JING Tong，TIAN Wei，HUANG Gaoming，and PENG Huafu

College of Electronic Engineering，Naval University of Engineering，Wuhan 430033，China

Abstract: Most existing studies about passive radar systems are based on the already known illuminator of opportunity (IO)states. However，in practice，the receiver generally has little knowledge about the IO states. Little research has studied this problem. This paper analyzes the observability and estimability for passive radar systems with unknown IO states under three typical scenarios. Besides，the directions of high and low estimability with respect to various states are given. Moreover，two types of observations are taken into account. The effects of different observations on both observability and estimability are well analyzed. For the observability test，linear and nonlinear methods are considered，which proves that both tests are applicable to the system. Numerical simulations confirm the correctness of the theoretical analysis.

Keywords: passive radar，passive coherent location (PCL)，observability，estimability，unknown illuminator states.

1. Introduction

Passive radar works with the help of non-cooperative illuminators of opportunity (IOs)，which is also called passive coherent location (PCL) system [1-6]. Compared with the conventional active counterparts that use their own dedicated transmitters，passive radar has a number of advantages，such as easy construction，spectrum saving，and significant performance improvement with multi-static configuration. However，since the receiver generically has no prior knowledge of the IO waveform，one may have difficulty gaining the aforementioned superiorities.Hence，intensive attention has been paid to the research filed on passive radar systems.

Target localization and tracking are hot research topics in passive radar. In [7]and [8]，two joint delay-Doppler estimators were proposed，where the direct-path interference (DPI) to the surveillance channel was taken into account. The direct-path delay was compensated based on the assumption that the IO location has been obtained for simplifying the problem model. Abdullah et al. [9]proved that the passive radar utilizing the stationary long-term evolution (LTE) communication station as IO could offer a satisfying performance on moving vehicle tracking. To the best of our knowledge，almost all existing studies assume that the IO location is fixed and exactly known.However，such an assumption deviates from the real situation. For instance，the passive radar may be required to promptly deploy in some unfamiliar areas. Therefore，little knowledge about the IO states，such as the IO location，could be provided to the receiver in advance. Hence，the system has to simultaneously estimate the states of both the IO and targets for its radar function. Herein，the problem is regarded as the simultaneous localization and mapping (SLAM) or opportunistic navigation (OpNav)problem [10]. In contrast to these two problems，the environment in the passive radar is more complex since the IO states are dynamic. Therefore，studying the target localization and tracking problems without the knowledge of the IO states is extremely necessary. First of all，we should find out whether such problems have solvability.This paper aims to make a detailed analysis of the observability in passive radar systems with unknown IO states.Simultaneously，the degree of observability，i.e.，estimability，is considered.

Observability is an important concept. When the system is observable，a unique solution about the system states can be obtained [11]. In general，a passive radar system is nonlinear. Several major methods for nonlinear observability analysis are concluded as follows. First，the geometric approach extracts the information associated with the parameters of target motion. Rao [12]adopted an elementary geometry method to analyze the observability of an observer with a multi-leg trajectory in a bearingsonly system. Based on the similar analysis，a method to calculate the target range using bearings-only measurement was given in [13]. Second，the linear observability test that converts the nonlinear system model into its pseudo-linear form may be exploited to nonlinear observability analysis. Jauffret et al. [14]proved that the necessary and sufficient condition for the system to be observable over a given finite time period is the so-called Gram matrix reversible. Since the Gram matrix is computationally intractable，Becker [15]gave an equivalent but more simple criterion. Song [16]extended such analysis to a broader target motion model. Finally，the nonlinear observability test is the most common analysis tool for nonlinear systems，in which two methods are typically used.One is the extension of the linear Gram matrix criterion，where relative Jacobian matrices are utilized to take the place of the state transition and observation matrices [17].The other is based on examining the Fisher information matrix (FIM) [18-21]. Besides，some research studied the observability by constructing the piece-wise constant system (PWCS) model [22-27].

In this work，the observability and estimability for passive radar systems are studied. The contributions of this paper are given as follows. First，different from most existing research with already known IO states，the observability with unknown and dynamic IO states in three typical scenarios is analyzed，which demonstrates whether the system in each scenario is observable. Second，associated estimability is also studied by decomposing the error covariance matrix of the extended Kalman filter(EKF). Third，we consider the effect of two different types of observations on both observability and estimability. Fourth，two observability analysis methods，linear and nonlinear observability tests，are applied. Theoretical analysis proves that both tests are applicable to the system. Finally，experimental results validate the correctness of the theoretical analysis and illustrate that the IO and target states can be simultaneously estimated. It is worth pointing out that Guo et al. [28]studied a similar problem about the observability without experimental validation，where the necessary and sufficient conditions for local observability are derived based on FIM. This paper extends the work of [28]in four different ways. First，the observability under more configurations is considered. Second，starting from the rigorous local properties of observability，both linear and nonlinear observability tests are used，which may have less computation than the method in [28]. Importantly，simulation experiments are given to confirm the correctness of our analysis.Third，we additionally analyze the corresponding estimability. Finally，the effect of different observations on the system is also studied.

