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TECHNICAL REPORT ARBRL-TR-02346

COMPUTER ALGORITHMS FOR THE DESIGN AND

IMPLEMENTATION OF LINEAR PHASE FINITE

IMPULSE RESPONSE DIGITAL FILTERS

James N. Walbert

July 1981

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMAND BALLISTIC RESEARCH LABORATORY ABERDEEN PROVING GROUND. MARYLAND

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TECHNICAL REPORT ARBRL-TR-n2.^46

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*. TITLE fand Sijfam;»;

COMPUTER ALGORITHMS FOR THE DESIGN AND IMPLEMENTATION OF LINEAR PHASE FINPTE IMPULSE RESPONSE DIGITAL FILTERS

7. AUTHORf*;

James N. Walbert

9. PERFORMING ORGANIZATION NAME AND ADDRESS U.S. Army Ballistic Research Laboratory ATTN: DRDAR-BLP Aberdeen Proving Ground, MD 21005

11. XONTRQLLING.OFFICE NAME AND ADDRESS , U.S. Army Armament Research 5 Development Command U.S. Army Ballistic Research Laboratory ATTN: DRDAR-BL

U. '\fe°NT#^lQGÂmî9SASiS^^P^R^|gr» Â°QS.^ 'rou, ConUoU.n, Office)

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18. SUPPLEMENTARY NOTES

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Software Digital Filter Linear Phase Finite Impulse Response Hilbert Transform

Differentiation Passband Stopband Transition band Convolution

Phase Delay

2Q. ABSTTÂCT (TCoirtfaue e«i r*-nr** aidm ft rretretraary arri IderUIfy by block number) 1 mk

A FORTRAN program, published in the open literature, for the design of linear phase finite impulse response digital filters has been installed on the BRL CDC computer. Portions of this program have been extracted and combined to form a subroutine for filter design. Ancillary subroutines have been developed to assist in the formulation of filter design parameters. A subroutinje for convolution of data with digital filters of finite odd length has also been written.

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TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS 5

I. INTRODUCTION 7

II. A DESCRIPTION OF THE DIGITAL FILTER DESIGN PROGRAM .... 7

III. A DESCRIPTION OF THE DIGITAL FILTER DESIGN SUBROUTINE. . . 8

IV. CONSIDERATIONS IN THE USE OF THE DESIGN SOFTWARE 8

V. IMPLEMENTATION OF DIGITAL FILTERS 11

VI. CONCLUSIONS 15

VII. SUMMARY 15

VIII. ACKNOWLEDGEMENTS 16

REFERENCES 17

APPENDIX A. A LISTING OF PROGRAM DESIGN 19

APPENDIX B. A LISTING OF SUBROUTINE FILTER 41

APPENDIX C. SAMPLE OUTPUT FROM PROGRAM DESIGN 53

DISTRIBUTION LIST 57

LIST OF ILLUSTRATIONS

Figure Page

1. Frequency response of a low pass filter 9

2. Frequency response of a low pass filter with narrow transition band 10

3. Graphical construction of new end points forix. i . . . . 13

I. INTRODUCTION

12 In 1973, McClellan and Parks ' published a listing of a computer

program for the design of finite-duration impulse-response digital filters. This program was unique in that the authors had developed a unified theory for the design of the four types of filters: bandpass, bandstop, Hilbert-transform, and differentiation. The resulting software is one of the most flexible digital filter design programs available.

In the analysis of ballistic data which has been converted from an analog voltage record to a digital time series, it is generally desirable to be able to isolate various signal components for individual study. Such components are usually identifiable by frequency content, and as a consequence, are ideally suited for separation or removal by digital filtering techniques. This report describes the adaptation of the filter design program to the CDC computer at BRL, the modification of a portion of this program into a subroutine, the development of subroutines to specify filter design parameters, and a convolution subroutine for filters of odd length. A complete description of design considerations for digital filters and definitions of related terms is beyond the scope of this report. Any of the cited references will provide the necessary information. This report does provide sufficient design information to allow the reader to implement digital filters; a subsequent BRL Technical Report will cover in greater detail specific application techniques.

II, A DESCRIPTION OF THE DIGITAL FILTER DESIGN PROGRAM

Only minor changes were made to the program as it appeared in reference 2. The program statement added was

PROGRAM DESIGN(INPUT,OUTPUT,TAPE5=INPUT,TAPE6=0UTPUT,TAPE?)

In the original program, when the value of the variable JPUNCH was input as 1, values of the filter coefficients were output to punched cards. In the program, as it exists on the CDC computer, TAPE? may be specified in the jobstream to be any suitable device or file. The constants PI and PI2 (TT and lit, respectively) were extended to the full double precision word length for the CDC. The free-field input form of the original program was replaced with formatted input. Finally, a test for end-of- file on input was added to allow for multiple designs per computer run. A listing of program DESIGN is in Appendix A.

