1,339 research outputs found
Atomic norm denoising with applications to line spectral estimation
Motivated by recent work on atomic norms in inverse problems, we propose a
new approach to line spectral estimation that provides theoretical guarantees
for the mean-squared-error (MSE) performance in the presence of noise and
without knowledge of the model order. We propose an abstract theory of
denoising with atomic norms and specialize this theory to provide a convex
optimization problem for estimating the frequencies and phases of a mixture of
complex exponentials. We show that the associated convex optimization problem
can be solved in polynomial time via semidefinite programming (SDP). We also
show that the SDP can be approximated by an l1-regularized least-squares
problem that achieves nearly the same error rate as the SDP but can scale to
much larger problems. We compare both SDP and l1-based approaches with
classical line spectral analysis methods and demonstrate that the SDP
outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix
Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.Comment: 27 pages, 10 figures. A preliminary version of this work appeared in
the Proceedings of the 49th Annual Allerton Conference in September 2011.
Numerous numerical experiments added to this version in accordance with
suggestions by anonymous reviewer
Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors
Compressed Sensing suggests that the required number of samples for
reconstructing a signal can be greatly reduced if it is sparse in a known
discrete basis, yet many real-world signals are sparse in a continuous
dictionary. One example is the spectrally-sparse signal, which is composed of a
small number of spectral atoms with arbitrary frequencies on the unit interval.
In this paper we study the problem of line spectrum denoising and estimation
with an ensemble of spectrally-sparse signals composed of the same set of
continuous-valued frequencies from their partial and noisy observations. Two
approaches are developed based on atomic norm minimization and structured
covariance estimation, both of which can be solved efficiently via semidefinite
programming. The first approach aims to estimate and denoise the set of signals
from their partial and noisy observations via atomic norm minimization, and
recover the frequencies via examining the dual polynomial of the convex
program. We characterize the optimality condition of the proposed algorithm and
derive the expected convergence rate for denoising, demonstrating the benefit
of including multiple measurement vectors. The second approach aims to recover
the population covariance matrix from the partially observed sample covariance
matrix by motivating its low-rank Toeplitz structure without recovering the
signal ensemble. Performance guarantee is derived with a finite number of
measurement vectors. The frequencies can be recovered via conventional spectrum
estimation methods such as MUSIC from the estimated covariance matrix. Finally,
numerical examples are provided to validate the favorable performance of the
proposed algorithms, with comparisons against several existing approaches.Comment: 14 pages, 10 figure
Compressive Estimation of a Stochastic Process with Unknown Autocorrelation Function
In this paper, we study the prediction of a circularly symmetric zero-mean
stationary Gaussian process from a window of observations consisting of
finitely many samples. This is a prevalent problem in a wide range of
applications in communication theory and signal processing. Due to
stationarity, when the autocorrelation function or equivalently the power
spectral density (PSD) of the process is available, the Minimum Mean Squared
Error (MMSE) predictor is readily obtained. In particular, it is given by a
linear operator that depends on autocorrelation of the process as well as the
noise power in the observed samples. The prediction becomes, however, quite
challenging when the PSD of the process is unknown. In this paper, we propose a
blind predictor that does not require the a priori knowledge of the PSD of the
process and compare its performance with that of an MMSE predictor that has a
full knowledge of the PSD. To design such a blind predictor, we use the random
spectral representation of a stationary Gaussian process. We apply the
well-known atomic-norm minimization technique to the observed samples to obtain
a discrete quantization of the underlying random spectrum, which we use to
predict the process. Our simulation results show that this estimator has a good
performance comparable with that of the MMSE estimator.Comment: 6 pages, 4 figures. Accepted for presentation in ISIT 2017, Aachen,
German
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