Statistical modeling of discrete-time chaotic processes-basic finite-dimensional tools and applications

The application of chaotic dynamics to signal processing tasks stems from the realization that its complex behaviors become tractable when observed from a statistical perspective. Here we illustrate the validity of this statement by considering two noteworthy problems-namely, the synthesis of high-electromagnetic compatibility clock signals and the generation of spreading sequences for direct-sequence code-division comunication systems, and by showing how the statistical approach to discrete-time chaotic systems can be applied to find their optimal solution. To this aim, we first review the basic mathematical tools both intuitively and formally; we consider the Perron-Frobenius operator, its spectral decomposition and its tie to the correlation properties of chaotic sequences. Then, by leveraging on the modeling/approximation of chaotic systems through Markov chains, we introduce a matrix/tensor-based framework where statistical indicators such as high-order correlations can be quantified. We underline how, for many particular cases, the proposed analysis tools can be reversed into synthesis methodologies and we use them to tackle the two above mentioned problems. In both cases, experimental evidence shows that the availability of statistical tools enables the design of chaos-based systems which favorably compare with analogous nonchaos-based counterparts.