FAS vs. ARIS: Which Is More Important for FAS-ARIS Communication Systems?

As shown in Fig. 1, we consider a FAS-ARIS downlink communication system comprising a BS, a UE, and an ARIS. The BS is equipped $N$ FAs, and UE has an FPA. Traditional communication systems assisted by passive RIS often suffer from severe attenuation due to multiple reflections. To address this issue, we deploy an ARIS with $M$ reflecting elements, where each element provides additional signal amplification compared to traditional passive RIS. The corresponding reflection coefficient matrix is represented by $\mathbf{E}=\mathrm{diag}\left(e_{1},e_{2},\cdots,e_{M}\right)\in\mathbb{C}^{M\times M}$, where $e_{m}=\beta_{m}\exp(j\theta_{m})$ for $m\in\mathcal{M}=\{1,\cdots,M\}$, and $\beta_{m}>0$ and $\theta_{m}\in[0,2\pi]$ denote the amplification factor and the phase shift of the $m$th element in the ARIS, respectively.

Within the considered system, we assume that the transmitted downlink signal from the BS is given by

where $\mathbf{w}\in\mathbb{C}^{N\times 1}$ represents the transmit beamforming vector, and $s\in\mathbb{C}^{1\times 1}$ is a data stream following a complex Gaussian distribution with a mean of zero and unit variance of, i.e., $s\sim\mathcal{CN}(0,1)$. Then, let us denote the channels between the BS and ARIS, the BS and UE, and the ARIS and UE as $\mathbf{H}_{\text{BR}}\in\mathbb{C}^{M\times N}$, $\mathbf{h}_{\text{BU}}\in\mathbb{C}^{1\times N}$ and $\mathbf{h}_{\text{RU}}\in\mathbb{C}^{1\times M}$, respectively. The details of modeling these channels will be provided in the following subsection. Accordingly, the signal received by the UE is written as

where $\mathbf{n}_{\text{R}}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}_{\text{R}}\mathbf{I}_{M})$ and $n_{\text{u}}\sim\mathcal{CN}(0,\sigma^{2}_{\text{u}})$ represent the Gaussian white noise generated by the infrared amplifier at the ARIS and the UE, respectively.

Initially, we assume that the mobility region of the FAs is substantially smaller than the distance between the transmitter and the receiver. This assumption allows us to utilize the far-field communication model. Under these conditions, the amplitude differences of the signals received by each array element are negligible, and the signals exhibit a simple time-delay relationship. Additionally, the angles of departure (AoDs) and angles of arrival (AoAs) for each channel path are predominantly influenced by the scatterers and the propagation environment, remaining invariant with respect to the antenna position [4, 5]. Specifically, we define $\mathbf{\overline{t}}=[\mathbf{t}_{1},\mathbf{t}_{2},\dots,\mathbf{t}_{N}]$ to represent the set of FAs positions at the BS, with each position $\mathbf{t}_{n}=[x_{n}^{t},y_{n}^{t}]^{T},n\in\mathcal{N}=\{1,\cdots,N\}$ denotes the position of the $n$-th FAs. Additionally, we assume that all FAs can move freely within a specified range, denoted as $S_{t}$.

Then, let us denote the elevation and azimuth angles at the transmit side and receive side as $\theta_{t,s_{i}}^{i},\varphi_{t,s_{i}}^{i},1\leq s_{i}\leq L_{t}^{i}$, and $\theta_{r,k_{i}}^{i},\varphi_{r,k_{i}}^{i},1\leq k_{i}\leq L_{r}^{i},i\in\{\text{BR},\text{BU},\text{RU}\}$, respectively. Here, $L_{t}^{i}$ represents the number of transmission paths, and $L_{r}^{i}$ represents the number of reception paths. Building upon this, in $s_{i}$-th path, we define the signal propagation difference between the $n$-th FA and the origin $O_{t}$ of the transmission region as $\rho_{t,s_{i}}^{i}(\mathbf{t}_{n})=x_{n}^{t}\sin\theta_{t,s_{i}}^{i}\cos\varphi_{t,s_{i}}^{i}+y_{n}^{t}\cos\theta_{t,s_{i}}^{i},i\in\{\text{BR},\text{BU},\text{RU}\}$. Accordingly, the transmit field response vectors of a single transmit FA in the BS-ARIS links and the BS-UE links can be expressed as follow [34]

for $i\in\{\text{BR},\text{BU}\}$, where $\lambda$ is the carrier wavelength. Subsequently, the transmit field response matrix of $N$ transmitting FAs in the BS-ARIS links and the BS-UE links can be expressed as

For the receive field response vector of a single reflecting element at ARIS is given by

where $\rho_{r,k_{\text{BR}}}^{\text{BR}}(\mathbf{r}_{m})=x_{m}^{r}\sin\theta_{r,k_{\text{BR}}}^{\text{BR}}\cos\varphi_{r,k_{\text{BR}}}^{\text{BR}}+y_{m}^{r}\cos\theta_{r,k_{\text{BR}}}^{\text{BR}}$ represents the signal propagation difference of the $k_{\text{BR}}$-th path between a single reflecting element and its reception origin $O_{r}$. Thus, the receive field response matrix at the receiver can be expressed as

Because the receive positions and the transmit positions of the reflect elements at ARIS are same, we can obtain that the transmit field response matrix in the BS-ARIS links as follows

where

Furthermore, we define the path response matrix $\bm{\Sigma}_{i}\in\mathbb{C}^{{L_{r}^{i}}\times{L_{t}^{i}}},i\in\{\text{BR},\text{BU},\text{RU}\}$ as the response of all transmit and receive paths between the transmitter and the receiver. Accordingly, we can represent the channels between the BS and ARIS, the BS and UE, the ARIS and UE as

respectively.

Based on the channel and signal model from the above subsections, the achievable rate from the BS to UE can be formulated as

Consequently, the amplification power of the ARIS can be expressed as

Accordingly, this paper aims at maximizing the achievable rate by jointly optimizing the transmit beamforming vector, the reflection coefficient matrix $\mathbf{E}$ at the ARIS, and the positions of the FAs at the BS $\mathbf{\overline{t}}$. Hence, we can formulate the related optimization problem as

where $D$ is the minimum distance between adjacent antennas to prevent coupling; $P_{0}$ is the maximum transmit power at the BS; $P_{1}$ is the maximum transmit power at the ARIS. However, due to the coupling of optimization variables within both the objective function and the constraints, along with the highly non-convex nature of Problem (14), addressing this issue poses significant challenges.

As the first step of the employed AO algorithm, our objective is to optimize $\mathbf{w}$ with given $\mathbf{\overline{t}}$ and $\mathbf{E}$. Accordingly, we can reformulate Problem (14) as

Then, due to the fact that $\log_{2}(1+x)$ is an increasing function with respect to $x$, maximizing $R$ can be regarded as maximizing $|(\mathbf{h}_{\text{RU}}\mathbf{E}\mathbf{H}_{\text{BR}}+\mathbf{h}_{\text{BU}})\mathbf{w}|^{2}$. Then, by letting $\bm{\varpi}=\mathbf{h}_{\text{RU}}\mathbf{E}\mathbf{H}_{\text{BR}}+\mathbf{h}_{\text{BU}}$ and $\mathbf{W}=\mathbf{w}\mathbf{w}^{H}$, Problem (15) can be further reformulated as

It can be observed that by removing the rank-one constraint, Problem (16) is transformed into a standard semi-definite programming (SDP) problem, which can be solved using CVX to obtain $\mathbf{\widehat{W}}$ [35]. If $\mathrm{rank}(\mathbf{\widehat{W}})=1$, we can get the optimal $\mathbf{w}$, i.e., $\mathbf{w}^{o}$ by the eigenvalue decomposition. If $\mathrm{rank}(\mathbf{\widehat{W}})>1$, the rank-one solution can be derived as follows:

where

whose the proof is shown in Appendix A.

