Linear filter of a stationary time series ($Y_t={\sum }_s{a}_s{X}_{t-s}$)

An absolutely summable sequence of coefficients defines $Y_t={\sum }_s{a}_s{X}_{t-s}$ in mean square. It remains weakly stationary and has covariance ${\gamma }_k^Y={\sum }_{s,u}{a}_s{a}_u{\gamma }_{k-s+u}$. Its frequency response is $\alpha (\omega )={\sum }_s{a}_s{e}^{is\omega }$.

For absolutely summable linear-filter coefficients, the generating function is the displayed Laurent series, absolutely convergent on the unit circle. If the filter is one-sided, it is a power series analytic inside the unit disk. A two-sided filter need not have a nonempty larger annulus of convergence. Substituting the backshift operator gives the formal notation $Y=A(B)X$.

The transfer function is the filter generating function evaluated on the unit circle, with a sign convention matching the Fourier representation. Filtering a sinusoid of angular frequency $\lambda$ multiplies its complex amplitude by $H_A(\lambda )$. Its modulus is the filter gain; filtering a weakly stationary process multiplies its time-series spectral density by $|H_A(\lambda ){|}^2$. Using the opposite Fourier sign conjugates the response for real coefficients and leaves that squared modulus unchanged.

Two absolutely summable linear filters of a stationary time series compose by convolution: $c_r={\sum }_j{b}_j{a}_{r-j}$ and ${\sum }_r|c_r|\le ({\sum }_r|a_r|)({\sum }_j|b_j|)$. Absolute convergence in the space of square-integrable random variables justifies exchanging the sums. The frequency responses multiply, giving the spectral density transformation under a linear filter twice.

For an absolutely summable linear filter of a stationary time series, the filter gain is the modulus of its frequency response $a(\lambda )={\sum }_r{a}_r{e}^{ir\lambda }$. It multiplies a sinusoidal amplitude, while its square multiplies the spectral density of a stationary process. A zero removes the corresponding frequency; the filter gain of a composition is the product of the two filter gains.

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.