[sdiy] Waveform analysis into non-sine components

[Neil Johnson]

Hi Tom,

> Can anyone give me some useful references for the above topic?

Well, there are many references, but they all fall foul of your later 
requirement:

"hideous pages of academic maths are really only intended for academic 
mathematicians and don't serve to teach the rest of us anything much"

Ironic, really, when that's *exactly* what you're asking.

> I've been looking into it, but there's not much from the synthesis
> world. I found a reasonably interesting discussion about using walsh
> functions to analyse seismic data, but that's about the size of it.

Try my Masters thesis:

http://www.milton.arachsys.com/nj71/index.php?menu=2&submenu=2&subsubmenu=5

> Principally, I'd like to know more about:
>
> 1) Basis functions - Sin/cos are a good basis for waveform analysis.
> Walsh functions are another. Are two phase-shifted squares another?
> If not, why not? What makes a set of functions a sound basis or not?

Well, for starters there's the property of orthogonality.  Ideally the 
set of basis functions should be orthonormal too, but that's not a 
strict requirement (it makes the maths easier).

> 2) Analysis - can the same method be used for all basis sets?

Can't say.  It depends on what your basis set comprises of.

> 3) Pros/cons of various basis function sets for various types of
> analysis, with particular emphasis on sound waveforms.

They all have pros and cons.  Some are easy to compute both for analysis 
and synthesis.  Some are mathematically cleaner.  Some are better at 
synthesis than analysis, and vice versa.  Some capture the low frequency 
content better, other do better at high frequency content.  Some need 
more windowing than others.

One word I didn't see in your email is "wavelet".

> I understand Fourier Analysis well enough these days (finally), but
> I'd like to extend this understanding and grok a bit more of the
> foundations.

Excellent!

 > Please bear in mind that hideous pages of academic maths
> are really only intended for academic mathematicians and don't serve
> to teach the rest of us anything much. I realise that's a big ask
> given the topic.

And then you go and spoil it.  The foundations *are* mathematical.  You 
can't escape it because that's exactly what you are talking about.

But I'll be interested to see what others have to add to this thread :)

Cheers,
Neil
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