Newtonized Orthogonal Matching Pursuit: Frequency Estimation over the Continuum (1509.01942v2)

Abstract: We propose a fast sequential algorithm for the fundamental problem of estimating frequencies and amplitudes of a noisy mixture of sinusoids. The algorithm is a natural generalization of Orthogonal Matching Pursuit (OMP) to the continuum using Newton refinements, and hence is termed Newtonized OMP (NOMP). Each iteration consists of two phases: detection of a new sinusoid, and sequential Newton refinements of the parameters of already detected sinusoids. The refinements play a critical role in two ways: (1) sidestepping the potential basis mismatch from discretizing a continuous parameter space, (2) providing feedback for locally refining parameters estimated in previous iterations. We characterize convergence, and provide a Constant False Alarm Rate (CFAR) based termination criterion. By benchmarking against the Cramer Rao Bound, we show that NOMP achieves near-optimal performance under a variety of conditions. We compare the performance of NOMP with classical algorithms such as MUSIC and more recent Atomic norm Soft Thresholding (AST) and Lasso algorithms, both in terms of frequency estimation accuracy and run time.

• The paper presents Newtonized OMP, which integrates initial detection with Newton-based refinements to accurately estimate sinusoidal frequencies and amplitudes from noisy data.
• It employs an iterative two-phase process and a CFAR termination criterion to approach the Cramér-Rao Bound performance under high SNR conditions.
• Comparative analysis shows that NOMP outperforms classical methods like MUSIC and modern techniques such as AST and Lasso in accuracy and computational efficiency.

Newtonized Orthogonal Matching Pursuit: Frequency Estimation over the Continuum

The paper describes an innovative approach to the classic problem of estimating frequencies and amplitudes from a noisy mixture of sinusoids. The suggested algorithm, called Newtonized Orthogonal Matching Pursuit (NOMP), extends traditional Orthogonal Matching Pursuit (OMP) by incorporating Newton refinements to accurately pursue solutions over a continuum, avoiding the basis mismatch typically caused by discretizing the parameter space.

Methodology Overview

NOMP leverages a two-phase iterative process: initial detection and subsequent refinement of sinusoidal parameters. Each iteration begins with the detection phase, which identifies a new sinusoid, and the refinement stage where Newton-based local refinements enhance previously detected sinusoidal parameters. This feedback mechanism is crucial, as it provides the ability to adapt the estimates of already detected sinusoids in light of new information obtained from each detected sinusoid.

The convergence of the algorithm has been rigorously characterized, and a Constant False Alarm Rate (CFAR) based criterion is used for termination. This allows the method to achieve near-optimal performance across various environmental conditions, as demonstrated through comparisons with classical methods like MUSIC and state-of-the-art algorithms such as Atomic Norm Soft Thresholding (AST) and Lasso.

Numerical Results

The paper highlights several numerical results, indicating that NOMP offers significant improvements in both frequency estimation accuracy and computational efficiency. For well-separated frequencies under high signal-to-noise ratio (SNR) conditions, the algorithm achieves performance close to the Cramér-Rao Bound (CRB). Importantly, NOMP can resolve closely spaced frequencies accurately provided their SNRs differ sufficiently, showcasing its robustness in challenging scenarios.

More specifically, the authors claim that in scenarios with well-separated frequencies, the NOMP algorithm maintains superior performance compared to both conventional and modern techniques. It has been shown to approach the CRB when the component sinusoids are adequately distinct and have high SNR.

Computational Complexity and Refinements

The computational complexity of NOMP is primarily influenced by the cyclic refinement and update steps. Despite potentially high complexity, empirical results demonstrate that NOMP's run-time is competitive or superior to classical algorithms like MUSIC and is substantially lower than the AST approach. The methodology includes a detailed convergence analysis, indicating that the refinement steps considerably enhance the algorithm's performance, importantly sustaining performance advantages without excessive computational burdens.

Implications and Future Directions

The introduction of NOMP has both practical and theoretical implications. Practically, it provides a powerful tool for communication and radar applications. Theoretically, it advances sparse approximation techniques in signal processing by effectively navigating the challenges presented by continuous frequency estimation. The ability to operate effectively without being restricted to discretized grids addresses a significant limitation in many traditional algorithms, and the use of Newton refinements sets a precedent for further research in algorithm refinement strategies.

Future developments could explore extending NOMP's principles to more complex or higher-dimensional scenarios or incorporating machine learning paradigms for even more robust estimations in real-world conditions.

In conclusion, the paper introduces a significant contribution to the field of frequency estimation by developing a method that seamlessly integrates detection over discrete grids with local refinement techniques, offering both enhanced accuracy and reduced computational load. Such innovations suggest promising directions for future research and applications in frequency estimation and related signal processing challenges.