Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications

1991 ◽  
Vol 79 (3) ◽  
pp. 278-305 ◽  
Author(s):  
J.M. Mendel
Keyword(s):  
System Theory ◽  
Signal Processing ◽  
Order Statistics ◽  
Higher Order ◽  
Higher Order Statistics ◽  
Theoretical Results

Nonlinear Techniques for Signals Characterization

2011 ◽  
pp. 1266-1272
Author(s):  
Jesús Bernardino Alonso Hernández ◽  
Patricia Henríquez Rodríguez
Keyword(s):  
Time Series ◽  
Signal Processing ◽  
Order Statistics ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Digital Signal ◽  
Higher Order ◽  
Higher Order Statistics ◽  
Three Body ◽  
Nonlinear Signal

The field of nonlinear signal characterization and nonlinear signal processing has attracted a growing number of researchers in the past three decades. This comes from the fact that linear techniques have some limitations in certain areas of signal processing. Numerous nonlinear techniques have been introduced to complement the classical linear methods and as an alternative when the assumption of linearity is inappropriate. Two of these techniques are higher order statistics (HOS) and nonlinear dynamics theory (chaos). They have been widely applied to time series characterization and analysis in several fields, especially in biomedical signals. Both HOS and chaos techniques have had a similar evolution. They were first studied around 1900: the method of moments (related to HOS) was developed by Pearson and in 1890 Henri Poincaré found sensitive dependence on initial conditions (a symptom of chaos) in a particular case of the three-body problem. Both approaches were replaced by linear techniques until around 1960, when Lorenz rediscovered by coincidence a chaotic system while he was studying the behaviour of air masses. Meanwhile, a group of statisticians at the University of California began to explore the use of HOS techniques again. However, these techniques were ignored until 1980 when Mendel (Mendel, 1991) developed system identification techniques based on HOS and Ruelle (Ruelle, 1979), Packard (Packard, 1980), Takens (Takens, 1981) and Casdagli (Casdagli, 1989) set the methods to model nonlinear time series through chaos theory. But it is only recently that the application of HOS and chaos in time series has been feasible thanks to higher computation capacity of computers and Digital Signal Processing (DSP) technology. The present article presents the state of the art of two nonlinear techniques applied to time series analysis: higher order statistics and chaos theory. Some measurements based on HOS and chaos techniques will be described and the way in which these measurements characterize different behaviours of a signal will be analized. The application of nonlinear measurements permits more realistic characterization of signals and therefore it is an advance in automatic systems development.

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