[sdiy] Waveform analysis into non-sine components

[James J. Clark]

(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