Complex Eigenfilter

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I1 ANALOG AND DIGITAL SIGNAL PROCESSING,VOL. 40, O. 1, JANUARY 1993

ComplexComplex Eigenfilter Design of ArbitraryCoefficient FIR Digital FiltersPei, Senior Member, IEEE, and Jong-Jy Shyu, Member, IEEE

Abstract-The real eigenfilter approach is extended to complexcases for designing arbitrary complex FIR filters. By minimizinga quadratic measure of the error in the passband and stopband,a complex eigenvector of an appropriate complex, Hermitiansymmetric, and positive-definite matrix is computed to get thefilter coefficients. Several arbitrary magnitude and phase FIRfilters, such as multiple passband complex filters and staircase-delay allpass phase equalizers, can be easily designed by thisapproach. This method can be easily extended to design 2-Dcomplex FIR filters. Also, if an appropriate iterative processis used, equiripple filters in the complex Chebyshev sense canbe obtained. Several numerical design examples are presented,which demonstrate the usefulness of the approach.

I. INTRODUCTIONAIDYANATHAN and Nguyen have recently introducedV he eigenfilter approach for designing linear phase realFIR filters [ l] . Comparison to the well-known McClel-lan-Parks algorithm for minimax equiripple filters shows thatboth are optimal in the sense of different minimum normsof the error function. The equiripple filters are optimal in theminimax sense, in which the maximum of the errors is smallestfor a given set of specifications, and the eigenfilters are optimalin the least squares sense that the total quadratic error in thepassband and stopband is the minimum. The unique advantageof the eigenfilter approach over the McClellan-Parks algorithmis that it is general enough to incorporate both time- andfrequency-domain constraints. Compared to the weightingleast squares method [2], the eigen-approach needs to computethe eigenvalue; however, large matrix inversion is necessaryfor the weighting least squares approach. Both methods havetheir respective features. The eigenfilter method involvesminimizing a quadratic measure of the error in which areal eigenvector of an appropriate matrix is computed toget the filter coefficients. Pei and Shyu have extended theeigen-approach to the design of FIR Hilbert transformers anddifferentiators [31, high-order digital differentiators [4], and 2-D FIR filters [5]. All the above filters deal with real filtercoefficient cases. Recently, Nguyen considered the designof complex-coefficient linear-phase and arbitrary-phase FIReigenfilters with arbitrary magnitude responses [6], [7]. In

Manuscript received August 21, 1991; revised August 7, 1992 and October27, 1992. This work was supported by the National Science Council ofthe Republic of China under Grant NSC 804404-EOO2-14. This paper wasrecommended by Associate Editor Y. C. Lim.S.-C. Pei is with the Department of Electrical Engineering, National TaiwanUniversity, Taipei, Taiwan, Republic of China.J.-J. Shyu is with the Department of Computer Science and Engineering,Tatung Institute of Technology, Taipei, Taiwan, R epublic of China.IEEE Log Number 9206248.

[7], he converted the complex-coefficient design problem intothe real-coefficient design problem. However, this real designproblem formulation will become twice as large as the complexdesign problem presented in this paper. Therefore, the matrixelements and the eigenvector computation will increase due toits doubled matrix size [7].In this paper, we propose a new eigen-approach to designingarbitrary complex FIR filters directly in the complex domain.By minimizing a quadratic measure of the error in the pass-band and stopband, a complex eigenvector of an appropriatecomplex, Hermitian symmetric, and positive-definite matrix iscomputed to get the complex-valued filter coefficients. Thiscomplex matrix size is the same as the real case; the onlydifference is that the matrix elements and the eigenvector arecomplex valued. In Section I1 we formulate the new approachfor the design of 1-D complex filters, and include severalarbitrary magnitude and phase FIR filter design examples.In Section 111, we extend the new approach to design 2-D complex filters; and in Section IV we present a methodfor designing the equiripple complex eigenfilters in whichthe complex errors are equiripple both in the passbands andstopbands. Finally, in Section V we give a summary.

11. EIGENFILTERORMULATIONOR THE DESIGNOF 1-D COMPLEX FIR DIGITAL ILTERS

For a complex coefficient FIR digital filter, its frequencyresponse can be characterized by

N - l~ ( w ) C a ( n ) e - j n w (1)n=O

where N is the filter length, and the filter coefficients a(.) arecomplex valued. Defining the column vectorsA = [a ( O) ,~ ( l ) ,(2),-..,a(N-)It (2)

and- j ( N - 1 ) w tC (W ) = [I, e-J w, e - - jZw . . . , e 1 > (3 )

then we can rewrite (1) asH ( w ) = AtC(w)= C t ( w ) A (4)

where t is the vector transpose operation. Now we wish to use(4) to approximate the following desired frequency response:~ ( w ) M ( u ) e j P ( w ) ( 5 )

where M ( w ) and P ( w ) are the desired filter magnitude andphase responses, respectively. As in the real eigen-approach1057-7130/93$03.OO 0 1993 IEEE


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PE1 AND SHYU: OMPLEX EIGENFILTER DESIGN 33

[4], a normalized factor H ( w o ) / D ( w o ) s added to D ( w ) suchthat the actual frequency response at the reference frequencyWO is approximately equal to the desired value [4]. Thepassband and stopband error functions become

in the passband with cutoff frequencieswp, and w p u , nd

in the stopband with cutoff frequencies w, , and w,,,. As tothe choice of the reference frequency WO,we will make adetailed discussion in Example 1 . However, in general, wechoose the reference frequency at the centerof the passband inour eigenfilter approach [4]. Equation (6) can be reformulatedinto

= A ~ Q , Aand (7) becomes

E, = l::A t C ( w ) ] * [ C t ( w ) A ]w= A H{ : C * ( w ) b ( w ) dw A1= A ~ Q , A

where

Q, = ~ Y , U C * ( ~ ) C t ( ~ )wI

and H denotes the Hermitian conjugate transpose operator.Hence, the elements of matrix Q p and Q , are given by

andq,(lc, 2 ) = : : Y e j kWe - j w dw 0 5 IC, 15 N - 1, (13).

respectively.Assume there are K passbands and L stopbands in thedesired filter, then the total error measure to be minimized isE = a i E p l + a z E P , + . ..+~ K E ~ ~,&Esl+ PZESZ . + P L E ~ L

= A H [ a i Q p l+wQpz . .+ Q ~ K Q ~ ~Pi&,,= A ~ Q A i,14)

where a%, = 1, 2 , - . . , K and Pi, = 1, 2 , . . - , L re theweighting constants, andQ = aiQpl+ a2Qpz+ . .+ ~ K Q ~ ~

+PzQs2+ . . + PLQ,,IA

+Pi&,, + P 2 Q s 2 + . .+ P L Q , ~ . ( 15)Notice that Q is an N x N complex, Hermitian symmetric andpositive-definite matrix (instead of a 2 N x 2N real positivedefinite-matrix as in Nguyens cases [6]) and can be computedby numerical integration using(12) and (13).Once Q is found,the solution vector is the complex eigenvector correspondingto the smallest eigenvalue of matrix Q in view of the well-known Rayleighs principle[8]. Since we are interested in onlyone eigenvector, this computation can be done efficiently bythe gradient technique [9] or the iterative power method [101.Example 1: Design of a Single-Passband Complex Filter

In this example, we consider the design of a length N = 41single-passband complex filter with passband [-0.05, 0.151,stopbands [-0.5, -0.091, [0.19, 0.51, and a constant groupdelay 16 in the passband, i.e.,-T 5 w 5-0.187~ and 0 . 3 8 ~ w 5 T .(16)D ( w )=When a1 = 1 and P1 = 82 = 2 are used for weighting in thepassband and stopband,Fig. 1 illustrates the total least squareerror curve with respect to the different reference frequencywo which varies within the passband. The 10 local minimumeigenfilter errors are almost equal to 3.29 x Compared

