THE UNIVERSITY OF MICHIGAN 5172-5-Q TARGET SIGNATURE STUDY Fifth Quarterly Report 1 June through 31 August 1963 Contract Nr. DA 36-039 SC-90733 Target Signature Research Department of Army Project Nr. 3A99-23-001 September 1963 OBJECT Conduct a Study and Investigation to Determine an Optimum Method of Identifying Military Targets by Radar Prepared by R. F. Goodrich, Z. A. Akcasu, B. A. Harrlson and O. G. Ruehr

THE UNIVERSITY OF MICHIGAN 5172-5-Q Qualified Requestors may obtain copies of this report from the Defense Documentation Center, Cameron Station, Alexandria, Virginla, 22314. DDC RELEASE TO OTS NOT AUTHORIZED. ii

THE UNIVERSITY OF MICHIGAN 5172-5-Q TABLE OF CONTENTS Section Page i PURPOSE 1 2 ABSTRACT 1 3 VISITS AND CONFERENCES 1 4 TECHNICAL WORK-1 June through 31 August 1963 3 4. 1 Introduction 3 4.2 A Mixed Filter 3 4. 3 Machine Simulation Using the Mixed Filter 15 4. 4 Representation of Target Signatures 15 5 PROGRAM FOR NEXT INTERVAL 17 Distribution List 18 ill

THE UNIVERSITY OF MICHIGAN 5172-5-Q 1. PURPOSE (This Section Is UNCLASSIFIED) The purpose of this program is to conduct an investigation to determine an optimum method of identifying military targets by radar means. 2. ABSTRACT (This Section is UNCLASSIFIED) A new filter is devised which depends on a parameter so that the filter varies continually from the inverse to the matched filter. This'mixed' filter then permits a compromise between the requirements of resolution and of discrimination. The mixed filter is now being used in the simulation program. A uniform method of describing targets in terms of their frequency signatures is proposed. By means of this scheme we hope to be able to arrive at quantitative discrimination criteria for different targets. 3. VISITS AND CONFERENCES (This Section is UNCLASSIFIED) 3. 1 Visit of Mr. Charles Eason to the Radiation Laboratory. The mixed filter and target signature description schemes were discussed with Mr. Eason on 29 August 1963. Present were R. E. Hiatt, B. A. Harrison, O. G. Ruehr, Z. A. Akcasu and R. F. Goodrich. 3.2 Visit of Radiation Laboratory Personnel to Fort Monmouth, N. J. On 7 June 1963, R. E. Hiatt, G.Rabson and R. F. Goodrich made an informal presentation of the work performed and planned under the contract at Fort Monmouth. Personnel contacted Included Leonard Hatkin, Charles Eason, William Flshbein and Ralph Dunn. The topics emphasized in the presentation and later discussions were the use of the inverse filter and the problem of 1

THE UNIVERSITY OF MICHIGAN 5172-5-Q finding frequency signatures for the various targets. A point emphasized by G. Rabson was that the inverse filter Is an excellent method of data reduction and should be considered such in conjectlon with the extremely short pulse research being done by General Dynamics. An important problem presented by the Radiation Laboratory group was that of finding a uniform method of target signature characterization so that a quantitative criterion of target discrimination could be found. (For a tentative resolution of this problem, see Sec.4.4). Mr. William FLshbein of the Signal Corps pointed out that the use of the Inverse filter Imposed rather extreme requirements on the signal-to-noise ratio. This observation led to our consideration of the matched filter and then the mixed filter of Sec.4. 2. In conjunction with the conference with the Signal Corps personnel, R. E. Hiatt and R. F. Goodrich attended the Ninth Annual Radar Sympos ium on June 4 - 6, 1963 at Fort Monmouth, New Jersey. 2

