Pierre Brémaud

An averaging principle for filtering a jump process with point process observations

A proof of the following result is given. Le X/sub t/ and Y/sub t/ be two jump processes which modulate the intensity of a multivariate point process N/sub t/, and suppose that the process X/sub t/ is a fast Markov chain with a unique invariant probability distribution. Then the filtering equations for Y/sub t/ can be obtained by considering, instead of the original problem, the averaged problem where the intensity is replaced by the averaged intensity. >

Power spectra of UWB time-hopping modulated signals: a shot noise approach

This paper presents a general method for obtaining the exact power spectra of generic time hopping modulated signals. Based on a point process approach, it provides simpler proofs for existing results and a powerful rigorous and at the same time systematic tool for computing the spectra of more complex time-hopping models. Spectrum formula are easy to understand and the contribution of each component of the model appears explicitly

Power spectral measure and reconstruction error of randomly sampled signals

We say that a signal is randomly sampled when the samples are taken at random instants of time. The study of random sampling and randomly sampled signals is motivated both by practical and theoretical interests. The first one includes spectral analysis (estimation of spectra from a finite number of samples) and quality of service (signal reconstruction), and the second one includes statistical analysis of reconstruction methods. The present paper focuses on the computation of the (theoretical) spectrum of randomly sampled signals and on the computation of the reconstruction error. Using a point process approach, we obtain general formulas for spatial random sampling, providing powerful tools for the analysis and the processing of randomly sampled signals.

The Wavelet Transform

We mentioned in the introduction to Part D the shortcomings of the windowed Fourier transform. This chapter gives another approach to the time-frequency issue of Fourier analysis. The role played in the windowed Fourier transform by the family of functions n n

Wavelet Orthonormal Expansions

{omega _{v,b}}(t) = omega (t - b){e^{ + 2ipi vt}},quad b,v in mathbb{R}

Pointwise Convergence of Fourier Series

n nis played in the wavelet transform (WT) by a family n n

Construction of an MRA

{psi _{a,b}}(t) = {left| a right|^{ - 1/2}}psi (frac{{t - b}}{a}),quad a,b in mathbb{R},;a ne 0