to the true least squares solution [ 111 , the frequency locations,in which the local minimum absolute complex errors occur forthe true least squares approach, are found to coincide with theabove 10 local minimum reference frequencies in the eigen-approach, and these are tabulated in Table I, respectively.The true least squares error is 3.285 x It is observedthat when the reference frequency wo is chosen just at thelocations in which the local minimum absolute errors occurfor the true least squares approach, the total least squareseigenfilter errors are also minimum with respect to the otherfrequencies, and almost equal to the true least squares error.This means that the proposed eigenfilter method can be usedto design filters with truly least squares errors at these localminimum reference frequency points. But these frequenciesare not known in advance, so we generally choose the central

e - j 1 6 ~ - 0 . 1 ~ w 5 0 . 3 ~{ 0,


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34 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I1 ANALOG A ND DIGITAL SIGNAL PROCESSING, VOL. 40, NO. 1 , JANUARY 1993

12

15

approach-0.044 -0.044-0.027 -0.026

NORMALIZED REFERENCE FREQUENCYFig. 1. Total least squares error curve with respect to several differentreference frequencies for the eigen-approach.

TABLE ITHEFREQUENCYOCATIONST WHICH THE LOCAL MINIMUM &SOLUTE

COMPLEX ERRORSCCUR FOR THE TRULY LEASTQUARE S APPROACH, COINCIDEWITH THE LOCAL MINIMUMEFERENCE FREQUENCIESN THE EIGEN-APPROACH

3 I -0.005 I -0.0060.0170.0390.0610.0830.1050.1270.144

0.0160.040.0620.0840.1060.1260.144

frequency of the passband as the reference frequency dueto its performance (the total least squares eigenfilter error is3.4 x is satisfactory and good enough. The frequencymagnitude response and group-delay response are shown inFig. 2(a) and (b), respectively, while the trace of complexerrors in the passband and stopband are shown in Fig. 2(c)and (d). The complex filter coefficients are given in Table 11,and the related results, such as peak magnitude of the complexerror, peak group delay error, etc., are tabulated in Table I11accompanying those of the other examples.Example 2: Design of a Staircase-DelayAllpass Phase Equalizergroup delay to be designed by the eigenfilter approach:

A length N = 39 allpass filter has the following desired18,16,

-T 5 w 5 - 0 . 7 ~- 0 . 7 ~< w 5 - 0 . 1 ~

19, -0.lT < w 5 0.2T (17)22,18,

0 . 2 ~ w 5 0 . 925~0 . 925~ w 5 T .

I

NORMALIZED FREQUENCY(a)

30

25

5 2oU' 15a0a IO

5

-8.5 -0 .25 0.0 0.25 0 . 5NORMALIZED FREQUENCY

Fig. 2. Example I: Design of single-passband complex FIR filter withN = 41, 7 = 16. (a) Magnitude response in decibels; (b) group delayresponse.The resultant magnitude response, group-delay response, andtrace of the complex error are all given in Fig. 3.

111. DESIGN F 2-D COMPLEX FIR DIGITAL ILTERSThe design method described in Section I1 for 1-D complexfilters can be extended to 2-D complex filter design. For a2-D complex FIR digital filter, its frequency response can becharacterized as

For simplicity, let Nl N2 N . DefiningA = [Ah,A:,*..,AL_1lt (19)

and

where


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PE1 AND SHYU COMPLEX EIGENFILTER DESIGN 35

N-%

E 3:+< 0 -t4z -3-

stopband region, respectively, and ( ~ 1 0 , po) is the referencefrequency point in the passband.in A, which will lead to the eigenformulation E = AtQA in

(26)

- This enables us to write the total error E as a quadraticwhich the matrix Q is-

Q = oQp+ PQ,where CY and P are the weighting constants.7V Example 3: Design of a 2 - 0 Circular-Pas sband Complex Filter1I In this example, we consider the design of an 11 x 112-Dfilter with the desired response& ' -3' ' I ' 0" ' IJ ' ' ' I 'R E A L P A R T 10**(-2)

N"

t4 -21 13

R E A L P A R T 10**(-2)Id \

2 0.5T.The design specification is illustrated in Fig. 4(a). If fk =P = 1 are used, Fig. 4(b) shows the magnitude responseand Fig. 4(c) and (d) show the group delay response alongw1- and wp-axis, respectively. Comparing this approach to thefrequency-shift method by multiplying the 2-D lowpass filtercoefficients by a sequence e - j 0 .2n1Te j0 .3n2T ,he frequency-shift method limits the shapes of the complex coefficient filtersthat can be designed, and often results in overconstraint andoverdesigning [121.

