Comb Filters Depending on applications, comb filters with other types of periodic magnitude responses can be easily generated by appropriately choosing the prototype filter For example, the M-point moving average filter H(z) 2 M(1-z has been used as a prototype Copyright C 2001, S K Mitra
6 Copyright © 2001, S. K. Mitra Comb Filters • Depending on applications, comb filters with other types of periodic magnitude responses can be easily generated by appropriately choosing the prototype filter • For example, the M-point moving average filter has been used as a prototype ( ) ( ) 1 1 1 − − − − = M z z M H z
Comb Filters This filter has a peak magnitude at o=0 and dM-1 notches at o=2/M1≤≤M The corresponding comb filter has a transfer function 1--LM G(z)=M-L whose magnitude has l peaks at o=2Tk/L 0≤k≤L-1andL(M-1) notches at 0=2兀MLM,1≤k≤L(M-1) Copyright C 2001, S K Mitra
7 Copyright © 2001, S. K. Mitra Comb Filters • This filter has a peak magnitude at w = 0, and notches at , • The corresponding comb filter has a transfer function whose magnitude has L peaks at , and notches at , M −1 w = 2p / M 1 M −1 ( ) ( ) L LM M z z G z − − − − = 1 1 w = 2pk/L 0 k L −1 L(M −1) w= 2pk/LM 1 k L(M −1)
Allpass Transfer Function Definition An IR transfer function A(z)with unity magnitude response for all frequencies, i.e A(e )=1, for all o is called an allpass transfer function An m-th order causal real-coefficient allpass transfer function is of the form A1(2)=±+d M+1 M M-1 z+…+a1z 1+l1=1++d10-12-M+1+lb=-M Copyright C 2001, S K Mitra
8 Copyright © 2001, S. K. Mitra Allpass Transfer Function Definition • An IIR transfer function A(z) with unity magnitude response for all frequencies, i.e., is called an allpass transfer function • An M-th order causal real-coefficient allpass transfer function is of the form = w w | ( )| 1, for all j 2 A e M M M M M M M M M d z d z d z d d z d z z A z − + − − − − − + − − + + + + + + + + = 1 1 1 1 1 1 1 1 1 ... ... ( )
pass Transter Function If we denote the denominator polynomial of AM(zas dm(z) D/()=1+d1= M+1 +…+c dmz then it follows that AM(z)can be written as M 2 2 Note from the above that if z=reJo is a pole of a real coefficient allpass transfer function, then it has a zero at z=le Copyright C 2001, S K Mitra
9 Copyright © 2001, S. K. Mitra Allpass Transfer Function • If we denote the denominator polynomial of as : then it follows that can be written as: • Note from the above that if is a pole of a real coefficient allpass transfer function, then it has a zero at AM (z) DM (z) M M M DM z d z dM z d z − + − − − = + + + + 1 1 1 1 1 ... ( ) AM (z) ( ) ( ) ( ) D z z D z M M M M A z − −1 = = j z re − = j r z e 1
pass I ransrer Function The numerator of a real-coefficient allpass transfer function is said to be the mirror- image polynomial of the denominator, and vice versa We shall use the notation DM(z) to denote the mirror-image polynomial of a degree-M polynomial DM(2),i.e M 10 Copyright C 2001, S K Mitra
10 Copyright © 2001, S. K. Mitra Allpass Transfer Function • The numerator of a real-coefficient allpass transfer function is said to be the mirrorimage polynomial of the denominator, and vice versa • We shall use the notation to denote the mirror-image polynomial of a degree-M polynomial , i.e., DM (z) ~ DM (z) D (z) z DM (z) M M − = ~