1. Lecture 13: Minimum-Phase Systems Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan Email: ghazi.alsukkar@ju.edu.jo ghazi.alsukkar@ju.edu.jo Spring 20141

2. Outline  Invertebility  Minimum-Phase System  Minimum-Phase All-pass decomposition  Frequency-Response Compensation  Properties of Minimum-Phase Systems Spring 20142

3. Invertibility and Minimum-phase systems Spring 2014 3

4. 4 Minimum-Phase System

5. Outline  Invertebility  Minimum-Phase System  Minimum-Phase All-pass decomposition  Frequency-Response Compensation  Properties of Minimum-Phase Systems Spring 20145

6. Minimum-phase and All-pass decomposition Spring 20146

7. Cont.. Spring 20147

8. 8 Example 1: Minimum-Phase System  Consider the following system  One pole inside the unit circle: Make part of minimum-phase system  One zero outside the unit circle: Add an all-pass system to reflect this zero inside the unit circle

9. Spring 20149 Example 2: Minimum-Phase System  Consider the following system  One pole inside the unit circle:  Complex conjugate zero pair outside the unit circle

10. Spring 201410 Example 2 Cont’d

11. Outline  Invertebility  Minimum-Phase System  Minimum-Phase All-pass decomposition  Frequency-Response Compensation  Properties of Minimum-Phase Systems Spring 201411

12. Spring 201412 Frequency-Response Compensation

13. Outline  Invertebility  Minimum-Phase System  Minimum-Phase All-pass decomposition  Frequency-Response Compensation  Properties of Minimum-Phase Systems Spring 201413

14. Spring 201414 Properties of Minimum-Phase Systems  Minimum Phase-Lag Property Continuous phase of a non-minimum-phase system All-pass systems have negative phase between 0 and  So any non-minimum phase system will have a more negative phase compared to the minimum-phase system The negative of the phase is called the phase-lag function The name minimum-phase comes from minimum phase-lag  Minimum Group-Delay Property Group-delay of all-pass systems is positive Any non-minimum-phase system will always have greater group delay

15. Spring 201415 Properties of Minimum-Phase System  Minimum Energy-Delay Property Minimum-phase system concentrates energy in the early part  Consider a minimum-phase system H min (z)  Any H(z) that has the same magnitude response as H min (z) has the same poles as H min (z) any number of zeros of H min (z) are flipped outside the unit-circle  Decompose one of the zeros of H min (z)  Write H(z) that has the same magnitude response as  We can write these in time domain

16. Spring 201416 Derivation Cont’d  Evaluate each sum  And the difference is  Since |q k |<1

17. Cont.. Spring 201417