**1. Introduction**

Passive radar technology, which allows operators to detect and localize potential targets using already existing transmitters such as commercial frequency modulation (FM) broadcast/digital audio broadcast (DAB)/terrestrial digital video broadcast (DVB-T) [1,2] and non-cooperative radar transmission [3,4], has been interesting to both civilian and military fields in the last few decades [5]. This sort of radar, compared to active radar technology, offers numerous advantages including lower cost, lower power usage and more covert surveillance capability, which suggests the possibility of employing passive radars on a wide range of concerned applications such as homeland security, costal surveillance and early warning system for vehicles detection, etc.

One of the remarkable characters of passive radar is the deployment of two receiving channels, with one for capturing the direct path signal from the transmitter and the other for collecting the potential target echoes [6]. By performing a cross-correlation operation between the direct path signal and the target echoes, the time delay (TD) could be measured, which holds information about the unknown target position. By multiplying with the signal propagation speed, the TD can be directly converted into the bistatic range (BR) [7]. Each BR measurement traces out an ellipsoid equation, with its foci located at the transmitter and the receiver positions. Theoretically, for multi-static passive radar (MPR), if enough BR measurements with respect to multiple transmitter-receiver pairs are available, the target position can be determined by solving the set of nonlinear ellipsoid equations. However, due to the high nonlinearity implied in the BR measurement equations, determining the target position from the BR measurements obtained at a single time instant is not a trivial topic.

In recent years, numerous algorithms have been developed to address this challenging topic, mainly including iterative methods [8,9] and closed-form solution methods [10–14]. Iterative methods [8,9] rely on an initial position guess close to the true solution but such a good guess may not always be available in reality. By contrast, closed-form solution methods [10–14] have always been more attractive to researchers due to their computational efficiency, independence on initial guess and absence of divergence problem. Illuminated by Ho and Xu's two-step weighted least squares (2WLS) idea [15], these closed-form solution methods [10–14] generally follow the basic two-step framework below: in the first step, the non-linear BR equations are linearized into pseudo-linear ones by introducing some proper nuisance parameters and a coarse estimate of target position can be obtained from the pseudo-linear equation set via weighted least squares (WLS) minimization; then in the second step, the function relation between the target position and the introduced nuisance parameters is explored to refine the initial estimate.

Nevertheless, all the aforementioned methods are designed based on the assumption that the positions of the transmitters and receivers are exactly known, but such exact priori knowledge may not be available in reality. In fact, the positions of the transmitters and receivers are inevitably perturbed by errors to some extent, and these errors (also referred to as position uncertainties) are often non-negligible, especially when the antennas are mounted on moving platforms [16,17] or the exploited transmitters are highly non-cooperative (such as the hostile radar radiation whose position could usually only roughly determined by electronic reconnaissance technique [18,19]). On the other hand, Rui and Ho [20] quantitatively analyze the influence of the transmitter and receiver position error on the localization accuracy, indicating that the target localization accuracy can be very sensitive to the transmitter/receiver position error and a slight error in transmitter/receiver position could remarkably deteriorate the localization accuracy. More recently, some novel methods [21,22] that take the statistical distributions of transmitter/receiver position error into consideration are developed to reduce the target localization error, and they are shown to attain the Cramér–Rao lower bound (CRLB) under small measurement noise and transmitter/receiver position error assumption. Nevertheless, these methods [21,22] only present the solutions when the transmitter/receiver position errors exist, but cannot fundamentally compensate the localization performance loss arising from transmitter/receiver position error at the CRLB level.

The use of calibration sensors has been a common technique in wireless sensor network self-localization, where each sensor broadcasts signals and receives signals from other sensors so as to determine their positions collaboratively [23,24]. Syldatk [25,26] considers the calibration of ground sensor networks where an accurate calibration of sensor positions and orientations is required for target tracking. For source localization problem, Hasan first considered in [27] utilizing calibration sensors to improve the angle-based source localization performance. Ho [28], Yang [29] as well as Li [30] et al. further expanded and applied the calibration technique to range difference (RD)-based source localization problem, where additional RD measurements from the calibration sensors were incorporated to reduce the receivers' position error and thus improve the source localization accuracy. The successful use of calibration techniques in these fields inspires us the possibility of employing calibration technique in the target localization for multi-static passive radar. When it comes to BR-based target localization in multi-static passive radar, using a 'calibration target' with known position may also be able to mitigate the target localization performance loss arising from the transmitter/receiver position error. In theory, any target appearing in the radar coverage area and meanwhile broadcasting its position can be taken as a calibration target. Typically, for example, to avoid potential accidents and collisions, the commercial aircrafts will report their positions and other information to the ground stations and other aircraft by using the automatic dependent surveillance broadcast (ADS-B) system [31]. Hence, the commercial aircraft broadcasting ADS-B signal can be regarded as an off-the-shelf calibration target. If no such off-the-shelf calibration targets are available in the radar coverage area, we can manually

launch some strong scatterers with known positions as calibration targets. However, despite that, up to now there exists no publication in the open literature that addresses refining inaccurate transmitter and receiver positions using calibration targets for target localization in multi-static passive radar.

Motivated by these facts, in this paper, taking the transmitter and receiver position error into consideration, we explore using calibration targets to counter the loss in BR-based target localization accuracy arising from the transmitter/receiver position error. We begin our work by deriving the CRLB for BR-based target position estimation when the BR measurements from the calibration targets are available. The interpretation on the CRLB demonstrates that the use of calibration targets can significantly mitigate the influence of the transmitter/receiver position error and dramatically enhance the localization accuracy, at least in the sense of CRLB. We then proceed to develop a novel localization method to alleviate the transmitter and receiver position error and enhance the localization accuracy using calibration targets. It mainly includes two processing stages, referred to as calibration stage and localization stage respectively. In the calibration stage, the BR measurements from the calibration targets are exploited to refine the inaccurate transmitter and receiver positions; in the localization stage, the refined transmitter and receiver positions and BR measurements corresponding to the unknown target are exploited to determine the target position. Both processing stages are closed-form, which brings the proposed method computational efficiency and high robustness. Furthermore, the accuracy of the proposed solution is shown analytically to reach the CRLB when the transmitters'/receivers'/calibration targets' position errors and the BR measurement noises are sufficiently small. Simulations will be conducted to verify the effectiveness and superiority of the proposed solution over existing methods.

*Notations*: There will be a lot of notations throughout this paper. Without exception, vectors (matrices) are denoted by bold lower (upper) case letters, respectively. Also, notations (·) ◦ , (·) T, · , (·) −1 , E(·), **O***p*×*q*, **I***p*×*p*, 0*p*×1, diag(·) and tr(·), represent the true value of a noisy or an estimated variable, transpose operation, Euclidean norm, inverse of matrix, statistical expectation, a *p*-by-*q* zero matrix, an identity matrix of size *p*, a *p*-by-1 zero vector, diagonal matrix and the trace of a matrix, respectively.

The remainder of this paper is organized as follows. Section 2 is about the localization scenario in the presence of transmitter/receiver position error and calibration targets. In Section 3, the corresponding CRLB is evaluated, indicating the potential of calibration targets in improve localization accuracy. In Section 4, a closed-form solution is developed for target localization in the presence of transmitter/receiver position error and calibration targets, and theoretical performance analysis is also given. Section 5 describes the results of Monte Carlo simulations that compare the proposed solution with existing methods. Section 6 is the conclusion.
