--- /dev/null
+%% Copyright (C) 2005 Julius O. Smith III <jos@ccrma.stanford.edu>
+%%
+%% This program is free software; you can redistribute it and/or modify it under
+%% the terms of the GNU General Public License as published by the Free Software
+%% Foundation; either version 3 of the License, or (at your option) any later
+%% version.
+%%
+%% This program is distributed in the hope that it will be useful, but WITHOUT
+%% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+%% FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
+%% details.
+%%
+%% You should have received a copy of the GNU General Public License along with
+%% this program; if not, see <http://www.gnu.org/licenses/>.
+
+%% -*- texinfo -*-
+%% @deftypefn {Function File} {[@var{r}, @var{p}, @var{f}, @var{m}] =} residued (@var{B}, @var{A})
+%% Compute the partial fraction expansion (PFE) of filter
+%% @math{H(z) = B(z)/A(z)}.
+%% In the usual PFE function @code{residuez},
+%% the IIR part (poles @var{p} and residues
+%% @var{r}) is driven @emph{in parallel} with the FIR part (@var{f}).
+%% In this variant (@code{residued}) the IIR part is driven
+%% by the @emph{output} of the FIR part. This structure can be
+%% more accurate in signal modeling applications.
+%%
+%% INPUTS:
+%% @var{B} and @var{A} are vectors specifying the digital filter @math{H(z) = B(z)/A(z)}.
+%% Say @code{help filter} for documentation of the @var{B} and @var{A}
+%% filter coefficients.
+%%
+%% RETURNED:
+%% @itemize
+%% @item @var{r} = column vector containing the filter-pole residues@*
+%% @item @var{p} = column vector containing the filter poles@*
+%% @item @var{f} = row vector containing the FIR part, if any@*
+%% @item @var{m} = column vector of pole multiplicities
+%% @end itemize
+%%
+%% EXAMPLES:
+%% @example
+%% Say @code{test residued verbose} to see a number of examples.
+%% @end example
+%%
+%% For the theory of operation, see
+%% @indicateurl{http://ccrma.stanford.edu/~jos/filters/residued.html}
+%%
+%% @seealso{residue residued}
+%% @end deftypefn
+
+function [r, p, f, m] = residued(b, a, toler)
+ % RESIDUED - return residues, poles, and FIR part of B(z)/A(z)
+ %
+ % Let nb = length(b), na = length(a), and N=na-1 = no. of poles.
+ % If nb<na, then f will be empty, and the returned filter is
+ %
+ % r(1) r(N)
+ % H(z) = ---------------- + ... + ----------------- = R(z)
+ % [ 1-p(1)/z ]^e(1) [ 1-p(N)/z ]^e(N)
+ %
+ % This is the same result as returned by RESIDUEZ.
+ % Otherwise, the FIR part f will be nonempty,
+ % and the returned filter is
+ %
+ % H(z) = f(1) + f(2)/z + f(3)/z^2 + ... + f(nf)/z^M + R(z)/z^M
+ %
+ % where R(z) is the parallel one-pole filter bank defined above,
+ % and M is the order of F(z) = length(f)-1 = nb-na.
+ %
+ % Note, in particular, that the impulse-response of the parallel
+ % (complex) one-pole filter bank starts AFTER that of the the FIR part.
+ % In the result returned by RESIDUEZ, R(z) is not divided by z^M,
+ % so its impulse response starts at time 0 in parallel with f(n).
+ %
+ % J.O. Smith, 9/19/05
+
+ if nargin==3,
+ warning("tolerance ignored");
+ end
+ NUM = b(:)';
+ DEN = a(:)';
+ nb = length(NUM);
+ na = length(DEN);
+ f = [];
+ if na<=nb
+ f = filter(NUM,DEN,[1,zeros(nb-na)]);
+ NUM = NUM - conv(DEN,f);
+ NUM = NUM(nb-na+2:end);
+ end
+ [r,p,f2,m] = residuez(NUM,DEN);
+ if f2, error('f2 not empty as expected'); end
+end
+
+%!test
+%! B=1; A=[1 -1];
+%! [r,p,f,m] = residued(B,A);
+%! assert({r,p,f,m},{1,1,[],1},100*eps);
+%! [r2,p2,f2,m2] = residuez(B,A);
+%! assert({r,p,f,m},{r2,p2,f2,m2},100*eps);
+% residuez and residued should be identical when length(B)<length(A)
+
+%!test
+%! B=[1 -2 1]; A=[1 -1];
+%! [r,p,f,m] = residued(B,A);
+%! assert({r,p,f,m},{0,1,[1 -1],1},100*eps);
+
+%!test
+%! B=[1 -2 1]; A=[1 -0.5];
+%! [r,p,f,m] = residued(B,A);
+%! assert({r,p,f,m},{0.25,0.5,[1 -1.5],1},100*eps);
+
+%!test
+%! B=1; A=[1 -0.75 0.125];
+%! [r,p,f,m] = residued(B,A);
+%! [r2,p2,f2,m2] = residuez(B,A);
+%! assert({r,p,f,m},{r2,p2,f2,m2},100*eps);
+% residuez and residued should be identical when length(B)<length(A)
+
+%!test
+%! B=1; A=[1 -2 1];
+%! [r,p,f,m] = residued(B,A);
+%! [r2,p2,f2,m2] = residuez(B,A);
+%! assert({r,p,f,m},{r2,p2,f2,m2},100*eps);
+% residuez and residued should be identical when length(B)<length(A)
+
+%!test
+%! B=[6,2]; A=[1 -2 1];
+%! [r,p,f,m] = residued(B,A);
+%! [r2,p2,f2,m2] = residuez(B,A);
+%! assert({r,p,f,m},{r2,p2,f2,m2},100*eps);
+% residuez and residued should be identical when length(B)<length(A)
+
+%!test
+%! B=[1 1 1]; A=[1 -2 1];
+%! [r,p,f,m] = residued(B,A);
+%! assert(r,[0;3],1e-7);
+%! assert(p,[1;1],1e-8);
+%! assert(f,1,100*eps);
+%! assert(m,[1;2],100*eps);
+
+%!test
+%! B=[2 6 6 2]; A=[1 -2 1];
+%! [r,p,f,m] = residued(B,A);
+%! assert(r,[8;16],3e-7);
+%! assert(p,[1;1],1e-8);
+%! assert(f,[2,10],100*eps);
+%! assert(m,[1;2],100*eps);
+
+%!test
+%! B=[1,6,2]; A=[1 -2 1];
+%! [r,p,f,m] = residued(B,A);
+%! assert(r,[-1;9],3e-7);
+%! assert(p,[1;1],1e-8);
+%! assert(f,1,100*eps);
+%! assert(m,[1;2],100*eps);
+
+%!test
+%! B=[1 0 0 0 1]; A=[1 0 0 0 -1];
+%! [r,p,f,m] = residued(B,A);
+%! assert({r,p,f,m},{[-j/2;j/2;1/2;-1/2],[-j;j;1;-1],1,[1;1;1;1]},100*eps);
+% Verified in maxima: ratsimp(%I/2/(1-%I * d) - %I/2/(1+%I * d)); etc.
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff