Allpass Transfer Function The unwrapped phase function of any arbitrary causal stable allpass function is a continuous function of o Properties 1)Acausal stable real-coefficient allpass transfer function is a lossless bounded real (LBR) function or, equivalently, a causal stable allpass filter is a lossless structure Copyright C 2001, S K Mitra
11 Copyright © 2001, S. K. Mitra Allpass Transfer Function • The unwrapped phase function of any arbitrary causal stable allpass function is a continuous function of 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(=)satisfies <1. for z>1 for >1, for z< BLet t(o denote the group delay function of an allpass filter A(2),1.e T()=-a(() 12 Copyright C 2001, S K Mitra
12 Copyright © 2001, S. K. Mitra Allpass Transfer Function • (2) The magnitude function of a stable allpass function A(z) satisfies: • (3) Let t() 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() = − [q ()] c d d
Allpass Transfer Function The unwrapped phase function 0(o)of a stable allpass function is a monotonically decreasing function of o so that t(o) is everywhere positive in the range0≤0<π The group delay of an M-th order stable real-coefficient allpass transfer function satisfies ∫τ(olo=M Copyright C 2001, S K Mitra
13 Copyright © 2001, S. K. Mitra Allpass Transfer Function • The unwrapped phase function of a stable allpass function is a monotonically decreasing function of so that t() is everywhere positive in the range 0 < < p • The group delay of an M-th order stable real-coefficient allpass transfer function satisfies: q () c t = 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(a)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 y cascading it with an pass filter A(z) so that the overall cascade has a constant group delay in the band of interest 14 Copyright C 2001, S K Mitra
14 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
Allpass Transfer Function G(=) A(=) Since A(e)=1, we have G(e/0)A(e0)=G(e0) Overall group delay is the given by the sum of the group delays of g(z)and A(z) Copyright C 2001, S.K. Mitra
15 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 j A e | ( ) ( )| | ( )| = j j j G e A e G e G(z) A(z)