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DESIGN OF MINIMUM-PHASE FIR DIGITAL-FILTERS BY DIFFERENTIAL CEPSTRUM

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570 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 5, MAY 1986

Circuits and Systems Letters

Design of Minimum-Phase FIR Digital Filters by

Differential Cepstrum

SOO-CHANG PEI AND SHEN-TAN LU

Absrruc~-The differential cepstrum is introduced to design equiripple minimum phase FIR filters by using cepstral deconvolution; this fast procedure only takes three FFT computation and avoids the complicated phase wrapping and polynomial root-finding algorithms.

I. INTR~DuC~~N

In recent years, there has been a considerable interest in the design of finite impulse response (FIR) linear phase digital filters; a powerful computer program [l] is available to efficiently design linear phase FIR filters. However, for some applications, the delay introduced by these linear phase filters is prohibitive; when the linear phase property is not desirable, much lower delay time can be achieved with minimum phase filters satisfying the same specifications on the amplitude. Optimum magnitude minimum phase filters are highly attractive in several applications, such as CTD transversal filters and communication channel filters, etc.

The optimum magnitude minimum phase FIR filter design has been theoretically described by Herrman and Schuessler [2], it transforms equiripple linear phase designs into equiripple mini- mum-phase designs with half the degree and with attenuation characteristics equal to the square root of the prototype, this method is limited to the numerical root-finding difficulty, Re- cently, Mian and Nainer proposes a design procedure to over- come this difficulty by using homomorphic deconvolution [3], however the complicated phase wrapping algorithm will be in- volved in the computation; In this paper, the differential cepstrum is introduced to design equiripple minimum phase FIR filters by using cepstral deconvolution; This fast procedure only takes three FFT computation and avoids the complicated phase wrapping and polynomial root-finding algorithms.

II. THE DIFFERENTIAL CEPSTRUM: DEFINITION, IMPLEMENTATION AND PROPERTY

Consider the convolution

x(n)=x,(n)*x,(n). Then

(1)

The differential cepstrum [4], [5] of x(n) is defined as

(3)

Equation (2) becomes

k!(n) =%(n)+%,(n) (4

which can be used as a new tool for homomorphic deconvolution.

‘-l dz. (5)

This allows for the following DFT implementation:

N-l X(k) = C x( n)e-j2nnk/N n-0 N-l xl(k) = - i C nx( n) e-Qnnk/N n-0 (6)

The differential cepstrum avoids the complicated logarithm and phase wrapping operations existed in the log cepstrum, and have some interesting properties, such as shift invariance, delay time measurement and scale standardization [l], [2]. The properties of the differential cepstrum can be summarized as follows:

PI: The differential cepstrum is shift and scale invariant. If y(n) = Ax( n - r), then

&(n)

= ;p

(

,.

for n #l

for n =l

where A is a scale factor, r is a delay and n,(n) is the differential cepstrum of a normalized exponential sequence x(n) without delay.

P2: The differential cepstrum z?~ (n) is related its correspond- ing log-cepstrum .?(n) and the original sequence x(n).

$x(r)

zX,w %,(n+l) + dz 2(n) = n ’ n#O -= X(z) 4(z) X2(z) * (2) and m Manuscript received March 16,198s.

S.-C. Pei is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, Republic of China.

S.-T. Lu is with the Institute of Computer Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China.

-(n-l)x(n-1)= c %,(k)x(n-k). (8)

k--m

P3: If x(n) is minimum phase (no poles or zeros outside the

(2)

P5: If x(n) is of finite duration, n,(n) will nevertheless have infinite duration.

III. MINIMUM PHASE FILTER DESIGN

Following Herrman and Schuessler’s method [2], the relation- ship between linear phase H(z), second-order zeros H,(z) and minimum phase H2(z) is presented as following:

H,(z) = H(z) + 82Z-(N-1)‘2 H,(z) =Z-(N-l)‘2H2(Z).H2(2-1)

H2( ej”) =\lH,( eJw) (11)

where 6, and 8, are the passband and stopband ripples of the filter H(z). Due to the very simple relationship between the linear phase and minimum phase cepstral sequences Z,(n) and c,(n) associated, respectively, with H,(z) and H2(z), the mini- mum phase filter impulse response h,(n) can be obtained from

:2(n) [31.

