## 1-D Stationary Wavelet Transform

This topic takes you through the features of 1-D discrete stationary
wavelet analysis using the Wavelet Toolbox™ software. For more
information see Nondecimated Discrete Stationary Wavelet Transforms (SWTs) in
the *Wavelet Toolbox User's Guide*.

The toolbox provides these functions for 1-D discrete stationary wavelet analysis. For more information on the functions, see the reference pages.

### Analysis-Decomposition Functions

Function Name | Purpose |
---|---|

Decomposition |

### Synthesis-Reconstruction Functions

Function Name | Purpose |
---|---|

Reconstruction |

The stationary wavelet decomposition structure is more tractable than the wavelet one. So the utilities, useful for the wavelet case, are not necessary for the stationary wavelet transform (SWT).

In this section, you'll learn to

Load a signal

Perform a stationary wavelet decomposition of a signal

Construct approximations and details from the coefficients

Display the approximation and detail at level 1

Regenerate a signal by using inverse stationary wavelet transform

Perform a multilevel stationary wavelet decomposition of a signal

Reconstruct the level 3 approximation

Reconstruct the level 1, 2, and 3 details

Reconstruct the level 1 and 2 approximations

Display the results of a decomposition

Reconstruct the original signal from the level 3 decomposition

Remove noise from a signal

### 1-D Analysis

This example involves a noisy Doppler test signal.

Load a signal.

From the MATLAB

^{®}prompt, typeload noisdopp

Set the variables. Type

s = noisdopp;

For the SWT, if a decomposition at level

`k`

is needed,`2^k`

must divide evenly into the length of the signal. If your original signal does not have the correct length, you can use the`wextend`

function to extend it.Perform a single-level Stationary Wavelet Decomposition.

Perform a single-level decomposition of the signal using the

`db1`

wavelet. Type[swa,swd] = swt(s,1,'db1');

This generates the coefficients of the level 1 approximation (swa) and detail (swd). Both are of the same length as the signal. Type

whos

Name Size Bytes Class `noisdopp`

`1x1024`

`8192`

`double array`

`s`

`1x1024`

`8192`

`double array`

`swa`

`1x1024`

`8192`

`double array`

`swd`

`1x1024`

`8192`

`double array`

Display the coefficients of approximation and detail.

To display the coefficients of approximation and detail at level 1, type

subplot(1,2,1), plot(swa); title('Approximation cfs') subplot(1,2,2), plot(swd); title('Detail cfs')

Regenerate the signal by Inverse Stationary Wavelet Transform.

To find the inverse transform, type

A0 = iswt(swa,swd,'db1');

To check the perfect reconstruction, type

err = norm(s-A0) err = 2.1450e-14

Construct and display approximation and detail from the coefficients.

To construct the level 1 approximation and detail (

`A1`

and`D1`

) from the coefficients`swa`

and`swd`

, typenulcfs = zeros(size(swa)); A1 = iswt(swa,nulcfs,'db1'); D1 = iswt(nulcfs,swd,'db1');

To display the approximation and detail at level 1, type

subplot(1,2,1), plot(A1); title('Approximation A1'); subplot(1,2,2), plot(D1); title('Detail D1');

Perform a multilevel Stationary Wavelet Decomposition.

To perform a decomposition at level 3 of the signal (again using the

`db1`

wavelet), type[swa,swd] = swt(s,3,'db1');

This generates the coefficients of the approximations at levels 1, 2, and 3 (

`swa`

) and the coefficients of the details (`swd`

). Observe that the rows of`swa`

and`swd`

are the same length as the signal length. Typeclear A0 A1 D1 err nulcfs whos

Name Size Bytes Class `noisdopp`

`1x1024`

`8192`

`double array`

`s`

`1x1024`

`8192`

`double array`

`swa`

`3x1024`

`24576`

`double array`

`swd`

`3x1024`

`24576`

`double array`

Display the coefficients of approximations and details.

To display the coefficients of approximations and details, type

kp = 0; for i = 1:3 subplot(3,2,kp+1), plot(swa(i,:)); title(['Approx. cfs level ',num2str(i)]) subplot(3,2,kp+2), plot(swd(i,:)); title(['Detail cfs level ',num2str(i)]) kp = kp + 2; end

Reconstruct approximation at Level 3 From coefficients.

To reconstruct the approximation at level 3, type

mzero = zeros(size(swd)); A = mzero; A(3,:) = iswt(swa,mzero,'db1');

Reconstruct details from coefficients.

To reconstruct the details at levels 1, 2 and 3, type

D = mzero; for i = 1:3 swcfs = mzero; swcfs(i,:) = swd(i,:); D(i,:) = iswt(mzero,swcfs,'db1'); end

Reconstruct and display approximations at Levels 1 and 2 from approximation at Level 3 and details at Levels 2 and 3.

To reconstruct the approximations at levels 2 and 3, type

A(2,:) = A(3,:) + D(3,:); A(1,:) = A(2,:) + D(2,:);

To display the approximations and details at levels 1, 2 and 3, type

kp = 0; for i = 1:3 subplot(3,2,kp+1), plot(A(i,:)); title(['Approx. level ',num2str(i)]) subplot(3,2,kp+2), plot(D(i,:)); title(['Detail level ',num2str(i)]) kp = kp + 2; end

To denoise the signal, use the

`ddencmp`

command to calculate a default global threshold. Use the`wthresh`

command to perform the actual thresholding of the detail coefficients, and then use the`iswt`

command to obtain the denoised signal.**Note**All methods for choosing thresholds in the 1-D Discrete Wavelet Transform case are also valid for the 1-D Stationary Wavelet Transform, which are also those used by the Wavelet Analysis app. This is also true for the 2-D transforms.

[thr,sorh] = ddencmp('den','wv',s); dswd = wthresh(swd,sorh,thr); clean = iswt(swa,dswd,'db1');

To display both the original and denoised signals, type

subplot(2,1,1), plot(s); title('Original signal') subplot(2,1,2), plot(clean); title('denoised signal')

The obtained signal remains a little bit noisy. The result can
be improved by considering the decomposition of `s`

at
level 5 instead of level 3, and repeating steps 14 and 15. To improve
the previous denoising, type

[swa,swd] = swt(s,5,'db1'); [thr,sorh] = ddencmp('den','wv',s); dswd = wthresh(swd,sorh,thr); clean = iswt(swa,dswd,'db1'); subplot(2,1,1), plot(s); title('Original signal') subplot(2,1,2), plot(clean); title('denoised signal')

A second syntax can be used for the `swt`

and `iswt`

functions, giving the same results:

lev = 5; swc = swt(s,lev,'db1'); swcden = swc; swcden(1:end-1,:) = wthresh(swcden(1:end-1,:),sorh,thr); clean = iswt(swcden,'db1');

You can obtain the same plot by using the same plot commands as in step 16 above.