Time-Series Models - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 112

completed by symmetry, to gain access to the expression of the PSD (using a
bilateral z transform of () γ k ):

xx

p

2

1 − z

∑

()

m

Sz = B [4.32]

( ) (

xx m

− 1

)

1 − zz 1 − zz

m = 1 m m

4.4. Non-linear models

There is a strong interest for this class of models, which help express a very large
morphological variability for the signals, much greater than the linear or exponential
models. The basic principle remains the same, with a finite dynamic equation:

() ( ) ( ) ( ) () ( )

xk = F xk − 1, xk − 2 … , xk P uk − , , … , uk Q θ − ,

⎡ ⎤

⎣ ⎦

excited by a neutral deterministic or random input.

The models can be characterized by the sub-classes of operators F [ . ] used, and
their relation to the properties of the model. The predictive qualities of these models
are mainly the objective of their study.

For the current topic, these models present a major difficulty: we generally do
not know the relation between the operator F [ . ], the parameters of the model θ and
the spectral characteristics (PSD, etc.). Thus, to date, they are not tools for spectral
analysis.

An inquisitive or avid reader can consult the references [PRI 91] and mainly
[TON 90].

4.5. Bibliography

[ABR 02] ABRY P., GONÇALVES P., LEVYVÉHEL J., Lois d'échelle, fractales et
ondelettes , IC2 series, 2 vols., Hermès Sciences, Paris, 2002.
[BEL 89] BELL ANGER M., Signal Analysis and Adaptive Digital Filtering, Masson, 1989.
[BOX 70] BOX G., JENKINS G., Time Series Analysis, Forecasting and Control, Holden-
Day, San Francisco, 1970.
[HLA 05] HLAWATSCH R., AUGER R., OVARLEZ J.-R., Temps-fréquence, IC2 series,
Hermès Sciences, Paris, 2005.
[KAY 88] KAY S. M., Modem Spectral Estimation: Theory and Applications, Prentice Hall,
Englewood Cliffs, (NJ), 1988.