1. 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.

2. Copyright (C) EEE413 Digital Signal Processing 2 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

3. Copyright (C) EEE413 Digital Signal Processing 3 Square Magnitude System Function Explore possible choices of system function of the form Restricting the system to be rational The square system function Given we can get C(z) What information on H(z) can we get from C(z)?

4. Copyright (C) EEE413 Digital Signal Processing 4 Poles and Zeros of Magnitude Square System Function For every pole d k in H(z) there is a pole of C(z) at d k and (1/d k ) * For every zero c k in H(z) there is a zero of C(z) at c k and (1/c k ) * 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

5. Copyright (C) EEE413 Digital Signal Processing 5 Example Two systems with Both share the same magnitude square system function

6. Copyright (C) EEE413 Digital Signal Processing 6 All-Pass Filters An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship between various frequencies. It does this by varying its phase shift as a function of frequency. Generally, the filter is described by the frequency at which the phase shift crosses 90° (i.e., when the input and output signals go into quadrature – when there is a quarter wavelength of delay between them). They are generally used to compensate for other undesired phase shifts that arise in the system, or for mixing with an unshifted version of the original to implement a notch comb filter. They may also be used to convert a mixed phase filter into a minimum phase filter with an equivalent magnitude response or an unstable filter into a stable filter with an equivalent magnitude response.

7. Copyright (C) EEE413 Digital Signal Processing 7 All-Pass System A system with frequency response magnitude constant Important uses such as compensating for phase distortion Simple all-pass system Magnitude response constant Most general form with real impulse response A: positive constant, d k : real poles, c k : complex poles

8. Copyright (C) EEE413 Digital Signal Processing 8 Phase of All-Pass Systems Let’s write the phase with a represented in polar form The group delay of this system can be written as For stable and causal system |r|<1 –Group delay of all-pass systems is always positive Phase between 0 and  is always negative