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DEPARTMENT OF COMMUNICATIONS C TEHNICAL NOTE,/NO. 691

MINISTERE 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 COMMUNICATIONS

CANADA

AN ITERATIVE TECHNIQUE FOR DESIGNING FINITE IMPULSE RESPONSE CHEBYSHEV MTI

FILTERS WITH NONLINEAR PHASE DELAY

by

R.W. Herring

(Radio and Radar Research Branch) 'o' 4

CRC TECHNICAL NOTE NO. 691 March 1978

OTTAWA

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

CAUTION

The use of this Information is permitted subject to recognition ofproprietaly and patent rights.

AA, .........

4 TABLE OF CONTENTS

ABSTRACT. .. ..... ...... ...... ....... ........ 1

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

2. BENEFITS OF NONLINEAR PHASE .. ...... ...... ...... .. 2

3. OUTLINE OF DESIGN PROCEDURE .. ..... ....... ...... .. .. 9

4. DESIGNING THE PROTOTYPE FILTER .. ... ...... ....... ... 10

5. GAIN NORMALIZATION .. ... ...... ...... ....... ... 13

6. COMPUTER PROGRAMS. ... ....... ...... ...... .... 13

7. SUMMARY .. ...... ...... ...... ...... ....... 14

8. ACKNOWLEDGEMENT. ... ....... ...... ...... ..... 14

9. REFERENCES. .. ..... ...... ....... ...... .... 14

APPENDIX A - PROGRAM MPMTIPSF. .... ...... ...... .......16

APPENDIX B - PROGRAM MPMTIPSFO .. ... ...... ....... ..... 25

APPENDIX C -SUBROUTINES. .. ..... ...... ....... ..... 34

Se-lon

AN ITERATIVE TECHNIQUE FOR DESIGNING FINITE IMPULSE RESPONSE CHEBYSHEV MTI

FILTERS WITH NONLINEAR PHASE DELAY

by

R.W. Herring

ABSTRACT

A technique for designing finite-impulse-response (FIR) equiripplehigh-pass digital filters with nonlinear phase, suitable for use in radarmoving-target-indication (MTI) systems, is described. Theoptimization 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 transition

A 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. INTRODUCTION

The object of this note is to describe a technique for designingfinite-impulse-response (FIR) digital filters with nonlinear phase-delaysfor 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

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 PHASE

The removal of the linear-phase constraint can result in improved MTIperformance in both the spectral and the time domains. The parameters ofinterest 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 detection

threshold 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 fSTOP

and 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. Thus

~enhanced incoherent integration gain can be achieved from a given fixednumber of radar pulses, since a greater number of independent filtered output

pulses are then available for integration [3]. 1

TABLE 1

Definitions of Equiripple MTI Filter Parameters

N number of weights in filter impulse response

h(n) (0 < n < N-i) filter impulse response

f Doppler frequency (Hz)

IH(f)I absolute magnitude of the MTI filter response

H(f) complex magnitude of the MTI filter response

fSTOP 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 RPBGPBM_

AS8 FILTER GAIN IN d8 VM~US DOPPLER

FREQUENCY IN Hz

I f (Hz)0 fSTOP fPASS 05 fPR

Figure 1. Definition of Equiripple MTI Filter Parameters

(I 4

TABLE 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

it

I 2

02OII " f (Hz)

0 fSTOP fPASS 05 fPR

Figure 2. Standard Definition of Equiripple High-Pass Filter Parameters

In Tables 3 and 4 there can be found examples showing each of the three

forms of benefit described above. The filters summarized in Table 3 were

designed 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 of

I fSTOP ranging from 100 Hz to 400 Hz. In both tables fPR = 2500 Hz. For

some sets of design parameters no results are shown, since such filters

K could not be designed subject to the constraints GWNP = 0 dB and GpBM = 0 dB

(vide Tables 3a and 3e).K The relaxation of the linear-phase constraint also allows some latitude

in the choice of the filter impulse-response function (IRF), because there

are several possible IRFs corresponding to a particular spectral amplitude

function. Each of these IRFs corresponds to a different phase-delay charac-

teristic, but since phase delay is of no concern, it now becomes possible to

select 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 selecting

that IRF having the smallest value of the ratio of the largest to smallest

weights; (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 the

IRF 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

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 3a

Comparative Examples of Equiripple Finite-Impulse-Response MTI Filter Characteristics

LINEAR PHASE NONLINEAR PHASE

Reduction 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 1.18

7 392 2.48 376 2.07 16 041

8 421 2.52 347 1.93 74 0.59

9 386 2.16 319 1.79 67 0.37

10 350 2.05 292 1.64 58 0.41

11 352 2.00 270 1.51 82 0.49

12 301 1.73 251 1.40 50 0.33

13 317 1.82 234 1.30 83 0.52

14 265 1.51 219 1.22 46 0.29

15 286 1.65 206 1.14 80 0.51

16 237 1.34 195 1.08 42 0.26

I STOP 25Hz f =2500 Hz A S8 50 dB GWNP OdB GPBM OdB

TABLE 3b

Comparative Examples of Equiripple Finite-Impulse-Response MTI Filter Characteristics

