mti filter

8/12/2019 MTI Filter

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NTIS Ri ,.,iCTIONBY PIRM ;0I N OFINFORMAIIN CANAOACommunicationsResearchoo Centre

< ANJTERATIVE JECHNIQUE FOR/DESIGNING _INITE IMPULSE RESPONSECHEBYSHEV W JILTERSWITH NONLINEAR DELAY /

byGo =.W.HerringZ

This document has bwon a.rovfor public relccyv. c-.d salI; iLa ( v /./-distribution is u dinted. . R"

DEPARTMENT OF COMMUNICATIONS C TEHNICAL NOTE,/NO. 691MINISTERE DES COMMUNICATIONS This work was sponsored by the Department of hational Defence,Research and Development Branch under Project No. 33C69.

CAN[~ OTTAWAMAR78


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/(COMMUNICATIONS RESEARCH CENTRE

DEPARTMENT OF COMMUNICATIONSCANADA

AN ITERATIVE TECHNIQUE FOR DESIGNING FINITE IMPULSE RESPONSE CHEBYSHEV MT IFILTERS WITH NONLINEAR PHASE DELAY

byR.W. Herring

Radio and Radar Research Branch) 'o' 4

CRC TECHNICAL NOTE NO. 91 March 1978OTTAWA

This wcrk wss sponsored by the Department of National Defence, Research and Development Branch undsr Project No. 33C69.

CAUTIONThe use of this Information ispermitted subject to recognition of

proprietaly and patent rights.


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AA, .........

4 TABLE OF CONTENTSABSTRACT. .. ..... ...... ...... ....... ........ 11. INTRODUCTION .. ... ...... ...... ....... ....... 12. BENEFITS OF NONLINEAR PHASE .. ...... ...... ...... .. 23. OUTLINE OF DESIGN PROCEDURE .. ..... ....... ...... .. .. 94. DESIGNING THE PROTOTYPE FILTER .. ... ...... ....... ... 105. GAIN NORMALIZATION .. ... ...... ...... ....... ... 136. COMPUTER PROGRAMS. ... ....... ...... ...... .... 137. SUMM RY .. ...... ...... ...... ...... ....... 148. ACKNOWLEDGEMENT. ... ....... ...... ...... ..... 149. REFERENCES. .. ..... ...... ....... ...... .... 14APPENDIX A - PROGRAM MPMTIPSF. .... ...... ...... ....... 16APPENDIX B - PROGRAM MPMTIPSFO .. ... ...... ....... ..... 25APPENDIX C SUBROUTINES. .. ..... ...... ....... ..... 34

Se-lon


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AN ITERATIVE TECHNIQUE FOR DESIGNING FINITE IMPULSE RESPONSE CHEBYSHEV MTIFILTERS WITH NONLINEAR PHASE DELAY

byR.W. Herring

ABSTRACTA technique for designing finite-impulse-response FIR) equiripple

high-pass digital filters with nonlinear phase suitable for use in radarmoving-target-indication MTI) systems, is described. Th eoptimization of such filters in both the time and frequency domainsis discussed, and comparisons are made between the spectralperf mnance vf nonlinear-phase filters and that of linear-phase filtersdesigned to meet the same stopband bandwidth and attenuationspecifications. In 92 out of the 94 cases exam ned, the transitionA bandwidth between the stopband is narrower fo. . e nonlinear-phasefilter. In the MTI application, this characteristic makes it possible todetect targets of low.r radial velocity. Listings of Fortran programsfor designing these nolinear-phase filters are included.

1. INTRODUCTIONThe object of this note is to describe a technique for designingfinite-impulse-response (FIR) digital filters with nonlinear phase-delays

for use in radar moving-target-indication (MTI) systems. The optimizationof such filters in both the time and frequency domains is discussed, andcomparisons are made between the spectral performance of nonlinear-phasefilters and that of linear-phase filters designed to meet the same stopbandbandwidth and attenuation specifications. In 92 out of 94 cases examined,the transition bandwidth between the filter stopband anu passband is narrowerfor the nonlinear-phase filter. Ir the MTI application, this characteristicmakes it possible to detect targets of lower radial velocity.

It is well known from digital signal theory (e.g., (1]) that delayinga sampled data sequence by a fixed amount imposLs a shift or delay in the


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2

relative phases of its spectral components which is linear with frequency.One of the great advantages of FIR filters is that they can be designed tohave such a linear phase-delay, so that their only effects are to modify themagnitudes of the spectral components and to delay the sequence by a fixedamount.

In contrast, infinite-impulse-response (IIR) filters and nonlinear-phaseFIR filters have nonlinear phase versus frequency characteristics. The use ofnonlinear-phase filters may increase the complexity of any coherent post MTIsignal processing, but since the phase characteristics of these filters aredeterministic, any undesirable effects due to the phase nonlinearities can becompensated.

Houts and Burlage [2] have described the advantages to be derived byusing Chebyshev eqiiiripple FIR filters in MTI systems, and they have publishedcomputer programs [3] for the design and evaluation of Chebyshev filtershaving linear phase-delay. Their design procedure is based on a computerprogram for designing equiripple FIR linear phase digital filters [4] usingthe Remez exchange algorithm.

2. BENEFITS OF NONLINEAR PHASEThe removal of the linear-phase constraint can result in improved MTIperformance in both the spectral and the time domains. The parameters of

interest in MTI filter design and defined in Table 1 and depicted in Figure1. The standard definitions of equiripple FIR filter parameters (e.g., 5])are defined in Table 2 and depicted in Figure 2. Note that the standarddefinitions refer to a filter designed to have unity gain in the passband,whereas for MTI applications, it may be desirable to have non-unity passbandgain in order to have 0 dB white noise power gain and/or 0 d3 minimum pass-band gain.

The improved spectral performance obtainable from nonlinear-phaseequiripple FIR filters relative to linear-phase equiripple FIR filters canbe realized in three different ways. First, smaller ripples in either thestopband or passband ripples (or both) can be achieved for given values offSTOP, fPASS and N. Smaller passband ripples mean increased ASB and thusgreater clutter suppression. Decreased RPB means that a higher detectionthreshold can be used without a loss of target visibility due to signalattenuation in the filter passband.a Second, the transition band, or the band of frequencies between fSTOPand fPASS, can be made narrower. Such a narrowing of the transition band isuseful when enhanced low-velocity target visibility is desired in the presenceof stationary clutter of finite bandwidth.

Third, the performance of a given linear-phase filter can be approxi-mated, except for phase, by a nonlinear-phase filter with smaller N. Thusincoherent integration gain can be achieved from a given fixednumber of radar pulses, since a greater number of independent filtered outputpulses are then available for integration [3].


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TABLE 1Definitions of Equiripple MTI Filter Parameters

N number of weights in filter impulse responseh(n) (0 < n < N-i) filter impulse responsef Doppler frequency Hz )IH(f)I absolute magnitude of the MTI filter responseH(f) complex magnitude of the MTI filter responsefSTOP frequency of upper edge of filter stopband (Hz)fPASS frequency of lower edge of filter passband (Hz)fPR radar pulse-repetition frequency (Hz)RpB peak-to-peak gain ripple in filter passband (dB)ASB minimum filter attenuation - measured from bottoms of passband ripples to peaks of stopbandripples (dB)GWNP white-noise-power gain (dB)GpBM minimum gain in filter passband (dB)

IH(f) (d)

0 RP BGPBM_

AS8 FILTER GAIN IN d8 VM~US DOPPLERFREQUENCY IN Hz

I f (Hz)0 fSTOP fPASS 05 fPR

Figure 1. Definition of Equiripple MTI Filter Parameters


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4TABLE 2

61 amplitude of passband ripple (linear)c2 amplitude of stopband ripple (linear)

(other definitions as in Table 1)

FILTER GAIN VERSUS DOPPLER-" , FREQUENCY IN Hz

2

02OII " f Hz)

0 fSTOP fPASS 05 fPR

Figure 2. Standard Definition of Equiripple High-Pass Filter ParametersIn Tables 3 and 4 there can be found examples showing each of the threeforms of benefit described above. The filters summarized in Table 3 weredesigned to have ASB 50 dB and values of fST0P ranging from 50 Hz to 400 Hz.

Those summarized in Table 4 were designed to have ASB = 30 dB and values offSTOP ranging from 100 Hz to 400 Hz. In both tables fPR = 2500 Hz. Forsome sets of design parameters no results are shown, since such filterscould not be designed subject to the constraints GWNP = 0 dB and GpBM = 0 dBvide Tables 3a and 3e).

