Feasible Regions Defined by Stability Constraints Based on the Argument Principle Ken KOUNO Masahide ABE Masayuki KAWAMATA Department of Electronic Engineering, Graduate School of Engineering, Tohoku University IIR 3 Abstract For the design of IIR digital filters, some methods which simplify the design problem with stability constraints by substituting simple constraints based on the argument principle (AP) for necessary and sufficient conditions for stability were recently proposed. However, it has not been shown how those simplifications will change the feasible region of the filter coefficients, and therefore, the effectiveness of a design method adopting those simplifications has not been proved yet. In this paper we evaluate the feasible regions defined by three possible AP-based constraints. Those constraints are expressed as equations with respect to the filter coefficients. As a result, those feasible regions are shown to be considerably narrow and it is also shown that the regions will not be guaranteed to be included in the stability region. IIR FIR []IIR [ 9] Chebyshev minimax PCLS [] [5] Rouché [7] [8] [9] 3
IIR IIR H(z) = B(z) A(z) = N n= b nz n M m= a mz m = bt φ b (z) + a T φ a (z) () a.5.5 A(z) = + a z + a z.5 b = [b b... b N ] T () a = [a a... a M ] T (3) φ b (z) = [ z... z N ] T (4) φ a (z) = [z z... z M ] T (5).5 3 3 a : IIR () A(z) z [] A(z) a A(z) Schur-Cohn-Fujiwara Jury-Marden [] Jury-Marden () A(z) 3 (I) A() > (II) A( ) > (III) a > a M, a () > a(), a (M ) > a (M ) a (k) l a (k) l = M, a (k ) a (k ) a (k ) M k+ M k+ l a (k ) l, l =,,,, M k, a() > a() M, a() l = a l, k =,,, M (6) M = (i) + a > a (ii) > a A(z) 3 3 3 M = 3 4 M = 4 3 (i) + a > a + a 3 (ii) > a 3 (iii) a 3 > a a a 3 4 (i) + a + a 4 > a + a 3 (ii) > a 4 (iii) a 4 > a 3 a a 4 (iv) ( a 4) (a 3 a a 4 ) > ( a 4 ) ( + a 4 )a (a a 3 a 4 )(a 3 a a 4 ) 4 a 4 = 3 3 a 3 = 3 3
A(z) = + a z + a z + a 3 z 3 A(z) = + a z + a z a 3 4 a 4 a 4 a a 3 : 3 3 IIR IIR M 3 3 Lu [8] Jiang Kwan [9] 3. K f(z) C K f(z) C f(z) q p [] q p = f (z) dz (7) πj f(z) f (z) = df(z)/dz A(z) = M m= a mz m M A(z) p = M A(z) C = {z : z = r}, r πj C C A (z) dz = (8) A(z) 3: (8) q = p p = M q = M A(z) C = {z : z = r}, r q = M A(z) p = M (8) (8) a A(z) (8) 3 a (8) 3 z 3 3. (8) A (z) πj C A(z) dz = d[ln A(z)] πj C = d[arg A(z)] (9) π (8) d[arg A(z)] = q M = () π C A(z) A R (z) A I (z) C ( ) arg A(z) = tan AI (z) () A R (z) arg A(re jθ ) ] = [ d AI (re jθ ) ] A R (re jθ ) A(re jθ ) A [ I(re jθ ) d AR (re jθ ) ] A(re jθ ) ()