J.H. McClellan, T.W. Parks, "A Unified Approach to the Design of Optimum FIR Linear-Phase Digital Filters," IEEE Trans. Circuit Theory, CT-20(6), 69?-?01 (19?3).

2 J.H. McClellan, T.W. Parks, L.R. Rabiner, "A Computer Program for Designing Optimum FIR Linear Phase Digital Filters, "IEEE Trans Audio Electroacoustics, AU-21[6), 506-526 (19?3).

III. A DESCRIPTION OF THE DIGITAL FILTER DESIGN SUBROUTINE

For most applications to analysis of ballistic data, optimum digital filter design specifications are the result of a systematic trial-and- error investigation. Frequently, the design specifications change from one data event to the next because certain aspects of the experiment were non-repeatable. In view of these factors, it seemed appropriate to formulate a filter design subroutine for use in interactive analysis computer programs, thereby permitting tailoring of the filter design on a round-by-round basis.

Subroutine FILTER, a listing of which appears in Appendix B, is extracted from program DESIGN. It will design bandpass filters of up to 10 bands, but will not design Hilbert Transformers or differentiators. The grid density (LGRID] has been fixed at 16, but the subroutine other- wise retains the full flexibility of program DESIGN. All variable names used in program DESIGN are also retained.

The subroutine statement is

SUBROUTINE FILTER(NFILT,NBANDS,EDGE,FX,WTX,IPRINT,H), where NFILT is the filter length; NBANDS is the number of pass/stop bands; EDGE is an array containing the band edges, expressed as fractions of the sampling frequency; FX is an array containing the desired filter shape, (1. in the pass bands and 0. in the stop bands); WTX is an array containing the desired relative weighting in each band; IPRINT is a control variable for printing the coefficients (0-print coefficients, 1-don't print coefficients); and H is the array containing the filter coefficients on output. The variables NFILT, NBANDS, AND IPRINT are integers; the arrays EDGE,FX,WTX, and H are real, dimensioned 2*NBANDS, NBANDS, AND CNFILT+l)/2, respectively. If NFILT is even, then the H array is dimensioned NFILT/2.

IV. CONSIDERATIONS IN THE USE OF THE DESIGN SOFTWARE

For the purposes of this discussion, assume that the data sequence X. " consists of points equally spaced in time; in particular, At will ^ i = l

denote the time between two consecutive samples. The sampling frequency, fs, is therefore 1/At, and the bandwidth of the data is .Sfg. The bandwidth of the data represents the highest unaliased frequency present in the data, provided due care has been given to the sampling process.

The essence of the design algorithm is to approximate the desired filter response function on the frequency-amplitude plane from -.5fs to +.5fs on the frequency axis. The coefficients are designed in a normalized form on the interval [-.5, .5]. Moreover, the frequency response has either odd or even symmetry about the origin on the frequency axis, so that the design problem is completely determined by specifying the desired response on the normalized frequency interval [0., .5].

In Figure 1, below, is shown the frequency response of a typical low pass filter. This is a two band filter: it has a pass band and a stop

band.

Figure 1, Frequency response of a low pass filter

The pass band is from 0. to .2, or to 40% of the bandwidth. .Zfg is termed the cutoff frequency of the filter. It is a since frequencies in this band are "passed" unaltered (i.e. by 1). The stop band is from .3 to .5; frequencies in this "stopped" (i.e. multiplied by 0).

The frequency "pass" band are multiplied band are

The frequency band Selection of the width o the design of a digital tion band narrows, the s roll-off) increases. As compensates by increasin pass and stop bands. Th in increases and decreas frequencies. An example tion band is shown in Fi

from .2 to .3 is termed the transition band. f this transition band is somewhat critical in filter, for the following reason: as the transi- lope of the frequency response (,t.e. the filter this slope increases, the design algorithm

g the deviation from the desired response in the is deviation is called the "ripple", and results es of amplitude in the signal at those particular of a filter designed with too narrow a transi- gure 2.

In any application software, it is advisable to have the capability of viewing the frequency response of the filter prior to its application, in order to be certain of its characteristics. The design program, as a part of its printed output, lists the normalized frequencies at which the maximum and minimum amplitudes of the ripple occur. Also listed are the deviations from the desired design, which provide a measure of the amplitude error to be expected as a result of applying the filter to the data. (See Appendix C).

Referring to the example of Figure 1, the input variables to design this filter were assigned the following values:

Figure 2. Frequency response of a low pass filter with narrow transition band

NFILT 33

NBANDS = 2

EDGE(l) = 0

EDGE(2) = .2

EDGE(3) = .3

EDGE(l) = .5

FX(1) =^1.

