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residued.m

%% Copyright (C) 2005 Julius O. Smith III
%%
%% 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 2, 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.  The GNU
%% General Public License has more details.
%%
%% You should have received a copy of the GNU General Public License
%% along with this program; see the file COPYING.  If not, see
%% <http://www.gnu.org/licenses/>.

%% -*- texinfo -*-
%% @deftypefn {Function File} { =} 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

%!test 
%! B=1; A=;
%! [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=; A=;
%! [r,p,f,m] = residued(B,A);
%! assert({r,p,f,m},{0,1,,1},100*eps);

%!test 
%! B=; A=;
%! [r,p,f,m] = residued(B,A);
%! assert({r,p,f,m},{0.25,0.5,,1},100*eps);

%!test 
%! B=1; A=;
%! [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=;
%! [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=; A=;
%! [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=; A=;
%! [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=; A=;
%! [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=; A=;
%! [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=; A=;
%! [r,p,f,m] = residued(B,A);
%! assert({r,p,f,m},{,[-1;1;-j;j],1,},100*eps);
%  Verified in maxima: ratsimp(%I/2/(1-%I * d) - %I/2/(1+%I * d)); etc.

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