Time-Series Models - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 97

Chapter 4

Time-Series Models

4.1. Introduction

The time series x ( k ), during the processing of a signal, generally comes from the
periodic sampling of a continuous time signal x ( t ), that is to say:

() ()

xk xt = ⎦

tkT .

c

The main idea of the models of such a series is to suggest a dynamic equation,
which takes into account the time dependence variation of x ( k ) most often in terms
of its past (

{ }

) ( ) ( )

− − − … . This equation can take into account any
future values of m > k , but the processing real time constraints can be limited to
causal dynamic equations:

xk 1, xk 2 , x

() ( ) ( ) ( )

xk = F xk ⎡ − 1, xk − 2 … , x −∞ ⎤ [4.1]

⎣ ⎦

The operator F [.] can be of a varied nature - linear, non-linear, of finite or non-
finite dimension, etc.

The practical use of such dynamic equations is to obtain compact expression
(thus F [.] in finite dimension) equipped with a set of parameters θ , which help adapt
this model

( ) ( ) ( )

⎡ ⎣ … has a certain variety of each actual
signal. The effectiveness of a model will be measured on the one hand in a range of

Fxk − 1, xk − 2 , x −∞ , θ