Monopulse radar track‐before‐detect using Bernoulli filter

IET Signal ProcessingVolume 16, Issue 8 p. 1011-1021 ORIGINAL RESEARCHOpen Access Monopulse radar track-before-detect using Bernoulli filter Fei Cai, Corresponding Author Fei Cai caifei@hunnu.edu.cn orcid.org/0000-0002-5452-5350 College of Information Science and Engineering, Hunan Normal University, Changsha, Hunan, China Correspondence Fei Cai, College of Information Science and Engineering, Hunan Normal University, Changsha, Hunan 410081, China. Email: caifei@hunnu.edu.cnSearch for more papers by this author Fei Cai, Corresponding Author Fei Cai caifei@hunnu.edu.cn orcid.org/0000-0002-5452-5350 College of Information Science and Engineering, Hunan Normal University, Changsha, Hunan, China Correspondence Fei Cai, College of Information Science and Engineering, Hunan Normal University, Changsha, Hunan 410081, China. Email: caifei@hunnu.edu.cnSearch for more papers by this author First published: 01 July 2022 https://doi.org/10.1049/sil2.12142AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Detection and tracking of a dim target with monopulse radar is a challenging problem. When the signal-to-noise ratio (SNR) is low, the angle estimation performance and the detection performance are poor, which makes the capture and close-loop tracking of the target very hard since the monopulse radar has a narrow beamwidth. This paper uses the track-before-detect (TBD) technique to address this problem. A measurement model is established using the Swerling II radar target model. A Bernoulli TBD filter for the monopulse radar is developed in a random finite set framework and is implemented approximately as a particle filter. The performance of the filter is studied via Monte Carlo simulations, which show that the proposed filter outperforms the classical detection and estimation methods, and it can efficiently detect and track the target at a SNR as low as −6 dB. 1 INTRODUCTION Monopulse is a widely used high-precision angle estimation technique in the tracking radar. In comparison with the conical scanning and sequential lobing tracking techniques, it is insusceptible to echo amplitude fluctuations and amplitude jamming [1]. The monopulse radar has a pencil beam and usually depends upon information from outside sensors to acquire the target. Because the number of range bins and the beamwidth are limited, the radar pointing vector should move dynamically to make the target within the sensor field of view, which relies on effective target detection and estimation. When the signal-to-noise ratio (SNR) is low, the poor detection and estimation performances will make the capture and close-loop tracking of the target very hard. The block diagram of monopulse radar signal processing and the tracking system is depicted in Figure 1. In classical monopulse systems, the detection, estimation and tracking operations are done separately. The detection is performed with the signal in the s channel only. After that, the angle is estimated using monopulse operation for the post-detection resolution bins, and the estimations in the form of point measurements are fed to the tracking block. In Refs. [2-5], joint detection and estimation (JDE) in the monopulse system is studied. The signal in the difference channel participates in the detection and brings some benefits in detection when the target is not at the