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 $K$-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

A fast and accurate basis pursuit denoising algorithm with application to super-resolving tomographic SAR

$L_1$ regularization is used for finding sparse solutions to an underdetermined linear system. As sparse signals are widely expected in remote sensing, this type of regularization scheme and its extensions have been widely employed in many remote sensing problems, such as image fusion, target detection, image super-resolution, and others and have led to promising results. However, solving such sparse reconstruction problems is computationally expensive and has limitations in its practical use. In this paper, we proposed a novel efficient algorithm for solving the complex-valued $L_1$ regularized least squares problem. Taking the high-dimensional tomographic synthetic aperture radar (TomoSAR) as a practical example, we carried out extensive experiments, both with simulation data and real data, to demonstrate that the proposed approach can retain the accuracy of second order methods while dramatically speeding up the processing by one or two orders. Although we have chosen TomoSAR as the example, the proposed method can be generally applied to any spectral estimation problems.Comment: 11 pages, IEEE Transactions on Geoscience and Remote Sensin