wavedec (Wavelet Toolbox)

Multi-level 1-D wavelet decomposition.

[C,L] = wavedec(X,N,'wname')
[C,L] = wavedec(X,N,Lo_D,Hi_D)

Description

wavedec performs a multi-level one-dimensional wavelet analysis using either a specific wavelet ('wname', see wfilters) or specific wavelet decomposition filters (Lo_D and Hi_D).

[C,L] = wavedec(X,N,'wname') returns the wavelet decomposition of the signal X at level N, using 'wname'. N must be a strictly positive integer (see wmaxlev). The output decomposition structure contains the wavelet decomposition vector C and bookkeeping vector L. The structure is organized as in this level-3 decomposition example:

[C,L] = wavedec(X,N,Lo_D,Hi_D) returns the decomposition structure as above, given the low- and high-pass decomposition filters you specify.

Examples

% Load original one-dimensional signal. 
    load sumsin; s = sumsin; 
% Perform decomposition at level 3 of s using db1. 
    [c,l] = wavedec(s,3,'db1');

Algorithm

Given a signal s of length N, the DWT consists of log₂ N stages at most. The first step produces, starting from s, two sets of coefficients: 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 (downsampling).

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 the coefficients cA₁ and cD₁ are of length

The next step splits the approximation coefficients cA₁ in two parts using the same scheme, replacing s by cA₁, and producing cA₂ and cD₂, and so on.

The wavelet decomposition of the signal s analyzed at level j has the following structure: [cA_j, cD_j, ..., cD₁].

This structure contains, for J = 3, the terminal nodes of the following tree: