pass Transter Function The unwrapped ed phase function of any arbitrary causal stable allpass function is a continuous function of o Pr roperties (1)A causal stable real-coefficient allpass transfer function is a lossless bounded real LBR)function or, equivalently, a causal stable allpass filter is a lossless structure 16 Copyright C 2001, S K Mitra
16 Copyright © 2001, S. K. Mitra Allpass Transfer Function • The unwrapped phase function of any arbitrary causal stable allpass function is a continuous function of w Properties • (1) A causal stable real-coefficient allpass transfer function is a lossless bounded real (LBR) function or, equivalently, a causal stable allpass filter is a lossless structure
Allpass Transfer Function (2)The magnitude function of a stable allpass function A(z)satisfies <1,.for|z>1 A(=)=1,r=1 1. for z<1 B3)Let to) denote the group delay function of an allpass filter A2),1.e T()=-o(0 17 Copyright C 2001, S K Mitra
17 Copyright © 2001, S. K. Mitra Allpass Transfer Function • (2) The magnitude function of a stable allpass function A(z) satisfies: • (3) Let t(w) denote the group delay function of an allpass filter A(z), i.e., = = 1 for 1 1 for 1 1 for 1 z z z A z , , , ( ) t(w) = − [q (w)] w c d d
pass I ransrer Function The unwrapped phase function 0c(o)of a stable all pass function is a monotonically decreasing function of o so that t(o)is everywhere positive in the range 0<O< T The group delay of an M-th order stable real-coefficient allpass transfer function satisfies ∫τ(0)do=M 18 Copyright C 2001, S K Mitra
18 Copyright © 2001, S. K. Mitra Allpass Transfer Function • The unwrapped phase function of a stable allpass function is a monotonically decreasing function of w so that t(w) is everywhere positive in the range 0 < w < p • The group delay of an M-th order stable real-coefficient allpass transfer function satisfies: q (w) c t w w= p p d M 0 ( )
Allpass Transfer Function A Simple application a simple but often used application of an allpass filter is as a delay equalizer Let g(z be the transfer function of a digital filter designed to meet a prescribed magnitude response The nonlinear phase response of g()can be corrected by cascading it with an allpass filter A(z)so that the overall cascade has a constant group delay in the band of interest Copyright C 2001, S K Mitra
19 Copyright © 2001, S. K. Mitra Allpass Transfer Function A Simple Application • A simple but often used application of an allpass filter is as a delay equalizer • Let G(z) be the transfer function of a digital filter designed to meet a prescribed magnitude response • The nonlinear phase response of G(z) can be corrected by cascading it with an allpass filter A(z) so that the overall cascade has a constant group delay in the band of interest
pass Transter Function G(-) Since jA(e)=1, we have (G(e/0)4(e)=(G(e0) Overall group delay is the given by the sum of the group delays of g(z)and a(z) 20 Copyright C 2001, S K Mitra
20 Copyright © 2001, S. K. Mitra Allpass Transfer Function • Since , we have • Overall group delay is the given by the sum of the group delays of G(z) and A(z) | ( )|=1 jw A e | ( ) ( )| | ( )| w w w = j j j G e A e G e G(z) A(z)