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