Joint Discrete Antenna Positioning and Beamforming Optimization in Movable Antenna Enabled Full-Duplex ISAC Networks

In this subsection, we primarily elaborate on the channel model for radar sensing and communication in the MA-enabled full-duplex ISAC system. Firstly, we introduce the radar sensing channel model of the system. Specifically, for the MA system, its MA elements can move among the candidate discrete positions within a two-dimensional area, leading to changes in the physical channel. Assuming that the channel sensing information is line-of-sight (LoS), taking the receiving MA as an example, the actual location vector of the ${n_{r}}$-th MA element is expressed as ${{\mathbf{t}}_{r{\text{,}}{n_{r}}}}={\left[{{t_{{r_{x}},{n_{r}}}},{t_{{r_{y}},{n_{r}}}}}\right]^{T}}$, which can be represented by a two-dimensional vector ${{\mathbf{b}}_{r,{n_{r}}}}$ as follows: ${{\mathbf{t}}_{r{\text{,}}{n_{r}}}}{\text{ = }}{{\mathbf{P}}_{r}}{{\mathbf{b}}_{r,{n_{r}}}}$, where ${{\mathbf{b}}_{r,{n_{r}}}}={\left[{{b_{r,{n_{r}}}}\left[1\right],...,{b_{r,{n_{r}}}}\left[M\right]}\right]^{T}}$, $\forall{n_{r}}\in\left\{{1,...,{N_{r}}}\right\}$, ${b_{r,{n_{r}}}}\left[m\right]\in\left\{{0,1}\right\}$, $\forall m\in\left\{{1,...,M}\right\}$ and $\sum\nolimits_{m=1}^{M}{{b_{r,{n_{r}}}}\left[m\right]=1}$. Similarly, the actual position of the ${n_{t}}$-th transmitting MA element is defined as ${{\mathbf{t}}_{t{\text{,}}{n_{t}}}}={\left[{{t_{{t_{x}},{n_{t}}}},{t_{{t_{y}},{n_{t}}}}}\right]^{T}}$, and a binary direction vector ${{\mathbf{b}}_{t,{n_{t}}}}$ is used to express it as ${{\mathbf{t}}_{t,{n_{t}}}}{\text{ = }}{{\mathbf{P}}_{t}}{{\mathbf{b}}_{t,{n_{t}}}}$. The binary direction ${{\mathbf{b}}_{t,{n_{t}}}}$ is given by ${{\mathbf{b}}_{t,{n_{t}}}}={\left[{{b_{t,{n_{t}}}}\left[1\right],...,{b_{t,{n_{t}}}}\left[N\right]}\right]^{T}}$, $\forall{n_{t}}\in\left\{{1,...,{N_{t}}}\right\}$ and it follows that ${b_{t,{n_{t}}}}\left[n\right]\in\left\{{0,1}\right\}$, $\forall n\in\left\{{1,...,N}\right\}$ and $\sum\nolimits_{n=1}^{N}{{b_{t,{n_{t}}}}\left[n\right]=1}$. Let the angles be the elevation angle and azimuth angle between the signal and the radar sensing target $\theta$ , $\phi\in\left[{{{-\pi}\mathord{\left/{\vphantom{{-\pi}2}}\right.\kern-1.2pt}2},{{-\pi}\mathord{\left/{\vphantom{{-\pi}2}}\right.\kern-1.2pt}2}}\right]$, respectively. Therefore, the steering vector for the transmitting MA array is expressed as

Similarly, the steering vector for the array receiving is expressed as

Then, we describe the communication channel model of the system. Based on the field response-based channel model, the channel between the transmitting and receiving MA can be modeled as ${{\mathbf{H}}_{{\text{SI}}}}\in{\mathbb{C}^{{N_{r}}\times{N_{t}}}}$, the channel between $U$ uplink users and the receiving MAs can be modeled as ${\mathbf{G}}\in{\mathbb{C}^{U\times{N_{r}}}}$, and the channel between $D$ downlink users and the transmitting MA can be modeled as ${\mathbf{H}}\in{\mathbb{C}^{D\times{N_{t}}}}$ as follows.

