Distance Estimation Using Wrapped Phase Measurements in Noise

Measuring a distance using phase measurements is a common practice in many areas of engineering. Almost inevitably these measurements are accompanied by noise, and are always subject to ambiguity resulting from the phase of modulo 2π. In the presence of phase ambiguity, where for instance the unknown distance is far longer than the wavelength of the signal carrying the phase measurement, the distance cannot be uniquely determined. One way to resolve this phase ambiguity is to measure the signal phase at multiple frequencies, converting the phase ambiguity problem into one of solving a family of Diophantine equations. Typically, under some reasonable assumptions, the Diophantine problems can be solved using the Chinese Reminder Theorem as documented in the literature. However, the existing algorithms can experience significant computational overhead for a given application because an exhaustive search is required. In this paper, a novel method addressing the phase ambiguity issue using lattice theoretic ideas is proposed and a closed-form algorithm is presented for the estimation of the number of wavelengths in the unknown distance using the phase measurements taken at multiple wavelengths. The algorithm is extremely efficient as the Diophantine equations are solved without searching. The unknown distance can then be estimated via a maximum likelihood method using the unwrapped phase measurement. A statistical bound of the measurement noise which ensures that the number of whole wavelengths in the unknown distance can be found with a probability close to unity is derived. The robustness, efficiency and estimation accuracy of the proposed method are demonstrated by the simulated results presented.

I. Introduction

Phase ambiguity is a common and significant problem in real world engineering problems where phase is measured in order to estimate a distance. This phase ambiguity issue arises when the distance to be measured by the phase of an electromagnetic signal is far longer than the signal wavelength. The resulting estimate of the distance is ambiguous by an integer multiple of signal wavelength. A simple example is the use of a phase only signal to measure the range of a target [1]. A more complex example is the use of a sum of signal phase differences measured from a Radio Interferometric Positioning System (RIPS) for wireless mote localization [2]. A single RIPS measurement involves three anchor motes and a free mote. Two motes act as transmitters, sending a pure sinusoidal waves at slightly different frequencies. This results in an interference signal at a low beat frequency that is received by the other two motes (acting as receivers). A sum of range differences between the four motes can be expressed by the phase difference of the received interference signals at the two receiver locations. In feasible RIPS setups, the measurement of phase difference are ambiguous because of hardware constraints. The problem of phase ambiguity also occurs in the signal frequency estimation and is known as the phase wrap problem [3].