LINEAR PHASE NONLINEAR PHASE

Reduction 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 0.82

6 539 3.33 482 2.87 57 0.46

7 556 3.44 416 2.45 140 0.99

8 423 2.54 367 2.14 56 0.409 455 2.80 332 1.90 123 0.90

10 352 2.07 300 1.72 52 0.35

11 386 2.35 282 1.60 104 0.75

12 303 1.76 280 1.60 23 0.16

13 337 2.03 274 1.57 63 0.46

14 288 1.66 263 1.52 25 0.14

i 300 1.79 252 1.46 48 0.3316 284 1.65 240 1.39 44 0.26

S T O P 5 0 H Z f R = 2 5 0 0 H z. . ....

PH A' S8=5d WPN dBGPBM 0d

TABLE 3c

Comparative~ Examples of Equiripple Finite-Impulse-Response M TI Filter Characteristics

LINEAR PHASE NONLINEAR PHASEReduction of Reduction

Number of Passband Lower Passband Ripple Passband Lower Passband Ripple Passband Lower in Passband

VWeights N Edge ( Hz) f PASS (dB) R PB Edge (Hz) fPASS (dB) R P8 Edge (Hz) Ripple (dB)

5 793 5.38 591 3.67 202 1.71

6 542 3.37 532 3.37 10 0.00

7 583 3.78 511 3.13 72 0.65

8 537 3.33 462 2.84 75 0.49

9 469 2.96 417 2.56 62 0.40 1

10 477 3.01 380 2.31 9707

11 397 2.47 349 2.11 48 036

12 419 2.64 332 2.01 87 0.63

13 346 2.13 330 2.00 16 0.13

14 372 2.33 321 1.95 51 0.28

15 340 2.09 308 1.88 32 0.21

16 335 2.09 293 1.79 42 0.30

1ST 100OHz fP =2500 Hz A S13 =50dB GWNOdB GPB OdB

TABLE 3d

Comparative Examples of Equiripple Finite-Impulse-Response MTI Filter Characteristics

Number of Passband Lower Passband Ripple Passband Lower Passband Ripple Passband Lower in Passband

Weiht N Ede (z)~P 58 (dB) R 8 Edge (Hzi IPA (dB) R FBEdge (Hz) Ripple (dB)

5 810 5.65 807 5.61 3 0.04

6 818 5.76 648 4.28 170 1.40

7 654 3.55 634 4.16 20 -0.61 '

V8 627 4.26 580 3.81 47 0.45

9 617 4.14 522 3.40 95 G. i4

10 519 3.45 479 3.30 40 0.15

11 536 3.60 479 3.10 57 0.50

12 493 3.25 458 2.97 35 0.28

13 471 3.14 431 2.80 40 0.34

14 472 3.14 407 2.64 65 050

15 423 2.80 405 2.62 18 0.18

16 436 2.90 397 2.58 39 0.32

~SO 200 Hi fP 2500 Hz A SB SdB GwNPOdB GPB 0 OdBZ

fSTOPPR S:, WP PB

TABLE 3e

Comparative Examples 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) R PB Edge (Hz) f PASS (dB3) R 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 0.08

10 716 5.28 684 5.58 32 -0.30

11 723 5.35 686 4.92 37 0.43

12 703 5.16 646 4.61 57 0.55

13 636 5.39 636 4.52 0 0.87

14 647 4.74 618 4.39 29 0.35

15640 4.68 596 4.07 44 0.61

16 607 3.97 593 4.20 14 -0.23

f =TP400 Hz f P=2500 Hz A S8 50 dB G WNP OdB GP8M OdB

TABLE 4a

Comparative Examples of Equiripple, Finite-Impulse-Response MTI Filter Characteristics

4.. Number of LINEAR PHASE NONLINEAR PHASE Rdcino euto

Nmeof Passband Lower Passb3nd Ripple Passband Lower Passband Ripple Passband Lower in Passband

Weiht N dg (*) PA8 dB) R PB Edge (Hz) fPS (dB3) RP 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 0.44

8 422 2.53 365 2.12 57 041

9 428 2.53 330 1.91 98 0.62

10 353 2.09 302 1.75 51 0.34

11 374 2.23 279 1.62 95 0.61

12 306 1.80 270 1.56 36 0.24

13 332 1.98 268 1.55 64 0.43

14 278 1.49 262 1.53 16 -0.04

15 298 1.78 253 1.48 45 0.30

16 275 1.61 243 1.43 32 0.18

fo, 100 Hz f =2500Hlz A8 30d(3 GWNP0 d8 G B OdB

rr

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

LINEAR PHASE NONLINEAR PHASE

Reduction of ReductionNumber of Passband Lower Passband Ripple Pasband Lower Pasband Ripple Passband Lower in PassbandWeights N Edge {H) I (dB) R Edge 1Hz) fPAS (dB) R Edge (Hz) Ripple (dB)1PASS 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 0.618 521 3.28 469 2.91 53 0.379 474 3.03 428 2.67 46 0.36