φ a (re jθ ) c a (re jθ ) s a (re jθ ) A R (re jθ ) = + a T c a (re jθ ) (3) A I (re jθ ) = a T s a (re jθ ) (4) AR (re jθ ) ] = a T Λs a (re jθ ) (5) AI (re jθ ) ] = a T Λc a (re jθ ) (6) Λ = diag[... M] () arg A(re jθ ) ] = at ΛΨ(re jθ )a + a T Λc a (re jθ ) A(re jθ ) (7) Ψ(re jθ ) = c a (re jθ )c T a (re jθ ) + s a (re jθ )s T a (re jθ ) (8) (7) θ () J AP (a) = a T G(a)a + a T g(a) = q M = (9) G(a) = π g(a) = π π π 3.. Lu ΛΨ(re jθ ) + a T φ a (re jθ () ) Λc a (re jθ ) + a T φ a (re jθ () ) â (9) J AP (a) Taylor J AP (a) = J AP (â) + δ T a J AP (â) â + δ T a J AP (â) â â T δ a + () δ a = a â â J AP (â) = â a J AP (a) = () Lu [8] () 3 δ a δ T a J AP (â) â = (3) a δ a : J AP (a) No. of (9) (6) L(4M + 6M + 4) L(5M + 9) + L(M + M) L(4M + ) L(M + ) L(M + ) e jθ L L 3.. Jiang Kwan Jiang Kwan [9] (9) G(a) g(a) G(â) g(â) â a a T G(â)a + a T g(â) = (4) a 3.3 (8) A (z) πj C A(z) dz = π a T ( Λ) φ a (re jθ ) π π + a T φ a (re jθ ) = π [ a T Λφ R a (re jθ ] ) π + a T φ a (re jθ ) π (5) (7) θ (8) (9) J AP (a) = a T k(a) = q M = (6) k(a) = π π [ R Λφ a (re jθ ) + a T φ a (re jθ ) ] (7) (9) (6) M A(z) L (6) (9) 5/4M /M (3) J AP (â)/ â (9) (6) Jiang Kwan (6) k(a) k(â) â a a T k(â) = (8) a
4 (8) (3) a δ a (8) (3) (3) (7) 3 (9) (6) J AP (a) J AP (a) J AP (a)/ a J AP (a)/ a = (3) (4) a (4) 5 (3) (4) (8) a 3 â (.85, ) â = [.85, ] T 4 3 4(a) 4(c) (3) (8) 4(b) (4).... A(z) = + a z + a z...4 a a a (a) A(z) = + a z + a z (b) a A(z) = + a z + a z 3.5..5 a a (c)..5 4: (a) Lu (b) Jiang Kwan (c) 3 (3) Lu 4(a) Lu
6 3 IIR [8,9] [] IIR [7] Rouché [5] [7] [] T. W. Parks and C. S. Burrus, Digital Filter Design, John Wiley & Sons, Inc., 987. [] A. T. Chottera and G. A. Jullien, A linear programming approach to recursive digital filter design with linear phase, IEEE Trans. Circuits Syst., vol. CAS-9, pp. 39 49, March 98. [3] W.-S. Lu, S.-C. Pei and C.-C. Tseng, A weighted least-squares method for the design of stable -D and -D IIR digital filters, IEEE Trans. Signal Processing, vol. 46, no., pp., Jan. 998. [4] J. W. Adams and J. L. Sullivan, Peak-constrained least-squares optimization, IEEE Trans. Signal Processing, vol. 46, no., pp. 36 3, Feb. 998. [5] M. C. Lang, Least-squares design of IIR filters with prescribed magnitude and phase responses and a pole radius constraint, IEEE Trans. Signal Processing, vol. 48, no., pp. 39 3, Nov.. [6] W.-S. Lu and T. Hinamoto, Optimal design of IIR digital filters with robust stability using conicquadratic-programming updates, IEEE Trans. Signal Processing, vol. 5, no. 6, pp. 58 59, June 3. [7] B. Dumitrescu and R. Niemistö, Multistage IIR filter design using convex stability domains defined by positive realness, IEEE Trans. Signal Processing, vol. 5, no. 4, pp. 96 974, April 4. [8] W.-S. Lu, An argument-principle based stability criterion and application to the design of IIR digital filters, IEEE Int. Symp. Circuits Syst., pp. 443 4434, 6. [9] A. Jiang and H. K. Kwan, IIR digital filter design with novel stability criterion based on argumunt principle, IEEE Int. Symp. Circuits Syst., pp. 339 34, 7. [] A. Antoniou, Digital Filters: Analysis, Design, and Applications, McGraw-Hill, Inc., 993. [] B. P. Palka, An Introduction to Complex Function Theory, Springer-Verlag New York Inc., 99. [] W.-S. Lu, Design of stable minimax IIR digital filters using semidefinite programming, IEEE Int. Symp. Circuits Syst., pp. I 355 I 358,.