[sdiy] Waveform analysis into non-sine components

[Tom Wiltshire]

Hi Neil,

>> 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.

What I meant was "resources that are aimed at teaching the material rather than showing off the awesome intelligence the author considers they have". A lot of academic output is for academic consumption in my view. This is fine, but it isn't suitable material for teaching. There are far simpler, more visual ways of explaining something like the fourier transform and related concepts like the dot product, orthogonality, etc using maths in context along with a diagram or plot that explains what the symbols mean. Anyone who explains something highly visual like a waveform and its analysis without using at the very least a waveform graph (instead preferring a "concise" mathematical definition) is utterly missing a trick in my view. But that's how my brain works, I guess. I think mostly in pictures, and I can understand these concepts if I see them as graphs and functions.

When I finally worked out how fourier analysis actually worked, I felt like I'd been robbed. So bloody simple! Someone could easily have explained that to me when I was fifteen and first came across it. Instead, I felt like a very simple and useful technique had been made unnecessarily mysterious for years so that mathematicians could understand it and use its magic power and no-one else could. It made me quite cross. If Feynman could explain quantum electrodynamics in a way I can understand, someone could easily do a much better job than they have with fourier analysis. And please don't think that only academics or mathematicians earn my ire for this - I see plenty of it in my own field of computing. People disguise fairly straightforward concepts in heaps of jargon and pretend its well over your head and they couldn't possibly explain it to justify the amount they charge and their exalted position. Grrr.

> Try my Masters thesis:
> 
> http://www.milton.arachsys.com/nj71/index.php?menu=2&submenu=2&subsubmenu=5

Thanks, I'll have a look.

> 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).

Ok, a follow-up question if I may; I understand that if two functions are orthogonal, their dot product will be zero  - over a given range, presumably. As 
we're talking about waveforms, let's say 0 to 2pi, or -pi to +pi or something - one wavecycle.

How do I know whether the set of orthogonal functions I've got is complete? For example, a group of half the walsh functions would all be orthogonal to each other, but wouldn't form a complete basis set. How do I tell if the set is a complete basis set?

>> 2) Analysis - can the same method be used for all basis sets?
> 
> Can't say.  It depends on what your basis set comprises of.

That sounds like a cautious "no"!

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

Yes, I came across wavelets. Have you got a good reference for that? I'll have to do some more searches with "wavelet" included.

> 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.

I'm not trying to avoid the maths. I just want it explained in a way I can understand, and for me that means pictures and not just reams of equations. At least people could *plot* the equations for me and save me doing that with every step!

Thanks,
Tom