Variational Bayes phase tracking for correlated dual-frequency measurements with slow dynamics

Highlights

• Model of correlated dual-frequency measurements with slow dynamics.

• Estimation of the signal absolute phase via a variational Bayes filtering method.

• Closed-form filtering distribution with known modes (number + location).

• Mean of the filtering distribution is a tractable estimator with low cycle slip rate.

• Benefit of the method is distinct at low SNR or for low correlation level.

Abstract

We consider the problem of estimating the absolute phase of a noisy signal when this latter consists of correlated dual-frequency measurements. This scenario may arise in many application areas such as global navigation satellite system (GNSS). In this paper, we assume a slow varying phase and propose accordingly a Bayesian filtering technique that makes use of the frequency diversity. More specifically, the method results from a variational Bayes approximation and belongs to the class of nonlinear filters. Numerical simulations are performed to assess the performance of the tracking technique especially in terms of mean square error and cycle-slip rate. Comparison with a more conventional approach, namely a Gaussian sum estimator, shows substantial improvements when the signal-to-noise ratio and/or the correlation of the measurements are low.

Introduction

Over the last few decades signal phase measurement has become an active area of research. Indeed, in many applications, the carrier phase of a transmitted or reflected wave conveys information of primary interest to the operator. Phase measurement is directly related to surface height in interferometric synthetic aperture radar (InSAR) [1] and to the target radial velocity in radar [2] whereas in navigation it provides a highly-precise range measurement between the satellites and the receiver [3], to name a few examples.

Phase measurement is however by nature ambiguous. For instance, estimating the phase via a conventional four-quadrant inverse tangent leads to a wrapped observation in the principal interval $[-\pi ,\pi ]$. Without additional information, retrieving the absolute phase is thus an ill-posed problem.

To remove ambiguity, prior knowledge about the phase dynamics is usually injected into the estimation problem. For instance, in the case of low value phase-gradient, an efficient unwrapping technique may be obtained simply by integrating the phase difference between two samples. However this method, which is based on restrictive assumptions [4], fails in the case of noisy samples or when the phase dynamics increases locally and causes aliasing. Phase jumps, known also as cycle slips, arise then in the estimation process. A thorough description of this phenomenon can be found in [5] for phase-locked loops (PLL). To avoid cycle slips, more advanced techniques are usually required. Among them statistical modeling offers a flexible way to proceed. In particular, Markov random fields (MRF) have been widely used since they can guarantee a certain continuity between phase samples [6], [7], [8].

In addition to injecting prior knowledge about the phase dynamics, frequency diversity has been advocated as a complementary means for phase disambiguation [9]. Frequency diversity is obtained when the sensing system is able to observe the scene with different frequencies. The phase is then measured, up to a known frequency ratio, as many times as there are frequencies. The redundancy in the observations helps therefore to reduce cycle slips while reconstructing the absolute phase. This principle can be found in many application areas: in InSAR with the use of multiple interferograms [10], in radar with OFDM waveforms [11] or with the use of multiple pulse repetition frequencies [12], in navigation with the use of multifrequency receiver [13], [14], [15], in robotics with time-of-flights cameras [16], etc. Interestingly, a very related approach to remove phase ambiguity consists in using jointly the information conveyed by the envelope and the carrier frequency of the signal. Phase unwrapping techniques have been accordingly developed, e.g., in GNSS [17], in wideband radar [18] and for communication systems [19].

In this paper, we restrict our attention to dual-frequency measurements and propose in Section 2 a model to estimate the absolute phase for in-line processing applications. Particularly, the phase dynamics is assumed to be smooth enough to be represented by a first order MRF while some correlation is introduced between the amplitudes of both frequencies. Based on this model a Bayesian filtering technique is developed in Section 3. The method uses a variational Bayesian approximation [20] and results into a nonlinear filtering algorithm [21]. Numerical results are provided in Section 4 to illustrate the performance of the proposed absolute phase estimator. The latter is compared to a benchmark algorithm that belongs to a more conventional nonlinear filtering approach, namely a Gaussian sum estimator.