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Bandpass Filter Design Example

The matlab code below designs a bandpass filter which passes frequencies between 4 kHz and 6 kHz, allowing transition bands from 3-4 kHz and 6-8 kHz (i.e., the stop-bands are 0-3 kHz and 8-10 kHz, when the sampling rate is 20 kHz). The desired stop-band attenuation is 80 dB, and the pass-band ripple is required to be no greater than 0.1 dB. For these specifications, the function kaiserord returns a beta value of $ \beta=7.85726$ and a window length of $ M=101$ . These values are passed to the function kaiser which computes the window function itself. The ideal bandpass-filter impulse response is computed in fir1, and the supplied Kaiser window is applied to shorten it to length $ M=101$ .

fs = 20000;                 % sampling rate
F = [3000 4000 6000 8000];  % band limits
A = [0 1 0];                % band type: 0='stop', 1='pass'
dev = [0.0001 10^(0.1/20)-1 0.0001]; % ripple/attenuation spec
[M,Wn,beta,typ] = kaiserord(F,A,dev,fs);  % window parameters
b = fir1(M,Wn,typ,kaiser(M+1,beta),'noscale'); % filter design

Note the conciseness of the matlab code thanks to the use of kaiserord and fir1 from Octave or the Matlab Signal Processing Toolbox.

Figure 4.6 shows the magnitude frequency response $ \vert B(\omega/2\pi)\vert$ of the resulting FIR filter $ b$ . Note that the upper pass-band edge has been moved to 6500 Hz instead of 6000 Hz, and the stop-band begins at 7500 Hz instead of 8000 Hz as requested. While this may look like a bug at first, it's actually a perfectly fine solution. As discussed earlier (§4.5), all transition-widths in filters designed by the window method must equal the window-transform's main-lobe width. Therefore, the only way to achieve specs when there are multiple transition regions specified is to set the main-lobe width to the minimum transition width. For the others, it makes sense to center the transition within the requested transition region.

Figure 4.6: Amplitude response of the FIR bandpass filter designed by the window method.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University