swt (Wavelet Toolbox)

Discrete stationary wavelet transform 1-D

Syntax

SWC = swt(X,N,'wname')
SWC = swt(X,N,Lo_D,Hi_D)
[SWA,SWD] = swt(X,N,'wname')
[SWA,SWD] = swt(X,N,Lo_D,Hi_D)

Description

swt performs a multilevel 1-D stationary wavelet decomposition using either a specific orthogonal wavelet ('wname' see wfilters for more information) or specific orthogonal wavelet decomposition filters.

SWC = swt(X,N,'wname') computes the stationary wavelet decomposition of the signal X at level N, using 'wname'.

N must be a strictly positive integer (see wmaxlev for more information) and length(X) must be a multiple of 2^N .

SWC = swt(X,N,Lo_D,Hi_D), computes the stationary wavelet decomposition as above, given these filters as input:

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

Lo_D and Hi_D must be the same length.

The output matrix SWC contains the vectors of coefficients stored row-wise:

For 1 i N, the output matrix SWC(i,:) contains the detail coefficients of level i and SWC(N+1,:) contains the approximation coefficients of level N.

[SWA,SWD] = swt(...) computes approximations, SWA, and details, SWD, stationary wavelet coefficients.

The vectors of coefficients are stored row-wise:

For 1 i N, the output matrix SWA(i,:) contains the approximation coefficients of level i and the output matrix SWD(i,:) contains the detail coefficients of level i.

Examples

% Load original 1D signal.
load noisbloc; s = noisbloc;

% Perform SWT decomposition at level 3 of s using db1.
[swa,swd] = swt(s,3,'db1');

% Plots of SWT coefficients of approximations and details
% at levels 3 to 1.

% Using some plotting commands,
% the following figure is generated.

Algorithm

Given a signal s of length N, the first step of the SWT 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.

More precisely, the first step is

Note cA₁ and cD₁ are of length N instead of N/2 as in the DWT case.

The next step splits the approximation coefficients cA₁ in two parts using the same scheme. But, with modified filters obtained by upsampling the filters used for the previous step and replacing s by cA₁. Then, the SWT produces cA₂ and cD₂. More generally,

See Also
dwt, wavedec

References

Nason, G.P.; B.W. Silverman (1995), "The stationary wavelet transform and some statistical applications," Lecture Notes in Statistics, 103, pp. 281-299.

Coifman, R.R.; Donoho D.L. (1995), "Translation invariant de-noising," Lecture Notes in Statistics, 103, pp. 125-150.

Pesquet, J.C.; H. Krim, H. Carfatan (1996), "Time-invariant orthonormal wavelet representations," IEEE Trans. Sign. Proc., vol. 44, 8, pp. 1964-1970.

shanwavf swt2