Relationship between Magnitude and Phase Quote of the Day Experience

Relationship between Magnitude and Phase Quote of the Day Experience is the name everyone gives to their mistakes. Oscar Wilde Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice Hall Inc.

Relation between Magnitude and Phase For general LTI system – Knowledge about magnitude doesn’t provide any information about phase – Knowledge about phase doesn’t provide any information about magnitude For linear constant-coefficient difference equations however – There is some constraint between magnitude and phase – If magnitude and number of pole-zeros are known Only a finite number of choices for phase – If phase and number of pole-zeros are known Only a finite number of choices for magnitude (ignoring scale) A class of systems called minimum-phase – Magnitude specifies phase uniquely – Phase specifies magnitude uniquely Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 2

Square Magnitude System Function Explore possible choices of system function of the form He j 2 H e j H e j H* 1 / z* H z z e j Restricting the system to be rational M M 1 c z b0 H z a0 k k 1 N 1 1 c z 1 dk z 1 * H 1/ z * b0 a0 k 1 M C z H z H* 1 / z* 2 1 dk* z 1 ck z 1 1 c*k z 1 dk z 1 1 d*k z k 1 N k 1 j k 1 N k 1 The square system function b0 a0 * k 2 Given H e we can get C(z) What information on H(z) can we get from C(z)? Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 3

Poles and Zeros of Magnitude Square System Function M * C z H z H 1 / z * b0 a0 2 1 c z 1 c z * k k 1 N 1 k 1 1 k dk z 1 1 d*k z For every pole dk in H(z) there is a pole of C(z) at dk and (1/dk)* For every zero ck in H(z) there is a zero of C(z) at ck and (1/ck)* Poles and zeros of C(z) occur in conjugate reciprocal pairs If one of the pole/zero is inside the unit circle the reciprocal will be outside – Unless there are both on the unit circle If H(z) is stable all poles have to be inside the unit circle – We can infer which poles of C(z) belong to H(z) However, zeros cannot be uniquely determined – Example to follow Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 4

Example Two systems with 1 z 1 2z H z 1 0.8e z 1 0.8e 2 1 z 1 1 0. 5z 1 H1 z 1 0. 8e j / 4z 1 1 0. 8e j / 4z 1 1 2 j / 4 1 1 j / 4 1 z Both share the same magnitude square system function H1 z Copyright (C) 2005 Güner Arslan H2 z 351M Digital Signal Processing H z H 1 / z C z H1 z H1* 1 / z* 2 * 2 * 5

All-Pass System A system with frequency response magnitude constant Important uses such as compensating for phase distortion Simple all-pass system z 1 a* Hap z 1 az 1 Magnitude response constant Hap e j * j e j a* j 1 a e e j 1 ae 1 ae j Most general form with real impulse response z 1 dk Mc z 1 ck* z 1 ck Hap z A 1 1 1 ck* z 1 k 1 1 dk z k 1 1 ck z Mr A: positive constant, dk: real poles, ck: complex poles Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 6

Phase of All-Pass Systems Hap e j * j e j a* j 1 a e e j 1 ae 1 ae j Let’s write the phase with a represented in polar form e j re j r sin 2 arctan j j 1 r cos 1 re e The group delay of this system can be written as e j re j 1 r2 1 r2 grd j j 2 2 j j 1 re e 1 2 r cos r 1 re e For stable and causal system r 1 – Group delay of all-pass systems is always positive Phase between 0 and is always negative arg Hap e j 0 for Copyright (C) 2005 Güner Arslan 0 351M Digital Signal Processing 7