THE UNIVERSITY OF MICHIGAN 5172-5-Q 4. TECHNICAL WORK - 1 June through 31 August 1963 (This Section is UNCLASSIFIED). 4. 1 Introduction In this report we describe In Sec.4. 2 a more versatile filter which we are now analyzing and using in our machine simulation program (Sec. 4.3). The filter is designeu to enalie us to aetermine mne oest compromise between the conflicting requirements of resolution and discrimination. In Sec. 4.4 we describe a proposed scheme for a uniform way of describing the various targets. 4. 2 A Mixed Filter The two problems of the target signature study, target discrimination and target resolution, are in a sense antithetical. Since, for best resolution an inverse filter is optimum; for best discrimination a matched filter is optimum, we propose a'mixed' filter. We parameterize a filter so that it varies continuously from the inverse to the matched filter as a function of the parameter. Although there are arbitrarily many ways of affecting the parameterization we have chosen one because it gives a simple representation of the filter and also the specific form of the parameterization is not Important for our purposes. 3

THE UNIVERSITY OF MICHIGAN 5172-5-Q We start with the return r(t) in the time domain: r(t) = aa sa(t-t)+n(t) (1) where the index a Indicates the type of the targets, and J3 distinguishes between the targets of the same type. Furthermore, ta3 is the time required for the signal to return from the 3-th target of type a, and aa, are constants depending on relative distances. The function s,(t) characterizes the return signal from a target of type a, and n(t) Is the noise (white, gausslan). Finite Fourier transform of r(t) gives: +T T-tal R(w)= r(t)e lWt dt=- aae ta sae(u)e-tWu du+ (w), -T (T-ta3) (2)

THE UNIVERSITY OF MICHIGAN 5172-5-Q where T -T Assuming that the Integration time 2T Is large compared to ta3, and defining T -[wu Sa(w)) \ s,(u) ei du (4) -T one obtains R(w) as follows: R(w)= 2 aa e 3S)(w)+E(W) (5) a,,B Note that R(-w)= R*(w), Sa(-e)= S~(w) and e(-W)=e(o). If we process R(w) with the'mixed' filter using the filter function sa (() xm(w)-= (6) we obtain a function D. (t) as follows: 0 I. Da g(t)R(w) Xm(w)et dw (7) 0D Aam where ACXm Is the normalization factor defined by 5

THE UNIVERSITY OF MICHIGAN 5172-5-Q Aa m| So )1 dw (8) where SI is a constant frequency determining the range of integration. The explicit form of Da (t) is obtained by substituting R(w) from (5) into (7): +S w S(t-tap) +1 () Ss (w) e a(t)=,_1~ ac3j. 1.. + A m5. n(u) du 0 S j (9) letting m - / mo: ( ow(t-tap) ao(t) A- E acyp Sa(w) S (w)e (+T iw(t-u) + A J n(u) du S (W)e dw (10) where Al imn mAn 2 dw (1) a0,m

THE UNIVERSITY OF MICHIGAN 5172-5-Q The expected value of Da (t) is the first term on the right-hand side of (9), because the mean value n(u) of the noise is zero. Let Da%(t) denote the expected (or mean) value of Da (t). One can express Da (t) as follows: S+f (Z) 12 e ) D (t)=Z a 0Ro dw 0 a30 L2+I Sa(o)I 7-' 1' g s (W) S,(w) e #+az,3 a+I I (12) Before beginning to discuss the various aspects of this expression, it seems to be in order to consider the variance of Da (t). Noise.-to-Signal Ratio: We define the noise-to-signal ratio Pa as 0 Pa- - (t). (13) Using (9), one obtains 2 2 2T(A~2 s IS (( 0)1 Po [m2 ) dow (14) In obtaining (14), we have used the fact that the noise is white, viz., n(u)n(u') = a2 5(u-u'), 71

THE UNIVERSITY OF MICHIGAN 5172-5-Q and the approximation +T +T -[i(c +~0')t e dt'72 2r (to+') -T It is noted that (p%/c), where C is the effective value of the noise, depends on the type of the target as well as on m. The variation of (Poo/o) with m is of Interest. Qualitatively, this variation is expected to be as shown in Fig. 1. (polc/i m m FIG. 1: 8

