Filter transfer function ($H_A(\lambda )=A(e^{-i\lambda })$)

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.