Definition:Linear Filter

Definition
Let $S$ be a stationary stochastic process governed by a white noise process:


 * $\map z t = \mu + a_t$

where:
 * $\mu$ is a constant mean level
 * $a_t$ is an independent shock at timestamp $t$.

A linear filter takes the terms of $S$ and applies a weighted sum of the past values so that:

where $B$ denotes the backward shift operator, hence:


 * $\map \phi B := 1 + \phi_1 B + \phi_2 B^2 + \cdots$