Periodogram spectrometry of a Gaussian random process.

kth spectral element is simply the mean square amplitude of the kth Fourier coefficient of a straight DFT of an unweighted sequence of data points. The Fourier coefficient from a single transform is itself a random variable; its variance is equal to its expected value. But when squared kth points from many transforms are averaged, the result converges to a spectral power estimate.

It is easy to see that these mean square should be a spectral estimator: The kth Fourier amplitude is formed by adding N successive down-converted samples. The multiplication by a (complex) LO signal down-converts frequency k to baseband. Adding the sum of N of these baseband amplitudes (I and Q separately) constitutes a low-pass filter operation (Box car filter), so the I and Q values from each transform are equivalent to successive I and Q signals from a Low

and turns out to be proportional to the total power falling within a narrow BPF window (having a sinx/x shape).