41 research outputs found
Frequency-Selective Vandermonde Decomposition of Toeplitz Matrices with Applications
The classical result of Vandermonde decomposition of positive semidefinite
Toeplitz matrices, which dates back to the early twentieth century, forms the
basis of modern subspace and recent atomic norm methods for frequency
estimation. In this paper, we study the Vandermonde decomposition in which the
frequencies are restricted to lie in a given interval, referred to as
frequency-selective Vandermonde decomposition. The existence and uniqueness of
the decomposition are studied under explicit conditions on the Toeplitz matrix.
The new result is connected by duality to the positive real lemma for
trigonometric polynomials nonnegative on the same frequency interval. Its
applications in the theory of moments and line spectral estimation are
illustrated. In particular, it provides a solution to the truncated
trigonometric -moment problem. It is used to derive a primal semidefinite
program formulation of the frequency-selective atomic norm in which the
frequencies are known {\em a priori} to lie in certain frequency bands.
Numerical examples are also provided.Comment: 23 pages, accepted by Signal Processin
Gridless Two-dimensional DOA Estimation With L-shaped Array Based on the Cross-covariance Matrix
The atomic norm minimization (ANM) has been successfully incorporated into
the two-dimensional (2-D) direction-of-arrival (DOA) estimation problem for
super-resolution. However, its computational workload might be unaffordable
when the number of snapshots is large. In this paper, we propose two gridless
methods for 2-D DOA estimation with L-shaped array based on the atomic norm to
improve the computational efficiency. Firstly, by exploiting the
cross-covariance matrix an ANM-based model has been proposed. We then prove
that this model can be efficiently solved as a semi-definite programming (SDP).
Secondly, a modified model has been presented to improve the estimation
accuracy. It is shown that our proposed methods can be applied to both uniform
and sparse L-shaped arrays and do not require any knowledge of the number of
sources. Furthermore, since our methods greatly reduce the model size as
compared to the conventional ANM method, and thus are much more efficient.
Simulations results are provided to demonstrate the advantage of our methods
2D phaseless super-resolution
In phaseless super-resolution, we only have the magnitude information of continuously-parameterized signals in a transform domain, and try to recover the original signals from these magnitude measurements. Optical microscopy is one application where the phaseless super-resolution for 2D signals arise. In this paper, we propose algorithms for performing phaseless super-resolution for 2D or higher-dimensional signals, and investigate their performance guarantees