*3.1. Performance Metric*

The BCRLB provides a lower bound on the mean square error (MSE) of parameter unbiased estimation, and compares to the posterior Cramer–Rao lower bound (PCRLB) [27,28]. In this paper, BCRLB is derived and used as an optimization criterion for the joint dwell time and bandwidth optimization strategy. At time index *k*, we use the observation vector **Z***<sup>q</sup> <sup>k</sup>* to estimate the state of *q*th target, which can be defined as **^** *q* , - , then the MSE of **^** *q* , -

**X** *k* **Z***q k* **X** *k* **Z***q k* satisfies the following equation:

$$\mathbb{E}\left\{ \left( \hat{\mathbf{X}}\_{k}^{q} \{ \mathbf{Z}\_{k}^{q} \} - \mathbf{X}\_{k}^{q} \right) - \left( \hat{\mathbf{X}}\_{k}^{q} \{ \mathbf{Z}\_{k}^{q} \} - \mathbf{X}\_{k}^{q} \right)^{T} \right\} = \mathbf{C}\_{k}^{q} \geq \mathbf{J}^{-1} \left( \mathbf{X}\_{k}^{q} \right) \tag{18}$$

where <sup>E</sup>{•} denotes mathematical expectation, **<sup>C</sup>***<sup>q</sup> <sup>k</sup>* is the *q*th target's BCRLB at time index *k*, and **J** , **X***q k* is the Bayesian information matrix (BIM), which can be written as:

$$\mathbf{J}\left(\mathbf{X}\_{k}^{q}\right) = -\mathbb{E}\_{\mathbf{X}\_{k}^{q}, \mathbf{Z}\_{k}^{q}} \left\{ \Delta\_{\mathbf{X}\_{k}^{q}}^{\mathcal{I}} \log p\left(\mathbf{Z}\_{k'}^{q}, \mathbf{X}\_{k}^{q}\right) \right\} \tag{19}$$

where Δ **X***q k* **X***q k* = ∇**X***<sup>q</sup> k* ∇T **X***q k* , here ∇**X***<sup>q</sup> k* denotes the first-order partial derivative vectors. In (19),

$$p\left(\mathbf{Z}\_{k'}^{q}, \mathbf{X}\_{k}^{q}\right) = p\left(\mathbf{X}\_{k}^{q}\right)p\left(\mathbf{Z}\_{k}^{q}|\mathbf{X}\_{k}^{q}\right) \tag{20}$$

is the joint probability density function (PDF) [11].

The BIM **J** , **X***q k* can be expressed as the sum of two matrices:

$$\mathbf{J}\left(\mathbf{X}\_{k}^{q}\right) = \mathbf{J}\_{\mathrm{P}}\left(\mathbf{X}\_{k}^{q}\right) + \mathbf{J}\_{\mathrm{D}}\left(\mathbf{X}\_{k}^{q}\right) \tag{21}$$

where **J**<sup>P</sup> , **X***q k* and **J**<sup>D</sup> , **X***q k* are the Fisher information matrix (FIM) of the priori information and the data, respectively.

$$\mathbb{J}\_{\mathbf{P}}\left(\mathbf{X}\_{k}^{q}\right) = \mathbb{E}\_{\mathbf{X}\_{k}^{q}}\left\{-\boldsymbol{\Delta}\_{\mathbf{X}\_{k}^{q}}^{\mathbf{x}\_{k}^{q}}\log p\left(\mathbf{X}\_{k}^{q}\right)\right\}\tag{22}$$

$$\mathbf{J}\_{\rm D} \{ \mathbf{X}\_k^q \} = \mathbb{E}\_{\mathbf{X}\_k^q, \mathbf{Z}\_k^q} \left\{ -\boldsymbol{\Delta}\_{\mathbf{X}\_k^q}^{\mathbf{x}\_k^q} \log p \left( \mathbf{Z}\_k^q | \mathbf{X}\_k^q \right) \right\} \tag{23}$$

