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Comparison to the Optimal Chebyshev FIR Bandpass Filter

To provide some perspective on the results, let's compare the window method to the optimal Chebyshev FIR filter4.10) for the same length and design specifications above.

The following Matlab code illustrates two different bandpass filter designs. The first (different transition bands) illustrates a problem we'll look at. The second (equal transition bands, commented out), avoids the problem.

M = 101;
normF = [0 0.3 0.4 0.6 0.8 1.0];  % transition bands different
%normF = [0 0.3 0.4 0.6 0.7 1.0]; % transition bands the same
amp = [0 0 1 1 0 0];              % desired amplitude in each band

[b2,err2] = firpm(M-1,normF,amp); % optimal filter of length M

Figure 4.7 shows the frequency response of the Chebyshev FIR filter designed by firpm, to be compared with the window-method FIR filter in Fig.4.6. Note that the upper transition band ``blows up''. This is a well known failure mode in FIR filter design using the Remez exchange algorithm [176,224]. It can be eliminated by narrowing the transition band, as shown in Fig.4.8. There is no error penalty in the transition region, so it is necessary that each one be ``sufficiently narrow'' to avoid this phenomenon.

Remember the rule of thumb that the narrowest transition-band possible for a length $ L$ FIR filter is on the order of $ 4\pi/L$ , because that's the width of the main-lobe of a length $ L$ rectangular window (measured between zero-crossings) (§3.1.2). Therefore, this value is quite exact for the transition-widths of FIR bandpass filters designed by the window method using the rectangular window (when the main-lobe fits entirely within the adjacent pass-band and stop-band). For a Hamming window, the window-method transition width would instead be $ 8\pi/L$ . Thus, we might expect an optimal Chebyshev design to provide transition widths in the vicinity of $ 8\pi/L$ , but probably not too close to $ 4\pi/L$ or below In the example above, where the sampling rate was $ 20$ kHz, and the filter length was $ L=101$ , we expect to be able to achieve transition bands circa $ (20,000/(2\pi))\cdot (8\pi/101) = 792$ Hz, but not so low as $ (20,000/(2\pi))\cdot (4\pi/101) = 396$ Hz. As we found above, $ 2000$ Hz was under-constrained, while $ 1000$ Hz was ok, being near the ``Hamming transition width.''

Figure 4.7: Amplitude response of the optimal Chebyshev FIR bandpass filter designed by the Remez exchange method.

Figure 4.8: Amplitude response of the optimal Chebyshev FIR bandpass filter as in Fig.4.7 with the upper transition band narrowed from 2 kHz down to 1 kHz in width.

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