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OPTIMIZATION OF EXTRAPOLATED IMPULSE RESPONSE FILTERS USINGYa Jun Yul, Guohui Zhao2,Kok Lay Teo2an d Yong Ching Lim3

SEMI-INFINITE PROGRAMMING

Temasek Labora tories, Nanyang Technological University, Singapore, 6397982Ap plied Mathem atics Department, The Ho ng Kong Polytechnic U niversity, Hong Kong

3Electrical & Electron ic Engineering, Nanyan g Technological U niversity, Singapore, 63979 8ABSTRACT

Narrow band lowpass or highpass digital finite impulseresponse filters can be synthesized by using extrapolatedimpulse response techniques to achieve reduced complex-ity. However, the non-linear optimization problem of theextrapolated impulse response filters was simplified to alinear programming problem in previous literature leadingto suboptimum. In this paper, a semi-infinite programmingis proposed to jointly optimize the filter coefficients andthe extrapolated scaling factors. A realization structuremakin g use of the coefficient sym metry is also presented. A nexam ple taken from literature is include d illustrating that thenumber o f multipliers for the resulting filter is less than 6 5percent of existin g results.

1. INTRODUCTIONLinear phase finite impulse response (FIR) digital filters arewell used in the signal processing applications due to theirguaranteed stability and exact linear phase property. A seri-ous disadvantage of FIR filters is their high implementationcomplexities. This problem becom es particularly acute insharp filters since the filter length is inversely proportionalto the filter transition width. In recent years, there hasbeen m uch effort devoted t o reducing the complexity of FI Rfilters. Am ong these techniqu es are extrapolated impulseresponse design [11, frequen cy-respo nse masking [2, 31,interpolated impulse response design [4], recursive runningsum pre-filtering [5], coefficient thinning [6], coefficientover-sampling [7] and predictive encoding of coefficientvalues [XI .The extrapolated impulse response approach [11 makesuse of the correlation between blocks of impulse responsesamples to synthesize narrow band lowpass and highpassfilters with reduced complexity. In this approach, smallerside lobes of impulse response samples are approximatedas scaled versions of larger ones so as to extrapolate thefilter length. Thus, the hardware comp lexity of the filteris reduced. An additional advantage of the extrapolatedimpu lse response filter is its low roundof f noise power due tothe reduced number of mu ltipliers compared with the directform structure.

0-7803-8379-6/04/$20.00 0 20 04 IEEE.

Fig. 1: A typical impulse response sequence of anarrow-band lowpass filter.However, the results obtained by using the extrapolatedimpulse response filters approach proposed in [l] may befurther improved: (i) The filter coefficients and the extrap-olated scaling factors of the extrapolated impulse responsesamples may be jointly optimized; (ii) coefficient sym metrymay be exploited in the realization. In this paper, a semi-

infinite programming is proposed to jointly optimize the filtercoefficients and the scaling factors. A realization structuremaking use of the coefficient sym metry is also presented.The results obtained represent a significant improvementover those achieved in [l].2. THE EX TRAPOLATED IMPULSE RESPONSE

This section reviews the extrapolated impulse response ap-proach to synthesizing narrow transition band lowpass filters.A typical impulse response sequence, h(n ) , f a narrowband lowpass filter is quasi-periodic, consisting of a centerlobe with the largest magnitude an d side lobes with decrea s-ing magn itude away from the center, as shown in Fig. 1. T hezero phase transfer function of h(n) s

NH ( z )= h(0)+ h(n)(zn+ z-") . (1 )

n= lAssum e that the impu lse response has lobe s at n = Lo +

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1= M through n = k l , n = k l +1 hrough n = k 2 , . . , an dn = + 1 hrough n = C p = N . H ( z )may be rewrittenas

