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First fix K=1 in (1). Then *w*_{0}=1 and corresponds to the familiar periodogram.
The periodogram is inconsistent and generally a biased estimate of the
true spectral density. It is inconsistent because its variance does
not approach zero as the number of observations increases. The remedy
is to ``smooth'' it, which is equivalent to multiplying the data by a
taper before computing the periodogram. Because of the
side-lobes of the Fejr kernel power is transferred
among different regions of the spectrum (leakage) and cause the
periodogram to be biased. The formula below illustrates the point:

where is the Fejr kernel.

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Stanford Exploration Project

10/9/1997