Spectral density transformation under a linear filter ($f_Y(\omega )=|\alpha (\omega ){|}^2{f}_X(\omega )$)

Filtering multiplies the time-series spectral density by the squared modulus of the frequency response. Insert the spectral integral into the covariance double sum and interchange sums using absolute summability of the filter. This holds even when the input covariances are not absolutely summable, provided the spectral measure has a density.

The symmetric filter with weights $(-1,2,4,2,-1)/6$ has unit gain at zero frequency, but its largest gain is $7/6$ at signed frequencies $\pm \pi /3$. Applied $k$ times, it multiplies a spectral density by $|H{|}^{2k}$. Thus repetition amplifies and selects a period-six oscillation instead of merely suppressing short-scale noise. On a two-sided frequency domain both peaks must be retained.