Off-the-Grid Line Spectrum Denoising and Estimation With Multiple Measurement Vectors

Abstract

Abstract-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.

Multi-pitch estimation using semidefinite programming

“…This relation was shown for m = 1 and various f in [7][8][9], and for the multiple measurement vector case m > 1 in [13]. From a solution (X 11 , X 12 , X 22 ) of (4), we have r = rank(X 11 ) and the frequencies of the atoms a k = (1/ √ n)z n (ω k ) may be extracted by various methods, see e.g.…”

Approximate support recovery of atomic line spectral estimation: A tale of resolution and precision

“…For instance, atomic norm minimization recovers a spectrally sparse signal from a minimal number of random signal samples [4], identifies and removes a maximal number of outliers [6,7], and perform denoising with an error approaching the minimax rate [5]. When multiple measurement vectors are available, a method of exploiting the joint sparsity pattern of different signals to further improve estimation accuracy are proposed in [33] and [34]. All these works draw inspirations from the dual polynomial construction strategy developed in the pioneer work [1].…”

A Sparse SAR Imaging Method Based on Multiple Measurement Vectors Model

“…MMV problems derive from many applications areas, such as magnetoencephalography, which is a modality for imaging the brain [13]. Similar conceptions were also developed in the context of array processing [14,15] equalization of sparse communication channels [16,17], and more recently line spectrum denoising [18] and cognitive radio communications [19]. In this paper, we want to incorporate this fast growing field into SAR imaging applications.…”