dwtper (Wavelet Toolbox)

Single-level discrete 1-D wavelet transform (periodized).

[cA,cD] = dwtper(X,'wname')
[cA,cD] = dwtper(X,Lo_D,Hi_D)

Description

dwtper is a one-dimensional wavelet analysis function.

[cA,cD] = dwtper(X,'wname') computes the approximation coefficients vector cA and detail coefficients vector cD, obtained by periodized wavelet decomposition of the vector X.

'wname' is a string containing the wavelet name (see wfilters).

Instead of giving the wavelet name, you can give the filters. When used with three arguments: [cA,cD] = dwtper(X,Lo_D,Hi_D), Lo_D is the decomposition low-pass filter and Hi_D is the decomposition high-pass filter.

If lx = length(X) then length(cA) = length(cD) = ceil(lx/2).

Examples

% Set initial signal and get filters. 
x = sin(0.3*[1:300]); lx = length(x)
lx =
    300

w = 'db9'; 
[Lo_D,Hi_D,Lo_R,Hi_R] = wfilters(w);

% DWT with zero-padding signal extension. 
[cazp,cdzp] = dwt(x,w);

% The transform uses some extra coefficients, 
% at most 2 if lx is odd. 
lxtzp = 2*length(cazp)
lxtzp =
    316

% Reconstruction. 
xzp = idwt(cazp,cdzp,w,lx); 

% Error with zero-padding. 
errzp = max(abs(x-xzp))
errzp =
    7.3231e-12

% Periodized DWT. 
[cap,cdp] = dwtper(x,w);

% The transform uses a minimum of extra coefficients. 
lxtp = 2*length(cap)
lxtp =
    300

% Reconstruction. 
xp = idwtper(cap,cdp,w,lx);

% Error with periodized DWT. 
errp = max(abs(x-xp))
errp =
 1.4588e-11

Algorithm

The algorithm is the same as in dwt but the signal X is extended assuming periodicity. More precisely, if lx = length(X) is even, the extended signal is
extX = [X(lx-lf+1:lx) X X(1:lf)] where lf is the length of the filter. Then, usual convolution and downsampling operations are done, followed by keeping the central part of length lx/2.