# Fitting data to sums of squares of sinusoids

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!!

• 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? May 19, 2021 at 16:01
• 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! May 19, 2021 at 16:17
• The FFT ought to produce the values directly and efficiently: this is a problem of Fourier analysis rather than of statistics.
– whuber
May 19, 2021 at 16:29
• I'm not entirely sure I understand, how would you obtain the phases using a Fourier transform? May 19, 2021 at 16:46
• The sines and cosines of twice the phases show up in the Fourier coefficients.
– whuber
May 19, 2021 at 17:44