?2(n) =:[~l(n)+~,(-n)l, n>O

h2(n)=c2(n)h2(0)+flfj1kc2(k)h(n-k),

k-0 n

O,<n<(N-1)/2. (12) The ?,(n) can be eificiently calculated by its corresponding differential cepstrum hdl( n), by Property 2

hd,(n

+I)

&(n) =

n

,

n+O

then

Z2(n) =-+-[i,,(n+l)-h,,(-n+l)]. (13)

In conclusion, the steps necessary to obtain h2( n) from h,(n) are the following:

1) Prepare h,(n) of length N (odd).

2) Choose p > 1 and p = 1 for avoiding the zeros on the unit circle.

(i) HI + (FFT),( p-“h,(n)}, L B N to reduce the aliasing error.

(4 Hl+ (FFT)L(np-“h(n)) (3 p(n) + (IFFT)L{ - HI/HI 1

(iv) hdl(n+l)+p”.d(n), -(N+1)/2<n<(N+1)/2. s 3) Calculate E2(n) from h,,(n) by (13).

C2(n) +--&[i,,(n+l)-i,,(-n+l)], O<n<- N+l 2 .

C t(n)

L n-o

6) Scaling the impulse response t(n)

N-l h2(n) + Ret(n), OdnG-

2

Then h 2 ( n) is the desired minimum phase filter impulse response of length (N + 1)/2.

7) The passband and stopband ripples 8; and 8; of minimum phase filter is obtained as follows [2]:

(14)

A choice L > 8N, L =1024, and p s 1.026 [3] worked quite well in all the examples considered.

IV. EXPERIMENTAL RESULTS

The proposed method has been tested on a large number of filters on a VAX-11 computer. The linear phase filter H(z) were first designed by the popular computer program given in [l]. The accuracy of the procedure has been measured b the deviation of 1 H,( eJw) 1 from its theoretical value li 1 H,( e”“) 1 . Tests carried out have shown’ that all examples obtain accurate results for 9 filters in Table I using single precision arithmetic; Some of design examples for equiripple minimum phase filters are shown in Figs. land2.

V. CONCLUSIONS

In this paper, the differential cepstrum is introduced to design equiripple minimum phase FIR filters by using cepstral decon- volution; This fast procedure avoids the complex logarithm, phase wrapping and polynomial root-finding algorithm, and re- quires only three FFT computation and a few other operations. Some design examples are illustrated to show this effective ap- proach using differential cepstrum.

Ill

121

[31

REFERENCES

J. H. McClellan, T. W. Parks, and L. R. Rabiner, “A computer program for designing optimum FIR linear phase digital filters,” IEEE Tram. Audio Electroncourt.. vol. AU-21. DD. 506-526. Dec. 1973.

0. Herrmann and-G. W. Schuessikr “Design df nonrecursive digital filters with minimum-phase,” Electron. Lett., vol. 6, no. 11, pp. 329-330, May

1970.

G. A. Mian and A. P. Nainer, “A fast procedure to design equiripple niinimum-phase FIR filters,” IEEE Trans. Circuits Syst., vol. CAS-29, pp. 327-331, May 1982.

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512 IEEE-TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 5, MAY 1986

. ..I- LINEAR PHASE SEPUENCP HI : Hl( 11=- .93,E3,9977E-oa YIS 2J=O.I9?95931?3E-01 “1, a,*-.!3690?90189E-02 Ll‘ +~=o.S236010133E-DI “14 3>.0.313¶D?*o34E-Dr. Ii,, f,1:D.93991ZPl~lE-~1 “1 a 7J ~D.lI6962~4IOE-Dl “t I B)~-.3?9~1416DbL-D1 H, , 9J =D.JIMII~KSBE-02 ,,l t1DJ.-.11399024J~E-01 HlclIJ~-.lD9939634sE-Dl

