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function sym=ref_gabmulappr(T,p2,p3,p4,p5);
%-*- texinfo -*-
%@deftypefn {Function} ref_gabmulappr
%@verbatim
%GABMULMAT Best Approximation by a Gabor multiplier.
% Usage: sym=gabmulappr(T,a,M);
% sym=gabmulappr(T,g,a,M);
% sym=gabmulappr(T,ga,gs,a,M);
%
% Input parameters:
% T : matrix to be approximated
% g : analysis/synthesis window
% ga : analysis window
% gs : synthesis window
% a : Length of time shift.
% M : Number of channels.
%
% Output parameters:
% sym : symbol
%
% GABMULAPPR(T,g,a,M) will calculate the best approximation of the given
% matrix T in the frobenius norm by a Gabor multiplier determined by the
% symbol sym over the rectangular time-frequency lattice determined by a
% and M. The window g will be used for both analysis and synthesis.
% IMPORTANT: The chosen Gabor system has to be a frame sequence!
%
% GABMULAPPR(T,a,M) will do the same using an optimally concentrated,
% tight Gaussian as window function.
%
% GABMULAPPR(T,gs,ga,a) will do the same using the window ga for analysis
% and gs for synthesis.
%
% SEE ALSO: GABMUL, GABMULINV
%
% In this algorithm first the 'lower symbol' is calcualted, then the
% so-called 'upper symbol'.
%
% The lower symbol is the inner product < T , P_lambda > where P_lambda
% are the projections gamma_lambda otimes g_lambda . These operators
% form a Bessel sequence and the lower symbol, the lower symbol is the
% analysis sequence of T using this Bessel sequence.
%
% The upper symbol is the inner product < T , Q_lambda > where
% Q_lambda are the dual projections operator. Therefore the upper
% symbol is the analysis with the dual sequence (if the P have formed a
% frame). Because the
%
% References :
% P. Balazs, Basic Definition and Properties of Bessel Multipliers,
% Journal of Mathematical Analysis and Applications 325(1):571--585,
% January 2007.
% P.~Balazs, Hilbert-Schmidt operators and frames - classification, best
% approximation by multipliers and algorithms, International Journal
% of Wavelets, Multiresolution and Information Processing, accepted,
% to appear.
% H. G. Feichtinger, M. Hampjes, G. Kracher, "Approximation of matrices
% by Gabor multipliers" , IEEE Signal Procesing Letters Vol. 11,
% Issue 11, pp 883-- 886 (2004)
%
% Author: P. Balazs (XXL)
%@end verbatim
%@strong{Url}: @url{http://ltfat.github.io/doc/reference/ref_gabmulappr.html}
%@end deftypefn
% Copyright (C) 2005-2016 Peter L. Soendergaard <peter@sonderport.dk>.
% This file is part of LTFAT version 2.2.0
%
% 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/>.
error(nargchk(3,5,nargin));
L=size(T,1);
if size(T,2)~=L
error('T must be square.');
end;
if nargin==3
% Usage: sym=gabmulappr(T,a,M);
a=p2;
M=p3;
ga=gabtight(a,M,L);
gs=ga;
end;
if nargin==4
% Usage: sym=gabmulappr(T,g,a,M);
ga=p2;
gs=p2;
a=p3;
M=p4;
end;
if nargin==5
% Usage: sym=gabmulappr(T,ga,gm,a,M);
ga=p2;
gs=p3;
a=p4;
M=p5;
end;
N=L/a;
b=L/M;
% gg = GBa(:,ii+jj*M); % Element of analysis frame
% hh = GBs(:,ii+jj*M); % Element of synthesis frame
% HS inner product : < T , g \tensor h> = < T h , g > =
part1=reshape(dgt(T',ga,a,M),M*N,L);
part2=reshape(dgt(part1',gs,a,M),M*N,M*N).';
lowsym = reshape(diag(part2),M,N);
GBa = frsynmatrix(frame('dgt',ga,a,M),length(ga));
% Gabor frame synthesis matrix
if ga ~= gs
GBs = frsynmatrix(frame('dgt',gs,a,M),length(gs));
else
GBs = GBa;
end
if ga == gs
% HS Gram matrix
Gram = abs(GBa'*GBa).^2;
else
Gram = (GBs'*GBs) .* conj(GBa'*GBa);
end
% The Gram matrix is square and Toeplitz.
%iGram=inv(Gram);
sym = reshape(Gram\lowsym(:),M,N);
% sym = involute(sym); % strange but true ?
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