Signal processing with wavelets

Since Joseph Fourier, a signal f(t) can be expanded into orthogonal basis functions.
Instead of giving the values at different date, the function is seen as a weighted sum of basis functions. But only a few terms of this sum are relevant, and you might neglect the others : this leads to data compression (e.g. graphic JPEG format) ...
With our program (you need java 1.1.x, i.e. Netscape 4.5 or later, to run it), you can experiment this filtering, with wavelets as basis functions :

First you build a signal, you can choose a predefined function with the menu (parameters are entered in the box), or draw your signal with the mouse, or read it after pasting it in the value frame (right).
Then you extract the wavelet Daubechies (of order 4, 12 or 20) coefficients (with the menu), you will see a representation of the coefficients,
Then you filter : i.e. you decide to keep a fraction of coefficients (adjust the scroll bar).
Then you synthesize the signal (invert ..) and compare it with the original.
You can try the same process with Fourier/filter/invert Fourier.
Alternately, coefficients can be entered, and a signal synthesis is performed with invert.
Right menu enables you to delete last plot, or place it it first position (roll curves).

The program uses wavelet expansion, the basis functions are characterized with :
- scale (like sines, where it is called frequency)
- as well as position.

Left, a simple wavelet : the Haar wavelet H(t) :
From this mother function, other functions can be obtained by translation and dilation:
If j represents scale and k translation : The function is H_j,k(t)=A * H(2j t - k)

Look at the two positions (green and red)
of wavelets at scale 1

Look at the four positions (four colors)
of wavelets at scale 2 :

Actually, our program uses Daubechies wavelets whose shapes are a little more complex : you can see the wavelet associated with the nth coefficient, with menu : xx values; coeff and in the parameter box xx;n, (this generates xx values, only the nth is not null), and then invertDaubechies

Once the "signal" f(t) is expanded into weights of the different wavelet functions, what can be done ?

The main use is the data reduction by neglecting the small weights. The signal is then defined only by few coefficients : a famous (and always quoted) example of data compression with wavelets is its use in the finger prints data base of the F.B.I.

Philosophical note ! :

The best compression is the "human compression" : a file with music recorded as a time signal e.g. in wave format is much bigger (even compressed with the best algorithm) than a file with "Midi events" like : C# played by a piano at a loudness of 64 follown by notes B, D#,F#, played after some time etc. :
The file Wave (a famous song by Carlos Jobim) in mid format is 38 k long, in MP3, it woulbe be around 5 Mb, wave format (uncompressed) 44 Mbytes !!

For documentation about wavelet site with many links.

For documentation about Fourier transform .

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