10 479 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 0.2114 386 2.48 349 2.21 37 0.27

y 15 384 2.46 334 2.12 so 0.341 16 353 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-Response MTI Filter Characteristics

LINEAR PHASE NONLINEAR PHASE

Reduction of ReductionNumber of Passband Lower Passband Ripple Passband Lower Passbaisd Ripple Passband Lower in PassbandWeights N Edge IHz) fPASS (d8) RPe 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 0.3814 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 400 Hz fPR 2

500 Hz A S=30 dB GWN, =dB - G PM 0d8

9

3. OUTLINE OF THE DESIGN PROCEDURE

The technique for designing nonlinear-phase FIR filters is based on aprocedure 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 andLm = (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 pairsof 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)-length

prototype 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 by

NB=2{ [(N-2)/2-1}

where (x] denotes the largest integer less than x, so it can be seen that the

set 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 FILTER

In order to make use of the standard filter design algorithms, it isfirst necessary to define the filter ripple parameters (RPB,ASB) of interestto radar KTI designers in terms of the standard linear ripple parameters

Ita

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 sB/20)

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

W C ei/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 transfoimation

H (f) = H (f) + 62" (4)

In the time domain this is equivalent to

h( =ho(n), 1 < Inl < N-Ih (n) (5)

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

H (f) is in turn related to H (f) by the transformationp

H (f) = K H (f) (6)pI

which becomes in the time domain

: 11

H0 (f)

"!

< I 0 - f (HZ)-- T2 ~PASS 05 fPR

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

H1 lf)

HIM

0- f(Hz)

0 f STOP f PASS 0.5 f PR

Figure 3(b) Intermediate Linear-Phase Prototype Filter Frequency Response

-- ------ ~-

12

; ; !Hp fM - !H (f )I

A

2_

-

2

)y. ~~C 1- -

---

. j STOP fPASS 0.5 f PR

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

?ii

h (n Kh1 (n), -N+1 <n <N-1 (7)p

where K is a scaling factor. It is in the determination of the factor K that

this development differs from that of [6], for it can be noted that a scaling

by the factor (1+62)-l does not produce a filter IH(f)I with equal (linear)

magnitude ripple excursions above and below unity gain.

Since it is the par ameters RPB and ASB and thus el and E2 which are

specified by the filter designer, it is necessary to invert the abovefunctional relationship to derive expressions for 61 and 62 in terms of cland E2. This being done, it is a straightforward matter to use the linear-

phase FIR filter algorithm to determine the (2N-1)-length IRF ho(n).

It can be seen from eqn. (6) and Figures 3(b) and 3(c) that

(1+ el2 K(l + 61 + 6,,) (8)

2(1c =K(l -6 +6) (9)

11 2

and

C 2 =2K 6 (10)2I 2

Some algebraic manipulation shows that

tv d

1<13

61 2 + 2E2 -E (11)

1 2

£2

2 2 + 2e2 E2 (12)1 2

and

1 22

It is now a simple matter to derive hp(n) from ho(n) in terms of these factorsas outlined above, and then to derive a nonlinear-phase N-length IRF h(n).

5. GAIN NORMALIZATION

It is often useful to design MTI filters which have 0 dB white-noisegain. This means that the white-noise power in the filtered signal is thesame as that in the unfiltered input signal, so that in the absence of clutter,the false-alarm probability PFA is not altered. Such normalization is easilyaccomplished (3] by invoking Parseval's theorem

! n-i 0f.5f

2 f2' PR f df (14)g2 = I h n)H j IHfn=O o

to compute the white noise power gain g2 by means of a simple summation.Scaling h(n) by g-i produces a filter with unity white noise power gain(GwNp = 0 dB) and leaves ASB and RPB unaltered. Note that this procedure isvalid for any FIR filter.

It can also be useful to design a filter which has both GWNP = 0 dB anda minimum gain of unity in its passband (GPBM = 0 dB). This means that theprobability of detection P in the passband will not be reduced by filterattenuation in these passband ripples. This goal can be achieved by aniterative procedure. In this case it is the passband ripple that is altereduntil the desired characteristics are obtained. The procedure is to specify

$ N, ASB and fSB, select an arbitrary reasonable value for RPB and design afilter using the iterative procedure described above. Based on whether thisfilter has GPBM greater or less than unity gain, the value for RPB isincreased or decreased respectively and another iteration is carried out.This procedure is repeated until convergence to GPBM = 0 dB is achieved towithin an acceptable tolerance.

6. COMPUTER PROGRAMS

The program MI,,1TlPSF generates nonlinear-phase FIR MTI filters forspecified values of N, fSTOP' fPR ASB and RpB. Parameters to specify the

* 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 withGpBM = 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. SUMMARY

It 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 whichoffer 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. ACKNOWLEDGEMENT

This work is supported by the Departnent of National Defence underResearch and Development Branch Project No 33C69.

9. REFERENCES

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

15

3. 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= m Phase. Electronics Letters 6, No. 11, 28 May 1970.

,'

A .

k

16

APPENDIX A

PROGRAM MPMTIPSF

iI

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