1.1. Problem Statement
Cognitive radar can basically be defined as: an intelligent radar system of hardware and software in which the transmit and receive parameters (i.e., pulse length, pulse repetition frequency (PRF), modulation, power, frequency, and polarization) are selected, in real-time, and use adaptation between the information extracted from the sensor/processor and the design and transmission of subsequent waveforms, in response to the observed scene to optimize the performance of a given application. The problem of target tracking in cognitive radar system has received considerable attention. It is well known that most of the state-space dynamical systems are nonlinear or non-Gaussian. For example, the multi-state transition of a drone from hovering to maneuvering is often nonlinear, and the measurement noise is mainly flicker noise or heavy-tailed noise [
1].
Despite the performance decline of extended Kalman filter (EKF) and unscented Kalman filter (UKF) in highly nonlinear problems, and that general analytical solutions are intractable in nonlinear or non-Gaussian systems, solutions continue to emerge from different viewpoints. However, most of the methods rely on the assumption that the noise has known statistics, or they require accurate mathematical representation of the dynamics of the system evolution; otherwise, it is almost impossible to directly approximate the true distribution. In practice, sometimes the assumption is in accordance with the actual situation, but sometimes not. The same situation occurs in cognitive radar systems. Without explicit mathematical models or a priori information, how cognitive radar can still exert its advantages is a concern of this article.
We briefly review various existing methods to cognitive radar tracking problem that involve particle filter (PF) in some relevant manner.
1.2. Related Works
For handling arbitrary nonlinear models and arbitrary noise distributions, methods based on Monte Carlo (MC) methodology have emerged in Bayesian estimation. MC is a simulation-based method aimed at estimating the posteriori pdf of the state given the observations. Markov Chain Monte Carlo (MCMC) and Sequential MC (SMC) are two main tools in it, and they can sample from high dimensional probability distributions. The performance of MCMC would be unreliable when the proposal distributions that are used to explore the space are poorly chosen and/or if highly correlated variables are updated independently. While, PF became popular for it is particularly suitable for real-time estimation.
The Sequential Importance Sampling (SIS) algorithm was used for the first time to solve the problem of nonlinear filtering [
2]. The formal establishment of PF was attributed to the proposal of resampling technology [
3,
4]. Meanwhile, the idea of Sampling Importance Resampling (SIR) was discovered and developed [
5]. PF is allowed to express a complete and precise state posterior distribution, so any statistical data such as mean, variance, and modulus can be easily calculated, and theoretically, the accuracy is higher than that of EKF and UKF in nonlinear systems [
6]. Hence, despite its computational pressure, it is still quite attractive to us. During the development of PF, there are several inevitable drawbacks need to be addressed: (1) How to approximate the optimal proposal distribution further; (2) How to overcome the problems of weight degeneracy, sample impoverishment, etc., to make the resampling more effective; (3) How to make the algorithm efficient and online.
Therefore, more variants of PF were produced [
7]. Efficient importance sampling techniques were studied in [
8,
9] for the first problem, but these algorithms require that the posterior distribution of the states are assumed as a priori known and can be approximated by a Gaussian distribution. An auxiliary variable particle filter was used to deal with the second problem [
10], but the filtering performance degrades when the state noise is strong, and because the likelihood function and weight value need to be calculated twice for each particle, the calculation amount increases. Adaptive PF (APF) can release the computation burden by adjusting the number of particles dynamically [
11,
12,
13,
14,
15]. This method chooses a small particle number if the density is focused on a small part of the state-space, and chooses a large number if the state uncertainty is high [
16]. KLD-APF is based on Kullback–Leibler (KL) information or KL distance (KLD) sampling, but it ignores any mismatch between the true and the proposal distribution [
17]. When the mismatch happens, the addition of particle number will only increase the computational load. On the other hand, reducing the particle number may aggravate the sample impoverishment and further weaken the effect of resampling.
