dwilt.m

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function [c,Ls]=dwilt(f,g,M,L)
%DWILT  Discrete Wilson transform.
%   Usage:  c=dwilt(f,g,M);
%           c=dwilt(f,g,M,L);
%           [c,Ls]=dwilt(f,g,M);
%           [c,Ls]=dwilt(f,g,M,L);
%
%   Input parameters:
%         f     : Input data
%         g     : Window function.
%         M     : Number of bands.
%         L     : Length of transform to do.
%   Output parameters:
%         c     : 2*M x N array of coefficients.
%         Ls    : Length of input signal.
%
%   DWILT(f,g,M) computes a discrete Wilson transform
%   with M bands and window g.
%
%   The length of the transform will be the smallest possible that is
%   larger than the signal. f will be zero-extended to the length of the 
%   transform. If f is a matrix, the transformation is applied to each column.
%
%   The window g may be a text string decribing the window. The following
%   types are recognized
% 
%   'gauss'     - Gaussian window.
%
%   'dualgauss' - Canonical dual of Gaussian window.
%
%   'tight'     - Tight window generated from a Gaussian.
%
%   'sqrthann'  - Square root of Hanning window. This is a tight, FIR
%                 window.  
% 
%   DWILT(f,g,M,L) computes the Wilson transform as above, but does
%   a transform of length L. f will be cut or zero-extended to length L before
%   the transform is done.
%
%   [c,Ls]=DWILT(f,g,M) or [c,Ls]=DWILT(f,g,M,L) additionally return the
%   length of the input signal f. This is handy for reconstruction:
% 
%                [c,Ls]=dwilt(f,g,M);
%                fr=idwilt(c,gd,M,Ls);
% 
%   will reconstruct the signal f no matter what the length of f is, provided
%   that gd is a dual Wilson window of g.
%
%   A Wilson transform is also known as a maximally decimated, even-stacked
%   cosine modulated filter bank.
%
%   Assume that the following code has been executed for a column vector f:
% 
%     c=dwilt(f,g,M);  % Compute a Wilson transform of f.
%     N=size(c,2)*2;   % Number of translation coefficients.
%
%   The following holds for m=0,...,M-1 
%   and n=0,...,N/2-1:
%
%   If m=0:
% 
%                    L-1 
%     c(m+1,n+1)   = sum f(l+1)*g(l-2*n*M+1)
%                    l=0  
% 
%   If m is odd and less than M
% 
%                    L-1 
%     c(m+1,n+1)   = sum f(l+1)*sqrt(2)*sin(pi*m/M*l)*g(k-2*n*M+1)
%                    l=0  
% 
%                    L-1 
%     c(m+M+1,n+1) = sum f(l+1)*sqrt(2)*cos(pi*m/M*l)*g(k-(2*n+1)*M+1)
%                    l=0  
% 
%   If m is even and less than M
% 
%                    L-1 
%     c(m+1,n+1)   = sum f(l+1)*sqrt(2)*cos(pi*m/M*l)*g(l-2*n*M+1)
%                    l=0  
% 
%                    L-1 
%     c(m+M+1,n+1) = sum f(l+1)*sqrt(2)*sin(pi*m/M*l)*g(l-(2*n+1)*M+1)
%                    l=0  
% 
%   if m=M and M is even:
% 
%                    L-1 
%     c(m+1,n+1)   = sum f(l+1)*(-1)^(l)*g(l-2*n*M+1)
%                    l=0
% 
%   else if m=M and M is odd
% 
%                    L-1 
%     c(m+1,n+1)   = sum f(l+1)*(-1)^l*g(l-(2*n+1)*M+1)
%                    l=0
%
%   SEE ALSO:  IDWILT, DGT, MDCT, WILDUAL, WILORTH
% 
%   REFERENCES:
%     H. Bölcskei, H. G. Feichtinger, K. Gröchenig, and F. Hlawatsch.
%     Discrete-time Wilson expansions. In Proc. IEEE-SP 1996 Int. Sympos.
%     Time-Frequency Time-Scale Analysis, june 1996.
%     
%     Y.-P. Lin and P. Vaidyanathan. Linear phase cosine modulated maximally
%     decimated filter banks with perfect reconstruction.
%     IEEETrans.SignalProcess., 43(11):2525-2539, 1995.

% 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(3,4,nargin));

if nargin<4
  L=[];
end;

assert_squarelat(M,M,1,'DWILT',0);

if isnumeric(g)
  if size(g,2)>1
    if size(g,1)>1
      error('g must be a vector');
    else
      % g was a row vector.
      g=g(:);
    end;
  end;
  Lwindow=size(g,1);
else
  Lwindow=0;
end;

% Change f to correct shape.
[f,Ls,W,wasrow,remembershape]=comp_sigreshape_pre(f,'DWILT',0);

[b,N,L]=assert_L(Ls,Lwindow,L,M,2*M,'DWILT');

g=comp_window(g,M,2*M,L,1,'DWILT');

f=postpad(f,L);  

% Call the computational subroutines.
a=M;

c2=comp_dgt(f,g,a,2*M,L,1);
c=comp_dwilt(c2,a);

% Get correct shape.
c=reshape(c,M*2,N/2,W);

% Clean coefficients if they are known to be real
% If input is real, and window is real, output must be real as well.
if (isreal(g) && isreal(f));
  c=real(c);
end;