beam centre. However, the benefits brought by the JDE are rather limited. Note that the tracking blocks in the approaches above are all based on thresholded measurements, which is not efficient in low SNR scenarios due to the information loss in the detection process. FIGURE 1Open in figure viewerPowerPoint Block diagram of monopulse radar signal processing and tracking system. JDE, joint detection and estimation; TBD, Track-before-detect Track-before-detect (TBD) is an efficient dim target detection and tracking technique. Different from the conventional target tracking method that is performed after detection, TBD performs detection and tracking simultaneously using raw sensor measurements, and the performance is improved in low SNR conditions at the expense of increased computational complexity. The commonly used TBD techniques include Hough transform [6], dynamic programming [7], particle filter based TBD (PF-TBD) [8-11] and the histogram probabilistic multi-hypothesis tracking (H-PMHT) algorithm [12]. The Bernoulli filter is also known as the JoTT filter and it is an optimal single target filter [13-15]. Different measurement models can be used in the Bernoulli filter, resulting in different update equations. The detector output model and the intensity measurement model are the two mainly used measurement models in the Bernoulli filter, with the latter resulting in the Bernoulli TBD filter. The Bernoulli TBD filter has been employed in several radar applications, such as the multiple-input multiple-output (MIMO) radars [16] and the scanning maritime radars [17, 18]. In this paper, the Bernoulli TBD filter is employed to address the dim target detection and tracking problem in monopulse radars. The detection, estimation and tracking are done jointly as shown in Figure 1. The spreading of target energy to adjacent range bins is exploited to improve the detection and estimation performances. Swerling II radar target model [19] is employed in the measurement model. The particle filter implementation of the Bernoulli TBD filter for monopulse radar is developed, in which a limiting operation to the estimated probability of existence is proposed to avoid performance degradation. Simulation results show that the estimation performance as well as the detection performance are improved significantly upon the classical methods in low SNR scenarios. The main contribution is as follows: First, a Bernoulli TBD filter for the monopulse radar is developed, which can efficiently detect and track the target in low SNR scenarios. To the best of the author's knowledge, this is the first time the Bernoulli TBD filter is employed in the monopulse radar. Second, a comparative performance analysis of the proposed filter with the classical detection and estimation method is provided. The remainder of the paper is organised as follows: The target and measurement models are formulated in Sections 2 and 3, respectively. The conceptual Bernoulli TBD filter and its particle filter implementation is described in Section 4. Numerical analysis of the filter's performance using simulated data is presented in Section 5. Conclusions are made in Section 6. 