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 16 Copyright C 2001, S K Mitra
16 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 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 G()=H(z) Copyright C 2001, S K Mitra
17 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 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 op, then G(eo) will exhibit L peaks at ok/L,0≤k≤L-1 in the frequency range0≤o<2兀 Likewise, if ( H(e/o)l has a notch at Oo then G(ejo )l will have L notches at ok/L 0≤k≤L-1 in the frequency range0≤o<2π a comb filter can be generated from either an Fir or an iir prototype filter 18 Copyright C 2001, S K Mitra
18 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 | ( )| j H e | ( )| j H e | ( )| j G e | ( )| j p G e o p k/L o k/L 0 k L −1 0 k L −1 0 2p 0 2p
Comb Filters For example, the comb filter generated from the prototype lowpass FIR filter Ho(z) (+z )has a transfer function G0()=H0(x2)=1(1+22) Go(eo )l has L notches Comb filter from lowpass prototype at o=2K+1)T/L and L ok peaks at o=2兀ML2 0≤k≤L-1, in the frequency range 0≤0<2丌 0.5 1.5 19 Copyright C 2001, S K Mitra
19 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 = (2k+1)p/L and L peaks at = 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 2p | ( )| 0 j G e 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 /p Magnitude Comb filter from lowpass prototype
Comb Filters For example, the comb filter generated from the prototype highpass FIR filter H1(z) l(-2)has a transfer function G(e/o)l has L peaks Comb filter from highpass prototype at @=(2k+1)/L andL L/ notches at=2πk/L 0.6 0≤k≤L-1, in the frequency range 0≤0<2丌 20 0.5 Copyright C 2001,S K Mitra
20 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 = (2k+1)p/L and L notches at = 2p k/L, | ( )| 1 j 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 2p 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 /p Magnitude Comb filter from highpass prototype