【多视角】Multi-Point Integrated Sensing and Communication: Fusion Model and Functionality Selection

• Use K dual-functional radars (DFR) to sense one target object and communicate with one user.

• A functionality selection module, which is represented by a binary variable \mathbf{x}= [x_1,\ldots,x_K]^T\in\{0,1\}^K,is adopted to determine the working states of all the DFRs. Specifically, x_i=1, \forall i\in\{1,\ldots,K\}, represents that the i-DFR operates in the sensing mode and x_i=0 denotes that the i-DFR operates in the communication mode.

• we aim to to use the target information of sensing signal for classification problem. Effective DFRs are all the sensing DFRs, i.e. x_i=1. Meanwhile, SINR of sensing signal of effective DFRs should be greater than a threshold \gamma. The set of effective DFRs is thus \mathcal{E}=\{i:x_i=1, SINR_i^s\geq\gamma\} whose cardinality is |\mathcal{E}|.

• we aim to fuse the outputs of the effective DFRs \mathcal{E} via voting.

• Fusion accuracy with the voting threshold n is \Theta(\mathcal{E}|n)=\frac12\sum_{l=0}^{n-1}\sum_{F\in\mathcal{F}_l}\prod_{i\in F}P_i\prod_{i\in F^c}(1-P_i)+\frac12\sum_{l=0}^{|\mathcal{E}|-n}\sum_{F\in\mathcal{F}_l}\prod_{i\in F}Q_i\prod_{i\in F^c}(1-Q_i), where P_i and Q_i are the false negative (false alarm) and false positive (missing alarm) rates at the i-th DFR, respectively, are estimated from the experimental data; note that different DFRs may have different (P_i,Q_i) under the same SINR due to different observation angles; (2) \mathcal{F}_l contains all the {ll}\mathrm{subsets~with~}l\mathrm{unique~DFRs~from~}\mathcal{E}\mathrm{and~its~cardinality~is}\\ |\mathcal{F}_l|=(\frac{|\mathcal E|}l)=\frac{|\mathcal E|!}{l!(|\mathcal E|-l)!};(3)\text{ For any }F\in\mathcal{F}_l,\text{ its comple-} ment is denoted as F^c=\mathcal{E}\backslash F. It can be seen that the fusion accuracy under optimal voting is \max_n\Theta(\mathcal{E}|n).
  I can not understand the meaning of the Fusion accuracy with the voting threshold

• x_i=1 denotes that the i-DFR operates in the communication mode. Hence, the communication spectral efficiency is R(\mathbf{x},\{\mathbf{w}_i\})=\log_2\biggl(1+\frac{|\sum_{i=1}^K(1-x_i)\mathbf{f}_i^H\mathbf{w}_i|^2}{\sigma^2+\sum_j|x_j\mathbf{f}_j^H\mathbf{w}_j|^2}\biggr), where \mathbf{f}_j\in\mathbb{C}^M is the channel from the j-th DFR to the communication receiver.

• The MPISAC system aims to maximize the sensing and communication performance under the mode selection, transmit power, and effective sensing constraints, which results in the following multi-objective optimization problem.
  \begin{aligned} \mathcal{P}0:\max_{\mathbf{x},\{\mathbf{w}_{i}\},\mathcal{E}}& \left\{\max_{n}\Theta(\mathcal{E}|n),R(\mathbf{x},\{\mathbf{w}_{i}\})\right\} \\ \mathrm{s.t.}& x_i\in\{0,1\}, \forall i,\quad(9\text{а}) \\ &\sum_{i=1}^K\|\mathbf{w}_i\|^2\leq P_{\mathrm{sum}}, \|\mathbf{w}_i\|^2\leq P_T, \forall i, (9\mathbf{b}) \\ &\mathcal{E}=\{i{:}x_{i}=1,\mathrm{~SINR}_{i}^{s}\geq\gamma\}.\quad(9\mathrm{c}) \end{aligned}

• The optimal x* and \mathcal{E}^* satisfy \mathcal{E}^*=\{i:x_i^*=1\}. Therefore, replacing \mathcal{E} with a new variable \mathcal{S}=\{i:x_i= 1\} (with its cardinality |S| being the number of sensing DFRs) in \mathcal{P} 0 would not change the problem solution. why \mathcal{P} 0

• To decouple the nonlinear between \mathbf{x} and \mathbf{w}_i , we use zero-forcing (ZF) beamforming \mathbf{w}_i=\sqrt{p_i}\operatorname{e}^{\mathrm{j}\phi_i}\mathbf{w}_i^{\mathrm{ZF}} as
  \begin{aligned} \mathcal{P}2:\max_{\mathbf{x},\{p_i\},\mathcal{S}}& (1-\mu)\max_n\Theta(\mathcal{S}|n)+\mu R(\mathbf{x},\{p_i\}) \\ \mathrm{s.t.}& x_i\in\{0,1\}, \forall i,\quad\mathcal{S}=\{i{:}x_i=1\}, (11\text{a}) \\ &\sum_{i=1}^Kp_i\leq P_\text{sum}, 0\leq p_i\leq P_T, \forall i, (11\text{b}) \\ &p_{i}|\mathbf{g}_{ii}^{H}\mathbf{w}_{i}^{\mathrm{ZF}}|^{2}\geq\sigma^{2}\gamma, \forall i\in\mathcal{S}.\quad(11\mathrm{c}) \end{aligned}
  where the vector {g}_{ji}\in\mathbb{C}^M denotes the two-hop channel that is the product of the channel from the j-th DFR to the target and the channel from the target to the i-th DFR.

