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" t Statistical Properties of a jSt~aggered-PRF MTI System,

Z~j~eS40 K.SIAO

Target Characteristics BranchRadar Division

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SNAVAL RESEARCH LABORATORY

Washigton, D.C.

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SECURITY CLAIIFICATIO'4 OF THIS PGt (Whei Date Knferod)

REPORT DOCUMENTATION PAGE RECA MNSTRUCTIONS

_T7 KPORT NUMSQI R. GOVT ACCISSION NO. B•FIRNTS COAMLOETUIMiERM

. m~enmreport on a continuingNRL Report 8219 NRL problem,4, TITLEI (ow' SubIdlia) S. TYPE OF MCPrMT A f[EIOO COVEMRO

STATISTICAL PROPERTIES OF A STAGGERED.PRF MTISYSTEM 6, P1,FORMING ORG. EPORT NUMSER

ST. AUYNOR(o) S. CNYRACY ON GRAN• NUMBER(e)

James K. Hulao

ý. PERFORMING ORGANIZATION NAME AND ADDRESS SO. PROGRAM ELKMENT. PROJECT, TASKAREA 6 WORK UNIT NU"aIR[11Naval Reseach Laboratory NRL Problem R12.61

Washington, D.C. 20375 Proj" RR014-0941JElel~pt 611163N.21

II. CONTROLLING OFFICE NAME AND ADDRESS t3 REPORT OAYS

Office of Naval Research April 26, 1978Arlington, VA 22217 is NurN or. PAGQSV

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I SCNEOULL

I$. WSTRJIUTION STATEMENT (of this Neaomi)

Approved for public releae; distributlon unlimited

17. 4ISTRIOUTION STATEMENT (of the S&it&Iftt on..ted In BIoch JO, If .lll nl. ho" Rl.,ft)

it SUPPLEMENTARY NOYES

IS. KEY WORDS (coIm.. t evea aIde Id noeeewmy md Iadentify by btlock nubot)

Moving-target indicatorsClutter rejectionStaggered.PRF MTI

0 PIUSTRACT (Cot'un"e on rovotO old* If ner.aowV 4R4 IdonltIf1 b•o htek mauwb.,)

In this report the statistical properties of improvement factors of a staggered.PRF MTI systemare prmnted for caes in which optimal filter weights are used and in which, 1,momial filter weights areused. It is shown that the degree of deviation of samples is a function of both thki standard deviation ofthe clutter spectrum density function arnd the amount of variation of interpulse 4urations.

DD IJoN 1473 EOITONOf I NOV *S 19 OGSOL•ET

SECURITY CLASSIFICATION OF TNIS PAGE (When Owe, tmwoo.d

CON'rENTS

INTRODUCTION ..................................... 1

IMPROVEMENT FACTOR AND INTERPULSE DURATION... 1

OPTIMAL MTI PERFORIK41i'CE ......................... 3

BINOMIALLY WEIGHTED FILTER ...................... 4

REFERENCES ....................................... 5 I

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STATISTICAL PROPERTIES OF A STAGGERED-PRF MTI SYSTEM

INTRODUCTION

To reject unwanted clutter, a radar usually transmits a sequence of pu.lses. When thereturns of these pulses are properly weighted and summed, stationary clutter can be filteredout. In a conventional radar system, the interpulse durations (or sampling frequencies) areheld constant. Targets having a doppler frequency which is an integer multiple of this eam.-pling frequency will be seen as a stationary target and be filtered out. This target is said tohave a blind velocitbr. To alleviate this problem, a staggered-PRIF system has been proposed.In that system the interpulse durations are varied from pulse to pulse; hence this blindvelocity phenomenon is avoided. A number of papers dealt with the design problem of thissystem [1-41. However no known analytic method can be used tcu select a set of interpulsedurations to achieve a desired MTI performance. In this report the effects of variation ofthe interpulse duratiors on the MTI improvement factor are investigated. A Monte Car.approach is used to derive the statistical properties of this improvement factor in a stag-gered-PRF MTI system.*

IMPROVEMENT FACTOR AND INTERPULSE DURATION

To set up a common reference for the convenience of comparison, a criterion tomeasure the performance of an MTI system will be presented here. One widely acceptedmeasurement parameter is the so-called improvement factor, which is defined as the ex-pected value of the ratio of the output target-signal-to-clutter ratio to the input target-signal-to-clutter ratio. This improvement factor is

a21= i ,(1)

ii

where the ai's are the MTI filter weights and Ri, is the clutter correlation function at timest1 and tp. This correlation function is the Fourier transform of the clutter spectrum densityfunction G (f):

Ri fG(f)e j2v f(T-T)d. (2)

In deriving Eq. (1) it is assumed that the target doppler has a uniform distribution function.

