Importance sampling based direct maximum likelihood position determination of multiple emitters using finite measurements

Highlights

• We resort to Pincus’ theorem and the IS technique to approximate the exact ML solution of the problem of direct position determination of multiple emitters with finite measurements.

• We design a factorable importance function to generate the required parameter realizations efficiently.

• Compared with the exponential complexity of the exhaustive multidimensional ML searching, the complexity of the new approach is moderate.

Abstract

The exact Direct Maximum Likelihood Position Determination (ML-DPD) of multiple emitters requires a multidimensional searching with exponential complexity. The subspace-based DPD technique and the filtering-based DPD technique are recently proposed to reduce the complexity by transforming the original high-dimensional optimization problem into several low-dimensional ones. However, these techniques approximate the exact ML solution only if the measurements are assumed infinite. In this paper, we model the received signals in the nonlinear regression form and resort to the multidimensional optimization framework, consisting of the Pincus’ theorem and the Importance Sampling (IS) technique, to approximate the exact ML solution using finite measurements. The proposed IS based ML DPD approach also reduces the complexity by decoupling the multidimensional optimization problem into several three-dimensional ones. Moreover, the new non-iterative ML DPD approach guarantees global optimality and does not suffer from the off-grid problems inherent to most DPD techniques. The numerical simulations show that the proposed ML DPD estimator acquires the exact multidimensional ML solution even using a narrow bandwidth or being at low Signal to Noise Ratio (SNR).

Introduction

The source localization technique has been studied for several decades and applied in many practical scenarios, such as the Location-Based Services (LBS) in communication systems, objects tracking in Internet of Things (IoT) and targets detection in radar system. The classical localization methods are two-step processing. Firstly, intermediate parameters that rely on the locations of the sources, like the Angle of Arrival (AOA) and the Time of Arrival (TOA), are estimated from the measurements of different receivers respectively. Then, the locations of sources are estimated based on these intermediate parameters. The position methods based on different intermediate parameters have been proposed. The array processing methods for position by AOA are reviewed by Krim and Viberg comprehensively [1]. The classical closed-form Least Squares (LS) solution using TDOA was derived by Smith and Abel [2]. Other TOA based methods are Semi-Definite Relaxations (SDR) [3], [4], [5] and Multidimensional Scaling (MDS) approach [6]. The two-step methods are sub-optimal as they measure the intermediate parameters at each receiver separately and independently. This ignores the constraint that the measurements from different receivers correspond to the same source position.

Recently, the DPD approaches have been proposed as a single-step localization technique where the locations of sources are estimated by minimizing a single cost function, into which all received data enter jointly. It has been verified that the DPD methods outperform the two-step methods at low SNR. Besides, an additional data association step required by two-step methods is inherently avoided by DPD methods. Nevertheless, the DPD approaches, which usually require exhaustive searching, obtain the above advantages at the expense of increasing computational complexity significantly. To decrease the computational complexity, the efficient DPD techniques based on the Expectation-Maximization (EM) algorithm replaced the multidimensional searching with several one-dimensional ones [7], [8]. A sequential Monte Carlo-based method was used to solve the multivariate optimization problem that results from the ML solution in an iterative way [9]. The localization problem of coherent sources has been addressed by combining the DPD with the iterative adaptive approach (IAA) [10].

In the scenario of multiple sources, the exact multidimensional ML estimator can be derived but requires a multidimensional searching, which is usually impractical for real-time implementations. To sidestep the multidimensional searching in the exact ML estimation, several existing DPD methods resort to the subspace-based technique or the filtering technique such as the Multiple Signal Classification (MUSIC) [11], [12], the Match Filter (MF) [13] and the Minimum Variance Distortionless Response (MVDR) filter [14], [15]. These techniques transform the exact multidimensional ML problem into several single ones that require merely two or three-dimensional space searching. However, these methods require sufficient observation samples to achieve the ML performance asymptotically. To be specific, the MUSIC-DPD [12] and MVDR-DPD [14], [15] both need lots of observation samples to approximate the covariance matrix by the sample covariance matrix. The MF-DPD [13] also need sufficient observation samples to decouple the problem based on the independence among transmitted signals. Unfortunately, the length of observation window is usually limited for the real-time position request or the stationary assumption. Thus, the number of observation samples is finite and the above estimators can not achieve the ML performance in practice. Besides, the iterative ML method [16] is proposed to achieve the exact ML solution efficiently but its convergence to the global optimum relies on the initial estimate.

In order to achieve the exact ML solution with finite observation resources, in this paper we resort to the optimization framework consisting of the Pincus’ theorem [17] and the IS technique [18]. We design a factorable probability density function (pdf) on model parameters. This pdf allows a very easy sampling in the vicinity of the exact ML solution. Numerical simulations show that the proposed IS-based multidimensional ML estimator outperforms the subspace-based and the filtering-based DPD ones in the scenario of multiple sources.

The contributions of our paper are summarized as follow: (1) In order to solve the DPD of multiple emitters, we resort to Pincus’ theorem and IS technique to approximate the exact ML solution with finite measurements. (2) We design a factorable importance function to generate the required parameter realizations efficiently. Compared with the exponential complexity of the exhaustive multidimensional ML searching, the complexity of the new approach is moderate.

The paper is organized as follows: Section 2 states the DPD problem formulated for multiple emitters. The concentrated likelihood function is derived in Section 3 and maximized by Pincus’ theorem and IS technique in Section 4. The factorable importance function is designed in Section 5. The implementation details and the complexity of the proposed algorithm are stated in Sections 6 and 7 respectively, followed by numerical simulations in Section 8. Finally, a conclusion is drawn in Section 9.