Iv . DESIGNF 1-D EQUIRIPPLEOMPLEX EIGENFILTEKS,-, The eigenfilters described in Sections I1 and 111 are alloptimal in the least squares sense subject to the chosenreference frequency constraint. In order to achieve an optimalequiripple complexFIR filter in the complex Chebyshev sense,we can also use the eigen-approach iteratively by incorporating

Fig.2. (conrinued) (c) Trace of complex error in the passband [-0.05, 0.151;(d) trace of complex error in the stopband 10.19, 0.91).

ande - j i ~ l e - j ~ 2. e - j i ~ l j ( ~ - i ) ~ ~a suitable nonuniform weighting function W ( w ) into theintegrands as follows:i ( w 1 ,w2) [e-jzwl, 7 ,0 5 2 5 N - 1. (22)

Qp = W w ) ~ C ( W O )( w ) - (U)] *Then ( 1 8) can be rewritten asWP l

I l ( w 1 , ~ 2 ) A t C ( w l , u p ) = C t ( w l , wp )A. (23)Similarly, when the desired response D ( w 1 , w p) is to beapproximated by H ( w 1 , wg) , the error function is and

Q, = IW'"( w ) C * ( w ) C t ( w )w. (29)WQ

For uniform weighting, the eigenfilter's errors are usually largethe band edges. Vaidyanathan and Nguyen have proposed thatthe error responses at the kth iteration are composed to bethe weighting function for the ( I C + 1)th iteration [l]. Theresulting weighting is larger near the band edges than the otherfrequencies, and the error tends to get equalized. Once themore with further iterations and we can stop the iteration. Aserious drawback to the Vaidyanathan and Nguyen algorithm

. [E, = AH{JIw,*wl, w z ) C t ( w l ,w 2 )

D ( w l ' w 2 )D ( W l 0 , w20)

c(wlo,z o )- A(24)

near the band edges and tend to fall off at points away from= A ~ Q , A

in the passband, andbp errors become equiripple, the peak errors do not change any

= A ~ Q , A (25)


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Type of filter(length)Example Desired group Peak magnitude ofdelay in passband complex error inpassband(stouband)1

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 40,NO . 1, JANUARY 1993

TABLE I1FILTERCOEFFICIENTSN EXAMPLE AND 4n EXAMPLE 1 EXAMPLE 40 -1.04040523-03 +i3.20212053-03 -3.52114353-03 +i1.08084333-0212345678910111213141516171819202122232425262728293031323334353637383940

2.46098733-08 + -6.34191193-032.02949603-03 +j -6.24605783-031.29971543-03 +j -1.78883493-03

-4.90557123-03 +j 3.56413593-03-1.36924463-02 +j 4.44892193-03-1.7021395E-02 +j -4.12798763-08-8.93759453-03 +j -2.90403843-036.63775443-03 +j 4.82258483-031.69200323-02 +j 2.92884443-021.23673673-02 +j 3.80630383-02

-1.02688523-08 +j 2.91033933-023.34126803-03 +j -1.02833063-024.44648943-02 +j -6.12005743-020.1211845 +j -8.80455673-020.2005258 +] -6.51547543-020.2381226 +] -3.84835063-080.2104345 +j 6.83741943-020.1334420 +] 9.69511203-025.13321763-02 +3 7.06524853-023.97111573-03 +j 1.22217263-02-5.72921273-08 +j -3.75251253-021.67052393-02 +j -5.14138903-022.39627833-02 +j -3.29820923-029.63401613-03 +j -6.99956583-03-1.46935453-02 +j 4.77426083-03-2.90285423-02 +j 1.13247373-07-2.43987093-02 +j -7.92753043-033.49193253-03 +j 4.80608873-034.93966143-03 +j 1.52023653-021.47575653-07 +j 1.59661923-02-2.51658213-03 +j 7.74552073-031.10892943-03 +j -1.52630243-037.04672653-03 +j -5.11984533-039.29040743-03 +j -3.01879763-035.99433483-03 +j -1.23281103-074.77573783-04 +j 1.55133403-04-2.87572293-03 +j -2.08930323-03