2 TARGET MODEL The target state at time k is modelled by a Bernoulli random finite set (RFS) X, with X = ∅ denoting the target absent and X = {x} denoting the target with state x is presented. The posterior distributions of the random finite set X at time step k have the form f k | k ( X ) = 1 − p k | k i f X = ∅ p k | k f k | k ( x ) i f X = { x } 0 i f | X | > 1 ${f}_{k\vert k}(X)=\left\{\begin{array}{cl}\hfill 1-{p}_{k\vert k}\hfill & \mathrm{i}\mathrm{f}\ X=\varnothing \hfill \\ \hfill {p}_{k\vert k}{f}_{k\vert k}(\mathbf{x})\hfill & \mathrm{i}\mathrm{f}\ X=\left\{\mathbf{x}\right\}\hfill \\ \hfill 0\hfill & \mathrm{i}\mathrm{f}\ \vert X\vert > 1\hfill \end{array}\right.$ (1)where fk|k(x) is the target's track density and pk|k is the probability that it exists. Assume that if there is no target in the scene, there is a probability pb that a target will enter. If this occurs, assume that the track distribution of the 'birth target' is bk|k−1(x). In this case the Markov density has the form: f k | k − 1 ( X | ∅ ) = 1 − p b i f X = ∅ p b b k | k − 1 ( x ) i f X = { x } 0 i f | X | > 1 ${f}_{k\vert k-1}(X\vert \varnothing )=\left\{\begin{array}{cl}\hfill 1-{p}_{b}\hfill & \mathrm{i}\mathrm{f}\ X=\varnothing \hfill \\ \hfill {p}_{b}{b}_{k\vert k-1}(\mathbf{x})\hfill & \mathrm{i}\mathrm{f}\ X=\left\{\mathbf{x}\right\}\hfill \\ \hfill 0\hfill & \mathrm{i}\mathrm{f}\ \vert X\vert > 1\hfill \end{array}\right.$ (2)On the other hand, if a target with state x′ is in the scene at time step k − 1, assume that there is a probability ps and it will remain in the scene. If it does, its motion between time steps k − 1 and k is governed by fk|k−1(x|x′). In this case, the Markov density has the form: f k | k − 1 X | x ′ = 1 − p s i f X = ∅ p s f k | k − 1 x | x ′ i f X = { x } 0 i f | X | > 1 ${f}_{k\vert k-1}\left(X\vert \left\{{\mathbf{x}}^{\prime }\right\}\right)=\left\{\begin{array}{cl}\hfill 1-{p}_{s}\hfill & \mathrm{i}\mathrm{f}\ X=\varnothing \hfill \\ \hfill {p}_{s}{f}_{k\vert k-1}\left(\mathbf{x}\vert {\mathbf{x}}^{\prime }\right)\hfill & \mathrm{i}\mathrm{f}\ X=\left\{\mathbf{x}\right\}\hfill \\ \hfill 0\hfill & \mathrm{i}\mathrm{f}\ \vert X\vert > 1\hfill \end{array}\right.$ (3) For simplicity, a 2-D case that the radar estimates the direction of arrival (DOA) in one angular dimension is considered. The target state vector x k ≜ [ x k , x ̇ k , y k , y ̇ k ] T ${\mathbf{x}}_{k}\triangleq {[{x}_{k},{\dot{x}}_{k},{y}_{k},{\dot{y}}_{k}]}^{T}$ consists of the target position and position-rate in the x-direction, xk and x ̇ k ${\dot{x}}_{k}$ , and the y-direction, yk and y ̇ k ${\dot{y}}_{k}$ . A nearly constant velocity (NCV) model is adopted with the state transition equation given by: x k = 1 T 0 0 0 1 0 0 0 0 1 T 0 0 0 1 x k − 1 + T 2 2 0 T 0 0 T 2 2 0 T w x w y ${\mathbf{x}}_{k}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill T\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill T\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]{\mathbf{x}}_{k-1}+\left[\begin{array}{cc}\hfill \frac{{T}^{2}}{2}\hfill & \hfill 0\hfill \\ \hfill T\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{T}^{2}}{2}\hfill \\ \hfill 0\hfill & \hfill T\hfill \end{array}\right]\left[\begin{array}{c}\hfill {w}_{x}\hfill \\ \hfill {w}_{y}\hfill \end{array}\right]$ (4)where T is the time duration between the two consecutive time steps, and wx and wy are zero-mean Gaussian white noise with the same variance σ w 2 ${\sigma }_{w}^{2}$ . 3 MEASUREMENT MODEL 3.1 The monopulse returns In this paper, the Swerling II radar target model is employed, which can be easily achieved via frequency diversity among the pulses [20]. In the Swerling II model, the target amplitude has pulse-to-pulse independence and follows a Rayleigh distribution. More specifically, the target radar cross section (RCS) σ has an exponential distribution p ( σ ) = 1 σ ¯ exp − σ σ ¯ $p(\sigma )=\frac{1}{\bar{\sigma }}\mathrm{exp}\left(-\frac{\sigma }{\bar{\sigma }}\right)$ (5)where σ ¯ $\bar{\sigma }$ is the average RCS (ARCS) of the target, which is assumed to be constant and known in this study. This assumption could be relaxed by estimating the ARCS along with the target motion, see Refs. [9, 21] for unknown cases. The signal amplitude α2 ∝ σ and thus α has a Rayleigh distribution p ( α ) = α α 0 2 exp − α 2 2 α 0 2 $p(\alpha )=\frac{\alpha }{{\alpha }_{0}^{2}}\mathrm{exp}\left(-\frac{{\alpha }^{2}}{2{\alpha }_{0}^{2}}\right)$ (6)where E α 2 = 2 α 0 2 ∝ σ ¯ $E\left[{\alpha }^{2}\right]=2{\alpha }_{0}^{2}\propto \bar{\sigma }$ . Assume that the radar waveform has a rectangular envelope of width τ. Consequently, the matched filter output of the signal has a triangular envelope of width 2τ. Assume that the matched filter sampling rate is once per pulse width (i.e. the range bin size ΔR = cτ/2, here c is the speed of light), meaning that the target can affect only two contiguous range bins. Consider a set containing L range bins for each pulse, and denote the first and second range bins affected by the target by l1 and l2, respectively. The contributions of the target at range r to the two range bins are, h l 1 ( r ) = 1 − mod ( r , Δ R ) / Δ R ${h}_{{l}_{1}}(r)=1-\mathrm{mod}(r,{\Delta}R)/{\Delta}R$ and h l 2 ( r ) = mod ( r , Δ R ) / Δ R ${h}_{{l}_{2}}(r)=\mathrm{mod}(r,{\Delta}R)/{\Delta}R$ , respectively. Figure 2 shows a visualisation of our bin spreading model. Note that this is no loss of generality, and a more general pulse shape would serve only to complicate the notation. FIGURE 2Open in figure viewerPowerPoint Visualisation of one target between two range bins The matched filter outputs comprise four samples in each range bin, namely in-phrase and quadrature parts of sum and difference signals. The matched filter outputs at the two range bins l1 and l2 can be written as s I l 1 = α 1 cos ϕ + n s I l 1 s I l 2 = α 2 cos ϕ + n s I l 2 s Q l 1 = α 1 sin ϕ + n s Q l 1 s Q l 2 = α 2 sin ϕ + n s Q l 2 d I l 1 = γ α 1 cos ϕ + n d I l 1 d I l 2 = γ α 2 cos ϕ + n d I l 2 d Q l 1 = γ α 1 sin ϕ + n d Q l 1 d Q l 2 = γ α 2 sin ϕ + n d Q l 2 $\begin{array}{rl}\hfill & {s}_{I}\left({l}_{1}\right)={\alpha }_{1}\mathrm{cos}\,\phi +{n}_{sI}\left({l}_{1}\right)\hfill \\ \hfill & {s}_{I}\left({l}_{2}\right)={\alpha }_{2}\mathrm{cos}\,\phi +{n}_{sI}\left({l}_{2}\right)\hfill \\ \hfill & {s}_{Q}\left({l}_{1}\right)={\alpha }_{1}\mathrm{sin}\,\phi +{n}_{sQ}\left({l}_{1}\right)\hfill \\ \hfill & {s}_{Q}\left({l}_{2}\right)={\alpha }_{2}\mathrm{sin}\,\phi +{n}_{sQ}\left({l}_{2}\right)\hfill \\ \hfill & {d}_{I}\left({l}_{1}\right)=\gamma {\alpha }_{1}\mathrm{cos}\,\phi +{n}_{dI}\left({l}_{1}\right)\hfill \\ \hfill & {d}_{I}\left({l}_{2}\right)=\gamma {\alpha }_{2}\mathrm{cos}\,\phi +{n}_{dI}\left({l}_{2}\right)\hfill \\ \hfill & {d}_{Q}\left({l}_{1}\right)=\gamma {\alpha }_{1}\mathrm{sin}\,\phi +{n}_{dQ}\left({l}_{1}\right)\hfill \\ \hfill & {d}_{Q}\left({l}_{2}\right)=\gamma {\alpha }_{2}\mathrm{sin}\,\phi +{n}_{dQ}\left({l}_{2}\right)\hfill \end{array}$ (7)where αi (i = 1, 2) is the signal amplitude at range bin li and γ is the 'electronic angle' of the target. They are related to the radar beam pattern and target state as follows: α i = κ σ h l i ( r ) S 2 ( θ ) r 2 γ = D ( θ ) S ( θ ) r = x 2 + y 2 θ = arctan ( y / x ) − θ p $\begin{array}{l}\!