1. Lecture 14: Generalized Linear Phase 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  Linear Phase System  Generalized Linear Phase System  Causal Generalized Linear-Phase System  Location of Zeros for Symmetric Cases  Relation of FIR Linear Phase to Minimum-Phase Spring 20142

3. 3 Linear Phase System  Ideal Delay System  Magnitude, phase, and group delay  Impulse response  If =n d is integer  For integer  linear phase system delays the input

4. Spring 20144 Cont..  For non-integer  the output is an interpolation of samples  Easiest way of representing is to think of it in continuous  This representation can be used even if x[n] was not originally derived from a continuous-time signal  The output of the system is Samples of a time-shifted, band-limited interpolation of the input sequence x[n]  A linear phase system can be thought as A zero-phase system output is delayed by 

5. Spring 20145 Symmetry of Linear Phase Impulse Responses =5 =4.5 =4.3

6. Outline  Linear Phase System  Generalized Linear Phase System  Causal Generalized Linear-Phase System  Location of Zeros for Symmetric Cases  Relation of FIR Linear Phase to Minimum-Phase Spring 20146

7. 7 Generalized Linear Phase System  Generalized Linear Phase  Additive constant in addition to linear term  Has constant group delay  And linear phase of general form

8. Spring 20148 Condition for Generalized Linear Phase  We can write a generalized linear phase system response as  The phase angle of this system is  Cross multiply to get necessary condition for generalized linear phase

9. Spring 20149 Symmetry of Generalized Linear Phase

10. Outline  Linear Phase System  Generalized Linear Phase System  Causal Generalized Linear-Phase System  Location of Zeros for Symmetric Cases  Relation of FIR Linear Phase to Minimum-Phase Spring 201410

11. Spring 201411 Causal Generalized Linear-Phase System  If the system is causal and generalized linear-phase  Since h[n]=0 for n<0 we get  An FIR impulse response of length M+1 is generalized linear phase if it is symmetric  Here M is an integer

12. Spring 201412 Type I FIR Linear-Phase System  Type I system is defined with symmetric impulse response M is an even integer  The frequency response can be written as  Where

13. Spring 201413 Type II FIR Linear-Phase System  Type II system is defined with symmetric impulse response M is an odd integer  The frequency response can be written as  Where

14. Spring 201414 Type III FIR Linear-Phase System  Type III system is defined with antisymmetric impulse response M is an even integer  The frequency response can be written as  Where

15. Spring 201415 Type IV FIR Linear-Phase System  Type IV system is defined with antisymmetric impulse response M is an odd integer  The frequency response can be written as  Where

16. Outline  Linear Phase System  Generalized Linear Phase System  Causal Generalized Linear-Phase System  Location of Zeros for Symmetric Cases  Relation of FIR Linear Phase to Minimum-Phase Spring 201416

17. Spring 201417 Location of Zeros for Symmetric Cases

18. Cont.. Spring 201418

19. Spring 201419 Location of Zeros for Antisymmetric Cases

20. Spring 201420 Typical Zero Locations Type I Type II Type IIIType IV

21. Outline  Linear Phase System  Generalized Linear Phase System  Causal Generalized Linear-Phase System  Location of Zeros for Symmetric Cases  Relation of FIR Linear Phase to Minimum-Phase Spring 201421

22. Spring 201422 Relation of FIR Linear Phase to Minimum-Phase