The relaxation of the linear-phase constraint also allows some latitudein the choice of the filter impulse-response function (IRF), because thereare several possible IRFs corresponding to a particular spectral amplitudefunction. Each of these IRFs corresponds to a different phase-delay charac-teristic, but since phase delay is of no concern, it now becomes possible toselect a particular IRF on the basis of its time domain characteristics.Three possible criteria for this selection are: (I) a mini-max criterion,which attempts to equalize the magnitudes of the IRF weights by selectingthat IRF having the smallest value of the ratio of the largest to smallestweights; (Ii) a criterion which minimizes susceptibility to numerical over-flow by selecting that IRF having the minimum absolute sum of its weights;or (III) another criterion which attempts to equalize the magnitude of theIRF weight by selecting that IRF having the minimum variance of the magni-tudes of its weights. Criteria I and III should therefore select IRFs which


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are less susceptible to disruption by impulsive noise, whereas Criterion IIis based on minimizing susceptibility to numerical overflow. The evaluationof the suitability of these criteria or the proposal of others remains as atopic requiring further investigation.

TABLE 3aComparative Examples of Equiripple Finite-Impulse-Response MTI Filter Characteristics

LINEAR PHASE NONLINEAR PHASEReduction of ReductionNumber of Passband Low.. Passband Ripple Passband Lower Passband Ripple Passband Lower in PassbandWeights N Edge (Hz) fPASS (dB) RPB Edgef(Hz) PASS (d8) RPB Edge (Hz) Ripple (dB)

5 477 2 71 - - - -6 537 3.30 390 2.12 147 1.187 392 2.48 376 2.07 16 0418 421 2.52 347 1.93 74 0.599 386 2.16 319 1.79 67 0.37

10 350 2.05 292 1.64 58 0.4111 352 2.00 270 1.51 82 0.4912 301 1.73 251 1.40 50 0.3313 317 1.82 234 1.30 83 0.5214 265 1.51 219 1.22 46 0.2915 286 1.65 206 1.14 80 0.5116 237 1.34 195 1.08 42 0.26

ISTOP 25Hz f =2500 Hz AS8 50 dB GWNP OdB GPBM OdB

TABLE 3bComparative Examples of Equiripple Finite-Impulse-Response MTI Filter Characteristics

LINEAR PHASE NONLINEAR PHASEReduction of ReductionNumber of Passband Lower Passband Ripple Passband Lower Passband Ripple Passband Lower in PassbandWeights N Edge (Hz) fPASS (dB) RPB Edge (Hz) IPASS (dB) RpB Edge (Hz) Ripple (dB),

5 718 4.30 576 3.48 142 0.826 539 3.33 482 2.87 57 0.467 556 3.44 416 2.45 140 0.998 423 2.54 367 2.14 56 0.409 455 2.80 332 1.90 123 0.9010 352 2.07 300 1.72 52 0.3511 386 2.35 282 1.60 104 0.7512 303 1.76 280 1.60 23 0.1613 337 2.03 274 1.57 63 0.4614 288 1.66 263 1.52 25 0.14i 300 1.79 252 1.46 48 0.3316 284 1.65 240 1.39 44 0.26

S T O P 5 HZ f R =2 5 00 Hz. . ....PH A' 8=5d WPN dBGPBM 0d


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TABLE 3cComparative~ Examples of Equiripple Finite-Impulse-Response M TI Filter Characteristics

LINEAR PHASE NONLINEAR PHASE Reduction of ReductionNumber of Passband Lower Passband Ripple Passband Lower Passband Ripple Passband Lower in Passband

N Edge (Hz) fPASS (dB) RPB Edge (Hz) fPASS (dB) RP8 Edge (Hz) Ripple (dB)5 793 5.38 591 3.67 202 1.716 542 3.37 532 3.37 10 0.007 583 3.78 511 3.13 72 0.658 537 3.33 462 2.84 75 0.499 469 2.96 417 2.56 62 0.40

10 477 3.01 380 2.31 970711 397 2.47 349 2.11 48 03612 419 2.64 332 2.01 87 0.6313 346 2.13 330 2.00 16 0.1314 372 2.33 321 1.95 51 0.2815 340 2.09 308 1.88 32 0.2116 335 2.09 293 1.79 42 0.30

1ST 100OHz fP =2500 Hz AS13=50dB GWNOdB GPB OdB

TABLE 3dComparative Examples of Equiripple Finite-Impulse-Response MTI Filter Characteristics

Number of Passband Lower Passband Ripple Passband Lower Passband Ripple Passband Lower in PassbandWeiht Nde (z)~P 58 (dB) R 8 Edge (Hzi IPA (dB) RFBEdge (Hz) Ripple (dB)

