[Paper Review] Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples

Michael Lexa, Mike E. Davies+1 more|arXiv (Cornell University)|Oct 12, 2011

Sparse and Compressive Sensing Techniques37 references19 citations

TL;DR

This paper proposes a novel power spectral density (PSD) estimator for band-limited, wide-sense stationary signals using multi-coset sampling, enabling sub-Nyquist sampling rates. It unifies compressive and noncompressive estimation via non-negative least squares (NNLS), achieving better tradeoffs in resolution, complexity, and sampling rate, especially when the PSD is sparse.

ABSTRACT

This paper presents a novel power spectral density estimation technique for band-limited, wide-sense stationary signals from sub-Nyquist sampled data. The technique employs multi-coset sampling and incorporates the advantages of compressed sensing (CS) when the power spectrum is sparse, but applies to sparse and nonsparse power spectra alike. The estimates are consistent piecewise constant approximations whose resolutions (width of the piecewise constant segments) are controlled by the periodicity of the multi-coset sampling. We show that compressive estimates exhibit better tradeoffs among the estimator's resolution, system complexity, and average sampling rate compared to their noncompressive counterparts. For suitable sampling patterns, noncompressive estimates are obtained as least squares solutions. Because of the non-negativity of power spectra, compressive estimates can be computed by seeking non-negative least squares solutions (provided appropriate sampling patterns exist) instead of using standard CS recovery algorithms. This flexibility suggests a reduction in computational overhead for systems estimating both sparse and nonsparse power spectra because one algorithm can be used to compute both compressive and noncompressive estimates.

Motivation & Objective

• Address the challenge of efficient power spectral density estimation for wideband signals with high sampling rate costs.
• Develop a consistent PSD estimator that operates at sub-Nyquist rates using periodic nonuniform sampling.
• Unify compressive and noncompressive estimation under a single algorithmic framework to reduce computational overhead.
• Enable accurate PSD estimation for both sparse and nonsparse spectra using the same recovery method.
• Characterize the tradeoffs between resolution, system complexity, and average sampling rate in the (L, q) parameter space.

Proposed method

• Utilizes multi-coset (MC) sampling, where signals are sampled at nonuniform, periodic intervals defined by offsets {c_i} over blocks of length LT.
• Models the sampling process as a linear system relating the true PSD to the sampled data, forming a matrix equation y = A x.
• Applies compressed sensing (CS) principles when the PSD is sparse, allowing underdetermined system solutions.
• For noncompressive estimation, treats the system as overdetermined or determined, solving via least squares.
• Employs non-negative least squares (NNLS) for both compressive and noncompressive cases when appropriate sampling patterns exist.
• Uses the invariance property of maximum likelihood estimators to show that noncompressive estimates are asymptotically efficient under Gaussian assumptions.

Experimental results

Research questions

• RQ1 Can a single estimation framework unify compressive and noncompressive PSD estimation under sub-Nyquist sampling?
• RQ2 What are the tradeoffs between estimator resolution, system complexity, and average sampling rate in multi-coset sampling?
• RQ3 How does the performance of compressive estimation compare to noncompressive estimation in terms of resolution and sampling rate?
• RQ4 Under what conditions can non-negative least squares (NNLS) be used to compute both compressive and noncompressive estimates?
• RQ5 Can the proposed estimator detect spectral holes (notches) in wideband signals at low sampling rates?

Key findings

• The proposed estimator produces consistent, piecewise constant PSD approximations whose resolution is controlled by the multi-coset parameters L and q.
• Compressive estimates achieve better tradeoffs among resolution, complexity, and average sampling rate compared to noncompressive estimates, especially when the PSD is sparse.
• For suitable sampling patterns, both compressive and noncompressive estimates can be computed using non-negative least squares (NNLS), eliminating the need for separate algorithms.
• Noncompressive estimates computed via NNLS are asymptotically efficient when the input signal is a zero-mean, wide-sense stationary Gaussian process.
• The method enables detection of spectral holes through thresholding of estimates, even at low average sampling rates (e.g., 781.25 MHz in simulations).
• The estimator achieves consistent performance across both sparse and nonsparse spectra, with resolution determined by the (L, q) parameter pair.