Self-Calibration direct position determination using a single moving array with sensor gain and phase errors

Highlights

• Formulates the DPD model with deterministic sensor gain/phase errors.

• Proposes a MUSIC based self-calibration DPD method (MuS-DPD) to eliminate these errors.

• Proposes a ML based self-calibrating DPD method (MLS-DPD) with higher positioning accuracy than MuS-DPD at low SNR.

Abstract

The problem of direct position determination (DPD) using a single moving array in the presence of deterministic sensor gain and phase errors is considered. To eliminate the localization bias caused by these errors, an eigenstructure based self-calibrating DPD method is first introduced, in which the sensor gain and phase errors and the emitter positions are jointly estimated by an iterative process. Considering the performance deterioration of eigenstructure methods when the signal to noise ratio or the number of samples is not sufficiently large, a maximum likelihood (ML) based two-step self-calibration approach for DPD is subsequently proposed. The sensor gain errors are provided using the diagonal of the covariance matrix of the array output by a closed form solution at the first step. Then, the phase errors and the emitter positions are jointly estimated by an iterative scheme based on ML, in which the phase errors are also determined by a closed form solution in each iteration. Besides, detailed analyses and discussions about the differences between the introduced eigenstructure based and the proposed ML based self-calibration DPD methods are also provided. At last, numerical simulations are involved to examine their performance.

Introduction

Localization of emitters from passive observations is a problem that attracts much interest for both civil and defense-oriented applications in the signal processing and underwater acoustics etc. The estimation of emitter position is generally performed by a two step procedure. These methods estimate the intermediate parameters e.g. Directions of Arrival (DOA) [1], [2], [3], Time of Arrival (TOA) [4], Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) [5], [6] etc in the first step, and then determine the emitter position using the previously estimated parameters in the second step [7], [8], [9], [10]. Nevertheless, the two-step methods are not guaranteed to yield optimal location results since they ignore the constraint that all measurements must correspond to the same location [11]. To avoid this problem, a novel localization method known as direct position determination (DPD) was presented in [12]. It is reported that the accuracy of DPD is superior to the two-step method especially at low signal to noise ratio (SNR).

In the recent years, multiple DPD algorithms have been presented such as in [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. In these publications, DPD approaches can be deemed as searching for the emitter position that best matches all the received data. This process is achieved by using the raw received signals from all the sensors together and ending with a localization objective function only related to the emitter position. Therefore, emitter position can be extracted in a single step in DPD. Besides, the DPD approaches also inherently avoid associating estimated parameters with their relevant emitters.

However, it should be noted that the superior performance of DPD is critically dependent on the knowledge of the sensor e.g. gains, phases, and positions etc. In practical application, the existence of various errors may result in the array model errors and further degrade the localization accuracy, which should be well considered. Compared with the aforementioned publications which focus on the DPD algorithms themselves, the problem of DPD under the presence of various errors has not been extensively discussed yet. Some existing studies only can be found in [11] and [21], [22], [23], [24], [25], [26]. Therein, [11], [23], [24] elaborately examine and analyze the performance of DPD approach in the presence of model errors due to imprecise array calibration, multipath propagation, and mutual coupling etc. Their results show that DPD is superior to the two step approaches in the presence of model errors since it inherently ignores the observations of the bad stations and determines the position based on the rest of the observations. Similar to these studies, dense multipath propagation from local scatterers is individually considered in [17], [22], and identical conclusion is obtained. All publications introduced above focus on verifying the superior performance of DPD in the presence of various errors. However, they do not provide suitable approach to eliminate the effects of the errors, if possible, to further improve the performance of DPD approach. The only three exceptions can be found in [21], [25], [26], in which the array uncertainties in DPD are modeled as Gaussian random variables. Therein, the first two compensate the errors by the classical auto-calibration (a.k.a self-calibration) and the third one mitigates the effects of model errors by an improved ML based estimator. Their estimators can reduce the array model errors considerably and reach the CRLB under the assumption that the prior knowledge of the uncertainties e.g. mean and covariance are available or the waveforms of intercepted signals are known. However, in most passive circumstances as considered in this paper, this prior knowledge and the signal waveforms are not easily available.

Among all above studies, different errors e.g. imprecise array position, multipath, gain and phase uncertainties are considered in DPD, and they are agilely modeled as deterministic parameters or random parameters depending on the practical applications of DPD. As remarked in [27], multipath propagation from local scatterers has significant impact on parameter estimation in the environments where the receiving channel is time-invariant during the observation period e.g. the global system for mobile telecommunication (GSM) [28] whose observation periods are short. In contrast, [29] concluded that local scattering has a minor effect in rapidly time-varying scenarios. Meanwhile, since the number of scattered signals is relatively large, the local scattering environment is usually modeled as a stochastic process [30]. Besides, the inaccuracy in array locations usually needs to be taken into account in some applications [31], [32]. For instant, in distributed sensor network applications [33], the sensors are deployed in a random way whose positions are not exactly known. Another more general example is that the position of an array also needs to be estimated which is subject to estimation system error. The position error is often modeled as a random parameter for the former, while the latter usually assume it is constant. Moreover, sensor gain and phase errors are accepted as fluctuant parameters if the surrounding environment (e.g. temperature) changes [34]. However, when the environment is steady, sensor gain and phase errors are generally deemed as invariant and deterministic parameters during a relatively long time (e.g. a total observation period) since they are mainly caused by mechanical manufacturing and array edge effects in this case [35].