Combined with the system model in Section 2, the *q*th target is tracked by a fixed number of radars at the time index *k*. Since the radar independently observes the target at the same moment, the BIM of the target state can be simply expressed as:

$$\mathbf{J}\{\mathbf{X}\_k^q\} = \mathbf{J}\_\mathbf{P}\{\mathbf{X}\_k^q\} + \sum\_{i=1}^N \boldsymbol{\mu}\_{i,k}^q \mathbf{J}\_\mathbf{D}^{(i)}\{\mathbf{X}\_k^q\} \tag{24}$$

where **J** (*i*) D , **X***q k* is the FIM of the *i*th radar's measurement on *q*th target. In (24), the term **J**<sup>P</sup> , **X***q k* can be calculated iteratively through the following formula:

$$\mathbf{J}\_{\mathbf{P}}\left(\mathbf{X}\_{k}^{q}\right) = \mathbf{D}\_{k-1}^{22} - \mathbf{D}\_{k-1}^{21}\left(\mathbf{J}\left(\mathbf{X}\_{k-1}^{q}\right) + \mathbf{D}\_{k-1}^{11}\right)^{-1}\mathbf{D}\_{k-1}^{12} \tag{25}$$

where,

$$\mathbf{D}\_{k-1}^{11} = \mathbb{E}\_{\mathbf{X}\_{k-1}^{\mathcal{J}} \mathbf{X}\_k^{\mathcal{J}}} \left\{ -\boldsymbol{\Delta}\_{\mathbf{X}\_{k-1}^{\mathcal{J}}}^{\mathcal{J}} \log p(\mathbf{X}\_k^{\mathcal{J}} | \mathbf{X}\_{k-1}^{\mathcal{J}}) \right\} \tag{26}$$

$$\mathbf{D}\_{k-1}^{12} = \mathbf{D}\_{k-1}^{21} = \mathbb{E}\_{\mathbf{X}\_{k-1}^{q}\mathbf{X}\_{k}^{q}} \left\{-\boldsymbol{\Delta}\_{\mathbf{X}\_{k}^{q}}^{\mathbf{x}\_{k-1}^{q}} \log p(\mathbf{X}\_{k}^{q}|\mathbf{X}\_{k-1}^{q})\right\} \tag{27}$$

$$\mathbf{D}\_{k-1}^{22} = \mathbb{E}\_{\mathbf{X}\_{k-1}^{q}\mathbf{X}\_{k}^{q}} \left\{-\boldsymbol{\Delta}\_{\mathbf{X}\_{k}^{q}}^{\mathbf{x}\_{k}^{q}} \log p\left(\mathbf{X}\_{k}^{q}|\mathbf{X}\_{k-1}^{q}\right) \right\} \tag{28}$$

Combined with the target dynamic model in Section 2.1, **J**<sup>P</sup> , **X***q k* can be written as:

$$\mathbf{J}\_{\mathbf{P}} \begin{pmatrix} \mathbf{X}\_{k}^{q} \end{pmatrix} = \left[ \mathbf{Q}^{q} + \mathbf{F} \mathbf{J}^{-1} \begin{pmatrix} \mathbf{X}\_{k-1}^{q} \end{pmatrix} \mathbf{F}^{\mathrm{T}} \right]^{-1} \tag{29}$$

For the *i*th radar, the FIM of the data can be given by:

$$\mathbf{J}\_{\rm D}^{(i)}\left(\mathbf{X}\_{k}^{q}\right) = \mathbb{E}\_{\mathbf{X}\_{k}^{q}\mathbf{Z}\_{i,k}^{q}}\left\{-\boldsymbol{\Delta}\_{\mathbf{X}\_{k}^{q}}^{\mathbf{x}\_{k}^{q}}\log p\left(\mathbf{Z}\_{i,k}^{q}|\mathbf{X}\_{k}^{q}\right)\right\} = \mathbb{E}\_{\mathbf{X}\_{k}^{q}}\left\{\mathbb{E}\_{\mathbf{Z}\_{i,k}^{q}|\mathbf{X}\_{k}^{q}}\left\{-\boldsymbol{\Delta}\_{\mathbf{X}\_{k}^{q}}^{\mathbf{x}\_{k}^{q}}\log p\left(\mathbf{Z}\_{i,k}^{q}|\mathbf{X}\_{k}^{q}\right)\right\}\right\}\tag{30}$$

According to [15], we can get:

$$\mathbf{J}\_{\rm D}^{(i)}\left(\mathbf{X}\_{k}^{q}\right) = \mathbb{E}\_{\mathbf{X}\_{k}^{q}}\left\{ \left(\mathbf{H}\_{i,k}^{q}\right)^{\rm T} \left(\mathbf{G}\_{i,k}^{q}\right)^{-1} \mathbf{H}\_{i,k}^{q} \right\} \tag{31}$$

where **H***<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* is the Jacobi matrix of h*<sup>i</sup>* , **X***q k* and can be expressed as:

