# Given f(t) = \sum_k C_k exp(2 pi i w_k t ) + noise. Need to estimate C_k and w_k .

## Simpliefied setup.

Assume I am given some function f(t). I know that it is constructed as $f(t) = \sum_{k=1...M} C_k exp(2 \pi~ i~ w_k t ) + noise(t)$. where $noise(t)$ is some random set of numbers depending on $t$ (white noise if you like). I need to estimate $C_k$ and $w_k$.

Solution: I will make Fourier Transform $\hat f(\lambda)$ and will pick up those $\lambda= w_k$ such that $\hat f(\lambda) > threshold$. How should I choose this threshold ? (The question is probably not 100% well-posed, but hope idea is clear).

## True setup.

Now assume that $t$ is discrete e.g. $t= l/N$, for some $N$ and $l=1...N$.

And the problem is that $w_k$ are do not of the form $l/N$.

Difficulty So the functions $exp(2 \pi~ i~ w_k t )$ are NOT orthogonal on the discrete set $t=l/N$.

So there are some problems with making Fourier transform approach.

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Shouldn't that be tagged statistics rather than functional analysis? – Vincent Beffara Mar 28 '12 at 20:53