Lecture 14 Generalized Linear Phase Instructor Dr Ghazi

Lecture 14: Generalized Linear Phase Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan Email: ghazi. alsukkar@ju. edu. jo Spring 2014 1

Outline Ø Ø Ø Spring 2014 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 2

Linear Phase System o Ideal Delay System o Magnitude, phase, and group delay o Impulse response o If =nd is integer o For integer linear phase system delays the input Spring 2014 3

Cont. . o o For non-integer the output is an interpolation of samples Easiest way of representing is to think of it in continuous o This representation can be used even if x[n] was not originally derived from a continuous-time signal The output of the system is o Samples of a time-shifted, band-limited interpolation of the input sequence x[n] o A linear phase system can be thought as A zero-phase system output is delayed by Spring 2014 4

Symmetry of Linear Phase Impulse Responses o =5 =4. 3 Spring 2014 5

Outline Ø Ø Ø Spring 2014 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 6

Generalized Linear Phase System o Generalized Linear Phase o Additive constant in addition to linear term o Has constant group delay o And linear phase of general form Spring 2014 7

Condition for Generalized Linear Phase o We can write a generalized linear phase system response as o The phase angle of this system is o Cross multiply to get necessary condition for generalized linear phase Spring 2014 8

Symmetry of Generalized Linear Phase o Spring 2014 9

Outline Ø Ø Ø Spring 2014 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 10

Causal Generalized Linear-Phase System o If the system is causal and generalized linear-phase o Since h[n]=0 for n<0 we get o An FIR impulse response of length M+1 is generalized linear phase if it is symmetric o Here M is an integer Spring 2014 11

Type I FIR Linear-Phase System o Type I system is defined with symmetric impulse response n M is an even integer o The frequency response can be written as o Where Spring 2014 12

Type II FIR Linear-Phase System o Type II system is defined with symmetric impulse response n M is an odd integer o The frequency response can be written as o Where Spring 2014 13

Type III FIR Linear-Phase System o Type III system is defined with antisymmetric impulse response n M is an even integer o The frequency response can be written as o Where Spring 2014 14

Type IV FIR Linear-Phase System o Type IV system is defined with antisymmetric impulse response n M is an odd integer o The frequency response can be written as o Where Spring 2014 15

Outline Ø Ø Ø Spring 2014 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 16

Location of Zeros for Symmetric Cases o Spring 2014 17

Cont. . o Spring 2014 18

Location of Zeros for Antisymmetric Cases o Spring 2014 19

Typical Zero Locations Type III Spring 2014 Type II Type IV 20

Outline Ø Ø Ø Spring 2014 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 21

Relation of FIR Linear Phase to Minimum-Phase o Spring 2014 22