$$\mathbf{H}\_{i,k}^{q} = \left[\nabla\_{\mathbf{X}\_{k}^{q}} \left(\mathbf{h}\_{i} \Big(\mathbf{X}\_{k}^{q}\Big)\right)^{\mathrm{T}}\right]^{\mathrm{T}} = \left[\nabla\_{\mathbf{X}\_{k}^{q}} \mathbf{R}\_{i,k'}^{q} \,\nabla\_{\mathbf{X}\_{k}^{q}} \theta\_{i,k}^{q}\right] \tag{32}$$

where

$$\nabla\_{\mathbf{X}\_{k}^{q}} \mathbf{R}\_{i,k}^{q} = \left[ \nabla\_{\mathbf{x}\_{k}^{q}} \mathbf{R}\_{i,k'}^{q} \nabla\_{\dot{\mathbf{x}}\_{k}^{q}} \mathbf{R}\_{i,k'}^{q} \nabla\_{\dot{\mathbf{y}}\_{k}^{q}} \mathbf{R}\_{i,k'}^{q} \nabla\_{\dot{\mathbf{y}}\_{k}^{q}} \mathbf{R}\_{i,k}^{q} \right]^{\mathbf{T}} \tag{33}$$

$$
\nabla\_{\mathbf{X}\_k^q} \theta\_{i,k}^q = \begin{bmatrix}
\nabla\_{\mathbf{x}\_k^q} \theta\_{i,k'}^q \nabla\_{\dot{\mathbf{x}}\_k^q} \theta\_{i,k'}^q \nabla\_{\dot{\mathbf{y}}\_k^q} \theta\_{i,k'}^q \nabla\_{\dot{\mathbf{y}}\_k^q} \theta\_{i,k}^q \\
\end{bmatrix}^T \tag{34}
$$

are the first-order partial derivatives of the target distance and azimuth to the position and velocity, respectively.

Substitute (29) and (31) into (24), we can get the BIM of the target state **X***<sup>q</sup> k* :

$$\mathbf{J}\left(\mathbf{X}\_{k}^{q}\right) = \left[\mathbf{Q}^{q} + \mathbf{F}\right]^{-1} \left(\mathbf{X}\_{k-1}^{q}\right) \mathbf{F}^{T}\right]^{-1} + \sum\_{i=1}^{N} u\_{i,k}^{q} \mathbb{E}\_{\mathbf{X}\_{k}^{q}} \left\{ \left(\mathbf{H}\_{i,k}^{q}\right)^{T} \left(\mathbf{G}\_{i,k}^{q}\right)^{-1} \mathbf{H}\_{i,k}^{q} \right\} \tag{35}$$

The first prior information FIM of **J** , **X***q k* is only related to the BIM of the target state at the time index *<sup>k</sup>* <sup>−</sup> 1 and the target dynamic model in Section 2.1. According to (7) and (8), the **<sup>G</sup>***<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* in the second item is related to the *i*th radar's bandwidth on the *q*th target and the radar echo SNR at time index *k*. Meanwhile, SNR is a function of the dwell time. As a result, **J** , **X***q k* is related to the bandwidth and the dwell time at time index *k*, thus laying the foundation for the joint dwell time and the bandwidth optimization strategy. Furthermore, in order to satisfy the demand of real-time, we can approximate (35) as:

$$\mathbf{J}\left(\mathbf{X}\_{k}^{q}\right) = \left[\mathbf{Q}^{q} + \mathbf{F}\right]^{-1}\left(\mathbf{X}\_{k-1}^{q}\right)\mathbf{F}^{\mathrm{T}}\right]^{-1} + \sum\_{i=1}^{N} u\_{i,k}^{q} \left(\mathbf{H}\_{i,k}^{q}\right)^{\mathrm{T}} \left(\mathbf{G}\_{i,k}^{q}\right)^{-1} \mathbf{H}\_{i,k}^{q} \tag{36}$$

According to (18), the corresponding BCRLB matrix of the target state estimation error can be calculated as:

$$\mathbf{C}\_{\text{BCRLB},k}^{q} = \mathbf{J}^{-1} \{ \mathbf{X}\_{k}^{q} \} = \left[ \left[ \mathbf{Q}^{q} + \mathbf{F} \mathbf{J}^{-1} \{ \mathbf{X}\_{k-1}^{q} \} \mathbf{F}^{\text{T}} \right]^{-1} + \sum\_{i=1}^{N} u\_{i,k}^{q} \left( \mathbf{H}\_{i,k}^{q} \right)^{\text{T}} \left( \mathbf{G}\_{i,k}^{q} \right)^{-1} \mathbf{H}\_{i,k}^{q} \right]^{-1} \tag{37}$$