-2.72142633-03 +j -3.74561923-03-9.77169143-04 +j -3.00722523-03

-8.70706703-03 +j -6.32608223-03

2.72504993-06 +j -1.10857943-023.05758213-03 +j -9.37990473-031.77064503-03 +j -2.41270013-03

-i.63001463-02 +j 5.31647593-03-1.9100901E-02 +j 1.91521833-05-9.05945623-03 +j -2.92498133-038.15046763-03 +j 5.93864683-031.84755253-02 +j 2.54481923-021.28689023-02 +j 3.96508803-02-9.97167083-06 +j 2.84606533-024.27636273-03 +j -1.31628363-024.69713663-02 +j -6.46293013-020.1242412 +j -9.02520423-020.2024797 +j -6.57793883-020.2383711 +] 9.84221703-060.2099234 +] 6.82207353-020.1335258 +j 9.70173783-025.22963333-02 +j 7.19734653-024.88967503-03 +j 1.50307623-022.13552273-06 +j -3.43954043-021.61139563-02 +j -4.96035073-022.40146413-02 +j -3.30646863-021.10923813-02 +j -8.07846523-03

-1.24105453-02 +j 4.00794443-03-2.72692703-02 +j -2.53859913-05-2.39119343-02 +j -7.78846443-033.25269003-03 +j 4.44632353-035.08833163-03 +j 1.56109333-026.68177383-06 +j 1.74253753-02

-3.07402673-03 +j 9.46625233-037.46755443-03 +j -5.43537553-031.13038953-02 +j -3.68730963-038.87902173-03 +j -1.52918513-052.78988363-03 +j 8.93657333-04

-2.02928693-03 +j -1.48483943-03-3.01974663-03 +j -4.16712933-03-2.44586913-03 +j -7.51021923-03

-6.21333303-03 +j 4.53489093-03

-9-04520883-03 +j -6.59457363-03

4.86179663-04 +j -6.73708333-04

TABLE 111ARBITRARYOMPLEXIR FILTERDESIG N XAMPLES

Single-passband(41)S taircase-delayallpass (39)2-D circularpassband (1 1 x11)Equiripple singlepassband (41 )

1618, 16.19,22, 18w1 - x i s : 4w2 - axis : 416

0.05758(0.0367 1)0.06 1720 1 3.77- -4.45W2 : 3.78- -4.440.03420(0.01137)

Actual groupdelay in passband(peak delay error)15.64-17.23(1.23)15.74-22.36(1.43)0.08774(0.09671)15.38-17.12

Design time inseconds on VAX87001.6855.64525

306

Figure

6

is that if the errors at certain frequencies vanish after someiterations, then the weights at those frequencies will become 4) 6,: max { T ~ % } ,5) p s : min { T ~ ~ } .zero at subsequent iterations, which will induce large errorsand lead to divergence afterwards. Hence, some modifications[13] are needed to improve the above algorithm.

Before describing the modified algorithm, some notationsare defined as below:1) rP;:he ith absolute complex error ripple in the passbandwith ripple interval ( w ~ , - ~ ,,,],\thinspace \thinspace

r,, : he ith absolute complex error ripple in the stopbandwith ripple interval (w,,-~,w,,],2) 6,: max { T ~ . } ,3) P p : min { T p , } ,

The proposed iterative method for single-passband filterdesign is illustrated in Fig. 5 and is described in detail below.Step 1: Initiate the weighting functionw E passband{ R 2 = ( % ) 2 , w E stopband (30)1 1W(w) =

where R is the desired passband to stopband ripple error ratio.Step 2: Find the coefficient vector A using the eigen-approach.Step 3: Search for rp , ,T, , , S,, 6,,pp , and p, .