The rest of the paper is organized as follows. In Section 2，the problem model is given. In Section 3，the linear and nonlinear observability test tools are studied.Besides，the observability analysis with unknown IO states using different observations is deduced. Section 4 introduces a method to analyze system estimability. Simulation results are illustrated in Section 5.

2. Problem formulation

We consider three typical cases in passive radar systems:(i) single receiver with single target and single IO(SRSTSIO)，(ii) single receiver with multiple targets and single IO (SRMTSIO)，and (iii) single receiver with single target and multiple IOs (SRSTMIOs). Both target and IO states are dynamic and unknown to the receiver.Herein，only one receiver with the stateis considered，whereare associated location and velocity states，respectively. We assume that the receiver states are exactly known. Moreover，the system will not lose the measurements of the targets and IOs during the receiver observation period.

2.1 State model

2.2 Observation model

In this problem，all the observations from the target and the IO should be centrally processed at the receiver with an appropriate filter. Herein，for clarity，we define the observation only produced by the IO as the IO observation(IOO) and the observation related to the target as the target observation (TO)，respectively. Hence，the whole system observations are composed of IOOs and TOs. Assume that the IOO is only the angle of arrival (AOA). Let αibe the AOA observation generated by theith IO，wherei=1，2，···，M. Whereby，

Two types of TOs are considered in this paper. One is based on single observation from the target (SOT). The other utilizes joint observation from the target (JOT). In the conventional passive location that only collects signals emitted by the target，single observation with Doppler difference can determine both the target's range and velocity. Compared with other observations，such as AOA that can only give the direction of the target motion，Doppler difference is more attractive in practice. Hence，for the SOT case，the TO only contains the Doppler difference information. Therefore，letfd，ijbe the Doppler difference in terms of thejth target associated with theith IO [29]，wherei=1，2，···，M，j=1，2，···，N. Then

3. Observability test and analysis

If the initial state of a system can be uniquely determined by the system output with a given input function，such a system will be treated as observable [32]. In this paper，linear and nonlinear observability tests [33]are applied to determining whether the system in each scenario with dif-ferent observations is observable.

3.1 Observability test

3.1.1 Nonlinear observability test

For a nonlinear system，the global observability is difficult to establish since the system may distinguish between initial conditions over a long period of time.Hence，local properties are more applicable [33，34]. Particularly，a system is supposed to be instantaneously observable in a certain neighborhood of the state trajectories [35]. Besides，it is suitable to utilize the nonlinear observability test to analyze nonlinear systems，which may better reflect the characteristics of the systems. Therefore，an observability algebraic test based on establishing local weak observability of the nonlinear system with its control affine form is applied [36]. The control affine form of a continuous-time (CT) nonlinear system can be written as

3.2 Observability analysis with unknown IO states

The observability of passive radar systems with unknown IO states using different observations under three typical cases is analyzed in this subsection. Based on the two observability tests，we will prove whether the system in each scenario is observable.

3.2.1 Geometry singularity

We assume that the receiver，target，and IO are not collinear in the forthcoming analysis since the observability matrix will lose rank. A simple example will be given to prove this point.

Consider a non-collinear SRSTSIO case，where the system has prior knowledge of the IO initial states.

4. Estimablity analysis

The observability can only qualitatively but not quantitatively reflect the characteristics of the system. However，the degree of observability，i.e.，estimability，also needs to be considered. The estimability can assess whether the system has good or poor observability. Herein，the method in [40]is adopted，where the estimability of different states of the system is assessed by decomposing the normalized error covariance matrix of the filter. The purposes of the normalization are twofold: (i) transforming the estimation error covariance matrix to be dimensionless，and (ii) bounding the eigenvalues between zero andn. The first purpose is based on a congruent transformation given as follows:

5. Simulation results

Table 1 Simulation parameter settings for SRMTSIO case

Fig. 1 Estimation errors and variance bounds for SRMTSIO case

Fig. 2 Estimation errors and variance bounds for SRSTMIOs case

Fig. 3 The least and the most observable directions according to the system states for SRMTSIO case

Fig. 4 The least and the most observable directions according to the system states for SRSTMIOs case

6. Conclusions

In this paper，we study the problem of observability and estimability analysis for passive radar systems with unknown IO states in three typical scenarios. Different observations are considered in each case. Linear and nonlinear observability tests are both considered to examine whether the system is observable. The estimability is also studied.

We observe that the systems in all scenarios are observable. The two observability tests that have the same conclusions are both applicable to the system. For the effect of different observations，the linear observability test with SOT generally needs more time steps to make the observability matrix full rank than that with JOT. Furthermore，increasing the number of IOs can decrease such time steps for SOT. Besides，the estimability，especially the directions of good and poor observability for the various states，is significantly dependent on the observations.In future work，we will consider the case，where the IOs and targets have various dynamic models.