Function File: [n, Wn, beta, ftype] = kaiserord (f, m, dev)
Function File: […] = kaiserord (f, m, dev, fs)

Return the parameters needed to produce a filter of the desired specification from a Kaiser window. The vector f contains pairs of frequency band edges in the range [0,1]. The vector m specifies the magnitude response for each band. The values of m must be zero for all stop bands and must have the same magnitude for all pass bands. The deviation of the filter dev can be specified as a scalar or a vector of the same length as m. The optional sampling rate fs can be used to indicate that f is in Hz in the range [0,fs/2].

The returned value n is the required order of the filter (the length of the filter minus 1). The vector Wn contains the band edges of the filter suitable for passing to fir1. The value beta is the parameter of the Kaiser window of length n+1 to shape the filter. The string ftype contains the type of filter to specify to fir1.

The Kaiser window parameters n and beta are computed from the relation between ripple (A=-20*log10(dev)) and transition width (dw in radians) discovered empirically by Kaiser: 

          / 0.1102(A-8.7)                        A > 50
   beta = |0.5842(A-21)^0.4 + 0.07886(A-21)     21 <= A <= 50
          \ 0.0                                  A < 21

   n = (A-8)/(2.285 dw)

Example: 

[n, w, beta, ftype] = kaiserord ([1000, 1200], [1, 0], [0.05, 0.05], 11025);
b = fir1 (n, w, kaiser (n+1, beta), ftype, "noscale");
freqz (b, 1, [], 11025);

See also: fir1, kaiser.

Demonstration 1

The following code

 Fs = 11025;
 for i=1:4
   if i==1,
     subplot(221); bands=[1200, 1500]; mag=[1, 0]; dev=[0.1, 0.1];
   elseif i==2
     subplot(222); bands=[1000, 1500]; mag=[0, 1]; dev=[0.1, 0.1];
   elseif i==3
     subplot(223); bands=[1000, 1200, 3000, 3500]; mag=[0, 1, 0]; dev=0.1;
   elseif i==4
     subplot(224); bands=100*[10, 13, 15, 20, 30, 33, 35, 40];
     mag=[1, 0, 1, 0, 1]; dev=0.05;
   endif
   [n, w, beta, ftype] = kaiserord(bands, mag, dev, Fs);
   d=max(1,fix(n/10));
   if mag(length(mag))==1 && rem(d,2)==1, d=d+1; endif
   [h, f] = freqz(fir1(n,w,ftype,kaiser(n+1,beta),'noscale'),1,[],Fs);
   hm = freqz(fir1(n-d,w,ftype,kaiser(n-d+1,beta),'noscale'),1,[],Fs);
   plot(f,abs(hm),sprintf("r;order %d;",n-d), ...
        f,abs(h), sprintf("b;order %d;",n));
   b = [0, bands, Fs/2]; hold on;
   for i=2:2:length(b),
     hi=mag(i/2)+dev(1); lo=max(mag(i/2)-dev(1),0);
     plot([b(i-1), b(i), b(i), b(i-1), b(i-1)],[hi, hi, lo, lo, hi],"c;;");
   endfor; hold off;
 endfor

 %--------------------------------------------------------------
 % A filter meets the specifications if its frequency response
 % passes through the ends of the criteria boxes, and fails if
 % it passes through the top or the bottom.  The criteria are
 % met precisely if the frequency response only passes through
 % the corners of the boxes.  The blue line is the filter order
 % returned by kaiserord, and the red line is some lower filter
 % order.  Confirm that the blue filter meets the criteria and
 % the red line fails.

Produces the following figure

Figure 1

Package: signal