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PE1 AND SHYU: COMPLEX EIGENFILTER DESIGN 31

,.m e! 63 4P6iw r; -2

i -=n

zUIIW" 0L*O F3 -4

I --ea0.21 0.1

N O R M A L I Z E D F R E Q UE N C Y

N O R M A L I Z E D F R EQ U E N C Y

(b)

I-6n

(9I

-4 I V(b)

Fig.4. Example 3: Design of 2-D circular-passband complex FIR filter withthe center at ( - 0 . 2 ~ , . 3 ~ )nd group delays ~~1 = 4, T,Z = 4. (a) Designspecification; (b) magnitude response.and

Fig. 3. Example 2: Design of stair-case-delay allpass phase equalizer withfive different group delays: T = 18, 16, 19,22, 8 in each band. (a)Magnitude response in dB; @) group delay response; (c) trace of complexerror.

where Ep and ES are the Preassigned Ve rY small Positiveconstants. If the condition is satisfied, then the largest errorripples are almost equal to the smallest error ripples in boththe passband and stopband, and then go to Step 6; otherwisego to the next step.Step 5: Compute the unnormalized weighting functionStep 4: Check whether the complex error is nearly equirip-Ple by

I p6 - P6


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38 IEEE TRANSAC TIONS ON CIRCUITS AN D SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 40, NO. 1 , JANUARY 1993

( 4Fig. 4. continued) (c) Group delay along wl-axis; (d) group delay alongwz-axis.

where I, and I, are the number of ripples in the passband andstopband, respectively, and find its maximum and minimumvalue

Sw,= max { @ ( w ) , w E passband), (34)

6ws = m a x { @ ( w ) , w E stopband}. (35)Then update the weighting function by

(36)w E passbandR W ( w ) / 6 w s w E stopband,and go to Step 2.

Initialize Weighting Function1, w E passbandR, w E stopband

Find A using eigen-approachwearch for r pi, rsi. Bp3 6s Update W ( w)

using (36).

NO . I \AR = R,r

Fig. 5. Flowchart for the design of equiripple single-passband complex filter.

Step 6: Check whether the actual absolute complex errorripple is nearly the same as R by(37)

where t is also a predetermined very small positive constant.If the condition is met, then stop the process; otherwise updatethe value of R by R = R whereR = ($)'(?)'R,

and go to Step 5 .Example 4: Design of an EquirippleSingle-Passband Com plex Filter

The specification of the single-passband filter is the sameas Example 1, and the ripple error ratio R = 32 is required.When e p = 6 , = E = 0.02 are used, the design takes nineiterations to converge. Fig. 6(a) and (b) shows the magnitude


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PE1 AND SHYU: OMPLEX EIGENFILTER DESIGN 39

.mU

Wat;

- 7 0

-J& - b !Zb b!o 0 . 2 5 0 . 5N O R M A L I Z E D F R E Q U E N C Y

c I

-549.0 -2.5 0.0 2 . 5 5 . 0R E A L P A R T 1O** (-2)

2s30; l:k 1

I S 5 0- -4r

a I0 t.U 1 0LI1 Q

0z - 7..

5

-

-8.6 - 0. 2 5 0 . 0 0 . 2 5 0 . 5N O R M A L I Z E D F R E Q U E N C Y

(b)Fig. 6. Example 4: Equiripple single-passband complex FIR filter withN = 41, 7 = 16, R = 3. (a) Magnitude response in decibels; @) groupdelay response. [0.19,.911.

Fig. 6. (continued) (c) Trace of equiripple complex error in the pass-band 1-0.05, 0.151; (d) trace of equiripple complex error in the stopband

response and group delay response, while the traces of thecomplex errors in the passband and stopband are shown inFig. 6(c) and (d), respectively. Notice that the trace curves areall equiripple in the complex Chebyshev sense.V. CONCLUSIONS

In this paper, arbitrary complex coefficient FIR digital filtershave been designed by the eigen-approach in the optimalleast squares sense subject to the chosen reference frequencyconstraint. This method is based on the computation of aneigenvector of an N x N complex, Hermitian symmetric,and positive-definite matrix. The resultant complex eigenvec-tor corresponding to the smallest eigenvalue is the desiredfilter coefficients. Several design examples, including constantgroup-delay complex FIR filters, allpass phase equalizers, 2-Dcomplex FIR filters, and 1-D quiripple complex FIR filters,have been presented to show the effectiveness of this approach.REFERENCES

[ I] P. P. Vaidyanathan and T. Q. Nguyen, Eigenfilter: A new approachto least-squares FIR filter design and applications including Nyquistfilters, IEEE Trans. Circuits Syst., vol. CAS-34, pp. 11-23, Jan. 1987.