FX(2) =0.

WTX(l) = 10.

WTX(2) = 100.

While NFILT is specified as 33, only 17 distinct coefficients are returned, since the design is symmetric about 0. The sample output in Appendix C indicates the ordering of the 17 coefficients, although this is not the filter of Figure 1.

As can be seen in this example, the EDGE array specifies the normalized band edges. The FX array specifies the desired amplitude

10

response, which is usually (but not necessarily) 1 in the pass bands and 0 in the stop bands. The WTX array specifies a relative scaling of the magnitude of the deviation between the pass band and the stop band. In this example, the design algorithm allows 10 times less deviation in the stop band than in the pass band. This relative weighting may be adjusted arbitrarily to suit a particular need. For example, by relaxing the pass band weighting, say WTX(1)=1., one could design a filter with a more narrow transition band.

For additional information concerning the design of digital filters, ' the reader is referred to references 1,2, and 3.

V. IMPLEMENTATION OF DIGITAL FILTERS

A digital filter is applied to a data sequence by convoluting the filter weights, or coefficients, with the data points. Specifically, if

/x.t is a sequence of data points equally spaced in time, and if ih I Mi=l ( '')k=l are the filter coefficients, where N£n, then the filtered data sequence

has values given by K. N N z k=l

One notes that if i£N-l, then i+l-k£N-k, so that some subscripts of x in the summation may have values less than 1; we have no corresponding x values. There are two choices: either start the convolution process at i=N, or start at i=l and modify k to avoid subscripts of x less than 1 until we get to the Nth point. In the first case, N-1 data points at the beginning are unused, and the output sequence starts at i-N. In the second case, the first N-1 output points have not been transformed by the same set of filter coefficients as have the rest of the data; the first N-1 points are of questionable value. It will be noted that the same problem occurs for i>n-N. In what follows, we will discuss a method to avoid these diffi- culties. In particular, it will be shown that an n-point input sequence can be modified to provide n useful output points.

For nonreal-time applications, i.e.: for post-processing of data, one is in the admirable position of knowing in advance what is going to occur. That is, the convolution process can be numerically manipulated so as to provide one output point corresponding to each input point, with no lag. (Only filters of finite odd length, say N = 2M + 1, M = 1,2,...,63, will be discussed here.) This is accomplished simply by moving the filter coefficients M indeces in Eq. (1), so that

2M+1 z ^i = Z^ Vi-M-l+k • (2) k=l

Eq. (2) implies that the filter coefficients are centered at the i*" data

11

point. If the coefficients are re-indexed as {h \ , then Eq. (2) is I ^/k=-M

more simply written as

M

k=-M

where h^ = h^^^^ .

Now, for the values i=l,2,...,M,n-M+l,n-M+2,...,n-l,n, Eq. (3) still has some values of i-k for which there is no corresponding x.

It is necessary to provide M values at the beginning and M values at the

end of the sequence f^X^ • This can be done with a minimum of frequency

distortion by using an oJd reflection of the first M and last M points. Speci- fically, for i-k<l, define x._, by

X. , = 2x,-x^ . „ . ^^^ i-k 1 k-x+2

Similary, for i-k>n, define x^_^ by

X. , = 2x -X, . (^\ i-k n k-i . v-'-'

Graphically, Eq. (4) reflects X2,X3,...xi4 about the vertical line through xi and then about the horizontal line through x^. The points X X ...X are reflected in a like manner about x , as shown n-M+1' n-M+2' n-1 ^ in Figure 3. '

This reflection process can be trivially incorporated into the convolution algorithm, as will be explained below.

The types of digital filters being considered here have an additional property which simplifies the convolution process: they are of either even or odd symmetry about their midpoint. That is,

h, = ± h , , k=l,2,...M . (6) k -k

12

X.

+ + -^ 1

o o

Figure 3. Graphical construction of new end points for x.

As a consequence, Eq. (3) may be written as

M

y.=hx.+2 h, (x., ±x.,,) 1 ox k i-k i+k

k=l

(7)

Whereas Eq. (3) requires 2M + 1 multiplications and 2M additions to implement, Eq. (7) requires only M + 1 multiplications and M additions.

