Allpass Transfer Function The expression -M DM(E AM(2)=+DM(2) implies that the poles and zeros of a real coefficient allpass function exhibit mirror image symmetry in the z-plane -0.2+0.18z-1+04z-2+z A3(=) 1+0.4z-1+0.18z2-0.2z E-0.5 Real part Copyright C 2001, S K Mitra
11 Copyright © 2001, S. K. Mitra Allpass Transfer Function • The expression implies that the poles and zeros of a realcoefficient allpass function exhibit mirrorimage symmetry in the z-plane ( ) ( ) ( ) D z z D z M M M M A z − −1 = 1 2 3 1 2 3 3 1 0.4 0.18 0.2 0.2 0.18 0.4 ( ) − − − − − − + + − − + + + = z z z z z z A z -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 Real Part Imaginary Part
pass I ransrer Function To show that AM(e))=1 we observe that (z-1)= M DM Therefore AM(2AM(z) z-MD/(二-1)zMD/(z) ()DM(二 Hence Mle@ 12=AM(=)AM(z z=e Copyright C 2001, S K Mitra
12 Copyright © 2001, S. K. Mitra Allpass Transfer Function • To show that we observe that • Therefore • Hence ( ) 1 ( ) 1 ( ) − = − D z z D z M M M M A z ( ) ( ) ( ) 1 ( ) 1 1 ( ) ( ) − − − = − D z z D z D z z D z M M M M M M M M A z A z | ( )|=1 jw AM e | ( )| ( ) ( ) 1 2 1 = = w = w − j M M z e j AM e A z A z
pass Transter Function Now. the poles of a causal stable transfer function must lie inside the unit circle in the z-plane Hence, all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle Copyright C 2001, S K Mitra
13 Copyright © 2001, S. K. Mitra Allpass Transfer Function • Now, the poles of a causal stable transfer function must lie inside the unit circle in the z-plane • Hence, all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle
Allpass Transfer Function Figure below shows the principal value of the phase of the 3rd-order allpass function 0.2+0.18z-1+0.42-2+z A3()= 1+0.4z-1+0.18z 2 0.2z note the discontinuity by the amount of 2n in the phase e(o) Principal value of phase 14 0.4 0.6 0.8 Copyright O 2001, S.K. Mitra
14 Copyright © 2001, S. K. Mitra Allpass Transfer Function • Figure below shows the principal value of the phase of the 3rd-order allpass function • Note the discontinuity by the amount of 2p in the phase q(w) 1 2 3 1 2 3 3 1 0.4 0.18 0.2 0.2 0.18 0.4 ( ) − − − − − − + + − − + + + = z z z z z z A z 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 w/p Phase, degrees Principal value of phase
Allpass Transfer Function If we unwrap the phase by removing the discontinuity, we arrive at the unwrapped phase function Ac(o)indicated below Note: The unwrapped phase function is a continuous function of o Unwrapped phase 15 -10 0.2 0.4 0.8 Copyright C 2001, S K Mitra
15 Copyright © 2001, S. K. Mitra Allpass Transfer Function • If we unwrap the phase by removing the discontinuity, we arrive at the unwrapped phase function indicated below • Note: The unwrapped phase function is a continuous function of w q (w) c 0 0.2 0.4 0.6 0.8 1 -10 -8 -6 -4 -2 0 w/p Phase, degrees Unwrapped phase