For the FD-DFRC-BS, the channel between the transmitting and receiving MA is denoted as ${{\mathbf{H}}_{{\text{SI}}}}$, where the channel from the ${n_{r}}$-th receiving MA element to the ${n_{t}}$-th transmitting MA element is expressed as ${\left[{{\mathbf{H}}_{{\text{SI}}}}\right]_{{n_{r}},{n_{t}}}}=\sqrt{\eta_{{n_{r}},{n_{t}}}^{\text{SI}}}{e^{-j2\pi\frac{{{d_{{n_{r}},{n_{t}}}}}}{\lambda}}}$, which represents the element in the ${n_{r}}$-th row and ${n_{t}}$-th column of matrix ${{\mathbf{H}}_{{\text{SI}}}}$. Here, $\eta_{{n_{r}},{n_{t}}}^{\text{{SI}}}$ and ${d_{{n_{r}},{n_{t}}}}$ denote the path loss and distance between the ${n_{r}}$-th receiving MA element and the ${n_{t}}$-th transmitting MA element.

For $U$ users in the uplink and the receiving MA, the channel matrix $\bf{G}$ is expressed as ${\mathbf{G}}{\text{ = }}\left[{{{\mathbf{g}}_{1}}\left({{{\mathbf{t}}_{r{\text{,1}}}}}\right){\text{,}}...{\text{,}}{{\mathbf{g}}_{{N_{r}}}}\left({{{\mathbf{t}}_{r{\text{,}}{N_{r}}}}}\right)}\right]\in{\mathbb{C}^{U\times{N_{r}}}}$, where ${{\mathbf{g}}_{{n_{r}}}}\left({{{\mathbf{t}}_{r{\text{,}}{n_{r}}}}}\right)$ represents the uplink channel between $U$ users and the ${n_{r}}$-th MA element. It is expressed as ${{\mathbf{g}}_{{n_{r}}}}\left({{{\mathbf{t}}_{r{\text{,}}{n_{r}}}}}\right)={\left[{{g_{{n_{r}}}}_{,1}\left({{{\mathbf{t}}_{r{\text{,}}{n_{r}}}}}\right),...,{g_{{n_{r}}}}{{{}_{,}}_{U}}\left({{{\mathbf{t}}_{r{\text{,}}{n_{r}}}}}\right)}\right]^{T}}\in{\mathbb{C}^{U\times 1}}$, where ${g_{{n_{r}}}}_{,u}\left({{{\mathbf{t}}_{r{\text{,}}{n_{r}}}}}\right)\in\mathbb{C}$ denotes the channel coefficient between the ${n_{r}}$-th receiving MA element and the $u$-th uplink user. To facilitate subsequent processing, we define the channel between the $U$ uplink users and the $M$ candidate discrete positions of the receiving MA as ${\mathbf{\hat{G}}}{\text{ = }}\left[{{{{\mathbf{\hat{G}}}}_{1}},...,{{{\mathbf{\hat{G}}}}_{{N_{r}}}}}\right]\in{\mathbb{C}^{U\times M{N_{r}}}}$, where ${{\mathbf{\hat{G}}}_{{n_{r}}}}{\text{ = }}\left[{{{\mathbf{g}}_{{n_{r}}}}\left({{{\mathbf{p}}_{r,1}}}\right),...,{{\mathbf{g}}_{{n_{r}}}}\left({{{\mathbf{p}}_{r,M}}}\right)}\right]\in{\mathbb{C}^{U\times M}}$ represents the channel vector of the $M$ candidate discrete positions from $U$ uplink users to the ${n_{r}}$-th receiving MA element. The channel vector at the candidate discrete position ${{\mathbf{p}}_{r,m}}$ for the $U$ uplink users and the ${n_{r}}$-th receiving MA element is given by ${{\mathbf{g}}_{{n_{r}}}}\left({{{\mathbf{p}}_{r,m}}}\right)={\left[{{g_{{n_{r}}}}_{,1}\left({{{\mathbf{p}}_{r{\text{,}}m}}}\right),...,{g_{{n_{r}}}}{{{}_{,}}_{U}}\left({{{\mathbf{p}}_{r{\text{,}}m}}}\right)}\right]^{T}}\in{\mathbb{C}^{U\times 1}}$. Here, ${g_{{n_{r}}}}_{,u}\left({{{\mathbf{p}}_{r{\text{,}}m}}}\right)\in\mathbb{C}$ denotes the channel coefficient between the ${n_{r}}$-th receiving MA element and the $u$-th uplink user, when the latter is located at the candidate discrete position ${{\mathbf{p}}_{r,m}}$. Suppose that ${l_{p}}\in\left\{{1,...,{L_{p}}}\right\}$ represents the number of paths from the ${n_{r}}$-th receiving MA element to the $u$-th uplink user, let $\theta_{{}_{u,{l_{p}}}}^{U}$ and $\phi_{{}_{u,{l_{p}}}}^{U}$ respectively represent the elevation angle and azimuth angle of the ${l_{p}}$-th path of the $u$-th uplink user, then ${g_{{n_{r}}}}_{,u}\left({{{\mathbf{p}}_{{N_{r}}{\text{,}}m}}}\right)$ can be expressed as