\!{\alpha }_{i}=\frac{\kappa \sqrt{\sigma }{h}_{{l}_{i}}(r){S}^{2}(\theta )}{{r}^{2}}\hfill \\ \,\gamma =\frac{D(\theta )}{S(\theta )}\hfill \\ \,r=\sqrt{{x}^{2}+{y}^{2}}\hfill \\ \theta =\mathrm{arctan}(y/x)-{\theta }_{p}\hfill \end{array}$ (8)where κ is a constant of the radar determined by the radar equation [19], σ is the RCS of the target, θ is the off-boresight angle of the target, S(θ) and D(θ) are sum and difference beam patterns at θ, respectively, and θp is the direction the radar pointed at. The phase ϕ in Equation (7) is the received phased angle of the target, which is assumed to be uniformly distributed on [0, 2π) and independent with each other. The matched filter outputs at the other range bins contain noises only. The noises nsI(l), nsQ(l), ndI(l) and ndQ(l) (l = 1, …, L) are independent, zero mean, and Gaussian, with variances E n s I ( l ) 2 = E n s Q ( l ) 2 = σ s 2 E n d I ( l ) 2 = E n d Q ( l ) 2 = σ d 2 $\begin{array}{rl}\hfill & E\left[{n}_{sI}{(l)}^{2}\right]=E\left[{n}_{sQ}{(l)}^{2}\right]={\sigma }_{s}^{2}\hfill \\ \hfill & E\left[{n}_{dI}{(l)}^{2}\right]=E\left[{n}_{dQ}{(l)}^{2}\right]={\sigma }_{d}^{2}\hfill \end{array}$ (9) It can be shown that the summands in Equation (7) are Gaussian random variables. Hence the signal vector in a single pulse is given as follows: g = s I l 1 , s I l 2 , d I l 1 , d I l 2 , s Q l 1 , s Q l 2 , d Q l 1 , d Q l 2 T $\mathbf{g}={\left[{s}_{I}\left({l}_{1}\right),{s}_{I}\left({l}_{2}\right),{d}_{I}\left({l}_{1}\right),{d}_{I}\left({l}_{2}\right),{s}_{Q}\left({l}_{1}\right),{s}_{Q}\left({l}_{2}\right),{d}_{Q}\left({l}_{1}\right),{d}_{Q}\left({l}_{2}\right)\right]}^{T}$ , has a Gaussian probability density function (PDF) given the target state x. That is, p ( g | x ) = N ( g ; 0 , P ) $p(\mathbf{g}\vert \mathbf{x})=\mathcal{N}(\mathbf{g};0,\mathbf{P})$ (10)where P = E g g T = I 2 ⊗ P 1 $\mathbf{P}=E\left[\mathbf{g}{\mathbf{g}}^{T}\right]={\mathbf{I}}_{2}\otimes {\mathbf{P}}_{1}$ (11)and P 1 = α 01 2 α 01 α 02 α 01 2 γ α 01 α 02 γ α 02 2 α 01 α 02 γ α 02 2 γ α 01 2 γ 2 α 01 α 02 γ 2 α 02 2 γ 2 + diag σ s 2 , σ s 2 , σ d 2 , σ d 2 ${\mathbf{P}}_{1}=\left[\begin{array}{cccc}\hfill {\alpha }_{01}^{2}\hfill & \hfill {\alpha }_{01}{\alpha }_{02}\hfill & \hfill {\alpha }_{01}^{2}\gamma \hfill & \hfill {\alpha }_{01}{\alpha }_{02}\gamma \hfill \\ \hfill \hfill & \hfill {\alpha }_{02}^{2}\hfill & \hfill {\alpha }_{01}{\alpha }_{02}\gamma \hfill & \hfill {\alpha }_{02}^{2}\gamma \hfill \\ \hfill \hfill & \hfill \hfill & \hfill {\alpha }_{01}^{2}{\gamma }^{2}\hfill & \hfill {\alpha }_{01}{\alpha }_{02}{\gamma }^{2}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\alpha }_{02}^{2}{\gamma }^{2}\hfill \end{array}\right]+\mathrm{diag}\left({\sigma }_{s}^{2},{\sigma }_{s}^{2},{\sigma }_{d}^{2},{\sigma }_{d}^{2}\right)$ (12)Here, α0i is the Rayleigh parameter corresponding to αi. I2 denotes a 2 × 2 identity matrix and ⊗ denotes the Kronecker product. The matrix P1 is symmetric, so only the upper triangular items are given. Normally, radars send more than one pulse in each dwell to get more samples (hence higher