[2] Y. C. Lim, J. H. Lee, C. K. Chen, and R. H. Yang, A weighted leastsquares algorithm for quasi-equirippleFIR and W igital filter design,IEEE Trans. Signal Processing, vol. 40, pp. 551-558, Ma r. 1992.[3] S. C. Pei and J. J. Shyu, Design of FIR Hilbert transformers anddifferentiators by eigenfilters, IEEE Trans. Circuits Syst., vol. CAS-35,[4] S. C. Pei and J. J. Shyu, Eigenfilter design of higher order digitaldifferentiators, IEEE Trans. Acoust., Speech, Signal Processing, vol.ASSP -37, pp. 505-511, A pr. 1989.[5] -, 2-D FIR eigenfilters: A least-squares approach, IEEE Trans.Circuits Syst., vol. 37, pp. 2 4 3 4 Jan. 1990.[6] T. Q. Nguyen, The eigenfilter for the design of linear-phase filters witharbitrary magnitud e response, in Pro c. IEEE Int. Con$ A cous t., Speech,SignaE Processing, pp. 1981-1984, May 1991.[7] -, Tbe design of arbitrary FIR digital filters using the eigenfiltermethod, IEEE Trans. Signal Proces sing, to appear.[8] B. Nobel and J. W. Daniel, Applied Linear Algebra. Englewood Cliffs,N J Prentice-Hall, 1977.[9] Y. H. Hu and P. K. Chou, Effective adaptive Pisarenko spectrumestimate, in Proc. IEEE Int. Con$ Aco ust., Speech, Signal Processing,pp. 577-580, Apr. 1986.[lo] J. N. Franklin, Matrix Theory. Englewood Cliffs, NJ: Prentice-Hall,1968.[ l l ] S. M. Kay, Modem Spectral Estimation. Englewood Cliffs, NJ:Prentice-Hall, 1988, Ch. 3.[12] M. T. Mccallig, Design of digital FIR filters with complex conju-gate pulse response, IEEE Trans. Circuits Syst., vol. CAS-25, pp.1103-1105, Dec. 1978.[13] C. Y. Chi and Y. T. Kou, A new self-initiated optimum WLSapproxim ation method for the design of linear phase FIR digital filters,in Proc. IEEE Int. Symp. Circuits Syst., June 1991, pp. 168-171.

pp. 1457-1461, NO V. 1988.


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Soo-Chang Pei (SM90) received the B.S. degreefrom National Taiwan University in 1970 and theM.S. and Ph.D. degrees from the University ofCalifomia, Santa Barbara, in 1972 and 1975, re-spectively, all in electrical engineering.He was an Engineering Officer in the ChineseNavy Shipyard at Peng Fu Island from 1970 to1971, and a Research Assistant at the U niversity ofCalifomia, Santa Barbara, from 1971 to 1975. Hewas Professor and Chairman in the Department ofElectrical Engineering at Tatung Institute of Tech-nology from 1981 to 1983. He is now a Professor in the Department ofElectrical Engineering at National Taiwan University. His research interestsinclude digital signal processing, digital picture processing, optical informationprocessing, laser, and holography.Dr. Pei is a m ember of Eta Keppa Nu and the Optical Society of America.

Jong-Jy S h y (S88-M93) received the B.S. de-gree from the Tatung Institute of Technology, Taipei,Taiwan, in 1983 and the M.S. and Ph.D. degreesfrom the National Taiwan University, Tapei, in1988 and 1992, respectively, all in electrical engi-neering.He was a Research Assistant at the NationalTaiwan University, Taipei, from 1986 to 1991. He iscurrently an Associate Professor in the Departmentof Computer Science and Engineering, Tatung In-stitute of Technology, Taipei. HISesearch interestsinclude filter design, digital signal processmg, and image processing.

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Complex Eigenfilter