THE UNIVERSITY OF MICHIGAN 5172-5-Q For m= 0, one has 2 ~n2 (Pao(O) =2 R2 i SaJ(w) (15) which corresponds to the'inverse' filter. When m -' co, the'matched' filter limit is obtained: L 0'' P(C)2e~()2 d (16) The fact that the above value of the noise-to-signal ratio represents the smallest possible value is proven in the theory of matched filters. Pulse Width: We now turn to the first term in (12), and investigate the quantity iW(tt cda 3(t) - 1 d w Aam m2+ ISa (W)l 0 -SQ 0 which Is a typical term in the summation on (3. It represents a peaked function at t=taoo with a height 1. The half-width W(m) of this peak Is defined by I Sa (4) iwW(m)/2 Aa om 0 S(ed= _dw 0' pm2+1S, (U~ 2 2e(

THE UNIVERSITY OF MICHIGAN 5172-5-Q The dependence of W(m) on m is also of interest. Fig. 2 depicts the predicted qualitative variation of W(m) with m. W(m) Wmax w. L/m m FIG. 2: The minimum value of the width is obtained from (17) as the root of sin (Wminl/2) = (SnWmin/4) (18) which is approximately given by Wmint * (19) This Is the minimum width attainable by any filter. The asymptotic value of W(m) as m -c oo follows from (17) as the solution of;+ Wmax S-Saw)I do 5 = ISa (w)2 e do, (20) 10

THE UNIVERSITY OF MICHIGAN 5172-5-Q which can be approximated by ry 1 Sa (w)t2 dwo Wmax (21) s" S |S (W)| dw 0 0 Counting the Number of Identical Targets: When there is only one type of target, (12) reduces to D(t)2 a Sf3? ]() dw (22) where we have dropped the subscript ao which is now unnecessary. if one Is interested primarily in the number of targets, and if the ranges determined by to do not have to be determined with the maximum possible accuracy, one may choose m - co, viz.,'matched' filter, since it provides the minimum noise-to-signal ratio. The possibility of overlapping peaks can be handled by forming the product of the pulse height and the pulse width. When the two pulses representing two different targets coincide completely, the height of the resultant peak is 2, and the width is just Wmax given by (20). Therefore the product height x width'~~ ~~11=~~~~~~~~~~ w (23) max 11

THE UNIVERSITY OF MICHIGAN 5172-5-Q Is also 2. On the other hand, if the two peaks are adjacent and the tip of the resultant curve is obscured by the noise, one would interpret the result as one peak of height 1, but with a width 2Wmax. Thus, again the product ry would be 2. It follows that the number of targets corresponding to an observ ed peak can be predicted from the values of r. If. 5' V1 c 1. 5 there Is one target, and if 1. 5r<v2. 5 there are two targets, etc. This procedure can be applied, of course, to the finite values of m, also. In that case, the q will be normalized to the width W(m) which Is obtained by (17) for the chosen value of m. Whether one can increase the accuracy in the determination of t3, and thus of the range, by choosing a smaller m, will depend on the relative shape of the curves giving the m-dependence of the pulse width and the noise-to-signal ratio. Therefore, it is desirable first to plot these curves, for a few target types, by computing (14) and (17) numerically. Distinguishing Between Two Different es of Tarets: Suppose that there are two different targets at the same distance. Then, (12) gives -- Sal(w) Sc2 () Dal(t)=+ 1 - 2 dw, (24) 1iAa1,m m +1 Sa ()2 ] where we denoted the targets by cral and ca2. If one processes the same return 12

THE UNIVERSITY OF MICHIGAN 5172-5-Q by the mixed filter corresponding to the target a2, one obtains +( S2(W) Sa1(W) Da2(t)= 1 + S dw. (25) QAam[m2+Sa(w)I 2] On the other hand, if the return Is processed by the filter correspondting to a third filter a3, the outcome will be DaESa2(w)+Sal(w Sa,3(W) dw (26) D3 (t) 2: Is ] (26) In the case of a matched filter, (24) takes the following form: (+n Da(t)i SSS(c) Sa (W) dw )1 + ~Fr~S (w)l dw Similar expressions are obtained for (25) and (26) when m -' o. The foregoing formulas indicate that the different targets can be identlfied by ustng a matched filter if the following inequality is satisfied for all target pairs: (W) S(W) dw 4 ISa (u)lI dW (27) 13

