Attenuation and Decay with General Loss Filter

The filter accounting for frequency-dependent loss may be other than a two-point averager.

A general presentation of the attenuation factor for the $k^{th}$ harmonics is given by

$$\alpha_k(t) = G_a(f_k)^{\frac{tf_s}{N + P_a(f_k)}},$$

and the decay for each harmonic becomes

$$\tau_k = -\frac{N + P_a(f_k)}{f_s \ln G_a(2\pi f_kT_s)},$$

where $G_a(f_k)$ and $P_a(f_k)$ are the gain and phase delays, respectively, of the filter used.