Eric Chassande-Mottin

Discrete time and frequency Wigner-Ville distribution: Moyal's formula and aliasing

In this letter, we propose a new definition of the discrete time and frequency Wigner-Ville distribution. The proposed distribution not only displays a readable representation (small aliasing) but also exhibits unitarity and is easy to compute. We compare the time-frequency representation associated with this proposed definition with other existing ones.

Time-Frequency/Time-Scale Reassignment

This chapter reviews the reassignment principle, which aims at “sharpening” time-frequency and time-scale representations in order to improve their readability.

On Phase-Magnitude Relationships in the Short-Time Fourier Transform

A complete evaluation of first-order, second-order and mixed derivatives is proposed for both the (log-)magnitude and the phase of a given Short-Time Fourier Transform (STFT), leading to equivalent expressions based on additional STFTs with specific windows. Consequences are drawn in terms of phase-magnitude relationships, resulting in new formulations of time-frequency techniques such as reassignment, as well as new insights in the structure of admissible STFTs in some special cases.

On the stationary phase approximation of chirp spectra

The use of the stationary phase principle is often advocated for evaluating the spectrum of a chirp. This issue is considered here in detail, especially with respect to the quantitative control of the corresponding approximation error. A careful analysis leads to the introduction of a refined criterion, which turns out to be much more complicated than the heuristic conditions which are usually considered in this context. It is moreover evidenced, by means of two counterexamples belonging to the important class of power-law chirps, that-as opposed to a common belief-usual heuristic conditions are by themselves neither necessary nor sufficient for validating a stationary phase approximation.

Best network chirplet chain: Near-optimal coherent detection of unmodeled gravitational wave chirps with a network of detectors

The searches of impulsive gravitational waves (GW) in the data of the ground-based interferometers focus essentially on two types of waveforms: short unmodeled bursts from supernova core collapses and frequency modulated signals (or chirps) from inspiralling compact binaries. There is room for other types of searches based on different models. Our objective is to fill this gap. More specifically, we are interested in GW chirps ‘‘in general,’’ i.e., with an arbitrary phase/frequency vs time evolution. These unmodeled GW chirps may be considered as the generic signature of orbiting or spinning sources. We expect the quasiperiodic nature of the waveform to be preserved independently of the physics which governs the source motion. Several methods have been introduced to address the detection of unmodeled chirps using the data of a single detector. Those include the best chirplet chain (BCC) algorithm introduced by the authors. In the next years, several detectors will be in operation. Improvements can be expected from the joint observation of a GW by multiple detectors and the coherent analysis of their data, namely, a larger sight horizon and the more accurate estimation of the source location and the wave polarization angles. Here, we present an extension of the BCC search to the multiple detector case. This work is based on the coherent analysis scheme proposed in the detection of inspiralling binary chirps. We revisit the derivation of the optimal statistic with a new formalism which allows the adaptation to the detection of unmodeled chirps. The method amounts to searching for salient paths in the combined time-frequency representation of two synthetic streams. The latter are time series which combine the data from each detector linearly in such a way that all the GW signatures received are added constructively. We give a proof of principle for the full-sky blind search in a simplified situation which shows that the joint estimation of the source sky location and chirp frequency is possible.

Data analysis challenges in transient gravitational-wave astronomy

Gravitational waves are radiative solutions of space-time dynamics predicted by Einstein’s theory of General Relativity. A world-wide array of large-scale and highly sensitive interferometric detectors constantly scrutinizes the geometry of the local space-time with the hope to detect deviations that would signal an impinging gravitational wave from a remote astrophysical source. Finding the rare and weak signature of gravitational waves buried in non-stationary and non-Gaussian instrument noise is a particularly challenging problem. We will give an overview of the data analysis techniques and associated observational results obtained so far by Virgo (in Europe) and LIGO (in the US), along with the prospects offered by the upcoming advanced versions of those detectors.

Joint searches for gravitational waves and high-energy neutrinos

Many of the astrophysical sources and violent phenomena observed in our Universe are potential joint emitters of gravitational waves and high-energy cosmic radiation, in the form of photons, hadrons, and also neutrinos. This has triggered a collaborative analysis project between gravitational wave detectors and high-energy neutrino telescopes. In this article, we review some of the motivations for having pursuing science jointly and present the efforts status.

Chains of chirplets for the detection of gravitational wave chirps

A worldwide collaboration attempts to confirm the existence of gravitational waves predicted by Einsteins theory of General Relativity, through direct observation with a network of large-scale laser interferometric antennas. This paper is a contribution to the methodologies used to scrutinize the data in order to reveal the tiny signature of a gravitational wave from rare cataclysmic events of astrophysical origin. More specifically, we are interested in the detection of short frequency modulated transients or gravitational wave chirps. The amount of information about the frequency vs. time evolution is limited: we only know that it is smooth. The detection problem is thus non-parametric. We introduce a finite family of template waveforms which accurately samples the set of admissible chirps. The templates are constructed as a puzzle, by assembling elementary bricks (the chirplets) taken a dictionary. The detection amounts to testing the correlation between the data and the template family. With an adequate time-frequency mapping, we establish a connection between this correlation measurement and combinatorial optimization problems of graph theory, from which we obtain efficient algorithms to perform the calculation. We present two variants. A first one addresses the case of amplitude modulated chirps and the second allows the joint analysis of the data from several antennas. Those methods are not limited to the specific context for which they have been developed. We pay a particular attention to the aspects that can be source of inspiration for other applications.