Utilizing Eqs. (4), (5), and (7), the following are three examples of convolution subroutines. The first, in FORTRAN, is in use on the BRL CDC system. The second is in standard BASIC. The third, in an enhanced BASIC, is in use on several BRL systems. In each case, x is the input/ output array of length N. The K=M+1 filter coefficients are stored in the array H. -

Example 1: FORTRAN Convolution Subroutine

SUBROUTINE CONVOL (H,K,X,N) DIMENSION H(K),X(N),S(127),SAVE(63) M = K-1 L = K + M IF(L.GT.127) STOP DO 5 1=1,M S(I) = 2.''>X(1) - X(K -I)

13

S(L+1-I)=X(I) SAVE(I) = 2.*X(N) -X(N-I)

5 CONTINUE S(K)=X(1) LAST=N-M DO 20 1=1,N X(I) = 0. DO 10 J=1,M X(I)=X(I)+H(J)*(S(J)+S(L+1-J)

10 CONTINUE X(I)=X(I)+H(K)AS(K) DO 15 J=2, L S(J-1)=S(J)

15 CONTINUE IF(I.LE.LAST) S(L)=X(I+K) IF(I.GT.LAST) S(L)=SAVE(I-LAST)

20 CONTINUE RETURN END

Example 2: BASIC Convolution Subroutine

100 SUBROUTINE Convolution (H,K,X,N) 110 DIM H(K),X(N),S(127),SAVE(63) 120 M=K-1 130 L=K+M 140 IF L>127 THEN STOP 150 FOR 1=1 TO M 160 S(I) = 2.*X(1)-X(K-1) 170 S(L+1-I)=X(I) 180 SAVE(I)=2.*X(N)-X(N-I) 190 NEXT I 200 S(K)=X(1) 210 Last-N-M 220 FOR 1=1 TO N • 230 X(I)=0 240 FOR J=l TO M 250 X(I)=X(I)+H(J)*(S(J)+S(L+1-J)) 260 NEXT J 270 X(I)=X(I)+H(K)*S(K) 280 FOR J=2 TO L 290 S(J-1=S(J) 300 NEXT J p. 310 IF I<=Last THEN S(L)-X(I+K) ' 320 If I> Last THEN S(L)=SAVE(I-Last) 330 NEXT I . 340 RETURN ^ 350 SUBEND

14

■,rWTKf!fT*?riW<hi ■_

I

In the third example, use is made of several matrix operations available in enhanced BASIC. The function DOT returns the dot product of the two input arrays. The function MAT REORDER rearranges the elements of one array according to the index order specified by another. In this example, the array B has the values 2,3,4,...,L,1, where L=2M+1, the filter length. Implementation of the routine in example 3 represents a decrease in execution time by a factor of 15 over the routine in example 2.

Example 3: BASIC Matrix convolution Subroutine

100 SUBROUTINE Convolution(H,L,X,N,B) 110 DIM H(L),X(N),S(127), Save(63),B(L) 120 IF L>127 THEN STOP 130 REDIM S(L) 140 M= INT(L/2) 150 FOR 1=1 TO M 160 S(I)=2.AX(1)-X(M+1-I) 170 S(L+1-I)=X(I) 180 Save(I)=2.*X(N)-X(N-I) 190 NEXT I 200 S(M+1)=X(1) 210 Last=N-M 220 FOR 1=1 TO N 230 X(I)=DOT(H,S) 240 MAT REORDER S BY B 250 IF I<=Last THEN S(L)=X(I+M+1) 260 IF I>Last THEN S(L)=SAVE(I-Last) 270 NEXT I 280 RETURN ' , 290 SUBEND

VI. CONCLUSIONS

Digital filters have a wide range of application for numerical analysis of time-series data. The filter design program presented here has been found to be one of the most versatile available. The reflection principle described in this report seems to introduce the least additional frequency content into the data of any of the methods available. This same technique has been used to produce periodic continuation of essentially transient phenomena, facilitating the use of numerical filters in their analysis.

In a forthcoming BRL Technical Report, the author will discuss specific techniques for the application of digital filters to the analysis of ballistic data. The report will also develop in greater detail the theory and applicability of digital filters to analysis of time series.

VII. SUMMARY

An open literature FORTRAN computer program for the design of finite

15

impulse-response digital filters has been implemented on the BRL CYBER system. Algorithms have been developed and coded for the convolution of digital filters with time series data. These algorithms include a method for the removal of the filter delay, as well as elimination of the loss of data at the beginning and end of the particular data set being filtered.

VIII. ACKNOWLEDGEMENTS

The author is indebted to Mrs. Emma Wineholt, who made the necessary coding changes in program DESIGN and subroutine FILTER to convert them from IBM to CDC FORTRAN.

16

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REFERENCES

1. J.H. McClellan, T.W. Parks, "A Unified Approach to the Design of Optimum FIR Linear-Phase Digital Filters," IEEE Trans. Circuit Theory, CT-20(6), 697-701 (1973).

2. J.H. McClellan, T.W. Parks, L.R. Rabiner, "A Computer Program for Designing Optimum FIR Linear Phase Digital Filters," IEEE Trans Audio Electroacoustics, AU-21(6), 506-526 (1973).

17

APPENDIX A

A LISTING OF PROGRAM DESIGN

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