Wavelet Toolbox

Compressing a Signal Using Wavelet Packets

Selecting a Threshold for Compression.   

1. Click the Compress button.

1. The Wavelet Packet 1-D Compression window appears with an approximate threshold value automatically selected.

   

   The leftmost graph shows how the threshold (vertical yellow dotted line) has been chosen automatically (1.482) to balance the number of zeros in the compressed signal (blue curve that increases as the threshold increases) with the amount of energy retained in the compressed signal (purple curve that decreases as the threshold increases).

   This threshold means that any signal element whose value is less than 1.482 will be set to zero when we perform the compression.

   Threshold controls are located to the right (see the red box in the figure above). Note that the automatic threshold of 1.482 results in a retained energy of only 81.49%. This may cause unacceptable amounts of distortion, especially in the peak values of the oscillating signal. Depending on your design criteria, you may want to choose a threshold that retains more of the original signal's energy.

2. Adjust the threshold by typing 0.8938 in the text field opposite the threshold slider, and then press the Enter key.

   

1. The value 0.8938 is a number that we have discovered through trial and error yields more satisfactory results for this analysis.

   After a pause, the Wavelet Packet 1-D Compression window displays new information.

   

   Note that, as we have reduced the threshold from 1.482 to 0.8938,

   • The vertical yellow dotted line has shifted to the left.
   • The retained energy has increased from 81.49% to 90.96%.
   • The number of zeros (equivalent to the amount of compression) has decreased from 81.55% to 75.28%.

Compressing a Signal.   

3. Click the Compress button.

1. The Wavelet Packet 1-D tool compresses the signal using the thresholding criterion we selected.

   

   The original (red) and compressed (yellow) signals are displayed superimposed. Visual inspection suggests the compression quality is quite good.

Looking more closely at the compressed signal, we can see that the number of zeros in the wavelet packets representation of the compressed signal is about 75.3%, and the retained energy about 91%.

If you try to compress the same signal using wavelets with exactly the same parameters, only 89% of the signal energy is retained, and only 59% of the wavelet coefficients set to zero. This illustrates the superiority of wavelet packets for performing compression, at least on certain signals.

You can demonstrate this to yourself by returning to the main Wavelet Packet 1-D window, computing the wavelet tree, and then repeating the compression.

One-Dimensional Wavelet Packet Analysis

De-Noising a Signal Using Wavelet Packets