Ad

H ( 2 )= h(0)+ h(n)(z"+ z- )n= lk i k z

+ 1 (n ) ( z"+z-") + h ( n ) ( P + z - " )n=k,, +1 n=k1+l

+ . .+ 5 ( n ) ( P+ Z-). (2)n= C 1 +l

The durations of side lobes, k l - ko, c ~ k l , . . . , andIC , - 5-1,ay or may not be all equal. For any two sidelobes having the same number of impulse response samples,the smaller lobe may be approximated by a scaled version o fthe larger one. For example, if k m + l - k m = kl+l - Cl = d ,then

h ( k m + n )= oh(kl + n ) fo r n = 1 , 2 , . . , . (3 )where Q is an approximate scaling factor. For expositoryconvenience, we assume that all the side lobes are of thesame duration, d . Extensions to other cases are simple andstraightforward extensions. In the case considered in thispaper, (2) may be rewritten asA4

H ( z )= h(0)+ h(n)(z"+ Z-")n = l

R d

r=Om=l(4)

where R+ 1 s the number o f lobes. If the lobes for T 2 1areapproximated a s scaled versions of the lobe fo r r = 0, H ( z )can be approximated byM

H ( z ) M fi(2) = h(0)+c (n)(z"+ z-")n= l

d R

m= 1 T= o

where aT s the rth scaling factor and a0 = 1.3. A REALIZATION OF EXTRAPOLATED

IMPULSE RESPONSEIn this section, a realization structure for the extrapolatedimpulse response filter by making use of the coefficientsymmetry is presented.

Equation (5 ) may be realized in hardware in three sep-arate sections. The center section, which consists of thefirst two terms, is the well-known direct form FIR filter.Terms associated with Q ~ z ~ + ~ + ~ ~an be realized usingthe structure shown in Fig. 2(a) and terms associated witha T ~ - ( h 4 + m + T d )an be realized using the structure shownin Fig. 2(b). Fig. 2(c) shows a realization of the transferfunctionH ( Z )= h(0)+ h ( l ) ( z+ 2-1) + h ( 2 ) ( 2 + 2-2)

2 2

This structure is a transposed form qf the structureproposed in [l]. The advantage of this structure is that thecoefficient symmetry can be used leading to a saving of dmultipliers. An additional advantage of this structure is thatthere is a saving of R d- + 1delay elements when comparedwith the structure proposed in [l].

4. OPTIMIZATION USING SEMI-INFINITEPROGRAMMINGThe frequency response I?(&") of the extrapolated impulseresponse filter is given by

Therefore, the optimization of the extrapolated impulseresponse filter is to find out the filter coefficients h(n) ,for n = 0 , 1 , . . ,M + d , and scaling factors aT , orr = 1 , 2 , . . . R , by solving the following semi-infiniteprogramming problem.

where W ( w ) s the ripple weighing and H d ( w ) s the desiredgain at frequency w.To design a lowpass filter with passband and stopbandedges at w p and w s , an initial value for the problem can beobtained by using linear programming as proposed in [l].Starting from this initial value, a semi-infinite programmingproblem with M + d +R+ 1 inite unknow n variables subject

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Fig. 2: Realization structures for the extrapolated impulse response filter by making use ofthe coefficient symmetry, where T represents a single delay element and dT represents dcascaded delay elements.

to 4 infinite constrains can be formulated as follows: 5. EXAMPLES

where

an d

The above semi-infinite programming problem can besolved by fseminf function in MATLAT, or similar softwarepackages. Imp rovem ents in the frequency response ripplemagnitude can be achieved since semi-infinite programmingjointly optimiz es the filter coefficients and the scaling factors.

This section illustrates, by means of an example, the effi-ciency of the filters obtained by using the proposed optimiza-tion and realization techniques compared w ith those obtainedusing the earlier techniques [l].Consider the specifications [ l ] of a lowpass filter withstopband edge and passband edge at, respectively, 0.1 and0.13 times the sampling frequency. With equal passban dripple and stopband ripple, a length 95 filter results in astopband attenuation of 52.8 dB. Notice from the impulseresponse of the filter that the durations of the lobes have therhythm 5,4,4,5,4,4,5,4,4 from h( 9) through h(47). ChooseM in (7) to be 8 an d d to be 13 , i.e., h(n) or In1 5 21 are re-alized accurately and the period o f side lobes are 13 samples.Hence, the impulse response sequences h(22) through h(34)and h(35) through h(47) may be approximated as scaledversions of h( 9) through h(21).

