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 uses a weight function $\psi$ to apply a weighted sum of the past values so that:

 $\ds \map z t$ $=$ $\ds \mu + a_t + \psi_1 a_{t - 1} + \psi_2 a_{t - 2} + \cdots$ $\ds$ $=$ $\ds \mu + \map \psi B a_t$

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

$\map \psi B := 1 + \psi_1 B + \psi_2 B^2 + \cdots$

Transfer Function

The operator:

$\map \psi B := 1 + \psi_1 B + \psi_2 B^2 + \cdots$

is the transfer function of $L$.

Stable

Consider the sequence $\sequence {\psi_k}$ formed by the weight function $\psi$ of $L$.

Suppose that:

$\ds \sum_k \size {\psi_k} < \infty$

Then $L$ is said to be stable, and the model for $S$ is stationary.

Hence $\mu$ is the mean about which $S$ varies.

Sources

$1$: Introduction:
$1.2$ Stochastic and Deterministic Dynamic Mathematical Models
$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Linear filter model: $(1.2.1)$