dwt (Wavelet Toolbox)

dwt	Examples See Also

Single-level discrete 1-D wavelet transform.

Syntax

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

Description

The dwt command performs a single-level one-dimensional wavelet decomposition with respect to either a particular wavelet ('wname', see wfilters) or particular wavelet decomposition filters (Lo_D and Hi_D) you specify.

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

[cA,cD] = dwt(X,Lo_D,Hi_D) computes the wavelet decomposition as above, given these filters as input:

Lo_D is the decomposition low-pass filter and
Hi_D is the decomposition high-pass filter.

Lo_D and Hi_D must be the same length.

If lx is the length of X and lf is the length of the filters Lo_D and Hi_D, then
length(cA) = length(cD) = floor((lx+lf-1)/2).

For the different signal extension modes, see dwtmode.

Examples

% Construct elementary original one-dimensional signal. 
    randn('seed',531316785) 
    s = 2 + kron(ones(1,8),[1 -1]) + ...
        ((1:16).^2)/32 + 0.2*randn(1,16);

% Perform single-level discrete wavelet transform of s by haar.
    [ca1,cd1] = dwt(s,'haar'); 
    subplot(311); plot(s); title('Original signal'); 
    subplot(323); plot(ca1); title('Approx. coef. for haar'); 
    subplot(324); plot(cd1); title('Detail coef. for haar');

% For a given wavelet, compute the two associated decomposition 
% filters and compute approximation and detail coefficients 
% using directly the filters. 
    [Lo_D,Hi_D] = wfilters('haar','d'); 
    [ca1,cd1] = dwt(s,Lo_D,Hi_D);

% Perform single-level discrete wavelet transform of s by db2
% and observe edge effects for last coefficients.
% These extra coefficients are only used to ensure exact 
% global reconstruction.
    [ca2,cd2] = dwt(s,'db2');
    subplot(325); plot(ca2); title('Approx. coef. for db2'); 
    subplot(326); plot(cd2); title('Detail coef. for db2');

Algorithm

Starting from a signal s, two sets of coefficients are computed: approximation coefficients CA₁ and detail coefficients CD₁. These vectors are obtained by convolving s with the low-pass filter Lo_D for approximation, and with the high-pass filter Hi_D for detail, followed by dyadic decimation.

More precisely, the first step is:

The length of each filter is equal to 2N. If n = length(s), the signals F and G are of length n + 2N - 1 and then the coefficients CA₁ and CD₁ are of length

Note: In order to deal with signal-end effects involved by convolution based algorithm, a global variable managed by dwtmode is used. The possible options are: zero-padding (used in the previous example, this mode is the default), symmetric extension, and smooth extension. It should be noted that dwt has the same single inverse function idwt for the three extension modes.

Limitations

Periodized wavelet transform is handled separately (see dwtper and idwtper).