overall SNR) of the return signal. Assume that the radar sends M pulses in each dwell (or time step). Recall that in the Swerling II model, there is pulse-to-pulse fluctuation, that is, the returns from different pulses are independent and identically distributed (IID). Let us denote G = g m m = 1 M $\mathbf{G}={\left\{{\mathbf{g}}_{m}\right\}}_{m=1}^{M}$ , where gm is the signal vector of the mth pulse. Then we have p ( G | x ) = ∏ m = 1 M p g m | x = ∏ m = 1 M N g m ; 0 , P $p(\mathbf{G}\vert \mathbf{x})=\prod\limits _{m=1}^{M}p\left({\mathbf{g}}_{m}\vert \mathbf{x}\right)=\prod\limits _{m=1}^{M}\mathcal{N}\left({\mathbf{g}}_{m};0,\mathbf{P}\right)$ (13) Denote the observation of the mth pulse as z m = z m , l l = 1 L ${\mathbf{z}}_{m}={\left\{{\mathbf{z}}_{m,l}\right\}}_{l=1}^{L}$ , where z m , l = s I ( l ) , d I ( l ) , s Q ( l ) , d Q ( l ) T ${\mathbf{z}}_{m,l}={\left[{s}_{I}(l),{d}_{I}(l),{s}_{Q}(l),{d}_{Q}(l)\right]}^{T}$ is the return of the l-th range bin. If the target is present, the two range bins affected by the target (l = l1 or l = l2) contain signal plus noise as shown in Equation (7) and only noises otherwise. If the target is not present, all the range bins contain noises only. All the observations in the M pulses constitute the measurement at time k for the TBD filter: z k = z m m = 1 M ${\mathbf{z}}_{k}={\left\{{\mathbf{z}}_{m}\right\}}_{m=1}^{M}$ (14) 3.2 The likelihood ratio Since the range bins not affected by the target have the same PDF under the two hypotheses, the target is either present (X = {x}) or not present (X = ∅), and the likelihood ratio l r z k | x ≜ p z k | { x } p z k | ∅ ${l}_{r}\left({\mathbf{z}}_{k}\vert \mathbf{x}\right)\triangleq \frac{p\left({\mathbf{z}}_{k}\vert \left\{\mathbf{x}\right\}\right)}{p\left({\mathbf{z}}_{k}\vert \varnothing \right)}$ is determined only by the two bins (l1 and l2) affected by x, that is l r z k | x = p ( G | { x } ) p ( G | ∅ ) = ∏ m = 1 M N g m ; 0 , P ∏ m = 1 M N g m ; 0 , P ′ = ∏ m = 1 M 1 | 2 π P | 1 / 2 exp − 0.5 g m T P − 1 g m ∏ m = 1 M 1 | 2 π P ′ | 1 / 2 exp − 0.5 g m T P ′ − 1 g m = | P ′ | M / 2 | P | M / 2 exp − 0.5 ∑ m = 1 M g m T P − 1 − P ′ − 1 g m $\begin{array}{l}{l}_{r}\left({\mathbf{z}}_{k}\vert \mathbf{x}\right)=\frac{p(\mathbf{G}\vert \left\{\mathbf{x}\right\})}{p(\mathbf{G}\vert \varnothing )}\hfill \\ \qquad \quad=\,\frac{{\prod }_{m=1}^{M}\mathcal{N}\left({\mathbf{g}}_{m};0,\mathbf{P}\right)}{{\prod }_{m=1}^{M}\mathcal{N}\left({\mathbf{g}}_{m};0,{\mathbf{P}}^{\prime }\right)}\hfill \\ \qquad \quad=\,\frac{{\prod }_{m=1}^{M}\frac{1}{\vert 2\pi \mathbf{P}{\vert }^{1/2}}\mathrm{exp}\left(-0.5{\mathbf{g}}_{m}^{T}{\mathbf{P}}^{-1}{\mathbf{g}}_{m}\right)}{{\prod }_{m=1}^{M}\frac{1}{\vert 2\pi {\mathbf{P}}^{\prime }{\vert }^{1/2}}\mathrm{exp}\left(-0.5{\mathbf{g}}_{m}^{T}{\mathbf{P}}^{\prime -1}{\mathbf{g}}_{m}\right)}\hfill \\ \qquad =\,\frac{\vert {\mathbf{P}}^{\prime }{\vert }^{M/2}}{\vert \mathbf{P}{\vert }^{M/2}}\mathrm{exp}\left(-0.5\sum\limits _{m=1}^{M}{\mathbf{g}}_{m}^{T}\left({\mathbf{P}}^{-1}-{\mathbf{P}}^{\prime -1}\right){\mathbf{g}}_{m}\right)\hfill \end{array}$ (15)where the covariance matrix P for the case X = {x} is as shown in Equation (11), and the covariance matrix P′ for the case X = ∅ is P ′ = diag σ s 2 , σ s 2 , σ d 2 , σ d 2 , σ s 2 , σ s 2 , σ d 2 , σ d 2 ${\mathbf{P}}^{\prime }=\mathrm{diag}\left({\sigma }_{s}^{2},{\sigma }_{s}^{2},{\sigma }_{d}^{2},{\sigma }_{d}^{2},{\sigma }_{s}^{2},{\sigma }_{s}^{2},{\sigma }_{d}^{2},{\sigma }_{d}^{2}\right)$ (16) 3.3 The sum and difference beam patterns The beam patterns of the monopulse radar are often well approximated to within 0.1 dB of the true pattern out to the peak of the first sum beam sidelobe [2]. Taking phase monopulse for example, the sum and difference beam patterns can be approximated by the following expressions [22]: S ( θ ) = cos ( θ ) log ( 1 / 2 ) log ( cos ( β / 2 ) ) cos π C λ sin ( θ ) $S(\theta )=\mathrm{cos}\,{(\theta )}^{\frac{\mathrm{log}(1/\sqrt{2})}{\mathrm{log}(\mathrm{cos}(\beta /2))}}\mathrm{cos}\left(\frac{\pi C}{\lambda }\mathrm{sin}(\theta )\right)$ (17) D ( θ ) = cos ( θ ) log ( 1 / 2 ) log ( cos ( β / 2 ) ) sin π C λ sin ( θ ) $D(\theta )=\mathrm{cos}\,{(\theta )}^{\frac{\mathrm{log}(1/\sqrt{2})}{\mathrm{log}(\mathrm{cos}(\beta /2))}}\mathrm{sin}\left(\frac{\pi C}{\lambda }\mathrm{sin}(\theta )\right)$ (18)where β is the half-power beamwidth of each antenna, C is the distance between the two antennas, and λ is the wavelength. These expressions are used for all numerical analysis throughout this paper, with β = 5°, C = 7.62 cm and λ = 8 mm. The corresponding sum-channel beamwidth is BW = 2.4°. Note that this approximation is only for simulation. For real monopulse systems, the true sum and difference beam patterns can be measured through experiments. 3.4 The case for a more general pulse shape For simplicity, a simple rectangular pulse shape, as described in Section 3.1, is employed in this paper. Here the case for a more general pulse shape is discussed. Assume that the transmitted signal has pulse shape ξ(t), and the impulse response11 The matched filter response should be h(t) = ξ(−t)* if we are not interested in Doppler information, where the asterisk superscript means complex conjugation. of the matched filter is h(t), then the matched filter output of the signal is q(t) = ξ(t) ⋆ h(t), where ⋆ denotes convolution. Accordingly, the contribution of the target at range r to range bin l is hl(r) = q(2(lΔR − r)/c). Note that the number of bins affected by the target may be more than two. The corresponding measurement model for this case can be easily derived following the steps in Section 3.1. 4 THE BERNOULLI TRACK-BEFORE-DETECT FILTER 4.1 The analytical expressions The goal of the track-before-detect filter is to detect the existence of a target, and, in the case the target is present, to estimate its state xk. In the Bayesian framework, this means that the goal is to estimate two posteriors: (1) the posterior probability of target existence pk|k = P{|X| = 1|z1:k} and (2) the posterior spatial PDF of the target, fk|k(x) = p(x|z1:k). The detection of target existence is then done using pk|k as the test statistic, and, if the target is detected, the kinematic state is estimated using the fk|k(x). The optimal Bayes filter in this context is known as the Bernoulli (or JoTT) filter. This filter has been derived using the tools of the finite set statistics (FISST), under the assumption that the target state is a Bernoulli RFS. The standard Bernoulli filter [13, 23, 24] is developed with thresholded measurements. The measurement model in the TBD context is different from the one in the standard Bernoulli filter. Because there is no thresholding applied, there is only one measurement (rather than