I+1 C Ii, 90.3 II&?46,EtDD HI t ISlfD.Sbb2963006E-01 Hlclb~~O.*Ja967JJS9Etno II(I t L7J=b-SKbZS~3DD9E-D2 ~llI~J.-.L934lILO3ZE-D1 H1~?00)=D.**~3Sl6E9IE-D1 u HI c~LJ=-.37S990~4lEE-DI HI l?3J~D.34W~I+LS.SE-DZ H, (?*)a-. 379JS+ILDLS-DL WJllSJrD.LlbSCZ?~JD6-01 I41 ~~4J~0.43481??70IE-D1 I+: SE11 ~0.313.5024024E-DZ HI f11J=D.3~3COI~1351-DI “1 tag,=-. 569629Dl&!JE-02 “,,SDJ*D.19ZSKO3rlJI-DI HltJlJ:-.*j?2379877E-02 FREOUENCY (a) z ‘0.00 Lt.20 0.40 0.60 0.m I .oo I FREBUENCY rc: d. 4 00 0’. 90 0: YO 0.60 0.80 I.90 FREPUENCY (b)

Fig. 1. (a) Impulse and magnitude responses of an equiripple linear phase bandstop filter of length 31. (b) Log magnitude and phase responses of an equiripple linear phase bmdstop filter of length 31.

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IIWLWJPI PHAJE SIPUE”CE HZ : HZ, IJaO.II3OLS&I34E+OD Hi!' ~J=D.l3+9133953C-01 WC, 3J=D.b3903&?I7,E+DD “2’ l ,=0.,1 z bJ 9 -01 “Z, JJ=D.ObLZ?QDJ34C+DD HLS LI,-. l6EIC1788BE-01 ,ll' 7J=D.9969749~36i-0, I421 .9J=-.2666386?66i-*I n/1 9J=-.,4rorlzz6zE+aO ~~~;~~.,.333~0976DlE-o? =-.931b35J39?E-DL llZ,i~J=D.ISSCyuII34Z-D, ~1l131=D.$819791~2~E-01 W~,IYJ=D.9189+75,28L-02 W3llJJ=3.9093?81331E-Ol )I~~l6J=-.aO33E9WSJc-o1 P

‘0 .a0 0. 20 O!VO 0.60 0.90 I.00 1

FREQUENCY

00

FREQUENCY

Cc) (4

(c) Impulse and magnitude responses of an equiripple minimum phase band stop filter of length 16. (d) Log magnitude and phase responses of an equiripple minimum phase bandstop filter of length 16.

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514 IEEE-TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 5, MAY 1986 ,.... wtIb4a PMM s64ttWE H) ,I,, IJ=-.6,29 ",a ZJ-0.1409 WI, 3J=D.3f47 Hl' 4J=O.49?4 HII 51'0.336L NII 61 =-.I112 ac-4 ZJ-&bW WI' 9J=-.37C ‘,I* 9J*D.I+3I ", , IO, ~0. JYZ! ",'lo=O.Z66I J,I'lZJ-O.,IBJ -wct3iurbu I, (10 *0.1414 Hl',5,=0.,~# jj,cIrJ=-.3761 N1',7J:-.3$75 HIcrgl:-.IIlJ IuIIsJ~n.s3u Hl,3DJ=D.+9,4 ,J,'ZlJ=D.9541 ",,,.P.J*~.,‘sD~ "t 'ZlJ=-.6JZf 646ZDE-Dt I I7SsE-OI '555336-bJ 89ICl3t-01 '191U?E-OL ', IWDE-DJ I4734 x-0, ISSCDIE-Dl :JlOI41-61 WbZOCE+D6 1775466400 '3s17ai*ou tw4bc+ao I+S2OW*DO JIOt4E-Dl IPZLOIE-0) !P13li~L-D1 rll43OE-01 ‘,z+aac-aJ 161*61E-0, u3BJJe’oI !, I’I+SE-DJ rb4OZOPOL CI 00 0.20 o.qo 0.60 0.80 I.00 FREQUENCY d

‘a! 00 0.20 o.uo 0.60 0: 00 I.00 1

FREQUENCY

z j

‘p. 00 0: a0 O.YO 0.60 0.80 1.00 I

FREQUENCY

Fig. 2. Impuse and magnitude responses of an equiripple linear phase low- pass filter of length 23. (b) Log magnitude and phase responses of an equiripple linear phase low-pass filter of length 23. (c) Impulse and magni- tude responses of an eq&pple minimum phase low-pass filter of length 12. (d) Log magnitude and phase responses of an equiripple minimum phase low-pass filter of length 12.

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FREOUENCY

i

Il.00 0’. 10 b.,P d. b0 P.IO I.00 F~~EDUENCT

01 FREOUE~C~

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516 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 5, MAY 1986

TABLE I

[4] A. Polydoros and A. T. Fan, “The differential cepstmm: definition and properties,” in Proc. IEEE Int. Symp. on Circuits and Systems, pp. 77-80, Apr. 1981.