Let ${{\mathbf{\hat{g}}}_{u}}$ denote the $u$-th row of ${\mathbf{\hat{G}}}$, which can represent the channel loss of the $u$-th uplink user and the $M$ candidate discrete positions of the ${N_{r}}$ receiving MA elements. The channel matrix $\bf{G}$ between $U$ uplink users and receiving MA can be represented as

where ${{\mathbf{B}}_{r}}\in{\mathbb{C}^{{N_{r}}M\times{N_{r}}}}$. It can also be specifically expressed as

In addition, for the wireless channel of this system between $D$ downlink users and the transmitting MA, we have ${\mathbf{H}}{\text{ = }}\left[{{{\mathbf{h}}_{1}}\left({{{\mathbf{t}}_{t{\text{,1}}}}}\right){\text{,}}...{\text{,}}{{\mathbf{h}}_{{N_{t}}}}\left({{{\mathbf{t}}_{t{\text{,}}{N_{t}}}}}\right)}\right]\in{\mathbb{C}^{D\times{N_{t}}}},$ where ${{\mathbf{h}}_{{n_{t}}}}\left({{{\mathbf{t}}_{{\text{t,}}{n_{t}}}}}\right)$ represents the channel coefficient between the ${n_{t}}$-th MA element and D downlink user. It can be expressed as ${{\mathbf{h}}_{{n_{t}}}}\left({{{\mathbf{t}}_{t{\text{,}}{n_{t}}}}}\right)={\left[{{h_{{n_{t}}}}_{,1}\left({{{\mathbf{t}}_{t{\text{,}}{n_{t}}}}}\right),...,{h_{{n_{t}}}}{{{}_{,}}_{D}}\left({{{\mathbf{t}}_{t{\text{,}}{n_{t}}}}}\right)}\right]^{T}}\in{\mathbb{C}^{D\times 1}}$, where ${h_{{n_{t}}}}_{,d}\left({{{\mathbf{t}}_{t{\text{,}}{n_{t}}}}}\right)\in\mathbb{C}$ represents the channel coefficient between the ${n_{t}}$-th transmitting MA element and the $d$-th downlink user. In order to facilitate subsequent processing, we define the channel between $D$ different users and $N$ candidate discrete positions of transmitting MA as ${\mathbf{\hat{H}}}{\text{ = }}\left[{{{{\mathbf{\hat{H}}}}_{1}},...,{{{\mathbf{\hat{H}}}}_{{N_{t}}}}}\right]\in{\mathbb{C}^{D\times N{N_{t}}}}$. ${{\mathbf{\hat{H}}}_{{n_{t}}}}{\text{ = }}\left[{{{\mathbf{h}}_{{n_{t}}}}\left({{{\mathbf{p}}_{t,1}}}\right),...,{{\mathbf{h}}_{{n_{t}}}}\left({{{\mathbf{p}}_{t,N}}}\right)}\right]\in{\mathbb{C}^{D\times N}}$ is the channel matrix from the $D$ downlink users to the $N$ candidate discrete channels of the ${n_{t}}$-th transmitting MA element. Additionally, ${{\mathbf{h}}_{{n_{t}}}}\left({{{\mathbf{p}}_{t,n}}}\right)={\left[{{h_{{n_{t}}}}_{,1}\left({{{\mathbf{p}}_{t,n}}}\right),...,{h_{{n_{t}}}}{{{}_{,}}_{D}}\left({{{\mathbf{p}}_{t,n}}}\right)}\right]^{T}}\in{\mathbb{C}^{D\times 1}}$ represents the channel vector of $D$ downlink users and the ${n_{t}}$-th transmitting MA element at the candidate discrete position ${{\mathbf{p}}_{t,n}}$. Here, ${h_{{n_{t}}}}_{,d}\left({{{\mathbf{p}}_{t{\text{,}}n}}}\right)\in\mathbb{C}$ represents the channel coefficient between the ${n_{t}}$-th transmitting MA element and the $d$-th downlink user. Let $\forall{l_{p}}\in\left\{{1,...,{L_{p}}}\right\}$ represent the number of paths between the ${n_{t}}$-th transmitting MA element and the $d$-th downlink user, where are a total of ${L_{p}}$ such paths. Denote by $\theta_{{}_{d,{l_{p}}}}^{D}$, $\phi_{{}_{d,{l_{p}}}}^{D}$ the elevation angle and azimuth angle for the ${l_{p}}$-th path to the $d$-th downlink user, respectively, then ${h_{{n_{t}}}}_{,d}\left({{{\mathbf{p}}_{{N_{t}}{\text{,}}n}}}\right)$ can be expressed as

