Definition:Linear Filter/Transfer Function

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 $t$.

Let $L$ be a linear filter on $S$:

\(\ds \map z t\)

\(=\)

\(\ds \mu + a_t + \psi_1 a_{t - 1} + \psi_2 a_{t - 2} + \cdots\)

\(\ds \)

\(=\)

\(\ds \mu + \map \psi B a_t\)

The operator:

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

is the transfer function of $L$.

Sources

• 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):

$1$: Introduction:

$1.2$ Stochastic and Deterministic Dynamic Mathematical Models

$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Linear filter model