[5] A. Polydoros, W. Au, and A. T. Fan, “Shift-invariant homomorphic filtering,” in Proc. 22nd Midwest Symp. on Circuits and Systems, pp. 140-145,1979.

Accurate F’redistorted Design of Single Op Amp Active Filters

JACEK M. ZURADA

Abstract-A general method of forcing the transfer function of a second-order active filter to have poles in the desired location is presented. Closed-form formulas for predistorted design of commonly used filters have been derived and tabulated for designer’s reference.

Most of the previous work on the finite gain-bandwidth (GB) effects in active filters pioneered by [l] has centered mainly on the analysis of the pole shift from its nominal position to an undesired position. In such an analytical approach, the loci of actual poles or incremental sensitivities of nominal pole due to finite GB may be found [2].

The technique outlined below is aimed toward the exact predis- torted design, and represents a synthetic approach. It is based on a general method of forcing a characteristic equation of order n to have n desired solutions by adjusting its polynomial coeffi- cients [3].

In recent literature, predistortion problem of active networks has been dealt with in [4]-[6]. Given the GB values of op amps and desired value of pole frequency o, and selectivity q, the

designed filter may be predistorted so that the actual filter has the desired pole as well as unavoidable parasitic pole(s). The approach in this paper yields accurate predistorted design for- mulas and does not involve numerical methods for solving result- ing nonlinear equations.

The pole defining equation for a practical filter has the form

n-1

s”+ c Ckpk(S)sk=O k-0

where the parasitic effects are represented by the nonideal multi-

Manuscript received October 22, 1985; revised January 2, 1986.

The author is with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115.

IEEE Log Number 8607792.

pliers pk( s) and the coefficients C, are to be chosen so as to

force (1) to have n desired solutions. The nonideal multipliers are unity for the case of ideal op amps having GB at infinity [3]. For a second-order filter with desired solutions at

ZI=ZT=-s+ jw, \i

I-'.

2q 4q2

Equation (1) may be replaced with two simultaneous equations: z,‘+P~(zi)c~zj+P~(zi)clJ=o~ i =1,2. (3) The set above can be used to calculate two required coefficients C,, C,,. For 2nd order filters discussed below, expressions for predistorted natural pole frequency w,, and selectivity q are

developed, rather than C,, C,,.

PREDISTORTIONOFA SINGLECOMFJLEXPAIROF ROOTS

If each filter op-amp is characterized by the transfer function G(s) = - GB/s, then a practical pole yielding equation of a 2nd order filter can be written as

5 hkSk + h,s2 + h,% + cd; = 0 (4)

k=3 %

where n - 2 is the number of op-amps used to implement the filter. Obviously, h, = h, . . . =Oandh,=h,=lfortheidealop

amp case.

Equation (4) can always be rewritten to the form similar to (3)

s2+p~(s)~s+po(s)+o (5)

where pi(s), p,(s) are nonideal multipliers which are defined as

follows p,(s) p h, + i hksk-2 i k=3 i -1 (64 p,(s) 4 h,po(s). (6b)

In all equations above wo, q. denote pole frequency and selectiv-

ity of an apparent pole which is not to be practically achieved for finite GB values. For a second-order filter with desired w,, q

values, one has to force two roots of the equation (5) to be at z and z* where

z=-0” /2 q + jw,J1-1/4qz.

Then (5) will hold for s = z, and

(7)

From the above equation, q. may be easily calculated as

aidOz

‘lo = - z2 +/3&J; (9) where aApl(z), /3spo(z).

Because q. is the real number, condition q. = q$ can be used

to solve for o. as follows:

0; =

Im( o*z)

Im( az/3*) w,‘.

(10)

Expressions (10) and (9) give the predistorted values of w,,, q,

being targeted pole parameters expressed in standard design values of wo, qo. These w. , q. values are describing the predis-

torted or apparent pole value of the system with desired poles z,

數據

Fig.  1.  (a)  Impulse  and  magnitude  responses  of  an  equiripple  linear  phase  bandstop  filter  of  length  31
Fig.  2.  Impuse  and  magnitude  responses  of  an  equiripple  linear  phase  low-  pass  filter  of  length  23

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