[sdiy] Waveform analysis into non-sine components

[Tom Wiltshire]

Hi Magnus,

>> I'm still curious about these bases though. How do I know whether a basis is incomplete, complete, or overcomplete?
> 
> If you have a set of these waveforms you can reduce them to orthogonal waveforms by first deciding one as basis, then for the next you find the correlation with the first one, then remove that amount from the second one. Then you have two waveforms which is orthogonal to each other. Then you do the same to the third, but using both of the orthogonals. As you do this, eventually you have orthogonalized all the waveforms. Now, consider that you have N samples, then you need N non-zero waveforms to be complete. If you have less than N, you have an incomplete set and well, if you started with more than N and have N remaining you have overcomplete set.

Ok, that seems pretty straightforward. The interesting thing about that is that it doesn't seem to impose any limitations on what could be used. Perhaps once I've played with removing the correlations from the waveforms in the set I might not feel that way, but on the face of it, you've got a lot of freedom.

>> 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?
> 
> If you try to analyze the base-vectors from one transform in another, they will not make much sense usually. This only shows that the transforms are different. It's thus expected.
> 
> Do you know what a phasor are?

It's a representation of a sine wave as an angle and a vector, isn't it? I can't remember how that deals with frequency though...
Where do the phasors come into it?

Thanks,
Tom