In addition to these drawbacks, PF would still be invalid when some issues are encountered. PF requires that the conditional pdf of the observed variable can be estimated, otherwise the weight of the particles cannot be calculated. Thus, particle MCMC (PMCMC) was proposed by combing SMC with MCMC as an efficient approach [
18]. Based on PMCMC, the SMC
2 algorithm is motivated to tackle the intractable problem of probable increments in state-space models [
19]. Similar to the SMC
2 scheme, nested particle filters for online parameter estimation were proposed but in a purely recursive manner, in order to address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system [
20]. On robustness, cooperative parallel particle filters were designed for the dual purpose of Bayesian inference and on-Line Model Selection, with the online adaptation for the particle number [
21]. Particle learning was proved in [
22] to be outperforming existing PF alternatives and a competitor to MCMC. In the presence of model uncertainty where discrete data are encountered, a new SMC method was proposed for the filtering and prediction of time-varying signals [
23]. A similar method with better predictive powers was proposed in [
24], wherein the resampling step could be dynamically adjusted and the predictive powers could be updated sequentially as more data were observed. Moreover, what deserves attention is that Bayes is not the only choice, as a neural filter based on GRNN is also outperforming numerical filters in state vector estimation during dynamic changes of target movement parameters [
25].
Inspired by the previous research, we might find a solution independent of the prior information through updating the proposal distribution and its stepwise approximation to the true distribution by iterative methods. The authors of [
26] designed a cost function in PF to iteratively update the state and variance when the prior information is unknown. Naturally, it has been applied to many areas such as target tracking, autonomous vehicle positioning, sensor network, orthogonal frequency division multiplexing (OFDM) systems, wireless local area network (WLAN), and tilt estimation [
27,
28,
29]. There are some improvements to the algorithm. In [
30], a particle selection algorithm was proposed and analyzed for implementation with parallel computing devices and to circumvent the main drawback of the conventional resampling techniques. Authors of [
31] melded random measures of two or more cost-reference particle filters to obtain a fused random measure that combines the information from the individual cost-reference particle filters. However, there is no further study on the optimization of the cost function, especially when the tracking structure can be adaptive. Cognitive radar designs the optimal estimator for Bayesian framework in [
32], but the majority of the existing Bayesian tracking methods for cognitive radar applications are based on the Kalman filter for linear systems [
33,
34], and Kalman-like solutions, e.g., cubature KF (CKF) and continuous-discrete (CD)-CKF for nonlinear problems [
35]. Few researches use PF. In [
36], a particle filter combined with probabilistic data association is used as a tracker. In [
37], a cognitive structure was designed as only a part of PF, namely, a parallel structure of PF, while EKF was used by waveform selection to adapt the particle number and reduce the computation cost.
1.3. Contributions and Organization of the Paper
When we specify PF as the nonlinear tracking method in a cognitive radar, we consider using the cognitive structure to expand the state-space of PF or its variants to another dimension where the waveform parameters can be changed from fixed to dynamic, and we can optimize the cost function design and corresponding lower bound of the estimation error. In turn, using the particle filter tracking method based on the optimal cost function can expand the scope of the application of cognitive radar. The main contributions of this paper will be presented by the following items:
(1) The cognitive radar tracking method based on PF is proposed and the mathematical model of which it is derived for the first time. Push the cognitive radar tracking framework from existing the Kalman-like to the developing SMC-like. Not only the data process but also the waveform design is in the PF iteration, that is, a fully cognitive PF.
(2) Refine the idea of cost-reference PF with cognitive framework. When the estimation of the parameters (i.e., signal-to-noise ratio (SNR)) in cognitive radar mismatches the real situation, a novel cognitive cost-reference particle filter algorithm is proposed to bring about robustness to the existing cognitive radar tracking methods and intelligence for the current adaptive method. The convergence is proofed mathematically.
(3) The Cramér–Rao Lower Bound (CRLB) of the proposed cognitive PF is derived, and the corresponding cost function is designed.
The rest of this paper is organized as follows: the standard PF, the mathematical model of cognitive radar, and the interface of PF to cognitive algorithm are presented, and the CRLB of the cognitive PF is derived in
Section 2. The principle of the cost-reference particle filter is restated, the proposed cognitive scheme is presented along with the implementation steps, and the convergence of the solution is proofed in
Section 3.
Section 4 shows the dynamic model, the maneuvering target tracking in unknown non-Gaussian noise, and the simulation results.
Section 5 presents the conclusions.