854 research outputs found
Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain
Recovering a sparse signal from its low-pass projections in the Fourier
domain is a problem of broad interest in science and engineering and is
commonly referred to as super-resolution. In many cases, however, Fourier
domain may not be the natural choice. For example, in holography, low-pass
projections of sparse signals are obtained in the Fresnel domain. Similarly,
time-varying system identification relies on low-pass projections on the space
of linear frequency modulated signals. In this paper, we study the recovery of
sparse signals from low-pass projections in the Special Affine Fourier
Transform domain (SAFT). The SAFT parametrically generalizes a number of well
known unitary transformations that are used in signal processing and optics. In
analogy to the Shannon's sampling framework, we specify sampling theorems for
recovery of sparse signals considering three specific cases: (1) sampling with
arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels
and, (3) recovery from Gabor transform measurements linked with the SAFT
domain. Our work offers a unifying perspective on the sparse sampling problem
which is compatible with the Fourier, Fresnel and Fractional Fourier domain
based results. In deriving our results, we introduce the SAFT series (analogous
to the Fourier series) and the short time SAFT, and study convolution theorems
that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie
Detection of variable frequency signals using a fast chirp transform
The detection of signals with varying frequency is important in many areas of
physics and astrophysics. The current work was motivated by a desire to detect
gravitational waves from the binary inspiral of neutron stars and black holes,
a topic of significant interest for the new generation of interferometric
gravitational wave detectors such as LIGO. However, this work has significant
generality beyond gravitational wave signal detection.
We define a Fast Chirp Transform (FCT) analogous to the Fast Fourier
Transform (FFT). Use of the FCT provides a simple and powerful formalism for
detection of signals with variable frequency just as Fourier transform
techniques provide a formalism for the detection of signals of constant
frequency. In particular, use of the FCT can alleviate the requirement of
generating complicated families of filter functions typically required in the
conventional matched filtering process. We briefly discuss the application of
the FCT to several signal detection problems of current interest
Fractional Fourier detection of L\'evy Flights: application to Hamiltonian chaotic trajectories
A signal processing method designed for the detection of linear (coherent)
behaviors among random fluctuations is presented. It is dedicated to the study
of data recorded from nonlinear physical systems. More precisely the method is
suited for signals having chaotic variations and sporadically appearing regular
linear patterns, possibly impaired by noise. We use time-frequency techniques
and the Fractional Fourier transform in order to make it robust and easily
implementable. The method is illustrated with an example of application: the
analysis of chaotic trajectories of advected passive particles. The signal has
a chaotic behavior and encounter L\'evy flights (straight lines). The method is
able to detect and quantify these ballistic transport regions, even in noisy
situations
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