239 research outputs found
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals
Wideband analog signals push contemporary analog-to-digital conversion
systems to their performance limits. In many applications, however, sampling at
the Nyquist rate is inefficient because the signals of interest contain only a
small number of significant frequencies relative to the bandlimit, although the
locations of the frequencies may not be known a priori. For this type of sparse
signal, other sampling strategies are possible. This paper describes a new type
of data acquisition system, called a random demodulator, that is constructed
from robust, readily available components. Let K denote the total number of
frequencies in the signal, and let W denote its bandlimit in Hz. Simulations
suggest that the random demodulator requires just O(K log(W/K)) samples per
second to stably reconstruct the signal. This sampling rate is exponentially
lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one
must use nonlinear methods, such as convex programming, to recover the signal
from the samples taken by the random demodulator. This paper provides a
detailed theoretical analysis of the system's performance that supports the
empirical observations.Comment: 24 pages, 8 figure
Compressive Imaging via Approximate Message Passing with Image Denoising
We consider compressive imaging problems, where images are reconstructed from
a reduced number of linear measurements. Our objective is to improve over
existing compressive imaging algorithms in terms of both reconstruction error
and runtime. To pursue our objective, we propose compressive imaging algorithms
that employ the approximate message passing (AMP) framework. AMP is an
iterative signal reconstruction algorithm that performs scalar denoising at
each iteration; in order for AMP to reconstruct the original input signal well,
a good denoiser must be used. We apply two wavelet based image denoisers within
AMP. The first denoiser is the "amplitude-scaleinvariant Bayes estimator"
(ABE), and the second is an adaptive Wiener filter; we call our AMP based
algorithms for compressive imaging AMP-ABE and AMP-Wiener. Numerical results
show that both AMP-ABE and AMP-Wiener significantly improve over the state of
the art in terms of runtime. In terms of reconstruction quality, AMP-Wiener
offers lower mean square error (MSE) than existing compressive imaging
algorithms. In contrast, AMP-ABE has higher MSE, because ABE does not denoise
as well as the adaptive Wiener filter.Comment: 15 pages; 2 tables; 7 figures; to appear in IEEE Trans. Signal
Proces
CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
Compressive sampling offers a new paradigm for acquiring signals that are
compressible with respect to an orthonormal basis. The major algorithmic
challenge in compressive sampling is to approximate a compressible signal from
noisy samples. This paper describes a new iterative recovery algorithm called
CoSaMP that delivers the same guarantees as the best optimization-based
approaches. Moreover, this algorithm offers rigorous bounds on computational
cost and storage. It is likely to be extremely efficient for practical problems
because it requires only matrix-vector multiplies with the sampling matrix. For
many cases of interest, the running time is just O(N*log^2(N)), where N is the
length of the signal.Comment: 30 pages. Revised. Presented at Information Theory and Applications,
31 January 2008, San Dieg
Reconciling Compressive Sampling Systems for Spectrally-sparse Continuous-time Signals
The Random Demodulator (RD) and the Modulated Wideband Converter (MWC) are
two recently proposed compressed sensing (CS) techniques for the acquisition of
continuous-time spectrally-sparse signals. They extend the standard CS paradigm
from sampling discrete, finite dimensional signals to sampling continuous and
possibly infinite dimensional ones, and thus establish the ability to capture
these signals at sub-Nyquist sampling rates. The RD and the MWC have remarkably
similar structures (similar block diagrams), but their reconstruction
algorithms and signal models strongly differ. To date, few results exist that
compare these systems, and owing to the potential impacts they could have on
spectral estimation in applications like electromagnetic scanning and cognitive
radio, we more fully investigate their relationship in this paper. We show that
the RD and the MWC are both based on the general concept of random filtering,
but employ significantly different sampling functions. We also investigate
system sensitivities (or robustness) to sparse signal model assumptions.
Lastly, we show that "block convolution" is a fundamental aspect of the MWC,
allowing it to successfully sample and reconstruct block-sparse (multiband)
signals. Based on this concept, we propose a new acquisition system for
continuous-time signals whose amplitudes are block sparse. The paper includes
detailed time and frequency domain analyses of the RD and the MWC that differ,
sometimes substantially, from published results.Comment: Corrected typos, updated Section 4.3, 30 pages, 8 figure
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