[sdiy] Waveform analysis into non-sine components

[Olivier Gillet]

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

The same approach as for Fourier analysis can be used for any
orthogonal basis. This means that for any pair of function from the
basis, their dot product (numerically: the sum of pairwise element
products) should be null.

Thus, you can decompose a periodic signal into a basis of saws,
squares, triangles, etc. Either by direct integration (dot product of
your input signal with each element of the basis), or starting from
the proper Fourier coefficient and with the appropriate matrix
transform (it's just changing the basis in which a vector is
represented).

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

I see two things that make the Fourier basis very special:
* It is the basis of eigenfunctions of LTI (linear time-invariant)
systems. The LTI bit: Electronic circuits built with ideal R, L, C
elements and linear gain elements (ideal op-amps, OTA, etc...) are LTI
systems - systems whose behavior is captured in a transfer function.
The eigenbasis bit: if you try to solve mathematically the question
"which representation of the input signal into simple elements is such
that those simple elements are simply affected by a complex
multiplication when they are processed through a LTI system", the
representation that will arise is the decomposition into complex
exponentials. Sorry for the hand-waving but there is this very deep
connection between LTI systems and the Fourier representation. All the
nice properties of the Fourier representation (product in one domain
is convolution in the other domain ; identities about
shifting/scaling) have no equivalent in other bases, and arise from
this connection.
* Very, very, roughly captures what's happening in the human ear.

Olivier