A couple demos:

http://math.hws.edu/eck/math371/applets/Haar.html

This is what's used in jpeg compression...

And is rather clever, when you think about it.

Take a signal, repeatedly break it up into pairs of low and high 
frequencies, subsample those segments, and place them back into the buffer.

Now, once that has been completed, discard the samples that have the 
lowest amplitudes.

Rebuild the signal with the remaining samples. You've dropped a ton of 
data, but still have a very close approximation of the original data.

Another use of wavelets: Fourier analysis of signals.

Break the signals down using wavelets, then take each section of the 
decomposition, and run a Fourier transform on each section. The lower 
frequencies will be pushed into the higher bands, allowing for higher 
frequency resolution of the lower frequencies.

(Typically, with a Fourier transform low frequencies don't have very 
good resolution, since the progression of frequencies are such:

1x 2x 3x 4x 5x 6x 7x 8x 9x ...

If the frequency band you're interested in is close to the fundamental 
of the transform; you have huge jumps in frequency. take the first two 
samples: Fundamental and double the fundamental. Nothing in between. 
Move that to the right:

32x 33x 34x 35x 36x 37x 38x 39x ...

Now you step between the fundamental at 32 and the next highest sample 
is 1.03x so, now you're looking at samples with this sort of stepping:

1x 1.03x 1.06x 1.09x 1.12x 1.15x ...

Much more precision in the frequency domain. The time domain, obviously 
will suffer for these lower frequencies. You can't pinpoint as 
accurately where they are in time, due to the slowness of their 
oscillation. There are ways, I think, of getting around this, but I 
haven't gotten that far in my understanding.

-- 
~Mike

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