华南理工大学电子与信息学院：《数字信号与处理》（英文版）Lecture 14 Comb Filters

Comb Filters The simple filters discussed so far are characterized either by a single passband and/or a single stopband There are applications where filters with multiple passbands and stopbands are d required The comb filter is an example of such filters Copyright C 2001, S K Mitra

1 Copyright © 2001, S. K. Mitra Comb Filters • The simple filters discussed so far are characterized either by a single passband and/or a single stopband • There are applications where filters with multiple passbands and stopbands are required • The comb filter is an example of such filters

Comb Filters In its most general form. a comb filter has a frequency response that is a periodic function of o with a period 2T/L, where L is a positive integer If Hz) is a filter with a single passband and/or a single stopband, a comb filter can be easily generated from it by replacing each delay in its realization with L delays resulting in a structure with a transfer function given by G(z)=H(2) Copyright C 2001, S K Mitra

2 Copyright © 2001, S. K. Mitra Comb Filters • In its most general form, a comb filter has a frequency response that is a periodic function of w with a period 2p/L, where L is a positive integer • If H(z) is a filter with a single passband and/or a single stopband, a comb filter can be easily generated from it by replacing each delay in its realization with L delays resulting in a structure with a transfer function given by ( ) ( ) L G z = H z

Comb Filters If H(e/o) exhibits a peak at O,, then G(e/o) will exhibit L peaks at o,k/L,0≤k≤L-1 in the frequency range0≤0<2π Likewise, if H(e/o) has a notch at o then G(elo) will have L notches at o,k/L 0≤k≤L-1 in the frequency range0≤0<2兀 a comb filter can be generated from either an FiR or an iir prototype filter Copyright C 2001, S K Mitra

3 Copyright © 2001, S. K. Mitra Comb Filters • If exhibits a peak at , then will exhibit L peaks at , in the frequency range • Likewise, if has a notch at , then will have L notches at , in the frequency range • A comb filter can be generated from either an FIR or an IIR prototype filter | ( )| jw H e | ( )| jw H e | ( )| jw G e | ( )| jw wp G e wo wp k/L wo k/L 0  k  L −1 0  k  L −1 0  w 2p 0  w 2p

Comb Filters For example the comb filter generated from the prototype lowpass FIR filter Ho(z) 1(1+z)has a transfer function L L (-)=H0(2)=;(1+2) Go(e )l has L notches Comb filter from lowpass prototype ato=(2k+1)T /L and L osk peaks at o=2兀M,1 O<k<L-1. in the frequency range 0<0<2 0.5 1.5 o/ Copyright C 2001, S K Mitra

4 Copyright © 2001, S. K. Mitra Comb Filters • For example, the comb filter generated from the prototype lowpass FIR filter has a transfer function • has L notches at w = (2k+1)p/L and L peaks at w = 2p k/L, ( ) 1 2 1 1 − + z H0 (z) = ( ) ( ) ( ) L L G z H z z − = = 1+ 2 1 0 0 0  k  L −1 , in the frequency range 0  w 2p | ( )| 0 jw G e 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 w/p Magnitude Comb filter from lowpass prototype

Comb Filters For example, the comb filter generated from the prototype highpass FIR filter H1(z) 1(1-z)has a transfer function G1()=H1(2)=(1-z1) Gie has L peaks Comb filter from highpass prototype at Q=(2K+1)I/L and L notches at a=2兀kL 0<k≤L-1. in the frequency range 0<0<2 Copyright C 2001,S K Mitra

5 Copyright © 2001, S. K. Mitra Comb Filters • For example, the comb filter generated from the prototype highpass FIR filter has a transfer function • has L peaks at w = (2k+1)p/L and L notches at w = 2p k/L, | ( )| 1 jw G e ( ) 1 2 1 1 − − z H1 (z) = ( ) ( ) ( ) L L G z H z z − = = 1− 2 1 1 1 0  k  L −1 , in the frequency range 0  w 2p 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 w/p Magnitude Comb filter from highpass prototype

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 mirror￾image 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 − = ~