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Nonlinear and Fractal Signal Processing Alan V.. Wornell Massachusetts Institute of Technology 71 Chaotic Signals and Signal Processing Alan V.. Cuomo Introduction •Modeling and Represen

Nonlinear and Fractal Signal Processing

Alan V Oppenheim

Massachusetts Institute of Technology

Gregory W Wornell

Massachusetts Institute of Technology

71 Chaotic Signals and Signal Processing Alan V Oppenheim and Kevin M Cuomo

Introduction •Modeling and Representation of Chaotic Signals•Estimation and Detection•Use

of Chaotic Signals in Communications •Synthesizing Self-Synchronizing Chaotic Systems

72 Nonlinear Maps Steven H Isabelle and Gregory W Wornell

Introduction •Eventually Expanding Maps and Markov Maps•Signals From Eventually Expanding

Maps •Estimating Chaotic Signals in Noise•Probabilistic Properties of Chaotic Maps•Statistics

of Markov Maps •Power Spectra of Markov Maps•Modeling Eventually Expanding Maps with

Markov Maps

73 Fractal Signals Gregory W Wornell

Introduction •Fractal Random Processes•Deterministic Fractal Signals•Fractal Point Processes

74 Morphological Signal and Image Processing Petros Maragos

Introduction •Morphological Operators for Sets and Signals•Median, Rank, and Stack Operators

•Universality of Morphological Operators•Morphological Operators and Lattice Theory•Slope

Transforms •Multiscale Morphological Image Analysis•Differential Equations for

Continuous-Scale Morphology •Applications to Image Processing and Vision•Conclusions

75 Signal Processing and Communication with Solitons Andrew C Singer

Introduction •Soliton Systems: The Toda Lattice•New Electrical Analogs for Soliton Systems•

Communication with Soliton Signals •Noise Dynamics in Soliton Systems•Estimation of Soliton

Signals •Detection of Soliton Signals

76 Higher-Order Spectral Analysis Athina P Petropulu

Introduction •Definitions and Properties of HOS•HOS Computation from Real Data•Linear

Processes •Nonlinear Processes•Applications/Software Available

TRADITIONALLY, SIGNAL PROCESSING as a discipline has relied heavily on a theoretical

foundation of linear time-invariant system theory in the development of algorithms for a broad range of applications In recent years a considerable broadening of this theoretical base has begun to take place In particular, there has been substantial growth in interest in the use

of a variety of nonlinear systems with special properties for diverse applications Promising new techniques for the synthesis and analysis of such systems continue to emerge At the same time, there has also been rapid growth in interest in systems that are not constrained to be time-invariant These may be systems that exhibit temporal fluctuations in their characteristics, or, equally importantly, systems characterized by other invariance properties, such as invariance to scale changes In the latter case, this gives rise to systems with fractal characteristics

In some cases, these systems are directly applicable for implementing various kinds of signal processing operations such as signal restoration, enhancement, or encoding, or for modeling certain kinds of distortion encountered in physical environments In other cases, they serve as mechanisms for generating new classes of signal models for existing and emerging applications In particular, when autonomous or driven by simpler classes of input signals, they generate rich classes of signals at their outputs In turn, these new classes of signals give rise to new families of algorithms for efficiently exploiting them in the context of applications

The spectrum of techniques for nonlinear signal processing is extremely broad, and in this chapter

we make no attempt to cover the entire array of exciting new directions being pursued within the community Rather, we present a very small sampling of several highly promising and interesting ones to suggest the richness of the topic

A brief overview of the specific chapters comprising this section is as follows

Chapters 71 and 72 discuss the chaotic behavior of certain nonlinear dynamical systems and suggest ways in which this behavior can be exploited In particular, Chapter 71 focuses on continuous-time chaotic systems characterized by a special self-synchronization property that makes them potentially attractive for a range of secure communications applications Chapter 72 describes a family of discrete-time nonlinear dynamical and chaotic systems that are particularly attractive for use in

a variety of signal processing applications ranging from signal modeling in power converters to pseudorandom number generation and error-correction coding in signal transmission applications Chapter 73 discusses fractal signals which arise out of self-similar system models characterized by scale-invariance These represent increasingly important models for a range of natural and man-made phenomena in applications involving both signal synthesis and analysis Multidimensional fractals also arise in the state-space representation of chaotic signals, and the fractal properties in this representation are important in the identification, classification, and characterization of such signals Chapter 74 focuses on morphological signal processing, which encompasses an important class

of nonlinear filtering techniques together with some powerful associated signal representations Morphological signal processing is closely related to a number of classes of algorithms including order-statistics filtering, cellular automata methods for signal processing, and others Morphological algorithms are currently among the most successful and widely used nonlinear signal processing techniques in image processing and vision for such tasks as noise suppression, feature extraction, segmentation, and others

Chapter 75 discusses the analysis and synthesis of soliton signals and their potential use in com-munication applications These signals arise in systems satisfying certain classes of nonlinear wave equations Because they propagate through those equations without dispersion, there has been longstanding interest in their use as carrier waveforms over fiber-optic channels having the appro-priate nonlinear characteristics As they propagate through these systems, they also exhibit a special type of reduced-energy superposition property that suggests an interesting multiplexing strategy for communications over linear channels

Finally, Chapter 76 discusses nonlinear representations for stochastic signals in terms of their higher-order statistics Such representations are particularly important in the processing of non-Gaussian signals for which more traditional second-moment characterizations are often inadequate The associated tools of higher-order spectral analysis find increasing application in many signal detection, identification, modeling, and equalization contexts, where they have led to new classes of powerful signal processing algorithms

Again, these articles are only representative examples of the many emerging directions in this active area of research within the signal processing community, and developments in many other important and exciting directions can be found in the community’s journal and conference publications