# Estimating parameters of a mixture of normal distributions.

I want to estimate the parameters $\mu_i$ and $\sigma^2_i$ of a countable mixture of Gaussians with assumed equal weights, variance and identically spaced means. I intially thought that the Fourier transforms of an iid sample from the mixture mentioned would give above another Gaussian in phase space, but when computed in matlab I get a sharp peak spectral distribution. I used a Gaussian windows function, so I don't understand why the power spectral density tends to be unbounded at zero phase.

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I do not really know whether this applies here, but some years ago I've come across some free software which was called "emmix", which was announced to solve exactly this problem (decomposition of a mix of normal distributions). It came with a good/lot of explanative material, so possibly, if this is still available, it might help to understand the method as well. – Gottfried Helms Aug 20 '12 at 17:30
A countable mixture with equal weights? Unless this "countable" is finite, no such thing exists in the realm of probability measures. – Robert Israel Aug 20 '12 at 18:55

If the Gaussians are constrained to be of equal shapes and distances, then you have a scaled theta function or the convolution of a Gaussian with a sha (Ш) function or scaled Dirac comb. Ш is a sum of equally spaced delta functions, and the Fourier transform of Ш is another Ш function, usually with different spacings and amplitudes depending on your conventions. Since the Fourier transform of a Gaussian is a Gaussian, the Fourier transform of your function is a sum of equally spaced delta functions whose amplitudes sample a Gaussian density, something like $$\beta \sum_{n=-\infty}^\infty e^{-\pi^2 z^2/\sigma - 2 \pi i x_0 z}\delta(z-n/\alpha)$$
where the parameters $\alpha, \beta, \sigma,$ and $x_0$ are determined by the parameters of your distribution and your Fourier conventions.
In particular, there is a delta function at $0$. You should expect this to happen when you have a periodic function whose average over a period isn't $0$.