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From: George Jefferson on 22 Apr 2010 14:14
Generally we can decompose certain functions into a sum of orthogonal
functions. I have a function which I can write as a sum of potentially
orthogonal functions except I would like to instead of requiring an
orthogonal decomposition to allow for only "almost orthogonality" or
"statistical orthogonality".

That is, the requirement of exact orthogonality is too stringent and gives
anomalous behavior.

Suppose I have

f(t) = sum(a_k(t))

where I could treat this as, say, a fourier series decomposition, I instead
would like to find some way to determine the a_k's in non-orthogonal but
meaningful way.

If the a_k's represent sinusoids then they are, in a sense, parameterized by
a containt w_k, so we have something like

f(t) = sum(a_k(t; w_k))

It may be that w_k represents the average frequency and a_k(t; w_k) varies
slightly around it... e.g., a*cos((w_k + e(t))*t) where |e(t)| << 1.


So the a_k(t; w_k)'s represent "bands" in the time-frequency domain.

In fact, my situation is more complex where the a_k's are non-stationary and
non-linear but are approximately stationary and linear(well, an assumption
on my part).

Anyone mind pointing me in a direction that can get me started?

Thanks



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