What is the reason that a periodogram is not a consistent method for obtaining information about frequency response of a system?
To be more precise, why even after increasing number of samples doesn't the variance of the estimator decrease?
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The Periodogram implicitly uses a window to get an idea of the spectrum. Implicitly this is either a rectangular window, with frequency response which is like $\dfrac{\sin(\omega n)}{N\sin(\omega)}$, or a triangular window $(N-k)/N$.
Roughly, the variance of the periodogram is the square of the Fourier transform multiplied by the square of the window. Note that neither windows decrease as N goes to infinity, so the periodogram isn't a consistent estimator.
