Repeated symmetric filtering can amplify an oscillation ($H(\omega )=[4+4\cos \omega -2\cos (2\omega )]/6$)

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.