Feedback Comb Filter Amplitude Response
Figure 2.26 shows a family of feedback-comb-filter amplitude responses, obtained using a selection of feedback coefficients.
![]() |
Figure 2.27 shows a similar family obtained using negated feedback coefficients; the opposite sign of the feedback exchanges the peaks and valleys in the amplitude response.
![]() |
As introduced in §2.6.2 above, a class of feedback comb filters can be defined as any difference equation of the form
![$\displaystyle y(n) = x(n) + g\,y(n-M).
$](http://www.dsprelated.com/josimages_new/pasp/img507.png)
![$ H(z)\isdef Y(z)/X(z)$](http://www.dsprelated.com/josimages_new/pasp/img508.png)
so that the amplitude response is
![$\displaystyle G(\omega) \isdef \left\vert H(e^{j\omega})\right\vert = \frac{1}{\left\vert 1 - g e^{-j\omega M}\right\vert}, \quad
-\pi \leq \omega \leq \pi .
$](http://www.dsprelated.com/josimages_new/pasp/img510.png)
![$ M=5$](http://www.dsprelated.com/josimages_new/pasp/img22.png)
![$ g=0.1$](http://www.dsprelated.com/josimages_new/pasp/img23.png)
![$ 0.5$](http://www.dsprelated.com/josimages_new/pasp/img24.png)
![$ 0.9$](http://www.dsprelated.com/josimages_new/pasp/img25.png)
For , the feedback-comb amplitude response
reduces to
![$\displaystyle G(\omega) = \frac{1}{2\left\vert\sin(\omega M/2)\right\vert},
$](http://www.dsprelated.com/josimages_new/pasp/img512.png)
![$ g=-1$](http://www.dsprelated.com/josimages_new/pasp/img513.png)
![$\displaystyle G(\omega) = \frac{1}{2\left\vert\cos(\omega M/2)\right\vert},
$](http://www.dsprelated.com/josimages_new/pasp/img514.png)
![$ g=1$](http://www.dsprelated.com/josimages_new/pasp/img511.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
Note that produces resonant peaks at
![$\displaystyle \omega_k = 2\pi\frac{k}{M}, \quad k=0,1,2,\dots,M-1,
$](http://www.dsprelated.com/josimages_new/pasp/img516.png)
![$ g<0$](http://www.dsprelated.com/josimages_new/pasp/img517.png)
Next Section:
Filtered-Feedback Comb Filters
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Feedforward Comb Filter Amplitude Response