The results for N = 69 and N = 95 extrapolated byusing our technique is shown in Fig. 3. The peak ripplemagnitude for N = 95 is -49.8 dB. The minimax optimumand the results extrapolated by using linear programmingfo r N = 69 and N = 95 are also shown in Fig. 3 fo rcomparison.

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-30 -

-35 -mU

-45

-50

U -40-

-

a

,Q Extrapolated rom N=43 %.*,*:>.by linear programming.ic Extrapolated rom N=43 ., ..?O

by semi-infinite programming **,., *b

*,* ..?...,- Minimax opitmum

Fig. 3: Peak ripple magnitude versus filter lengthplot.To meet the original specification s of the length 9 5 filter,more coefficients must be implemented accurately. Let Mbe equal to 1 1 an d d remain 13, a filter with N = 101extrapolated from a filter with N = 49 achieves -53.1dB peak ripple magnitude. The frequency response of theextrapolated filter is shown in Fig. 4. Twenty-nine multipliersand 52 adders are needed for the realization of this filter byusing the structure presented in S ection 3.By using the linear programming method proposed in [11,a filter with N = 109 extrapolated from N = 57 canachieve the original -52.8 dB peak ripple magnitude. Forty-six multipliers are needed for the realization of this filter byusing the structure presented in [13. Therefo re, the nu mbe r ofmultipliers of the filter extrapolated by using our techniqueis approximately 63 percent of that of the earlier design.Compared with the direct form design implemented byexploiting the coefficient symmetry, which requires 47 multi-pliers and 94 adders, ourtechnique needs only approximately62 percent multipliers and 55 percent adders. The price topay for is an increase of 6 percent in the filter order.

6. CONCLUSION

The arithmetic complexity of lowpass filters using the ex-trapolated impulse response technique has been reduced bythe joint o ptimization technique and the realization structureby m aking use of the coefficient symmetry. The num berof multipliers for the resulting filter is less than 65 percentscompared with the original results presented in [ l].

Normalized requency

Fig. 4: The frequency response of the filter withN = 102 extrapolated from N = 49 .

7. REFERENCES[11 Y.C. Lim, Ex trapolated impulse response FIR filters,IEEE Trans. Circuits, Syst., vol. 37, pp. 1548-1551,Dec. 1990.[2 ] -, Frequency-response masking approach for thesynthesis of sharp linear phase digital filters, IEEETrans. Circu its Syst., vol. CAS -33, pp.357-364, April.1986.[3] T. Saramaki and H. Johansson, Optimization of FIRfilters using the frequency-response masking tech-nique, in Proc. of IEEE international Conference onCircuits, Syst. vol. 11, pp.177-180, May 2001.[4 ] Y. Neuvo, C.Y. Dong, and S.K. Mitra, Interpolated

finite impulse response filters, IEEE Trans. Acoust.,Speech, Signal Processing, vol. ASSP-32, pp. 563-570,June 1984.[5] J.W. Adam s and A.N. Willson, Jr., A new appr oach toFIR digital filters with fewer multipliers and reducedsensitivity, IEE E Trans. Circuits Syst., vol. CAS-30,pp. 277-283, May 1983.[6] G.F. Boudreaux and T.W. Parks, Thinning digitalfilters: A piecewise-exponential approximation ap-proach, IEEE Trans. Acoust., Speech, Signal Process-

ing, vol. ASS P-31, pp. 105-113, Feb. 19 83.[7] M.R. Bateman and B. Liu, An approach to pro-grammable CTD filters using coefficients 0, +1, and-1, IEEE Trans. Circuits Syst., vol. CAS-27, pp.451-

456, June 1980.[8] Y.C. Lim, Pre dictive coding for FIR filter wordlengthreduction, IEE E T rans. Circuits Syst., vol. CAS-32,pp. 365-372, April 1985.

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