5 810 5.65 807 5.61 3 0.046 818 5.76 648 4.28 170 1.407 654 3.55 634 4.16 20 -0.61

V8 627 4.26 580 3.81 47 0.459 617 4.14 522 3.40 95 G. 4

10 519 3.45 479 3.30 40 0.1511 536 3.60 479 3.10 57 0.5012 493 3.25 458 2.97 35 0.2813 471 3.14 431 2.80 40 0.3414 472 3.14 407 2.64 65 05015 423 2.80 405 2.62 18 0.1816 436 2.90 397 2.58 39 0.32

~S 200 Hi fP 2500 Hz ASB SdB GwNPOdB GPB 0 dBZ

fSTOPPR S:, WP PB


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TABLE 3eComparativeExamples of Equiripple Finite-impulse-Resoonse MTI Filter Characteristics

LINEAR PHASE NONLINEAR PHASEReduction of Reduction

Number of Passband Lower Passband Ripple Passband Lower Passband Ripple Passband Lower in PassbandWeights N Edge (Hz) f PASS (dB3) PB Edge (Hz) f PASS (dB3) PB Edge (Hz) Ripple (dB)

I--871 6.66 -8 873 6.65 789 4.89 84 1.76

815 5.58 754 5.50 61 0.0810 716 5.28 684 5.58 32 -0.3011 723 5.35 686 4.92 37 0.4312 703 5.16 646 4.61 57 0.5513 636 5.39 636 4.52 0 0.8714 647 4.74 618 4.39 29 0.3515640 4.68 596 4.07 44 0.6116 607 3.97 593 4.20 14 -0.23

f =TP400Hz fP=2500 Hz A S8 50 dB GWNP OdB GP8M OdB

TABLE 4aComparative Examples of Equiripple, Finite-Impulse-Response MTI Filter Characteristics

4.. Number of LINEAR PHASE NONLINEAR PHASE Rdc ino eutoNmeof Passband Lower Passb3nd Ripple Passband Lower Passband Ripple Passband Lower in Passband

Weiht N g (*) PA8 dB) R B Edge (Hz) fPS (dB3) P Edge (Hz) Ripple (dI3)5492 2.65 489 2.78 3 -0.136534 3.27 455 2.61 79 0.66

7 485 2.79 407 2.35 78 0.448 422 2.53 365 2.12 57 0419 428 2.53 330 1.91 98 0.62

10 353 2.09 302 1.75 51 0.3411 374 2.23 279 1.62 95 0.6112 306 1.80 270 1.56 36 0.2413 332 1.98 268 1.55 64 0.4314 278 1.49 262 1.53 16 -0.0415 298 1.78 253 1.48 45 0.3016 275 1.61 243 1.43 32 0.18fo, 100 Hz f =2500Hlz A8 30d(3 GWNP0 d8 G B OdB


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TABLE 4bComparative Examples of Equiripple Finite Impulse Response MTI Filter CharacteristicsLINEAR PHASE NONLINEAR PHASE

Reduction of Reductionumber of Passband Lower Passband Ripple Pasband Lower Pasband Ripple Passband Lower in Passbandeights N Edge {H) I (dB) R Edge 1Hz)PAS (dB) R Edge (Hz) Ripple (dB)PASS RB AS PB5 766 4.97 594 3.71 172 1.266 549 3.46 503 3.64 46 -0.187 479 3.73 505 3.12 74 0.61 521 3.28 469 2.91 53 0.379 474 3.03 428 2.67 46 0.3610 479 3.05 394 2.47 85 0.5811 409 2.61 387 2.42 22 0.1912 428 2.74 380 2.39 48 0.3513 395 2.52 366 2.31 29 0.21

14 386 2.48 349 2.21 37 0.27y 15 384 2.46 334 2.12 so 0.3416 353 2.28 331 2.11 22 0.17

fSTOP 200 Hz fPR 2 500 Hz ASB 30 d8 GWNP 0dB G Od8STPPG8WPpBM

TABLE 4cComparative Examples of Equiripple Finite-Impulse-ResponseMTI Filter CharacteristicsLINEAR PHASE NONLINEAR PHASE