As the second sub-problem of Problem (14) when employing AO algorithm, our objective is to optimize the positions $\mathbf{\overline{t}}$ of the FAs with given $\mathbf{W}$ and $\mathbf{E}$. Accordingly, Problem (14) can be rewritten as

Then, by applying the monotonicity of $\log_{2}(1+x)$, we can rewrite the optimization problem as

Building upon this, the objective function (20a) is then transformed into a more tractable form as

where $\bm{\xi}^{H}=\mathbf{h}_{\text{RU}}\mathbf{E}\mathbf{F}_{\text{BR}}^{H}\mathbf{(\overline{r})}\bm{\Sigma}_{\text{BR}}$ and $\bm{\omega}^{H}=\mathbf{1}^{H}\bm{\Sigma}_{\text{BU}}$ are invariant to $\mathbf{\overline{t}}$. By define ${\mu}_{n}=\bm{\xi}^{H}\bm{\zeta}_{\text{BR}}(\mathbf{t}_{n})+\bm{\omega}^{H}\bm{\zeta}_{\text{BR}}(\mathbf{t}_{n}),n\in\mathcal{N}$, Eq. (III-B) can be further written as

where

Given that $\{\mathbf{t}_{l},l\neq n\}_{l=1}^{N}$ is known, $\widetilde{\alpha},\widetilde{\beta}$ and $\mu_{l}$ remain invariant to $\mathbf{t}_{n}$. Therefore, maximizing (20a) is equivalent to maximizing $g(\mathbf{t}_{n})$. However, although $g(\mathbf{t}_{n})$ is a linear function of $\bm{\zeta}_{\text{BR}}(\mathbf{t}_{n})$, it is non-convex and non-concave with respect to $\mathbf{t}_{m}$. To address this issue, we employ the second-order Taylor expansion to construct a surrogate function that locally approximates the objective function. We denote the gradient vector and the Hessian matrix of $g(\mathbf{t}_{n})$ as $\nabla g(\mathbf{t}_{n})$ and $\nabla^{2}g(\mathbf{t}_{n})$, respectively, with detailed derivations provided in Appendix B. Then, we introduce a scalar $\kappa_{n}$ such that $\kappa_{n}\mathbf{I}_{2}\succeq\nabla^{2}g(\mathbf{t}_{n})$, with the specific expression detailed in Appendix C. At this juncture, we can determine the global lower bound of $g(\mathbf{t}_{n})$ as follow

where $\mathbf{t}_{n}^{(q)}$ is the value of $\mathbf{t}_{n}$ at the $q$-th internal iteration.

Next, we address the non-convex constraint (20d). For the first term on the left side of (20d), we reformulate it as follows:

where $\bm{\Psi}=\bm{\Sigma}_{\text{BR}}^{H}\mathbf{F}_{\text{BR}}\mathbf{(\overline{r})}\mathbf{E}^{H}\mathbf{E}\mathbf{F}_{\text{BR}}^{H}\mathbf{(\overline{r})}\bm{\Sigma}_{\text{BR}}$. Similarly, since our optimization goal is the position $\mathbf{t}_{n}$ of the $n$-th antenna, we can expand (III-B) into the following form

where $\mathbf{w}(n)$ is the $n$-th element of $\mathbf{w}$ obtained from Problem (16). The detailed derivation process is provided in Eq. (III-B) at the top of the next page.

Then, we employ the MM iterative algorithm to address this non-convex constraint. Specifically, we denote the value of ${\overline{g}(\mathbf{t}_{n})}$ at the $q$-th iteration as ${\overline{g}(\mathbf{t}_{n}^{(q)})}$. Then, at the $(q+1)$-th iteration, we introduce $\varepsilon(\mathbf{t}_{n})$ as a surrogate function for ${\overline{g}(\mathbf{t}_{n}^{(q)})}$ that satisfies the following three conditions [36]:

Conditions (29) and (30) ensure that the surrogate function’s value and gradient at $\mathbf{t}_{n}^{(q)}$ match those of the original function. Meanwhile, Condition (31) guarantees that the surrogate function serves as an upper bound to the original function. This alignment allows us to represent the original function with $\varepsilon(\mathbf{t}_{n})$ at the $(q+1)$-th iteration.

