Spectral Measures of \(\alpha \)-Stable Distributions: An Overview and Natural Applications in Wireless Communications

Currently, we are witnessing the proliferation of wireless sensor networks and the superposition of several communicating objects which have a heterogeneous nature. Those are merely the beginnings of an evolution toward the so-called Internet of Things. The advent of these networks as well as the increasing demand for improved quality and services will increase the complexity of communications and put a strain on current techniques and models. Indeed, they must first adapt to the temporal and spatial evolutions and second, they must take into account the rare and unpredictable events that can have disastrous consequences for decision-making. This chapter provides an overview of the various spectral techniques used in signal processing and statistics literature to describe a communication channel having an impulsive behavior. This project is mainly motivated by the historical success of the interaction between probability, statistics and the world of communications, information theory and signal processing. The second motivation is the scarcity of references and literature summarizing mathematical developments on the application of alpha-stable process for channel modeling. This chapter will be divided into two parts: the first is devoted to the synthesis of various developments on alpha-stable variables and processes in a purely mathematical mind. The second part will be devoted to applications in the context of communications. The two sides will combine two fundamentally linked aspects: first, a theoretical approach, necessary for a good formalization of problems and identifying the best solutions. Second, the use of these models in real work of channel modeling.

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