Reduction of Reductionumber of Passband Lower Passband Ripple Passband Lower Passbaisd Ripple Passband Lower in PassbandWeights N Edge IHz) fPASS d8) Pe Edge (Hzj fPASS (dB) RpB Edge (Hz) Ripple (dB)5 822 5.85 822 5.84 0 0.016 829 5.95 759 4.66 70 1.297 773 5.06 697 4.85 76 0.218 600 4.66 630 4.97 -30 -0.319 669 4.75 630 4.35 39 0.4010 645 4.55 593 4.10 52 0.4511 688 4.17 579 4.00 9 0.17

12 597 4.24 567 3.92 30 0.3213 584 4.15 543 3.77 41 0.384 548 3.91 542 3.75 6 0.1615 555 3.96 530 1.68 25 0.2816 548 3.90 518 3.55 30 0.35

I STOP 4 00 Hz fPR 2 500 Hz AS=30 dB GWN, =dB - GPM 0d8


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9

3. OUTLINE OF THE DESIGN PROCEDUREThe technique for designing nonlinear-phase FIR filters is based on a

procedure first suggested in [6]. This procedure involves the design of aprototype linear-phase FIR filter having an IRF h (n) of length (2N-1) and anamplitude response in the frequency domain Hp(f) equal to the square of themagnitude of the desired frequency response. It is shown below that thisprototype filter cannot be designed directly from the desired parameters ofthe nonlinear-phase filter, but that an iterative technique must be used.

A property of linear-phase FIR filters with real-valued IRFs is that ifzo is a zero of the IRF, then so are zo-l, z* and zol)* where * denotescomplex conjugate [11; page 159). Note that if zo is real, then zo = zo andalso, if Izol = 1, then zo-1 = z*, so that roots can be real and single, oroccur in conjugate pairs (if IZol = 1) or in reciprocal pairs (if zo is real),or in conjugate-reciprocal quartets (if Izol 0 1 and zo complex). In general,of the 2(N-1) zeroes of the (2N-l)-length prototype filter, there are k pairsof reciprocal real zeroes, 9 quartets of conjugate reciprocal zeroes and= (N-k-29.-l) pairs of double zeroes on the unit circle. To extract an iRFh(n) of length A, it is necessary to discard one zero of each of the m pairs

of double zeroes, one zero of each of the k pairs of reciprocal zeroes, andone pair of conjugate 7eroes from each of the Z quartets of conjugate-reciprocal zeroes. This leaves a set of (N-i) zeroes which is expanded toproduce a real-valued IRF of length N.

A set of M IRFs of length N can thus be derived from the (2N-l)-lengthprototype filter, where

M =Note that one zero or pair of conjugate zeroes can be arbitrarily chosen tolie inside or outside the unit circle, since the only effect of this choiceis to reverse the IRFs in the time domain; i.e., h(n) is replaced by h(N-l-n)for 0 < n < N-I.

An upper bound on M as a function of N is given byNB=2{ (N-2)/2-1}

where x] denotes the largest integer less than x, so it can be seen that theset of IRFs to be examined can contain of order 210 for N=24. Hence theoptimization techniques described above can become quite time-consuming forfilters of such length. This limitation, however, should not preclude theuse of these procedures for designing filters of the lengths usually consid-ered for MTI applications (e.g., N < 20; so that M < 256).

4. DESIGNING THE PROTOTYPE FILTERIn order to make use of the standard filter design algorithms, it is

first necessary to define the filter ripple parameters (RPB,ASB) of interestto radar KTI designers in terms of the standard linear ripple parameters


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1.0

(EI, 2) It can easily be shown from Figures 1 and 2 that(RpB 20)10 . I.

_1 (R /20)10 +1

and

C ~ (2)2 :10(A B/20)

In the standard design procedure for linear-phase FIR filters [3], ifN, STOP' fPASS and fPR are specified, then only the ratio

W Ci/C 2 (3)

can be specified as a free parameter. This restriction can be circumventedby allowing fPASS to be varied in an iterative manner until that value forfPASS is found which gives the desired values for C2 and hence el [3).4

A similar iterative procedure is necessary in the design of nonlinear-phase FIR filters. For the design of the linear-phase prototype filter Ho(f)(see Figures 3(a)-3(c)), it is necessary to specify 2N-1, fSTOP' fPR, 61 and62, where 61 and 62 have to be specified in terms of el and C2. The outlineof the scheme for relating the V's and c's is pictured in Figures 3(a)-3(c)and is similar to that of [6] except for one detail pointed out below. Forclarity the scheme is described progressing from the original prototype filterHo f) (Figure 3(a)) to the intermediat.e filter HT(f) (Figure 3(b)) to thefinal prototype filter H (f) = IH f)Il (Figure 3(c)). In practice, the actual.progression is from specifying H (f) to specifying Ho(f) in terms of H (f),since 1o(f) is the filter which Ts actually designed using the algorithmof (4].

