[Paper Review] Polynomial phase estimation by phase unwrapping

Robby G. McKilliam, Barry G. Quinn+3 more|arXiv (Cornell University)|Nov 2, 2012

Synthetic Aperture Radar (SAR) Applications and Techniques32 references3 citations

TL;DR

This paper proposes a least squares unwrapping (LSU) estimator for polynomial phase signals, which unwraps noisy phase data in a least squares manner to estimate polynomial coefficients. The key contribution is proving the LSU estimator is strongly consistent and asymptotically normal, with coefficient estimates converging at rate o(N^{-k}) for the k-th coefficient, under general noise assumptions.

ABSTRACT

Estimating the coefficients of a noisy polynomial phase signal is important in fields including radar, biology and radio communications. One approach attempts to perform polynomial regression on the phase of the signal. This is complicated by the fact that the phase is wrapped modulo 2πand must be unwrapped before regression can be performed. In this paper we consider an estimator that performs phase unwrapping in a least squares manner. We describe the asymptotic properties of this estimator, showing that it is strongly consistent and asymptotically normally distributed.

Motivation & Objective

• To develop a statistically robust estimator for polynomial phase coefficients in noisy signals, particularly when phase is wrapped modulo 2π.
• To address the challenge of phase wrapping in polynomial phase signal estimation, which complicates direct regression on wrapped phase data.
• To establish the asymptotic statistical properties of the least squares unwrapping (LSU) estimator, including consistency and normality.
• To generalize prior results by allowing a wider class of noise distributions and proving convergence rates of o(N^{-k}) for the k-th coefficient.
• To provide a theoretical foundation for using phase unwrapping in polynomial phase estimation that supports applications in radar, sonar, and medical imaging.

Proposed method

• The LSU estimator performs phase unwrapping by minimizing a least squares objective function over the wrapped phase observations.
• The method involves transforming the wrapped phase into an unwrapped form using a nearest-point lattice projection, enabling regression on the unwrapped phase.
• The estimator is defined as the minimizer of a non-differentiable objective function, requiring advanced tools from empirical process theory and hyperplane arrangement analysis.
• Theoretical analysis uses results on arithmetic progressions in sets {1, ..., N} to control error terms in the convergence proof.
• Strong consistency is established via a covering argument using finite sets of points in a bounded region, leveraging hyperplane arrangements to bound the distance between parameter estimates.
• Asymptotic normality is proven using empirical process techniques and results on the behavior of the objective function near the true parameter, despite non-differentiability.

Experimental results

Research questions

• RQ1 Can a least squares approach to phase unwrapping yield a consistent and asymptotically normal estimator for polynomial phase coefficients?
• RQ2 What is the convergence rate of the k-th polynomial phase coefficient estimate under the LSU estimator?
• RQ3 How does the estimator perform under general noise distributions beyond Gaussian assumptions?
• RQ4 What are the theoretical guarantees for the LSU estimator in terms of strong consistency and asymptotic normality?
• RQ5 Can the non-differentiability of the objective function be overcome using empirical process and geometric techniques to derive asymptotic properties?

Key findings

• The LSU estimator is strongly consistent, meaning it converges almost surely to the true polynomial phase coefficients as the number of samples N increases.
• The estimator is asymptotically normally distributed, which supports the construction of confidence intervals and hypothesis tests for the coefficients.
• The k-th coefficient estimate converges to the true value at rate o(N^{-k}), a result that holds despite the nonlinearity and wrapping of the phase data.
• The proof of asymptotic normality relies on advanced tools from empirical process theory and hyperplane arrangement geometry due to the non-differentiability of the objective function.
• The convergence rate o(N^{-k}) is consistent with polynomial regression, but its validity in the wrapped phase setting is non-trivial and requires novel combinatorial arguments on arithmetic progressions.
• Monte Carlo simulations confirm the favorable statistical performance of the LSU estimator, aligning with the theoretical asymptotic predictions.