Load the electrical consumption signal nelec.

Now we use the discrete wavelet analysis at level 5 with the sym4 wavelet. We set the thresholding type to 's' (soft).

Denoise the signal using the function wdencmp with the threshold set at 4.5, which is the minimum value provided by the GUI tools.

The obtained result is not good. The denoising is very efficient at the beginning and end of the signal, but between 100 and 1100 the noise is not removed. Note that the perf0 value gives the percentage of coefficients set to zero and the perfl2 value gives the percentage of preserved energy.

Now, we denoise the signal with the maximum value provided by the GUI tools for the threshold, 19.5

The denoised signal is very smooth. It seems quite good, but if we look at the residual after position 1100, we can see that the variance of the underlying noise is not constant. Some components of the signal have likely remained in the residual, such as, near the position 1300 and between positions 1900 and 2000.

Now we will use an interval-dependent thresholding, as in the denoising GUI tools.

Perform a discrete wavelet analysis of the signal.

Using the GUI tools to perform interval-dependent thresholding for the signal nelec, and setting the number of intervals at three, we get the content of the denPAR variable, which can be interpreted as follows:

I1 = [1 94] with a threshold thr1 = 5.9

I2 = [94 1110] with a threshold thr2 = 19.5

I3 = [1110 2000] with a threshold thr3 = 4.5

Define the interval-dependent thresholds.

Show the wavelet coefficients of the signal and the interval-dependent threshold for each level of the discrete analysis.

For each level k, the variable thrParams{k} contains the intervals and the corresponding thresholds for the denoising procedure.

Threshold the wavelet coefficients level-by-level, and interval-by- interval, using the values contained in the thrParams variable.

Using the function wthresh, we threshold the wavelet coefficients values between the horizontal lines by replacing them with zeros, while other values are either reduced if sorh = 's' or remain unchanged if sorh = 'h'.

Show the thresholded wavelet coefficients of the signal.

Reconstruct the denoised signal.

Display the original, denoised, and residual signals.

Compare the three denoised versions of the signal.

Looking at the first half of the signals, it is clear that denoising using the minimum value of the threshold is not good. Now, we zoom on the end of the signal for more details.

We can see that when the maximum threshold value is used, the denoised signal is smoothed too much and information is lost.

The best result is given by using the threshold based on the interval- dependent thresholding method, as we will show now.

Instead of manually setting the intervals and the thresholds for each level, we can use the function utthrset_cmd to automatically compute the intervals and the thresholds for each interval. Then, we complete the procedure by applying the thresholds, reconstructing, and displaying the signal.

In command-line mode, we can use the function cmddenoise to automatically compute the denoised signal and coefficients based on the interval-dependent denoising method. This method performs the whole process of denoising using only this one function, which includes all the steps described earlier in this example.

The automatic procedure finds two intervals for denoising:

I1 = [1 1110] with a threshold thr1 = 17.6

I2 = [1110 2000] with a threshold thr2 = 4.5.

We can display the result of denoising and see that the result is fine.

Now, we look at a more complete example of the automatic denoising.

Rather than using the default values for the input parameters, we can specify them when calling the function. Here, the type of threshold is chosen as s (soft) and the number of intervals is set at 3.

For the output parameters, the variable thrParams{1} gives the denoising parameters for levels from 1 to 5. For example, here are the denoising parameters for level 1.

We find the same values that were set earlier this example. They correspond to the choice we have made by fixing the number of intervals to three in the input parameter: nb_Int = 3.

The automatic procedure suggests 2 as the best number of intervals for denoising. This output value BestNbOfInt = 2 is the same as used in the previous step of this example.

The variable int_DepThr_Cell contains the interval locations and the threshold values for a number of intervals from 1 to 6.

Finally, we look at the values corresponding to the locations and thresholds for 5 intervals.

This example shows how to use command-line mode to achieve the same capabilities as the GUI tools for denoising, while giving you more control over particular parameter values to obtain better results.