H1 (f) is related to H.(f) by the transfoimationH (f) = H (f) + 62 (4)

In the time domain this is equivalent toh( =ho(n), 1 < Inl < N-I(n) (5)

1ho(n) + 62, n = 0.

H (f) is in turn related to H (f) by the transformationpH (f) = K H (f) (6)pI

which becomes in the time domain


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: 11

H0(f)

< 0 - fHZ)-- T2 ~PASS 05 fPR

Figure 3 a). Original Linear-Phase Prototype Filter Frequency Response

H1 lf)

HIM

0- f Hz)0 f TOP f ASS 0 .5 PR

Figure3 b) Intermediate Linear-Phase Prototype Filter Frequency Response

-- ------ ~-


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12

; ; Hp fM - H (f )IA

2_

-

2

)y. ~~C 1- ----

. j STOP fPASS 0.5 f PR

Figure 3 b). Linear-Phase Prototype Filter Frequency Response

h (n Kh1 (n), -N+1


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14

grid density for the design of the prototype filter [4) and to select theimpulse-response design criterion must also be provided. This program islisted in Appendix A, and the detailed operating instructions are given there.Typical times required to design a filter using a Xerox Sigma 9 computer rangefrom 2.2 sec for N=5 to 28.4 sec for N-16.

The program MPMTIPSFO generates nonlinear-phase FIR 14TI filters withGpB M = 0 dB. The same parameters as for the program MPMTIPSF must be specifiedby the user, but here the parameter RPB is used only as an initial value whichis then modified by the program to converge to GPBM = 0 dB. Hence the choiceof a starting value for RpB near its final value can significantly reduce thetime required to design a filter. This program is listed in Appendix B.Typical running times on the Sigma 9 can be 1 to 10 or more times as long asfor the program MPMTIPSF, depending on how good an estimate for the initialvalue of R B is used.

Appendix C contains the listings of the subroutines required by theprograms MPMTIPSF and MPMTIPSFO. The subroutine REMEZ and its ancillarysubroutines have noz been included, since they are readily accessible else-wh re [4], but note the modifications required in the dimensions of theCOMMON block variables.

7. SUMMARYIt has been shown in Tables 3 and 4 that, for the typically narrow stop-bands used in radar MTI filters, nonlinear-phase filters can be designed which

offer superior visibility for low-velocity targets, relative to that offeredby linear-phase filters designed to the same stopband specifications. It hasaiso been pointed out that the time-domain response of such nonlinear-phaseNTI filters can be optimize~d without altering the filter power response inthe frequency domain. The details of the design procedure have been described,and listings of Fortran programs for implementing the procedure have beenprovided.

8. ACKNOWLEDGEMENTThis work is supported by the Departnent of National Defence under

Research and Development Branch Project No 33C69.

9. REFERENCES1. Oppenheim, A.V. and R.W. Schafer, Digital Signal Processing. Prentice-

Hall, Inc. Englewood Cliffs, New Jersey, 1975.2. Houts, R.C. and R.W. Burlage, Mfaximizing the Usable Bandwidth of MTT

SignaZ Processors, IEEE Trans. Aerospace Electronic Syst-ms AES-13,No. 1, January 1977, 48-55.


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153. Burlage, D.W. and R.C. Houts, Design Technique for Improved BandwidthIMoving Target Indica or Processors in Surface Radars. Report No.RE-TR-75-35, US Army 4issile Command, Redstone Arsenal, Alabama, June1975.4. McClellan, J.H., T.W. Parks and L.R. Rabiner, A Computer Program forDesigning 9ptimum FIR Linear Phase Digital Filters. IEEE Trans. AudioElectroacoust. AU-21, No. 6, December 1973, 506-526.5. Herrmann, 0., Design of Nonrecursive Digital Filters with Linear Phase.Electronics Letters 6, No. 11, 28 May 1970, 328-329.6. Herrmann, 0. and W. Schuessler, Design of Nonrecursive Digital Filters

with Minim Phase. Electronics Letters 6, No. 11 28 May 1970.

.


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16

APPENDIX A

PROGRAM MPMTIPSF

iI


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APPENDIX C

SUBROUTINES


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