wavedec :: Functions (Wavelet Toolbox™)

wavedec

Purpose

Multilevel 1-D wavelet decomposition

Syntax

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

Description

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

Note wavedec supports only Type 1 (orthogonal) or Type 2 (biorthogonal) wavelets.

[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 for more information). The output decomposition structure contains the wavelet decomposition vector C and the 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

% The current extension mode is zero-padding (see dwtmode).

% Load original one-dimensional signal. 
load sumsin; s = sumsin; 
% Perform decomposition at level 3 of s using db1. 
[c,l] = wavedec(s,3,'db1');
% Using some plotting commands,
% the following figure is generated.

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:

References

Daubechies, I. (1992), Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics. SIAM Ed.

Mallat, S. (1989), "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. and Machine Intell., vol. 11, no. 7, pp 674-693.

Meyer, Y. (1990), Ondelettes et opérateurs, Tome 1, Hermann Ed. (English translation: Wavelets and operators, Cambridge Univ. Press. 1993.)