132 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 1, JANUARY 1988

Circuits and Svstems Letters J

Arbitrary Phase Polynomials for Sampled-Data Systems

I. H. ZABALAWI AND B. Z. KAHHALEH

Abstract -A simple method is proposed for generating polynomial of degree n to match a prescribed phase at a specified set of frequencies on the unit circle in the 2-plane. This class of polynomials is very useful for designing recursive digital filters with flat group delay or linear phase. Furthermore, the proposed polynomials are easy to adopt for designing all-pass digital circuits.

INTRODUCTION The simplest method to obtain a close match between a

specified response and an approximating one is to obtain equality at the maximum number of possible points [l]. The approxima- tion problem for matching a desired amplitude response and that of an all-pole transfer function at n points on the unit circle in the Z-plane is commonly performed by using Lagrange interpo- lation [2]. However, very little is mentioned in the technical literature about the interpolation of the phase by using Lagrange interpolation [3].

This paper deals with the generation of an n th degree poly- nomial p , , (~) to match a prescribed phase characteristics at n frequencies on the unit circle in the Z-plane by using Lagrange interpolation. Furthermore, the utilization of this class of poly- nomials in sampled-data system will be demonstrated by an example.

GENERATION PROCEDURE

To generate the desired polynomial P,,(z) we consider the following sequence of polynomials:

PI( z) 3 P2 (z ) 3 P3 ( z) ,. . . 7 P, ( z ) . (1)

The properties of this sequence are given in Table I. To develop the general formula for the n th degree polynomial,

let us consider a special but relevent case when n = 4. P4(z) should match the phase at the frequencies

w = O , T w , , T 0 2 , T wg, T 0 4 . (2) By using Lagrange interpolation, we have

P4( z) = a3P3( z) + d,z-'( z - e'"))( z - e - J w 3 )P2(z)

+ d , ~ - ~ ( z - e l w ) ) ( z - e-Jw3)( z - eJm2)( z - e-J*2 )

> P O ( Z )

~ P l ( z ) + d , z - 3 ( z - e J " 3 ) ( z - e ~ J " 3 )

. ( z - e P 2 )( - e - l w 2 ) .( - e J m i ) ( - e - I w l

(3) where P 3 ( z ) , P2(z), Pl(z), and Po(z) match the desired phase at the frequencies listed in Table I.

Relation (3) demonstrates that the lowest power term in P4(z) will be less than - 2 unless the multiplier do = 0. Similarly the hghest power in (3) will be greater than 2 unless we set the multiplier d, = 0.

Manuscnpt received May 18, 1987 The authors are with the Electncal Engmeenng Department, Faculty of

IEEE Log Number 8717026 Engineenng and Technology, Umversity of Jordan, Amman, Jordan

By imposing the constraints do = 0

d, = 0

and without loss of generality we may assume

d, =l .

Consequently, the RHS of (3) becomes

P4( z) = a3 P3 ( z ) + [ z + 2-1 - 2cos( w 3 ) ] P2 ( 2 )

Thus if P2 (z) matches the desired phase at w - 0 , T wl, T w2

w = 0 , T w 1 , T w 2 , Tu3 .

0 = 0, T ol, T 0 2 , T w3.

and P3 (z) matches the desired phase at

Hence P4 (z) satisfies the phase constraints at

Furthermore, the parameter a3 can be freely selected to con- straint the phase of P4(z) to match the desired phase at T U,.

The proper value of a j can be determined as follows. Let

P3(eJU4) = R , ( w , ) + j X , ( w 4 )

P2( e"4) = R 2 ( w4) + j X 2 ( w.,). ( 7 ) The substitution of (7) in (6) at z = exp(jw4) yields

P4( elw4) = R4( w4) + jX4( w4)

= a3 R , ( w4) + 2{ cos w4 -cos w3 } R, ( w4)

tan(+4) = X4(m4)/R4(04) ( 9)

+ j [ a3X3( a) CO COS w4 -COS w3 } X 2 ( a4)] (8)

where + is the desired phase at w . Hence by rearranging (8) and (9) we get

Having determined o3 the polynomial P4 (z) can be generated by using (6).

Now the general case for any n can be deduced from (6) and it can be shown that the polynomial Pn+l(z) is related to P n ( z ) and P,-l(z) for n > 0 by (11):

P,+ 1 ( z) = Q,,P"( z ) + { z + z-1 - 2cos an} e,- 1 ( z ) (11)

where

R,- 1 ( a,+ 1) tan 4(+ 1 - X,- 1 ( U r + 1)

Rr(wr+l) tm+r+l+ X r ( ~ + 1 ) ' Q, = 2{ cos wr - cos or+ 1 }

for r = 1 , 2 , 3 , . . . , n (12)

and the initial polynomials P,(z) and Pl(z) are given by

P o ( z ) = l a n d P l ( z ) =a,,+z (13)

a, =sin(w,)cot(+,)-cos(w,). (14)

OO98-4094/88/01OO-0132$01 .OO 01 988 IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 1, JANUARY 1988

TABLE I

133

Polynomial Phase Matching at

P o ( z ) = 1 P1(Z) = a o + n1z p 2 ( z ) = U _ ~ Z - ’ + no + a l z

w = o w = 0, T w1 w = 0, T wl, T w2

TABLE I1

Parameter Value

Passband attenuation A, 0.1 dB Stopband attenuation A, 40dB Passband edge freq. w, 0.277 rad/s Stopband edge freq. 0.37 rad/s Sampling period T I s

2 c U

4

TABLE I11

no, + al,z + a Z t z 2

bo, + bl,z + b2,z2 H ( z ) =

1 1.oooO -0.4398 1.oooO 0.5965 -1.3593 1.oooO 2 1.oooO -1.1236 1.oooO 0.8708 -1.4644 1.oooO 3 1.oooO 1.oooO 0.oooO -0.6573 1.oooO 0.oooO

TABLE IV

P,(z ) = [ a o + a l z + a 2 z 2 + a 3 z 3 + a 4 z 4 + a5zs1/z3

i U,

0 0.00 1 0.0477 2 0.0877 3 0.1277 4 0.167 5 0.2077

+, 08

0.oooO 0.1304 1.2289 - 0.8124 2.4495 2.0775

4.7301 1.8439 3.6304 - 2.7289

5.6153 - 0.5135

APPLICATION EXAMPLE The proposed set of arbitrary phase polynomials is used to

equalize the group delay response of a fifth-order elliptic recur- sive digital low-pass filter. The specifications of the filter are given in Table 11, and the coefficients of its transfer function are listed in Table 111. The transfer function of the equalizer which

100 , 100

80 1

i 60 -

40 -

2 0 .

0

0.2 0.4 0 . 6 0.8 1.0 I.o F r e q u e n c y i n P i r a d / s e c

Fig. 1.

30 E q u a l i z e d Dealy

0 4 D 0 . 0 4 0.08 0 . 1 2 0.16 (

F r e q u e n c y i n P i r a d / s e c

Fig. 2.

2

has been used is digital filters with linear phase or flat group delay. Furthermore, it can be used as illustrated in the example for designing all-pass digital circuits with arbitrary phase or group delay characteristics. (15)

The parameters of Ps ( z ) are given in Table IV. The unequalized and the equalized responses are shown in Figs. 1 and 2. REFERENCES

[I] R. Gregorian and G. C. Temes, “Design techniques for digital and analog all-pass circuits,” IEEE Truns. Circuits Syst., vol. CAS-25, pp. 981-988, Dec. 1978.

[2] R. E. Chorchiere and L. R. Rabiner, “Internolation and decimation of

CONCLUSION A simple design algorithm is presented for the construction Of - -

arbi t rq- phase polynomials in the discrete domain. The al- gorithm requires evaluation Of a set POlYnOmialS at the interpolat- ing frequencies. The algorithm is well suited to designing all-pole

~

[3] digital signals,” Proc. IEEE, vol. 69, pp. 3001318, ~ a . 1981. I. H. Zabalawi, “Selective filters with finite transmission zeros and arbi- trary phase characteristics,” in Proc. 1986 IEEE Int. Symp. on Circuits and System, pp. 1089-1092, May 1986.