[sdiy] Waveform analysis into non-sine components

[Tom Wiltshire]

Thanks Jim.

Explanations I can understand are appreciated wherever they come from.

I'm still curious about these bases though. How do I know whether a basis is incomplete, complete, or overcomplete?

I've been looking into the wavelet analysis that Neil J mentioned, and it's interesting stuff, since it can be reasonably applied to transients and the attack portion of waveforms. Whilst I've tried doing fourier analysis on these, the results never really seem convincing - unsurprisingly.
I'd like to experiment with wavelets, but I'm not clear what constitutes a basis set of wavelet functions. Obviously I can dream up some functions at different scales and use cross-correlation to see whether that function is a good fit for my waveform, but it's a bit random. Presumably there's some reason why you'd use some and not others. For example, are wavelet scales related harmonically like the sine waves in fourier analysis?

Thanks,
Tom



On 14 Apr 2012, at 02:31, James J. Clark wrote:

> 
> (I thought I would add in my 2-cents here even though I am one of those academic types...)
> 
> Basis sets need not be orthogonal to be useful and can even have more elements than are stictly needed. It is often useful to have an "overcomplete basis" which has some redundant elements. These can be more robust to noise and can simplify analysis or decomposition. To take a perhaps not too useful example, you could combine a sinusoidal basis with a Walsh function basis. This might be good for representing signals that are mostly bandlimited but have some jumps like a sum of sine and square waves.
> 
> I think overcomplete bases might be good for analyzing and synthesizing granular waveforms, since the grains themselves can be considered as a large, sparse, overcomplete basis. Sparsity here is one of those "academic terms" which means that the basis waveforms are mostly zero. It is theorized that the human brain uses a sparse overcomplete representation to analyze images, where most neurons only respond to rather specific patterns and are silent most of the time.
> 
> One other item worth knowing is that perhaps the main reason that sinusoidal basis functions are so widely used is that sinusoids are eigenfunctions (another of them academic terms) of linear time-invariant systems. This basically means that if you put a sine wave into a linear time-invariant system, such as a linear filter, you get a sine wave out with the same frequency, and possibly a different amplitude. This fact is what makes analyzing linear systems with sine waves so straightforward. Trying to analyze a linear system using Walsh functions is much trickier. Of course, in making interesting sounds we don't always have to analyze systems but it would be nice to have some idea of what the effect is on our signals of passing them through various systems.
> 
> Note that this nice property of sine waves only holds for linear time-invariant systems. If you have a nonlinear system (such as a distortion box) or a time-varying system (such as a linear filter whose cutoff frequency is being modulated) then sine wave representations are no more useful than say Walsh signal representations.
> 
> I'd draw a picture but ASCII art is not my forte...
> 
> Jim
> www.cylonix.com
> 
> 
> 
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