DWILT Discrete Wilson transform
Usage: c=dwilt(f,g,M);
c=dwilt(f,g,M,L);
[c,Ls]=dwilt(...);
Input parameters:
f : Input data
g : Window function.
M : Number of bands.
L : Length of transform to do.
Output parameters:
c : 2M xN 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 vector of numerical values, a text string or a
cell array. See the help of WILWIN for more details.
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.
[c,Ls,g]=DWILT(...) additionally outputs the window used in the
transform. This is useful if the window was generated from a description
in a string or cell array.
A Wilson transform is also known as a maximally decimated, even-stacked
cosine modulated filter bank.
Use the function WIL2RECT to visualize the coefficients or to work
with the coefficients in the TF-plane.
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
References:
H. Boelcskei, H. G. Feichtinger, K. Groechenig, and F. Hlawatsch.
Discrete-time Wilson expansions. In Proc. IEEE-SP 1996 Int. Sympos.
Time-Frequency Time-Scale Analysis, june 1996.
I. Daubechies, S. Jaffard, and J. Journe. A simple Wilson orthonormal
basis with exponential decay. SIAM J. Math. Anal., 22:554--573, 1991.
Y.-P. Lin and P. Vaidyanathan. Linear phase cosine modulated maximally
decimated filter banks with perfectreconstruction. IEEE Trans. Signal
Process., 43(11):2525--2539, 1995.
Url: http://ltfat.github.io/doc/gabor/dwilt.html
See also: idwilt, wilwin, wil2rect, dgt, wmdct, wilorth.
Package: ltfat