Robustness of the Least Squares Range Estimator

We consider the problem of distance, or range, estimation by measuring the phase of a sinusoidal signal transmitted between two locations. The distance can only be unambiguously measured if it is contained in an interval of length less than the wavelength of the signal. To increase the length of this interval, multiple phase measurements at different wavelengths can be used. The identifiable range is then extended to an interval of length equal to the least common multiple of the wavelengths. Phase measurements are noisy in practice and the problem of estimating range from multiple noisy phase measurements is non trivial. Existing solutions are based on least squares, the method of excess fractions, and on noise resilient versions of the Chinese remainder theorem (CRT). These estimators all, either explicitly, or implicitly, make an estimate of so called wrapping variables related to the whole number of wavelengths that occur over the range. In this paper, we discover an upper bound such that if all absolute phase measurement errors are less than this bound, then the least squares range estimator is guaranteed to correctly estimate the wrapping variables. We compare this with a similar bound discovered for estimators based on the CRT. The bound for the least squares estimator is often larger. This corroborates with existing empirical evidence suggesting that the least squares estimator is often more accurate.