Let ${{\mathbf{\hat{h}}}_{d}}$ denote the $d$-th row of ${\mathbf{\hat{H}}}$, which represents the channel gain between the $d$-th downlink user and the $N$ candidate discrete positions of the ${N_{t}}$ transmitting MA elements. The channel matrix $\bf{H}$ between $D$ downlink users and transmitting MA can be represented as

where ${{\mathbf{B}}_{t}}\in{\mathbb{C}^{{N_{t}}N\times{N_{t}}}}$, specifically represented as

For the downlink transmission process of the MA-enabled full-duplex ISAC system, the FD-DFRC-BS transmits narrowband ISAC signals ${\mathbf{x}}\in{\mathbb{C}^{{N_{t}}\times 1}}$ via multi-antenna beamforming, which serves for radar target sensing and downlink communication for multiple users. According to [29, 30], it can be represented as

where ${{\mathbf{w}}_{d}}\in{\mathbb{C}^{{N_{t}}\times 1}}$ represents the beamforming vector associated with the $d$ downlink user. ${s_{d}}$ denotes the data transmitted to the $d$-th user for communication, satisfying $\mathbb{E}\left\{{{{\left|{{s_{d}}}\right|}^{2}}}\right\}=1$. ${{\mathbf{s}}_{0}}\in{\mathbb{C}^{{N_{t}}\times 1}}$ represents the dedicated sensing signal sent to the radar sensing target, determined by ${{\mathbf{W}}_{0}}\triangleq\mathbb{E}\left\{{{\mathbf{s}}_{0}}{{\mathbf{s}}_{0}^{H}}\right\}$ and is mutually orthogonal to $\left\{{{s_{d}}}\right\}_{d=1}^{D}$. Therefore, the signal received by the $d$-th user in the downlink can be expressed as

where ${\alpha_{d}}$ is the channel fading factor from the FD-DFRC-BS to the $d$-th downlink user, and ${n_{d}}$ represents the additive Gaussian white noise (AWGN) with a variance of $\sigma_{d}^{2}$ introduced at the receiving end of the $d$-th downlink user.

Then, considering the uplink transmission process of the system, when FD-DFRC-BS communicates and senses in the downlink, it will receive the data sent by the uplink users as well as the sensing echo signal of radar at the same time. Suppose that the $u$-th uplink user sends data ${d_{u}}\in\mathbb{C}$. Supposed to be detected sensing target is located at ${\theta_{0}},{\phi_{0}}$, then the reflection of target is ${\beta_{0}}{{\mathbf{a}}_{r}}\left({{\theta_{0}},{\phi_{0}}}\right){\mathbf{a}}_{t}^{H}\left({{\theta_{0}},{\phi_{0}}}\right){\mathbf{x}}$, where ${\beta_{0}}\in\mathbb{C}$ represents the gain of the target sensing channel, mainly influenced by path loss and radar. Using parameter estimation scheme [31, 32], we can obtain parameters ${\beta_{0}}$, ${\theta_{0}}$, ${\phi_{0}}$ at FD-DFRC-BS, so the received signal at FD-DFRC-BS can be expressed as

where ${\mathbf{A}}\left({\theta,\phi}\right)={{\mathbf{a}}_{r}}\left({\theta,\phi}\right){\mathbf{a}}_{t}^{H}\left({\theta,\phi}\right)$, ${\alpha_{u}}$ is the channel attenuation of the $u$-th uplink user to FD-DFRC-BS, ${\mathbf{n}}\in{\mathbb{C}^{{N_{r}}\times 1}}$ represents the AWGN with variance $\sigma_{r}^{2}{{\bf{I}}_{{N_{r}}}}$. ${\beta_{k}}$ is the amplitude of the interference signal introduced by the $k$-th non-target sensing radar.

