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# swt

Discrete stationary wavelet transform 1-D

## Syntax

SWC = swt(X,N,'wname')
SWC = 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 2N .

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.

 Note   swt is defined using dwt with periodic extension.

## Examples

```% Load original 1D signal.

% 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.
```

expand all

### Algorithms

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,

## 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.