Manuscript submitted February 28, 1978,*Part of this report has been presented as a paper at the 1977 IEEE International Conference on Acoustic,

Speech and Signal Processing, Hartford, Connecticut, May 9-11, 1977.

1

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JAMES K. HSIAO

For a constant-PRF MTI system, one may normalize the doppler frequency f by theradar PRF (the reciprocal of interpulse duration T), and Eq. (2) becormes

NY) fa-Ž ej2wf'('j) df'. (3)

Under this asumption the improvement factor I is not a function of the radar inter-z pulse duration T. However. the clutter spectrum density function may have to be modified

due to this tranformation. For example, if the clutter spectrum density function is aGauian function

GM .(4a)

then

Rij Z e-2 w2"t2(i-/•. (4b)

If one lots f - fT and o' - a/T, one has

G =f' 1 .-f(2/ 2 " "J t!a

and

Rij . e-2x 2a'11(1-j) 2 (5b)

One notices that the standard deviation o of the spectrum density is modified. How-ever, the spectrum density remains unchanged. This formulation has the advantage that theradar PRF is not directly involved in the computation of the improvement factor. In a stag-gered-PRF MTI system the interpulse durations vary from pulse to pulse. To accommodatethis situation and for the convenience of comparison, a basic interpulse duration T is definedwhich is the shortest interpulse time among all pulses in a staggerted PRF system. The inter-pulse time between any two successive pulses is then

Ti - Ti-I (1 + ai)T, (6)

where ai > 0 and

Gf) e, 2 12,). (7)

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NRL REPORT 8219

which is identical to Eq. (5a). In other words, as long as the basic interpulse time is thesame, the normalized apectrurn density functions are the same for both the constant-PRFand the staggered-PRF cases. The correlation function R however becomes

R ij Cxp {21r2 o'2 2(1 1. (8)

When this relation is inserted into Eq. (1), one sees that the variation of interpulseduration ak influences the MTI improvement factor. However, one may see intuitively thatRij reduces in the case of a staggered-PRF system, because the correlation time becomeslonger. Naturally, the MTI performance is degraded, and the improvement factor is reduced.

07TIMAL MTI PERFORMANCE

The conclusion has been drawn that the MTI filter can be so chosen that it yields abest improvement factor for a given clutter spectrum density. Hsiao [ 51 showed that for astaggered-PRF system this optimal improvement factor is bounded by two limits. The upperbound is the improvement factor of a constant-PRF system with a PRF that is equivalent tothe shortest interpulse duration of the staggered system, and the lower bound is the im-provement factor of a constant-PRF system which has a PRF equivalent to the longest, inter-pulse duration of the staggered system.

The preceding conclusion is drawn from investigations of a large number of samples.Each sample has a randomly chosen interpulse duration. However in each case the filterweights are so chosen that the improvement factor is optimized. This approach is useful indetermining the performance bounds. In practic,, however, one may be more interested inkeeping the filter weights fixed while varying the interpulse durations. Some statisticalproperties of such systems are as follows.

Figure I shows the statistical distribution of the improvement factor of a three-pulse,

staggered-PRF MTI system. The filter weights are initially chosen for optimal performancefor a constant-PRF system assuming that the clutter spectrum density function is Gaussianhaving a normalized standard deviation o (normalized with respect to PRF). The improve-ment factor of this MTI system is then computed -assuming that the interpulse durationvaries from T to T + ,vT where a is a random variable with a uniform distribution. In Fig. 1four sets of curves are plotted, for normalized standard deviations o - 0.03, 0.05, 0.07 and0.1. Within each set of curves the limit of the variation of the interpulse tim, varies from0.1 to 0.6. The improvement factor of each sample is computed when the intrpulse dura-tions of that sample are chosen randomly (with a uniform distribution) with a maximumlimit as mentioned above. The cumulative probability of the improvement factor of thesesamples is plotted for each differ-,nt o and ca.

Severel interesting points may be observed:

* Since the sample having the snua.liest interpulse duration is the one which has aconstant PRF, the highest improvement factor for various a values occurs at the same point(of the constant-PRF case) no matter what a is chosen.

3

I

JAMES8 K. 1181A0

Tile variation1 of t1Ie IimproVmilent factor is Sallw for Small a but increases as o:increases.

T lhe spreading of the stuliples is also a function of (1, the Standard deviation of theclatter Spectrumi dvinsity function. As o) increases, tile spreaiding of tile samples rt-duces.

The results shown in Fig. I are summarized in Tlable 1. When u 0.03, the differenceof he mprveentf~vorvaries from 1.5 dB t 8.5 dB asavaries from 0. o0.6. When

o- 0.1, thle variation is limited to 1.2 to 6.2 dB.

Figure 2 shows the salt'e curves for the case' of a four-pulse caniceler. These curves haveSimilar propeorties as those- shown in Figutre 1. 1 lowever, the spread of the saniples in goneralis more pronoutwed, piarticularly for htigh improvement factors. This means that if one has ahigh-performance NITI System, with four or more pulses for rejection of clutter with smnallspeoctral spread, one Should bie more careful in choosing thet interpulse time when a stp*g-gered-l'Ht F System 18 used, particularly wheii thlt variation of int erpulse duration is large. Onlthe other hand, if the maximum variation of interpulse duration is suwial and the designedNMTI systA'm has a smnaller improvement factor, the choice of interpu&,e duration is not imi-piortant. Probably any randomly Selected combination of interpulse durations maky yield justabout the samne result as that of a carefully Selected one.

Th'le result~s showni in Fig. 2 are Summarized in Table 2. Otte notices that in general thesp)readinxg of samples is more pronounced in this, case than in the thret-pulse ease.

Figure 3 shows the statistical propet-rties of the improvement factor of a thrtee-pulsestaggered-PR1 F NITI System. lit thle figure thle average value and the Standard deviation (orUNMS deviation fromt mleanl) are plotted as a function of the percrent of variation of intetr-pulse delay. Thle average improvement factor is almost a linear function of thle percent ofvariation of interpulse delay. As thlt percent, of delay variation increases, the improvementfactor reduces. This improvement factor is also very sensitive to thle o value. Thie RNISdvvia~t ion increases as theit perent o f variat ion c f delay increases, but its value remains Small(the deviation curves in Fig. 3 being plotted to anl expianded scale relative to average-valueScOale). Th'le Significance of this is that by a randenm choice, of any combination of interpulsedurations the amiount of iimprovt'ment-fact~or variultion is Small. For example, for a casetot) - 0.03, when thet delay variation of the Staggere(I I'l F sySteml is Set atL 0.56%, by anlychoice of a combination of interpulse duration, the RNMS deviation from the meoan of allthese, samiples is not more than 1 .8 dB.

Figure 4 shlows thle samie st~atistic pro~perties of the improvement factor for a four-pulse stagggered systemt. T'his figure exhibits properties similar to those exhibited in Fig. 3.

BINOMIlALLY WEIGHITED) FILTER

lIn tWe previous examples, optimal filter weights; are used. lin practice, however. filterweights are often set according to the binomial distribution. 'Ilierefore, it is of interest toinvestigate the efftxt oif stag4,gering onl the NITI systm~i for such ease's. Thle distribution ofimprovement factors for a three-pulse and four-pulse- sttaggered-PRF MTl System usingbinomial weighits are, respectively shown in Figs. 5 and 6. Thei clutter spiectrumi density func-tion is againl assumed to be Gaussian wvitli a normalized standard deviation o, wvith the

4

N NRL REPORT 8 2194L

variation of interpulse duration being randomly distributed from T to 7' + uT similar to thei')} varition in the previous examples. Comparing these two figures with Figs. I w~d 2, one sttesi that ý,hese curves have almost the same shape. Therefore the properties discussed in the pre-.

coding section apply to these cases. In genera•l, for the same o and ot, the improvement fac-

tor which can be achieved by a MTI system with optimal weights is slightly better than thatof a binomial case. However the difference is not that much.

k Igulre 7 and Figure 8 show respectivoly the statistic prolperties of the improvementfactor of a three-pulse and four-pulse staggered-PRF NITi system. These figures show asimilar properties of that of an optimally weighted MTI.

REFERENCES

1. 0. J. Jacomini, "Weighting Factor and 'hansmission Time Optimization in Video MTIRadar," IEEE Trans. on AES AFS-8, (July 1972).

2. P. J. A. Phinse.n, "Elimination of Blind Velovities of MTI Radar by Modulating theInterpulse Period," IEEE Trans. on AES AES-9, 714-72-4 (Sept. 1973).

3. J. K. lisiao and F. F. Kretschmer, Jr., "Design of a Staggered PRF Moving Target* Indication Mlter." Radio and Electronic Engineer 43 689-693 (Nov. 1973).

41. C. W. Ewell and A. M. Bush, "Constrained huprovement NITI Radar Procvesors," IEEE"Thins. or AS AE S-1 1, 768-780 (Sept. 1975).

5. J. K. ihiso, "On the Optimization of MITI Clutter Rejection," IEEE Tirns. on AESA4S-10, 622-629 (Sept. 1974).

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