Feasible Regions Defined by Stability Constraints Based on the Argument Principle

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1 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

2 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) 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) 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

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 ) ()

4 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

5 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 a a (c)..5 4: (a) Lu (b) Jiang Kwan (c) 3 (3) Lu 4(a) Lu

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