Allpass Transfer Function · The expression M AM/(=)=土 2 D M(2 2 implies that the poles and zeros of a real coefficient allpass function exhibit mirror image symmetry in the z-plane 02+0.18z1+0.42+2÷0s A(二)= 1+0.4z-1+0.18z-2-0.2z 0 Real part Copyright C 2001, S K Mitra
6 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
Allpass Transfer Function To show that AM(e)=1 we observe that Av(2-1)=±=MDv(=) v/( Therefore AM/(二)AM(二-)= - MDM 2 MDM 2 2 DM(=) 2 Hence Ar(e0)2=AM(=)A1(x1 Iz=e Copyright C 2001, S K Mitra
7 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 j AM e | ( )| ( ) ( ) 1 2 1 = = = − j M M z e j AM e A z A z
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 Copyright C 2001, S K Mitra
8 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+0.4z-2+ A3(=) 1+0.4x-1+0.18z-2-0.2z note the discontinuity by the amount of 2t in the phase e(o) Principal value of phase -2 0.8 o/π Copyright C 2001, S K Mitra
9 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() 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 /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 0c(o)indicated below Note: The unwrapped phase function is a continuous function of o Unwrapped phase 10 0.2 0.4 0.6 0.8 Copyright C 2001, S K Mitra
10 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 q () c 0 0.2 0.4 0.6 0.8 1 -10 -8 -6 -4 -2 0 /p Phase, degrees Unwrapped phase