Signal-to-interference-plus-noise ratio (SINR) can be used as a metric to measure the performance of communication and sensing systems, so we use communication SINR and sensing SINR as the metric for the communication and sensing capabilities of the system. In this paper, we assume that ${\mathbf{v}}\in{\mathbb{C}^{{N_{r}}\times 1}}$ is the receiving beamforming vector at FD-DFRC-BS to capture the desired reflection sensing signal from the target. For radar sensing target, its sensing SINR can be expressed as Eq. (12), where we define ${\bf{Q}}=\sum\nolimits_{k=1}^{K}{{\beta_{k}}{\bf{A}}\left({{\theta_{k}},{\phi_{k}}}\right)}+{{\bf{H}}_{{\rm{SI}}}}$. Similarly, we assume $\left\{{\bf{r}}_{u}\right\}_{{u=}1}^{U}\in{{\mathbb{C}}^{{N_{r}}\times 1}}$ is the receiving beamforming vector at FD-DFRC-BS to obtain uplink transmitted signal. For the uplink, the SINR of the communication data sent by the $u$-th user can be expressed as Eq. (13), where ${\bf{C}}=\sum\nolimits_{k=0}^{K}{{\beta_{k}}{\bf{A}}\left({{\theta_{k}},{\phi_{k}}}\right)}+{{\bf{H}}_{{\rm{SI}}}}$ represents the interference channel of the downlink path. In addition, for downlink communication, the SINR received by the $d$-th user can be expressed as Eq. (14). For downlink communication, it is assumed that users cannot eliminate the interference from dedicated radar sensing signal ${{\bf{s}}_{0}}$. In this paper, through the optimization of the beamforming vectors, it can suppress the non-target radar sensing interference and multi-user interference in Eq. (12), the radar snesing interference and multi-user interference in Eq. (13) and Eq. (14).

Since MA elements have a certain volume, and different MA elements cannot be in the same candidate discrete position, this paper assumes thatthe distance from the center of one MA element to the center of another must exceed the specified minimum distance ${D_{\min}}$. Define a matrix ${{\bf{D}}_{r}}\in{{\mathbb{C}}^{M\times M}}$, where the element at row $i$ and column $j$ represents the distance between the receiving MA’s $i$-th candidate discrete position and its $j$-th candidate discrete position. Hence, the distance between any two receiving MA elements can be articulated as ${\bf{b}}_{r,{n_{r}}}^{T}{{\bf{D}}_{r}}{{\bf{b}}_{r,{n_{r}}^{\prime}}}$, ${n_{r}}\!\neq\!{n_{r}}^{\prime}$, $\forall{n_{r}},{n_{r}}^{\prime}\in\left\{{1,...,{N_{r}}}\right\}$. According to the definition, let ${{\bf{D}}_{t}}\in{{\mathbb{C}}^{N\times N}}$ be defined such that its element in the $i$-th row and $j$-th column represents the distance between the $i$-th transmitting MA element’s position and the $j$-th receiving MA element’s position. Therefore, the distance between any pair of transmitting MA elements can be expressed as ${\bf{b}}_{t,{n_{t}}}^{T}{{\bf{D}}_{t}}{{\bf{b}}_{t,{n_{t}}^{\prime}}}$, ${n_{t}}\!\neq\!{n_{t}}^{\prime}$, $\forall{n_{t}},{n_{t}}^{\prime}\in\left\{{1,...,{N_{t}}}\right\}$. For the downlink, the transmit power of FD-DFRC-BS is $\sum\nolimits_{d=1}^{D}{||{{\bf{w}}_{d}}|{|^{2}}}+{\rm{Tr}}({{\bf{W}}_{0}})$. In addition, for the uplink, the transmit power of user is represented as ${p_{u}}{\text{ = }}{\mathbb{E}}\left\{{{{\left|{{d_{u}}}\right|}^{2}}}\right\}$, $\forall u$. Therefore, the total transmit power of the system can be expressed as $\sum\nolimits_{d=1}^{D}{||{{\bf{w}}_{d}}|{|^{2}}}+{\rm{Tr}}({{\bf{W}}_{0}})+\sum\nolimits_{u=1}^{U}{{p_{u}}}$. In this paper, while ensuring the minimum SINR requirements for uplink communication, downlink communication, and radar sensing, we jointly optimize ${{\bf{B}}_{r}}$, ${{\bf{B}}_{t}}$, $\left\{{{{\bf{w}}_{d}}}\right\}_{d=1}^{D}$, ${{\bf{W}}_{0}}$, $\left\{{{{\bf{r}}_{u}}}\right\}_{u=1}^{U}$, ${\bf{v}}$ and $\left\{{{p_{u}}}\right\}_{u=1}^{U}$ to minimize the total transmit power consumption of the system. The specific optimization problem can be formulated as follows