• \hat{\Theta}(\mathcal{S}|n)=\frac{1}{2}\sum_{l=0}^{n-1}\binom{|\mathcal{S}|}{l}P^{l}(1-P)^{|\mathcal{S}|-l}+\frac{1}{2}\sum_{l=0}^{|\mathcal{S}|-n}\binom{|\mathcal{S}|}{l}Q^{l}(1-Q)^{|\mathcal{S}|-l},where P= \frac 1{| S| }\sum _{i= 1}^{| S| }P_{i} and Q= \frac 1{| S| }\sum _{i= 1}^{| S| }Q_{i}. \mathcal{P}3:\max_{\mathbf{x},\{p_i\},\mathcal{S}}\Xi(\mathbf{x},\{p_i\},\mathcal{S}), where \Xi(\mathbf{x},\{p_{i}\},\mathcal{S})=(1-\mu)\hat{\Theta}\Big[\mathcal{S}|\min\Big(|\mathcal{S}|,\Big\lceil\frac{|\mathcal{S}|}{1+\alpha}\Big\rceil\Big)\Big]+ \mu R(\mathbf{x},\{p_{i}\}).

• In particular, we leverage the hybrid meta-heuristic and optimization \text{which starts from a feasible solution of x ( e. g. , x}^{[ 0] }= 1, 0, \ldots , 0] ^T), and randomly selects a candidate solution \mathbf{x} ^{\prime } from the neighborhood
  \mathcal{N}(\mathrm{x}^{[0]})=\{\mathrm{x}:||\mathrm{x}-\mathrm{x}^{[0]}||_0\leq L,\:x_i\in\{0,1\},\forall i\},
  where L\geq1 is the variable size of neighborhood. It can be seen that \mathcal{N}(\mathbf{x}^{[0]}) is a subset of the entire feasible space containing solutions“close”to \mathbf{x}^{[0]}.\mathcal{N}(\mathbf{x}^{[0]}) is generated b randomly flipping L elements inside \{x_i\} [12]. With the neighborhood \mathcal{N}(\mathbf{x}^{[0]}) defined above and the choice of x fixed to \begin{array}{l}\mathbf{x}=\mathbf{x}^{\prime}\in\mathcal{N}(\mathbf{x}^{[0]}),\text{ the set of sensing DFRs is }\mathcal{S}^{\prime}=\{i:x_i^{\prime}=1\}\\ \text{Then the problem }\mathcal{P}3\mathrm{~w.r.t.~}\{p_i\}\text{ keeping }\{\mathbf{x}=\mathbf{x}^{\prime},\mathcal{S}=\mathcal{S}^{\prime}\}\end{array} fixed is

\begin{aligned}\mathcal{P}4:\max_{\{p_i\}}&\sum_{i=1}^K(1-x_i')|\mathbf{f}_i^H\mathbf{w}_i^{\mathrm{ZF}}|\sqrt{p_i}\\ \mathrm{s.t.}&\sum_{i=1}^Kp_i\leq P_{\mathrm{sum}},\quad0\leq p_i\leq P_T,\:\forall i,\:(18\text{a})\\ &p_i|\mathbf{g}_{ii}^H\mathbf{w}_i^{\mathrm{ZF}}|^2\geq\sigma^2\gamma,\:\forall i\in\mathcal{S}^{\prime},&\text{(18b)}\end{aligned}
where we have removed the terms not related to \{p_i\} and the logarithm and quadratic functions due to their monotonicity Since the objective function is concave in \{p_i\} and the constraints are linear, the problem \mathcal{P}4 is a convex optimization problem w.r.t. \{p_{i}\}, which can be readily solved via the opensource software CVX. Let {p_i^\prime} denote the optimal solution of {p_i} to \mathcal{P}4.We consider \begin{array}{ll}{\text{two cases:(i) If }\Xi(\mathbf{x}^{{\prime},{p_{i}}{\prime}},\mathcal{S}^{{\prime})\geq\Xi(\mathbf{x}}{[0]},{p_{i}^{{[0]}},\mathcal{S}}{[0]}),\mathrm{~we}}\ {\text{update x}^{{[1]}\leftarrow\mathbf{x}}{\prime}.\text{By treating x}^{[1]}\mathrm{_asa_newfeasible~solu-}}\end{array} tion, we can construct the next neighborhood \mathcal{N}(\mathbf{x}^{[1]}); (ii) If \Xi(\mathbf{x}^{{\prime},{p_i}{\prime}},\mathcal{S}^{{\prime})<\Xi(\mathbf{x}}{[0]},{p_i^{{[0]}},\mathcal{S}}{[0]}), we re-generate anothen point within the neighborhood \mathcal{N}(\mathbf{x}^{[0]}) until \Xi(\mathbf{x}^\prime,{p_i{\prime}},\mathcal{S}^{\prime})\geq \Xi(\mathbf{x}^{[0]},{p_{i}{[0]}},\mathcal{S}^{[0]}).$

K=6
总传输功率为 P_{\mathrm{sum}} = 50 \text{mW}
噪声功率为 \sigma^2 = -50 \text{dBm}
请提供Matlab代码以实现仿真

提问

因为 x* 和 \mathcal{E}^* 是最优解的原因在于它们符合特定条件。具体而言，在感知模式下，
x* 必须符合 SINR 条件：对于所有节点 j，

SINR_j^{s} \geq \gamma.

然而，在实际系统中，
可能存在某些节点使得

SINR_j^{s} < \gamma.

对于这些节点而言，
将其设置为通信功能（即设其值为 1）
将有助于提升整体通信性能。
然而，
这与我们所寻求的最优解相矛盾。
因此，
为了达到最优状态，
所有满足 SINR 条件的节点都应被设为 1。