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I am given some time series data $(t, y(t))$ sampled at regular intervals $t=0,s,2s,3s,\ldots,1$ for some step size $s$, obtained from a function $$ y(t) = y(t, \vec A, \vec \delta) = \sum_{i=1}^N A_i \sin(\pi t + \delta_i)^2 $$ i.e. a linear combination of squares of sin's, all with the same frequency. $N$, the number of terms, is known. Assuming the step size $s$ is "small enough" ($1/s\gg N$) and the data is noise-free, I have the following questions:

  1. Is there a way of obtaining the parameter vectors $\vec A$ and $\vec \delta$ from the time series data, e.g. by a least squares fit? Is there a method that converges, and does so effiicently (say, in the number of data points/number of parameters)?
  2. Is the solution unique (where, for the $\delta_i$, I mean unique modulo $2\pi$)?
  3. What if all the $A_i$ were $1$ to start with, i.e. $y(t) = y(t, (1, \ldots, 1), \vec\delta)$? In this case I only need to find the phase shifts; does that alter answers for the above?

I am asking this because I suspect the solution would not be unique (as in, I tried fitting a few example cases with Mathematica, and didn't always get a unique answer, which might just mean I'm not doing it right though); but I can't find a simple argument for it.

Thanks!!

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  • $\begingroup$ When you write that your data is noise-free, does that mean that you know the true $N$? Solutions are definitely not unique, because we can add any integer multiple of $2\pi$ to each $\delta_i$ and get the exact same fit. For your point 3, are you still assuming noise-free observations when you fix $A_i=1$, i.e., are the $A_i=1$ the actual true values? $\endgroup$ May 19 '21 at 16:01
  • $\begingroup$ Indeed, $N$ is known beforehand; I'm only interested in uniqueness mod $2\pi$; and in the second part, indeed, noise-free means $A_i=1$ are the actual true values. I amended the question to reflect this, thanks! $\endgroup$
    – J Bausch
    May 19 '21 at 16:17
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    $\begingroup$ The FFT ought to produce the values directly and efficiently: this is a problem of Fourier analysis rather than of statistics. $\endgroup$
    – whuber
    May 19 '21 at 16:29
  • $\begingroup$ I'm not entirely sure I understand, how would you obtain the phases using a Fourier transform? $\endgroup$
    – J Bausch
    May 19 '21 at 16:46
  • $\begingroup$ The sines and cosines of twice the phases show up in the Fourier coefficients. $\endgroup$
    – whuber
    May 19 '21 at 17:44

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