c=dwilt(f,g,M); c=dwilt(f,g,M,L); [c,Ls]=dwilt(...);
f | Input data |
g | Window function. |
M | Number of bands. |
L | Length of transform to do. |
c | \(2M \times N\) 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 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 \(m=0,\ldots,M-1\) and \(n=0,\ldots,N/2-1\):
If \(m=0\):
If \(m\) is odd and less than \(M\)
If \(m\) is even and less than \(M\)
if \(m=M\) and \(M\) is even:
else if \(m=M\) and \(M\) is odd
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. [ http ]
I. Daubechies, S. Jaffard, and J. Journé. 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.