Power spectra of random spikes and related complex signals

"Random spikes" belong to the common language used by engineers, physicists and biologists to describe events associated with time records, locations in space, or more generally, space-time events. Indeed, data and signals consisting of, or structured by, sequences of events are omnipresent in communications, biology, computer science and signal processing. Relevant examples can be found in traffic intensity and neurobiological data, pulse-coded transmission, and sampling. This thesis is concerned by random spike fields and by the complex signals described as the result of various operations on the basic event stream or spike field, such as filtering, jittering, delaying, thinning, clustering, sampling and modulating. More precisely, complex signals are obtained in a modular way by adding specific features to a basic model. This modular approach greatly simplifies the computations and allows to treat highly complex model such as the ones occurring in ultra-wide bandwidth or multipath transmissions. We present a systematic study of the properties of random spikes and related complex signals. More specifically, we focus on second order properties, which are conveniently represented by the spectrum of the signal. These properties are particularly attractive and play an important role in signal analysis. Indeed, they are relatively accessible and yet they provide important informations. Our first contribution is theoretical. As well as presenting a modular approach for the construction of complex signals, we derive formulas for the computation of the spectrum that preserve such modularity: each additional feature added to a basic model appear as a separate and explicit contribution in the corresponding basic spectrum. Moreover, these formula are very general. For instance, the basic point process is not assumed to be a homogeneous Poisson process but it can be any second order stationary process with a given spectrum. In summary, they provide very useful tools for model analysis. We then give applications of the theoretical results: spectral formulas for traffic analysis, pulse based signals used in spread spectrum communications, and randomly sampled signal.