17,388 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
Approximation of L\"owdin Orthogonalization to a Spectrally Efficient Orthogonal Overlapping PPM Design for UWB Impulse Radio
In this paper we consider the design of spectrally efficient time-limited
pulses for ultrawideband (UWB) systems using an overlapping pulse position
modulation scheme. For this we investigate an orthogonalization method, which
was developed in 1950 by Per-Olov L\"owdin. Our objective is to obtain a set of
N orthogonal (L\"owdin) pulses, which remain time-limited and spectrally
efficient for UWB systems, from a set of N equidistant translates of a
time-limited optimal spectral designed UWB pulse. We derive an approximate
L\"owdin orthogonalization (ALO) by using circulant approximations for the Gram
matrix to obtain a practical filter implementation. We show that the centered
ALO and L\"owdin pulses converge pointwise to the same Nyquist pulse as N tends
to infinity. The set of translates of the Nyquist pulse forms an orthonormal
basis or the shift-invariant space generated by the initial spectral optimal
pulse. The ALO transform provides a closed-form approximation of the L\"owdin
transform, which can be implemented in an analog fashion without the need of
analog to digital conversions. Furthermore, we investigate the interplay
between the optimization and the orthogonalization procedure by using methods
from the theory of shift-invariant spaces. Finally we develop a connection
between our results and wavelet and frame theory.Comment: 33 pages, 11 figures. Accepted for publication 9 Sep 201
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