Iterative Memoryless Non-linear Estimators of Correlation for Complex-Valued Gaussian Processes that Exhibit Robustness to Impulsive Noise

Abstract

The autocorrelation function is a commonly used tool in statistical time series analysis. Under the assumption of Gaussianity, the sample autocorrelation function is the standard method used to estimate this function given a finite number of observations. Non-Gaussian, impulsive observation noise following probability density functions with thick tails, which often occurs in practice, can bias this estimator, rendering classical time series analysis methods ineffective.

This work examines the robustness of two estimators of correlation based on memoryless nonlinear functions of observations, the Phase-Phase Correlator (PPC) and the Median- of-Ratios Estimator (MRE), which are applicable to complex-valued Gaussian random pro- cesses. These estimators are very fast and easy to implement in current processors. We show that these estimators are robust from a bias perspective when complex-valued Gaussian pro- cesses are contaminated with impulsive noise at the expense of statistical efficiency at the assumed Gaussian distribution. Additionally, iterative versions of these estimators named the IMRE and IPPC are developed, realizing an improved bias performance over their non- iterative counterparts and the well-known robust Schweppe-type Generalized M-estimator utilizing a Huber cost function (SHGM).

An impulsive noise suppression technique is developed using basis pursuit and a priori atom weighting derived from the newly developed iterative estimators. This new technique is proposed as an alternative to the robust filter cleaner, a Kalman filter-like approach that relies on linear prediction residuals to identity and replace corrupted observations. It does not have the same initialization issues as the robust filter cleaner.

Robust spectral estimation methods are developed using these new estimators and impulsive noise suppression techniques. Results are obtained for synthetic complex-valued Guassian processes and real-world digital television signals collected using a software defined radio.