According to (29), (29), and (29), for any value of $\mathbf{t}_{n}$ at the $q$-th iteration, we can construct the surrogate function as the upper bound of $\widetilde{g}(\mathbf{t}_{n})$, which can be expressed as

where $\widetilde{\bm{\Psi}}=\bm{\Psi}\mathbf{w}(n)\mathbf{w}^{H}(n)$ and $\bm{\Phi}=\lambda_{\text{max}}\mathbf{I}_{L_{t}^{\text{BR}}}$, $\lambda_{\text{max}}$ is the maximum eigenvalue of $\widetilde{\bm{\Psi}}$. Furthermore, we can observe that $\bm{\zeta}^{H}_{\text{BR}}(\mathbf{t}_{n})\bm{\Phi}\bm{\zeta}_{\text{BR}}(\mathbf{t}_{n})=\lambda_{\text{max}}L_{t}^{\text{BR}}$, and the second term on the right side of (III-B) is known. We define

Thus, we can rewrite constraint (20d) in the following form

where

However, $\widehat{g}(\mathbf{t}_{n})$ remains a non-concave function of $\mathbf{t}_{n}$, precluding the use of the first-order Taylor expansion to obtain its upper bound. Therefore, as previously mentioned, we introduce a scalar $\widehat{\kappa}_{n}$ and employ the second-order Taylor expansion of $\widehat{g}(\mathbf{t}_{n})$ to construct a surrogate function that provides the upper bound of $\widehat{g}(\mathbf{t}_{n})$. Specifically, the formulation is as follows:

where

whose proofs are similar to those in the proof provided in the Appendix B and Appendix C.

Furthermore, we address the non-convex constraint (20c). It can be observed that $||\mathbf{t}_{n}-\mathbf{t}_{v}||_{2}$ is a concave function of $\mathbf{t}_{n}$, enabling us to directly obtain its lower bound through a first-order Taylor expansion at $\mathbf{t}_{n}^{(q)}$. Specifically, the expansion is given as

Finally, based on the above derivations, we rewrite the final optimization problem in the following form

Here, the objective function (44a) is a concave quadratic function of $\mathbf{t}_{n}$, constraints (44b) and (44c) are both linear, and constraint (20b) is a concave function. This aligns with the principles of CVX, allowing us to directly use the CVX toolbox for solving the problem [37].

As the third sub-problem of Problem (14) when employing AO algorithm, we fix $\overline{\mathbf{t}}$ and $\mathbf{W}$ and optimize $\mathbf{E}$, the sub-problem can be rewritten as follow

Let us define $\mathbf{e}=[e_{1},e_{2},\cdots e_{M}]^{H}$ as the vector consisting of the diagonal elements of $\mathbf{E}$ and $\widetilde{\mathbf{e}}=[\mathbf{e}^{H},1]^{H}$ as the vector containing $\mathbf{e}$ and $1$. Furthermore, since

We can transform Problem (45) into the following form

where $\widetilde{\mathbf{E}}=\widetilde{\mathbf{e}}\widetilde{\mathbf{e}}^{H}$ and the expressions for $\mathbf{V}$, $\overline{\mathbf{V}}$ and $\widehat{\mathbf{V}}$ can be found in (50), (51) and (52).

Next, we address Problem (45). First, for the non-convex constraint (49d), we adopt the sequential rank-one constraint relaxation (SRCR) method, equivalently transforming the rank-one constraint into [38]

where $\widetilde{\mathbf{E}}^{(p)}$ is the value at the $p$-th internal iteration, $\mathbf{u}_{\text{max}}\left(\widetilde{\mathbf{E}}^{(p)}\right)$ is the eigenvector corresponding to the largest eigenvalue $\lambda_{\text{max}}\left(\widetilde{\mathbf{E}}^{(p)}\right)$ of $\widetilde{\mathbf{E}}^{(p)}$ and $\vartheta^{(p)}$ represents the relaxation parameter and updated by following

where $o^{(p+1)}$ is a given scalar.

However, Problem (49) remains a complex fractional programming problem, which is challenging to solve. To reduce the complexity, we first introduce auxiliary variables to equivalently transform the problem into a more tractable form. Then, we use the successive convex approximation (SCA) algorithm to iteratively approximate these non-convex functions with linear functions, thereby solving the problem. Specifically, by introducing auxiliary variables $\chi$ and $\varrho$, we transform Problem (49) into the following form

It is worth noting that constraint (55b) is still non-convex. For this reason, we propose an SCA algorithm to find the a upper bound of $\chi\varrho$. Specifically , let

It can be seen that $\frac{1}{4}\left(\chi+\varrho\right)^{2}$ and $\frac{1}{4}\left(\chi-\varrho\right)^{2}$ are convex on both $\chi$ and $\varrho$. Thus, we have

Based on the above analysis, we can build the final optimization problem