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function sym=ref_gabmulappr_1(T,p2,p3,p4,p5);
%-*- texinfo -*-
%@deftypefn {Function} ref_gabmulappr_1
%@verbatim
%GABMULAPPR_1 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.
%
%
%
% References:
% P. Balazs. Hilbert-Schmidt operators and frames - classification, best
% approximation by multipliers and algorithms. International Journal of
% Wavelets, Multiresolution and Information Processing, 6:315 -- 330,
% 2008.
%
% P. Balazs. Basic definition and properties of Bessel multipliers.
% Journal of Mathematical Analysis and Applications, 325(1):571--585,
% January 2007.
%
% H. G. Feichtinger, M. Hampejs, and G. Kracher. Approximation of
% matrices by Gabor multipliers. IEEE Signal Procesing Letters,
% 11(11):883--886, 2004.
%
%@end verbatim
%@strong{Url}: @url{http://ltfat.github.io/doc/reference/ref_gabmulappr_1.html}
%@seealso{gabmul, demo_gabmulappr}
%@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/>.
% AUTHOR: P. Balazs (XXL)
% ---------- Verify the input -----------------
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;
if size(ga,2)>1
if size(ga,1)>1
error('Input g/ga must be a vector');
else
% ga was a row vector.
ga=ga(:);
end;
end;
if size(gs,2)>1
if size(gs,1)>1
error('Input g/gs must be a vector');
else
% gs was a row vector.
gs=gs(:);
end;
end;
% -------- Algorithm starts here --------------------------
% Calculate the lower symbol. This is basically linear algebra
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);
% Change from lower symbol to upper symbol. This is a quick calculation
% of a 2D convolution.
Gramfirst = conj(dgt(ga,ga,a,M)).*dgt(gs,gs,a,M);
Gramfirst = fft2(Gramfirst);
lowsym = fft2(lowsym);
lowsym = lowsym./Gramfirst;
sym = ifft2(lowsym);
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