr mst situatins, such a prcedure is grssly impractical, as real time estimates are required and these imply primarily analg data prcessing. Unfrtunately, a straightfrward analg implementatin f the maximum likelihd technique fr several targets requires a large amung f hardware, and the varius avenues which might lead t a practical analg implementatin have nt yet been explred. Infrmatin is f n value if it cannt be extracted. Let us nw summarize what Figs. 3 and k actually imply. They prvide a bund n pssible system perfrmance. Mre imprtant, there is gd reasn t believe that sme systems d r will exist fr which this perfrmance can be apprached. Thus, fr mre than a beamwidth separatin, multiple-target capability clse t that f single target mnpulse lbe cmpressin is a distinct pssibility, prvided a practical data prcessing prcedure can be develped and implemented. A prime purpse f this reprt is t mtivate investigatins n such data prcessing techniques. JB-
ACKNOWLEDGEMENT This study wes its existence t Dr. Eugene W. Pike. His cnvictin that effective "beam splitting" shuld be pssible in the presence f multiple targets led t the fllwing investigatins, -19-
REFERENCES 1. L. E. Brennan, "Angular Accuracy f a Phased Array Radar," IRE Transactins n Antennas and Prpagatin, AP-9 (May 1961), 268. 2. H. Cramer, Mathematical Methds f Statistics, Princetn University Press (19l+6Ti 3. S. Wilks, Mathematical Statistics, Jhn Wiley and Sns (I962). -20-