### Testing Filter Stability in Matlab

Figure 8.6 gives a listing of a matlab function
`stabilitycheck` for testing
the stability of a digital filter using the Durbin step-down
recursion. Figure 8.7 lists a main program for testing
`stabilitycheck` against the more prosaic method of factoring
the transfer-function denominator and measuring the pole radii. The
Durbin recursion is far faster than the method based on root-finding.

function [stable] = stabilitycheck(A); N = length(A)-1; % Order of A(z) stable = 1; % stable unless shown otherwise A = A(:); % make sure it's a column vector for i=N:-1:1 rci=A(i+1); if abs(rci) >= 1 stable=0; return; end A = (A(1:i) - rci * A(i+1:-1:2))/(1-rci^2); % disp(sprintf('A[%d]=',i)); A(1:i)' end |

% TSC - test function stabilitycheck, comparing against % pole radius computation N = 200; % polynomial order M = 20; % number of random polynomials to generate disp('Random polynomial test'); nunstp = 0; % count of unstable A polynomials sctime = 0; % total time in stabilitycheck() rftime = 0; % total time computing pole radii for pol=1:M A = [1; rand(N,1)]'; % random polynomial tic; stable = stabilitycheck(A); et=toc; % Typ. 0.02 sec Octave/Linux, 2.8GHz Pentium sctime = sctime + et; % Now do it the old fashioned way tic; poles = roots(A); % system poles pr = abs(poles); % pole radii unst = (pr >= 1); % bit vector nunst = sum(unst);% number of unstable poles et=toc; % Typ. 2.9 sec Octave/Linux, 2.8GHz Pentium rftime = rftime + et; if stable, nunstp = nunstp + 1; end if (stable & nunst>0) | (~stable & nunst==0) error('*** stabilitycheck() and poles DISAGREE ***'); end end disp(sprintf(... ['Out of %d random polynomials of order %d,',... ' %d were unstable'], M,N,nunstp)); |

When run in Octave over Linux 2.4 on a 2.8 GHz Pentium PC, the Durbin recursion is approximately 140 times faster than brute-force root-finding, as measured by the program listed in Fig.8.7.

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Geometric Series

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Step-Down Procedure