a finite set that can be empty or have multiple elements) in each time step: the observation vector comprised of the signals in all the resolution bins (or pixels); and thus we have Zk ≡ zk. The measurement model in the TBD context is called the intensity measurement model in Ref. [14]. The update equations for the intensity measurement model were derived using FISST in Refs. [14, 25], while the predict equations are the same as the standard Bernoulli filter. Here, the ultimate expressions are given as follows: Predictor: Given the posterior density fk−1|k−1(X|z1:k−1) at k − 1, which is completely characterised by pk−1|k−1 and fk−1|k−1(x), then pk|k−1 can be predicted as p k | k − 1 = p b 1 − p k − 1 | k − 1 + p s p k − 1 | k − 1 ${p}_{k\vert k-1}={p}_{b}\left(1-{p}_{k-1\vert k-1}\right)+{p}_{s}{p}_{k-1\vert k-1}$ (19)and fk|k−1(x) takes the form f k | k − 1 ( x ) c = p b b k | k − 1 ( x ) 1 − p k − 1 | k − 1 p k | k − 1 + p s p k − 1 | k − 1 ∫ f k | k − 1 x | x ′ f k − 1 | k − 1 x ′ d x ′ p k | k − 1 $\begin{array}{rl}\hfill {f}_{k\vert k-1}(\mathbf{x})\mathrm{c}=& \frac{{p}_{b}{b}_{k\vert k-1}(\mathbf{x})\left(1-{p}_{k-1\vert k-1}\right)}{{p}_{k\vert k-1}}\hfill \\ \hfill & +\frac{{p}_{s}{p}_{k-1\vert k-1}\int {f}_{k\vert k-1}\left(\mathbf{x}\vert {\mathbf{x}}^{\prime }\right){f}_{k-1\vert k-1}\left({\mathbf{x}}^{\prime }\right)d{\mathbf{x}}^{\prime }}{{p}_{k\vert k-1}}\hfill \end{array}$ (20) Corrector: Given the predicted density fk|k−1(X|z1:k−1), which is specified by pk|k−1 and fk|k−1(x), and the measurement vector zk, the correction function for pk|k is p k | k = p k | k − 1 ∫ l r z k | x f k | k − 1 ( x ) d x 1 − p k | k − 1 + p k | k − 1 ∫ l r z k | x f k | k − 1 ( x ) d x . ${p}_{k\vert k}=\frac{{p}_{k\vert k-1}\int {l}_{r}\left({\mathbf{z}}_{k}\vert \mathbf{x}\right){f}_{k\vert k-1}(\mathbf{x})d\mathbf{x}}{1-{p}_{k\vert k-1}+{p}_{k\vert k-1}\int {l}_{r}\left({\mathbf{z}}_{k}\vert \mathbf{x}\right){f}_{k\vert k-1}(\mathbf{x})d\mathbf{x}}.$ (21)and the correction function for fk|k(x) is f k | k ( x ) = l r z k | x f k | k − 1 ( x ) ∫ l r z k | x f k | k − 1 ( x ) d x . ${f}_{k\vert k}(\mathbf{x})=\frac{{l}_{r}\left({\mathbf{z}}_{k}\vert \mathbf{x}\right){f}_{k\vert k-1}(\mathbf{x})}{\int {l}_{r}\left({\mathbf{z}}_{k}\vert \mathbf{x}\right){f}_{k\vert k-1}(\mathbf{x})d\mathbf{x}}.$ (22)where the likelihood ratio l r z k | x ≜ f z k | x f z k | ∅ ${l}_{r}\left({\mathbf{z}}_{k}\vert \mathbf{x}\right)\triangleq \frac{f\left({\mathbf{z}}_{k}\vert \mathbf{x}\right)}{f\left({\mathbf{z}}_{k}\vert \varnothing \right)}$ . Note that in the correction step, only the likelihood ratio lr(zk|x) is needed. Finally, Equations (19) and (20) together with Equations (21) and (22) complete the recursion for the analytic TBD filter. 4.2 The particle filter implementation Approximate fk|k(x) by f k | k ( x ) ≈ ∑ n = 1 N w k ( n ) δ x k ( n ) ( x ) ${f}_{k\vert k}(\mathbf{x})\approx \sum\limits _{n=1}^{N}{w}_{k}^{(n)}{\delta }_{{\mathbf{x}}_{k}^{(n)}}(\mathbf{x})$ (23)where x k ( n ) ${\mathbf{x}}_{k}^{(n)}$ and w k ( n ) ${w}_{k}^{(n)}$ are the state and weight of the nth particle, respectively, with the weights normalised such that ∑ n = 1 N w k ( n ) = 1 ${\sum }_{n=1}^{N}{w}_{k}^{(n)}=1$ , and δb(x) is the Dirac delta function concentrated at point b. Predictor: Given pk−1|k−1 and particles w k − 1 ( n ) , x k − 1 ( n ) n = 1 N ${\left\{{w}_{k-1}^{(n)},{