139 research outputs found
Technical Report: Compressive Temporal Higher Order Cyclostationary Statistics
The application of nonlinear transformations to a cyclostationary signal for
the purpose of revealing hidden periodicities has proven to be useful for
applications requiring signal selectivity and noise tolerance. The fact that
the hidden periodicities, referred to as cyclic moments, are often compressible
in the Fourier domain motivates the use of compressive sensing (CS) as an
efficient acquisition protocol for capturing such signals. In this work, we
consider the class of Temporal Higher Order Cyclostationary Statistics (THOCS)
estimators when CS is used to acquire the cyclostationary signal assuming
compressible cyclic moments in the Fourier domain. We develop a theoretical
framework for estimating THOCS using the low-rate nonuniform sampling protocol
from CS and illustrate the performance of this framework using simulated data
Compressive Sensing of Analog Signals Using Discrete Prolate Spheroidal Sequences
Compressive sensing (CS) has recently emerged as a framework for efficiently
capturing signals that are sparse or compressible in an appropriate basis.
While often motivated as an alternative to Nyquist-rate sampling, there remains
a gap between the discrete, finite-dimensional CS framework and the problem of
acquiring a continuous-time signal. In this paper, we attempt to bridge this
gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a
collection of functions that trace back to the seminal work by Slepian, Landau,
and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's
form a highly efficient basis for sampled bandlimited functions; by modulating
and merging DPSS bases, we obtain a dictionary that offers high-quality sparse
approximations for most sampled multiband signals. This multiband modulated
DPSS dictionary can be readily incorporated into the CS framework. We provide
theoretical guarantees and practical insight into the use of this dictionary
for recovery of sampled multiband signals from compressive measurements
An Introduction To Compressive Sampling [A sensing/sampling paradigm that goes against the common knowledge in data acquisition]
This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use. To make this possible, CS relies on two principles: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality.
Our intent in this article is to overview the basic CS theory that emerged in the works [1]–[3], present the key mathematical ideas underlying this theory, and survey a couple of important results in the field. Our goal is to explain CS as plainly as possible, and so our article is mainly of a tutorial nature. One of the charms of this theory is that it draws from various subdisciplines within the applied mathematical sciences, most notably probability theory. In this review, we have decided to highlight this aspect and especially the fact that randomness can — perhaps surprisingly — lead to very effective sensing mechanisms. We will also discuss significant implications, explain why CS is a concrete protocol for sensing and compressing data simultaneously (thus the name), and conclude our tour by reviewing important applications
Signal Space CoSaMP for Sparse Recovery with Redundant Dictionaries
Compressive sensing (CS) has recently emerged as a powerful framework for
acquiring sparse signals. The bulk of the CS literature has focused on the case
where the acquired signal has a sparse or compressible representation in an
orthonormal basis. In practice, however, there are many signals that cannot be
sparsely represented or approximated using an orthonormal basis, but that do
have sparse representations in a redundant dictionary. Standard results in CS
can sometimes be extended to handle this case provided that the dictionary is
sufficiently incoherent or well-conditioned, but these approaches fail to
address the case of a truly redundant or overcomplete dictionary. In this paper
we describe a variant of the iterative recovery algorithm CoSaMP for this more
challenging setting. We utilize the D-RIP, a condition on the sensing matrix
analogous to the well-known restricted isometry property. In contrast to prior
work, the method and analysis are "signal-focused"; that is, they are oriented
around recovering the signal rather than its dictionary coefficients. Under the
assumption that we have a near-optimal scheme for projecting vectors in signal
space onto the model family of candidate sparse signals, we provide provable
recovery guarantees. Developing a practical algorithm that can provably compute
the required near-optimal projections remains a significant open problem, but
we include simulation results using various heuristics that empirically exhibit
superior performance to traditional recovery algorithms
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