*3.2. Problem Formulation*

This part our main task is to formulate the optimization problem, whose objective is minimizing the total dwell time of the radar network with the tracking performance meeting a predefined threshold.

In Section 3.1, we derived the BCRLB of the target tracking error, which can be used to measure the target tracking accuracy. Moreover, given the updated BIM **J** , **X***q k*−1 at the time index *k* − 1 and the radar radiation parameters, we can now determine the predictive BCRLB of the target *q* at time index *k* according to the formula (37):

$$\mathbf{C}\_{\text{BCRLB},k|k-1}^{q} = \left[ \left( \mathbf{Q}^{q} + \mathbf{F} \right)^{-1} \left( \mathbf{X}\_{k-1}^{q} \right) \mathbf{F}^{\text{T}} \right]^{-1} + \sum\_{i=1}^{N} u\_{i,k}^{q} \left( \mathbf{H}\_{i,k|k-1}^{q} \right)^{\text{T}} \left( \mathbf{G}\_{i,k|k-1}^{q} \right)^{-1} \mathbf{H}\_{i,k|k-1}^{q} \right]^{-1} \tag{38}$$

where **G***<sup>q</sup> <sup>i</sup>*,*k*|*k*−<sup>1</sup> and **<sup>H</sup>***<sup>q</sup> <sup>i</sup>*,*k*|*k*−<sup>1</sup> are the predicted values of **<sup>G</sup>***<sup>q</sup> <sup>i</sup>*,*<sup>k</sup>* and **<sup>H</sup>***<sup>q</sup> i*,*k* , respectively. The diagonal element of **C***<sup>q</sup>* BCRLB,*k*|*k*−<sup>1</sup> is the lower bound of the estimated MMSE of the target state estimation, which can be extracted as a measurement metric of target tracking accuracy:

$$F\_{k|k-1}^{q} = \sqrt{\mathbf{C}\_{k|k-1}^{q}(1,1) + \mathbf{C}\_{k|k-1}^{q}(3,3)}\tag{39}$$

*m*=1

where **C***<sup>q</sup> k*|*k*−1 (1, 1) and **C***<sup>q</sup> k*|*k*−1 (3, 3) are the first variable and the third variable on the diagonal **C***q k*|*k*−1 , respectively.

Since the tracking accuracy meets a predefined threshold *F*max, the constraint on the accuracy is:

$$F\_{k|k-1}^{q} \le F\_{\text{max}}, \qquad \forall q = 1, 2, \dots, Q \tag{40}$$

Then, with respect to the total bandwidth budget, if *u q <sup>i</sup>*,*<sup>k</sup>* = 1, the bandwidth of the *<sup>i</sup>*th radar's illumination on the *q*th target at time index *k* should satisfy an upper bound βmax and a lower bound <sup>β</sup>min: <sup>⎧</sup>

$$\begin{cases} \beta\_{i,\eta,k} = 0, & \boldsymbol{u}\_{i,k}^{\eta} = 0\\ \beta\_{\text{min}} \le \beta\_{i,\eta,k} \le \beta\_{\text{max}}, & \boldsymbol{u}\_{i,k}^{\eta} = 1 \end{cases} \tag{41}$$

Similarly, the dwell time constraints can be denoted as:

$$\begin{cases} \begin{array}{cc} T^{\mathrm{d}}\_{i,\eta,k} = 0, & \mathcal{u}^{\eta}\_{i,k} = 0\\ T^{\mathrm{d}}\_{\mathrm{min}} \le T^{\mathrm{d}}\_{i,\eta,k} \le T^{\mathrm{d}}\_{\mathrm{max}\prime} & \mathcal{u}^{\tilde{\eta}}\_{i,k} = 1 \end{array} \end{cases} \tag{42}$$

We define the data processing rate of fusion center as ε, and the total number of samples in the radar network should satisfy the following constraints:

$$\sum\_{q=1}^{Q} N\_k = \frac{1}{\varepsilon} \tag{43}$$

By fusing (40), (41), (42) and (43) together, we can formulate the optimization problem for the joint dwell time and bandwidth optimization strategy:

$$\begin{array}{ll} \min & \sum\_{m,q,k} \sum\_{i,j,q=1}^{Q} \sum\_{i=1}^{T} T\_{m,q,k}^{d} \\ & T\_{m,q,k}^{d} \le T\_{\max}, & \forall q = 1,2,\dots,Q \\ & \begin{cases} F\_{ik-1}^{d} \le F\_{\max}, & \forall q = 1,2,\dots,Q \\ \beta\_{i,q,k} = 0, & u\_{i,k}^{q} = 0 \\ \beta\_{\min} \le \beta\_{i,q,k} \le \beta\_{\max}, & u\_{i,k}^{q} = 1 \\ T\_{i,q,k}^{d} = 0, & u\_{i,k}^{q} = 0 \\ T\_{\min}^{d} \le T\_{i,q,k}^{d} \le T\_{\max}^{d}, & u\_{i,k}^{q} = 1 \\ \sum\_{q=1}^{Q} N\_{k} = \frac{1}{\tau'} \sum\_{q=1}^{Q} u\_{i,k}^{q} \le 1, \sum\_{m=1}^{N} u\_{i,k}^{q} = M \end{cases} \end{array} \tag{44}$$

where \$ *Q q*=1 *u q <sup>i</sup>*,*<sup>k</sup>* ≤ 1 represents that a single radar tracks at most one target in a revisit period. The term \$*N q*

*u <sup>i</sup>*,*<sup>k</sup>* = *<sup>M</sup>* represents that each radar is tracked by *<sup>M</sup>* radars at time index *<sup>k</sup>*.

Since *u q <sup>i</sup>*,*<sup>k</sup>* ∈ {0, 1} is a binary variable, the optimization problem described in (44) is a non-convex problem with three parameters: radar selection, dwell time and the transmitted signal's bandwidth. However, for a given **u***<sup>q</sup> k* , assuming that the radar *i* is assigned to *q*th target, the unique radar node selection scheme for the *q*th can be determined. Furthermore, in order to ensure that all targets have enough information, assuming that each target has the same amount of samples which needs to be

sent to the fusion center, then the optimization problem can be converted to the following formula, which only has the variables *T*<sup>d</sup> *<sup>m</sup>*,*q*,*<sup>k</sup>* and <sup>β</sup>*m*,*q*,*<sup>k</sup>* (<sup>1</sup> <sup>≤</sup> *<sup>m</sup>* <sup>≤</sup> *<sup>M</sup>*):

$$\begin{array}{c} \min\_{\begin{subarray}{c}T^{\mathsf{d}}\_{m,\rho},\beta\_{i,k,q},m=1\\m\end{subarray}} \sum\_{m=1}^{M} T^{\mathsf{d}}\_{m,q,k} \\ \text{s.t.} \begin{cases} F^{\mathsf{d}}\_{k|k-1} \le F\_{\max} \\ \sum\_{m=1}^{M} \beta\_{m,q,k} = \frac{\mathcal{c}}{Q\rho c\mathcal{V}} = \beta\_{\text{total}} \\ \beta\_{\text{min}} \le \beta\_{i,q,k} \le \beta\_{\text{max}} \\ T^{\text{d}}\_{\text{min}} \le T^{\text{d}}\_{i,q,k} \le T^{\text{d}}\_{\text{max}} \end{cases} \end{array} \tag{45}$$

where βtotal is the total bandwidth of the transmitted waveform of all radars that are assigned to the same target.