where constraints (15a)-(15c) are used to ensure the quality-of-service (QoS) for radar sensing, uplink communication and downlink communication. ${\gamma^{r}}$, $\gamma_{u}^{U}$ and $\gamma_{d}^{D}$ are constant thresholds for the minimum SINR requirement for sensing, the minimum SINR requirement for the uplink transmitting user $u$ and the minimum SINR requirement for the downlink transmitting user $d$. Constraints (15d) and (15e) restrict ${{\bf{B}}_{r}}$ and ${{\bf{B}}_{t}}$ to be binary matrices. Constraints (15f) and (15g) ensure that each MA element is only in one candidate discrete position. Constraints (15h) and (15i) ensure that the inter-center separation for any two MA elements is in excess of the minimum allowable distance ${D_{\min}}$. Note that any two different MA elements cannot be in the same candidate discrete position. Due to SINR constraints and the existence of binary variables, it is a mixed-integer non-convex optimization problem, which is hard to obtain the optimal solutions. Next, we will propose a joint optimization framework based on BPSO to solve it.

Traditional alternating position optimization methods involve fixing the positions of all other MA elements and only moving one to alternate. However, since the candidate positions of MA elements are discrete, there is a possibility of converging to an undesired suboptimal solution. Moreover, as the solution space grows, the computational load increases exponentially. The BPSO algorithm reduces the computational load through the collaborative efforts of discrete particles in a swarm, allowing for a faster approach to the suboptimal solution. Taking all factors into account, the BPSO algorithm framework is applied to solve the problem (P1) formulated in this paper. Specifically, taking the receiving MA as an example, BPSO algorithm workflow can be expressed as follows.

Initialization Position and Speed: In order to facilitate subsequent calculations, we first assume ${{\bf{\tilde{b}}}_{r,i}}={\left[{{\bf{b}}_{{}_{r,1}}^{T},{\bf{b}}_{{}_{r,2}}^{T},\ldots,{\bf{b}}_{{}_{r,{N_{r}}}}^{T}}\right]^{T}}\in{{\mathbb{C}}^{M{N_{r}}\times 1}}$. The BPSO algorithm initializes the positions of $I$ particles to ${{\cal B}^{\left(0\right)}}=\left\{{{\bf{\tilde{b}}}_{r,1}^{\left(0\right)},{\bf{\tilde{b}}}_{r,2}^{\left(0\right)},\ldots,{\bf{\tilde{b}}}_{r,I}^{\left(0\right)}}\right\}$, where each particle represents a possible distribution of the position of the ${N_{r}}$ receiving MA elements in $M$ candidate discrete positions. The velocity of $I$ particles is initialized as ${{\cal V}^{\left(0\right)}}=\left\{{{\bf{v}}_{1}^{\left(0\right)},{\bf{v}}_{2}^{\left(0\right)},\ldots,{\bf{v}}_{I}^{\left(0\right)}}\right\}$, where ${\bf{v}}_{i}^{\left(0\right)}={\left[{{\bf{v}}_{{}_{r,1}}^{T},{\bf{v}}_{{}_{r,2}}^{T},\ldots,{\bf{v}}_{{}_{r,{N_{r}}}}^{T}}\right]^{T}}\in{{\mathbb{C}}^{M{N_{r}}\times 1}}$ indicates that the speed at which the ${n_{r}}$-th receiving MA element moves in the $M$ candidate discrete positions, and ${v_{\min}}\leqslant{\bf{v}}_{{}_{r,{n_{r}}}}^{T}\left(m\right)\leqslant{v_{\max}}$.