Without loss of generality, we focus on the common applications of the passive DPD in which only a single moving passive array intercepts the signals and the emitters to be localized are located in the far-field without any reflector around. In this case, the effect of multipath and local scattering can be ignored. Meanwhile, considering the development of high precision navigation positioning technology, the position errors can be further omitted when the sensor array is properly installed. Finally, the remained sensor gain and phase errors are considered in this paper. In addition, the sensor gain and phase errors are assumed to be deterministic parameters, which is a general assumption when the surrounding environment is steady [35].

In fact, the deterministic sensor gain and phase errors are also encountered in other parameter estimation frameworks. Here we shall mainly introduce it in DOA estimation applications, which may be referential for the DPD applications. Paulraj and Kailath [36], Wylie et al. [37], Youming and Er [38] developed two-step calibration procedures based on the hypothesis that the true array covariance is Toeplitz. However, all of these methods are only effective for Uniform Linear Array (ULA). The arrays other than ULA are studied in [39], [40], in which the errors are calibrated by an iterative procedure based on eigenstructure method e.g. Multiple Signal Classification (MUSIC), namely, self-calibration method. The major deficiency of this method may be the necessary conditions for unique solution including the emitter number should be larger than 1, the number of array elements should be larger than 4, and the array is nonlinear. Besides, by constructing a covariance matrix by the dot product of the array output vector and its conjugate, an alternative self-calibration method is proposed in [41], and some similar methods can be found in [35], [42] and [43]. They can avoid the joint iteration but meanwhile lead to higher computational complexity which show more severe for the complicated DPD approaches.

Considering the facts introduced above, we shall tend to calibrate the deterministic sensor gain and phase errors for DPD without additional necessary conditions and extra known calibration source or signal, and meanwhile take the computational complexity into account. Inspired by Friedlander and Weiss [39], the intuitional idea to solve this problem is the eigenstructure based self-calibrating DPD. However, it is well known that the eigenstructure based method shows barely satisfactory performance when SNR or snapshot number is not sufficiently large. Motivated by this, the self-calibration DPD method based on ML is considered. Consequently, the major contributions of this paper can be concluded as follows

• The model of DPD with single moving array under the presence of sensor gain and phase errors is formulated, in which the sensor gain and phase errors are assumed invariant and deterministic in the total observation time.

• The MUSIC based self-calibration DPD method is introduced and analyzed, which is inspired by the eigenstructure based self-calibrating DOA estimation approach proposed in [39] (denoted as MuS-DOA for shot). In the detailed derivation, the MUSIC based DPD objective function under the presence of sensor gain and phase errors is first provided. Then MUSIC based self-calibrating DPD method is introduced to estimate the sensor gain and phase errors and the positions simultaneously by some iterations aiming to achieve the optimization of this objective function. Note that the estimate of the considered errors can be obtained by a closed form solution in each iterations, which reduces the computational complexity to some extent. For the convenience of expression, the MUSIC based self-calibration DPD method is denoted as MuS-DPD for short in the rest of this paper.

• To ensure the superior performance of DPD in the presence of sensor gain and phase errors when SNR or the number of snapshots is not sufficiently large, a ML based self-calibration method for DPD with single moving array is proposed, which is denoted as MLS-DPD for short in the following sections. Similar to MuS-DPD, the ML based DPD objective function with the considered errors are first derived. However, the sensor gain and phase errors can not be determined by a closed form solution in this case because of the difference between the MUSIC based and ML based objective functions. To reduce the computational complexity, two steps are involved in the ML based self-calibrating DPD. The first is to estimate the sensor gain errors using the diagonal of the covariance matrix of the array output, which can be provided by a closed form solution as expected. In the second step, the phase errors and the positions are estimated by an iterative scheme aiming to achieve the maximum of the ML based objective function. With the sensor gain errors given in the first step, the sensor phase errors can also be estimated by a closed form solution during each iteration.

• The relation between MLS-DPD and two eigenstructure based self-calibrating methods MuS-DPD and MuS-DOA is analyzed and discussed including the iterative formulas, the necessary conditions, the estimated parameters, and the computational complexity etc.

The rest of this paper is organized as follows. Section 2 introduces the notations used throughout the paper and formulates the data model. Section 3 presents the MUSIC based self-calibrating DPD method. In Section 4, we propose the ML based self-calibrating DPD method. Subsequently, Section 5 discusses the relation between MLS-DPD and other existing self-calibrating methods. Numerical simulations are provided in Section 6, and the conclusions are made in Section 7.