THE UNIVERSITY OF MICHIGAN 5172-5-Q The procedure to be followed Is to process the return with the matched filters corresponding to the various targets, one at a time. The targets which are present in the return will be identified by a peak of a height greater than or equal to 1 If the left-hand side of (27) is positive. Those that are not present in the return will not give rise to a peak of any appreciable magnitude if (27) holds. Since the effective value of the noise at the output of the matched filter is given by (16) as L 2 2 I Sai()Ml do one finds that the false peaks corresponding to the missing targets will be buried in the noise if +f _) d 1/2 S a.(~) S.(0) d -w r d|i 1 (28) -n -A Our conclusion from the foregoing discussions is that the matched filter will be the best choice insofar as, a) counting the number of targets of the same type, b) Identifying of different types of targets, are concerned. However, a computer study is needed for more definite conclusions. 14

THE UNIVERSITY OF MICHIGAN 5172-5-Q 4. 3 Machine Simulation Using the Mixed Filter We have completed a number of machine simulations using the mixed filter for various values of the parameter. The target and noise simulations are the same as some of the previous ones In which we used the inverse filter. We intend to make a detailed comparison of results for different values of the parameter, noise level, and targets. This comparison has not been completed by the end of this reporting period but will appear In the next report. 4.4 Representation of Target Signatures A basic consideration in the target signature problem is the efficient organization and utilization of known target data. Presumably this information will be stored in some kind of analog or digital memory. Since processing time, signal quality, and memory size will be limited It is desirable to optimize the memory with regard to the class of targets expected and the signal processing. Taking the prime function of the system to be the identification and counting of targets chosen from a fairly small class of similar targets, it becomes apparent that relative information Is important. That is, a system of memory storage and signal processing which seeks out and magnifies the differences in target characteristics is likely to be the most efficient. One mathematical technique which holds considerable promise in this direction is the use of Wiener's translation expansion.+ N. Wiener, "Tauberian Theorems," Annals of Math., Second Series, 3 No. 1, 1-25 (January 1932). 15

THE UNIVERSITY OF MICHIGAN 5172-5-Q Wiener considers the set of all real translations of a complex-valued square-integrable function, f(x), and gives conditions under which the set is closed. That is, if the set of real zeros of the Fourier transform of f(x) is of zero measure, then a general square Integrable function, F(x), may be approxtmated with arbitrary precision: F(x)lv ak f(x+Xk) k-1 The numbers { kX are real and the ~ak~ are in general complex. We may think of X ={Xk\ and a = ak as'vectors' which describe the function F(x) relative to f(x. Unfortunately, no recipe is given by Wiener for determining these vectors, although their existence is clearly establtshed. Moreover, it seems natural that a dependence between a and X must hold since In general one such'vector' characterizes a function. Suppose, now, that a class of targets, characterized by their frequency signatures Sa(w) is being Investigated. For any particular target, described by S(w), we have the expansions Sa() = ak, a S(w + k, a) Questions of identification and resolution can be answered by comparlng vectors aa}. In particular, we hope to find frequency Intervals (described essentially by 16

THE UNIVERSITY OF MICHIGAN 5172-5-Q values of k) for which the variation of amplitude values over the class of targets (i) Is sufficiently great to afford a high probability of discrimination and identiflcation. 5. PROGRAM FOR NEXT INTERVAL (This Section is UNCLASSIFIED) We will continue our studies of the mixed filter in order to determine the discrlmination vs resolution criterion as a function of the filter parameter. This study will include both theoretical analysis and the use of simulation programs. Making use of our previously computed signature of avehicle model, we will perform a machine simulation study using in the filter the various component signatures of thevehicle model as well as various values of the filter paramneter. The study of the signature representation scheme (Sec. 4. 4) will be continued. 17

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