Related papers: Theory of the Hilbert Spectrum
The phenomenon of wave packet diffraction in space and time is described. It consists in a diffraction pattern whose spatial location progresses with time. The pattern is produced by wave packet quantum scattering off an attractive or…
We introduce the joint time-frequency scattering transform, a time shift invariant descriptor of time-frequency structure for audio classification. It is obtained by applying a two-dimensional wavelet transform in time and log-frequency to…
In this paper, we consider signals with intra-wave frequency modulation. To handle this kind of signals effectively, we generalize our data-driven time-frequency analysis by using a shape function to describe the intra-wave frequency…
This paper makes 3 contributions. First, it generalizes the Lindeberg\textendash Feller and Lyapunov Central Limit Theorems to Hilbert Spaces by way of $L^2$. Second, it generalizes these results to spaces in which sample failure and…
It is shown that the phenomenon of irreversibility in many-body and few-body systems can be explained and described within the framework of the concept of direct (not instantaneous) interaction of particles without using probabilistic…
We study Hilbert spaces $H$ interpreted, in an appropriate sense, in a first-order theory. Under a new finiteness hypothesis that we call {\em scatteredness} we prove that $H$ is a direct sum of {\em asymptotically free} components, where…
Spectrum sensing is a fundamental operation in cognitive radio environment. It gives information about spectrum availability by scanning the bands. Usually a fixed amount of time is given to scan individual bands. Most of the times,…
In this paper, we propose a new method to estimate instantaneous frequency using a combined approach based on the discrete linear chirp transform (DLCT) and the Wigner distribution (WD). The DLCT locally represents a signal as a…
We address the problem of estimating time and frequency shifts of a known waveform in the presence of multiple measurement vectors (MMVs). This problem naturally arises in radar imaging and wireless communications. Specifically, a signal…
In audio signal processing, probabilistic time-frequency models have many benefits over their non-probabilistic counterparts. They adapt to the incoming signal, quantify uncertainty, and measure correlation between the signal's amplitude…
This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence…
We solve the problem of resonance statistics in systems with broken time-reversal invariance by deriving the joint probability density of all resonances in the framework of a random matrix approach and calculating explicitly all n-point…
We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose…
We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space.) The models allow for a scale of complexity, indexed by the natural numbers…
This paper reports, by way of introduction, on the advances made by our group and the broader signal processing community on the concept of sample abundance; a phenomenon that naturally arises in one-bit and few-bit signal processing…
The origin of the "theoretical limit of time-frequency resolution of Fourier analysis" is from its numerical implementation, especially from an assumption of "Periodic Boundary Condition (PBC)," which was introduced a century ago. We…
We propose a simple method to estimate the parameters of a continuously measured quantum system, by fitting correlation functions of the measured signal. We demonstrate the approach in simulation, both on toy examples and on a recent…
In this paper, we obtain new observability inequalities for the vibrating string. This work was motivated by a recent paper by A. Szij\'art\'o and J. Heged\H{u}s in which the authors ask the question of determining the initial data by only…
We propose a stable version of Principal Component Analysis (PCA) in the general framework of a separable Hilbert space. It consists in interpreting the projection on the first eigenvectors as a step function applied to the spectrum of the…
This article introduces a nonparametric approach to spectral analysis of a high-dimensional multivariate nonstationary time series. The procedure is based on a novel frequency-domain factor model that provides a flexible yet parsimonious…