Time-Series Models - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 98

signal classes of which it can model the variation with considerable accuracy, and on
the other hand on the number of characteristics of the signal which it can model.

If equation [4.1] represents no physical reality (which will practically always be
the case), the adjustment of the model to a physical signal can only be done in the
case of a precise criterion of accuracy applied to a characteristic of the signal, which
is also precise and limited. We will see later in this paragraph that if a model is fitted
to respect the characteristics at the M order of the signal (moments for example), it
will not adjust itself to characteristics of a different order or laws, etc.

Thus, the choice of a model is very important and never easy: it affects the actual
success of methods, which result from its use.

As part of the current topic, the model must also provide a relation between the
parameters θ and the spectral properties to be measured. If, for example, we are
interested in the power spectral density (PSD) S x ( f ) of x ( t ), the model retained
( ) ( ) ( )

Fxk ⎡ − 1, xk − 2 … , x −∞ , θ ⎤ must satisfy the relation:

⎣ ⎦

( )

[4.2]

θ ⇒

x Sf

, θ

All estimators θ of θ can thus provide a corresponding estimator of S x ( f ) from a
relation of the type:

() ( )

ˆ ˆ ˆ

[4.3]

Sf Sf θ ,

x x

This approach must be verified for all the spectral characteristics of interest:
stationary or non-stationary PSD and ESD, multispectra, etc.

The fundamental problem that such an approach poses in the context of spectral
analysis is to evaluate how the estimation errors on

ˆ x Sf . The tools
present in section 3.4 make it possible to tackle it as a particular case of an estimator
function, and by a limited expansion to evaluate the bias, variance, covariances, etc.
of ()

θ will affect ()

ˆ x Sf from the equivalent characteristics of θ . The application to the model
known as autoregressive given in section 3.4.1 is a paradigm of this type of
problem.

From among the models, the most classical ones are strongly linked to “system”
representation of the signal: the signal observed x ( k ) is conceptually represented as
the output of a system, fed by an input u ( k ). We select as neutral an input as
possible, which is described by a restricted number of parameters, the largest
number of parameters possible being carried by the transfer function such as the one
represented in Figure 4.1.