Local Optimal Position and Global Optimal Position: Let ${\bf{\tilde{b}}}_{r,i}^{*}$ be the local optimal position of the $i$-th particle. For the $i$-th particle, this position is the optimal solution to the fitness function that it has encountered in its history. Let ${\bf{\tilde{b}}}_{r}^{*}$ be the global optimal position, which is the optimal solution to the fitness function found among the local optimal positions of all $I$ particles in the swarm.

Velocity Update Criterion: In the $j$-th iteration, the velocity update of each particle is given by

where $j$ is the number of iterations and $0\leq\textit{j}\leq\textit{J}$. $\omega$ is the inertia weight, which can be expressed as

with ${\omega_{\max}}=1.2$, ${\omega_{\min}}=0.4$ generally taken. ${c_{1}}$ and ${c_{2}}$ are local and global learning factors that push each particle towards the local and global optimal positions, respectively. ${e_{1}}$ and ${e_{2}}$ are uniformly distributed random numbers in the range $[0,1]$. They are used to increase randomness and reduce the possibility of converging to an unexpected local optimum.

Position Update Criterion: Given the method of probability mapping, we use the sigmoid function to map speed to $[0,1]$ as the probability. This probability is the likelihood that the particle will take a value of 1 in the next step and can be expressed as

Absolute Probability of Position Change: The position of the receiving MA element is updated according to the probability of the candidate discrete position. Since different MA element positions cannot be at the same candidate discrete position, the position update can be expressed as

Fitness Function: During each iteration, the local and global optimal positions are updated based on the fitness function value. Assuming the positions meet the constraint conditions, but considering constraints (15f) and (15h). A penalty term is added in the problem (P1), which can be represented as the problem (P2) as follows

where $F\left({{\bf{\tilde{t}}}_{i}^{\left(j\right)}}\right)$ is a function that returns the number of MA elements that violate the minimum MA distance constraint at position ${\bf{\tilde{t}}}$, ${\bf{\tilde{t}}}_{i}^{\left(j\right)}=\left[{{\bf{t}}_{{}_{r,1}}^{\text{T}},\ldots,{\bf{t}}_{{}_{r,{N_{r}}}}^{\text{T}}}\right]\in{{\mathbb{C}}^{2{N_{r}}\times 1}},$ and $\kappa$ is a very large number. Because of the complex SINR constraints, it remains a non-convex optimization problem. When two different MA elements are in the same candidate discrete position, their push particles satisfy constraints (15f) and (15h). Through iteration, we can generally obtain a suboptimal solution. Generally, as the number of iterations increases, it becomes easier to find a suboptimal solution that is closer to the optimal solution. However, the computational load also increases. How to balance this trade-off is also a potential direction for the future research.

In this paper, objective function of the problem (P2) is independent of $\left\{{{{\bf{r}}_{u}}}\right\}_{u=1}^{U}$ and $\bf{v}$. When ${{\bf{B}}_{r}}$, ${{\bf{B}}_{t}}$, $\left\{{{{\bf{w}}_{d}}}\right\}_{d=1}^{D}$, ${{\bf{W}}_{0}}$ and $\left\{{{p_{u}}}\right\}_{u=1}^{U}$ are given, $\left\{{{{\bf{r}}_{u}}}\right\}_{u=1}^{U}$ and $\bf{v}$ only affect ${\text{SINR}}_{u}^{U}$ and ${\text{SIN}}{{\text{R}}^{r}}$ respectively. In order to better realize (P2), the constraints (15a) and (15b) are transformed into

to solve for $\left\{{{{\bf{r}}_{u}}}\right\}_{u=1}^{U}$ and $\bf{v}$. In order to facilitate expression, we define

The solutions to optimization problems (21) and (22) are expressed as

By introducing the $\left\{{{{\bf{r}}_{u}}}\right\}_{u=1}^{U}$, ${\bf{v}}$, ${{\bf{B}}_{r}}$ and ${{\bf{B}}_{t}}$ into the corresponding constraints, the problem is transformed into minimizing the total transmit power of the system by optimizing variables $\left\{{{{\bf{w}}_{d}}}\right\}_{d=1}^{D}$,${{\bf{W}}_{0}}$ and $\left\{{{p_{u}}}\right\}_{u=1}^{U}$. We can rewrite the problem (P2) into the problem (P3) as follows