Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors (1408.2242v2)

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.

• The paper presents atomic norm minimization to accurately denoise line spectra without relying on a predefined grid.
• It employs structured covariance estimation to efficiently recover frequencies, bypassing full signal reconstruction.
• The methods achieve robust theoretical guarantees and enhanced accuracy compared to traditional compressed sensing techniques.

Overview of "Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors"

The paper by Li and Chi presents methods for denoising and estimation of line spectra from an ensemble of spectrally-sparse signals with continuous frequencies. It highlights the challenges in reconstructing signals that are sparse in a continuous dictionary rather than a predefined discrete basis and explores solutions for this problem through atomic norm minimization and structured covariance estimation.

Problem Context and Motivation

In many signal processing applications such as super-resolution imaging and remote sensing, signal ensembles can be represented as sparse superpositions of sinusoids sharing the same set of frequencies. Traditional compressed sensing methodologies assume signals are sparse on a predefined grid, which can lead to performance degradation due to basis mismatch. To address the limitation, Li and Chi propose novel methods that do not require such assumptions, focusing on parameter estimation principles instead.

Methodological Contributions

1. Atomic Norm Minimization: The paper adapts atomic norm minimization to leverage multiple measurement vectors (MMV) for line spectrum estimation. The atomic norm, defined without grid restrictions, offers a continuous counterpart to sparsity promotion on grids, allowing for efficient reconstruction via semidefinite programming (SDP). This method provides theoretical performance guarantees under certain separation conditions and proves effective when frequencies are spread non-uniformly and not confined to a grid.
2. Structured Covariance Estimation: In scenarios where only the frequencies are of interest, rather than the full signal ensemble, the authors propose a covariance estimation approach. This method utilizes the structured Toeplitz nature of the covariance matrix to estimate frequencies efficiently using conventional techniques like MUSIC after recovering the covariance structure from partial observations. The algorithm circumvents the need to reconstruct entire signals, hence considerably reducing computational demands, especially suitable for contexts with a large number of measurement vectors.

Theoretical and Numerical Results

Theoretically, it is established that the proposed methods offer robust performance guarantees and are less susceptible to noise when the sparsity of the frequency set is properly utilized. The atomic norm minimization approach, for instance, can exactly recover signals given enough samples, provided the frequencies adhere to a minimum separation condition.

Numerically, the authors provide substantial experimental validation. They present comparisons against traditional compressed sensing techniques and demonstrate the superior performance of their methods, both in terms of reconstruction accuracy and computational efficiency.

Implications and Future Directions

Practically, the ability to efficiently estimate line spectra from spectrally-sparse signals without grid constraints has notable implications across multiple domains like radar, wireless communications, and biomedical imaging, where continuous frequency spectra are common.

Theoretically, this paper advances the understanding of off-the-grid reconstruction problems and expands the potential application of compressed sensing in continuous domains. Future research may delve into optimizing parameter estimation techniques further, exploring scenarios with partial and noisy datasets, and relaxing the separation conditions to enhance real-world applicability. The development of algorithms that can handle higher-dimensional frequency estimation and adapt to non-uniform sampling schemes could also form a logical progression from the work presented by Li and Chi.