A comparative study of two discrete-time phase delay estimators

Two simple discrete-time methods for unbiased estimation of the phase-shift between two real sinusoidal signals in white noise are proposed. The first estimator removes the bias of the recently proposed quadrature delay estimator (QDE) by utilizing all in-phase and quadrature-phase components of the received signals. The phase estimate of the second method is given by the phase difference of the discrete-time Fourier transforms (DTFTs) of two complex sinusoids derived from the real signals. In particular, we have derived the mean square phase-shift error of the DTFT-based estimator as well as the minimum achievable variance for phase-shift estimation, which are identical. Computer simulations show that the two proposed estimators are superior to the biased QDE and can attain optimum performance, although the second estimator has a larger signal-to-noise ratio operation range.

I. Introduction

Accurate estimation of propagation delay between signals received at two spatially separated sensors is a problem of considerable interest in many areas such as radar, underwater acoustics, biomedicine and ultrasonics [1], [2]. Many of the methods devised for time delay estimation are related through a generalized cross correlation (GCC) approach [3]. The delay estimate of the GCC is found by locating the cross correlation peak of the filtered versions of the two received signals, and it has been proved that optimum performance can be attained when the signals and noises are Gaussian distributed. When the source signal is deterministic, namely, a single complex sinusoid, a discrete-time Fourier transform (DTFT)-based approach has been devised [4] for optimum delay estimation. If the source is a real-valued sinusoid, a quadrature delay estimator (QDE) has been recently proposed [5] for phase delay estimation. The technique utilizes the in-phase and quadrature-phase components of one of the receiver outputs and provides a high-resolution phase-shift estimate. Using the idea of [5], two new discrete-time phase delay estimators for real signals, which are superior to the QDE and can attain optimum accuracy, are developed in this paper.