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

Question

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.

Answer 1

Looks like could take a look at Prony's method.

Answer 2

You have what's called a 'predictable' process (sum of shifted diracs) and a simple regular process (white noise in this case). You can just average out signal as much as you like, the harmonies will stay the same and the noise will diminish. You can actually get as low a noise as you like.

If you just want to threshold the coefficients, you can try it by seeing the FFT (white noise has a constant frequency response and the exponents will be peaks), you will immediately see the threshold. You can also calculate it via the power (or variance) of the noise, but this also relies on you knowing the amplitude of the C's.