First fix K=1 in (1). Then w0=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.