Quasi-Distributed Antenna Selection for Spectral Efficiency Maximization in Subarray Switching XL-MIMO Systems

AS strategies for MIMO systems are extensively discussed in the literature. One AS algorithm to improve capacity in low rank matrix channels on point-to-point MIMO was first introduced in [26]. Later, the capacity distribution of systems with receive AS has been derived in [27]. These results were extended to massive MIMO regime in [28] and [29]. In these papers, the authors derived capacity bounds for systems with transmit and receive AS, respectively.

The authors in [13, 14] proposed AS procedures respectively for the channel capacity and downlink (DL) sum-capacity maximization based on the convex optimization framework. One technique based on the branch-and-bound algorithm is used in [8]. Considering linearly-precoded systems, the problems of AS for SE and sum-SINR maximization are addressed respectively in [15, 30]. Differently, the work in [31] analyzed one joint AS and power allocation (PA) procedure in a system with spatially distributed antennas. The proposed procedure runs at each antenna with side-information shared within its neighborhood. Besides, AS considering limited connections in the RF transceivers switching matrices is examined in [7].

On the other hand, there are only a few works that consider the AS problem for the XL-MIMO systems. A spatial users mapping procedure to maximize SE implemented with convolutional neural networks (CNN) is proposed in [16]. The aim is to determine each effective subarray window to precode the users signals using ZF. Results demonstrate that the CNN-based procedure achieves SE values comparable to the optimal mapping algorithm. In [17], several transmit AS procedures to maximize the energy efficiency (EE) from the long-term fading coefficients are proposed. Asymptotic SINR expressions for the received signal with AS are derived. Since the derived optimization problem is NP-hard, three of the proposed procedures are implemented by metaheuristic techniques, one being the genetic algorithm (GA). The GA is a powerful evolutionary metaheuristic that was used in different contexts to solve AS problems, as it is considered in [18, 19, 20].

Motivated by the benefits of large numbers of antennas at the BS and the restricted number of RF transceivers, this work examines the joint AS and PA problem on the DL of a linearly-precoded XL-MIMO system. Differently from other papers adopting AS strategy, a distributed BS signal processing architecture is considered and the AS procedures are characterized in terms of the exchanged information between the processing nodes. Furthermore, we extend part of the results of [17] with the proposition of AS algorithms for XL-MIMO that use the short-term fading coefficients instead of the long-term ones. Additionally, we address the problem of joint AS and PA in XL-MIMO sub-arrays using a decentralized RA algorithm. The proposed RA algorithm uses the Sherman-Morrison-Woodbury (SMW) formula to perform optimal power allocation (OPA) and AS in a decentralized fashion.

The BS is constituted by multiple non-overlapping subarrays with dedicated remote processing units (RPUs), which perform independently channel estimation, precoding calculation and RA, mainly AS and PA. Each subarray is equipped with a fixed number of antenna elements and RF transceivers. Using the ZF precoding, the optimization goal is to maximize the SE subjected to the constraints of subarrays connections and maximum transmitted power.

The contribution of this work is fourfold. i) Description of a distributed transceiver design for XL-MIMO based on a subarray switching architecture; ii) proposition of a centralized procedure based on the evolutionary heuristic GA to perform joint AS and PA to maximize the SE with subarray connection and maximum transmitted power constraints; iii) proposition of a distributed version of the GA procedure for joint AS and PA which achieves performance tight to the centralized one but with low-size coordination data and less number of executed operations; iv) extensive analyses of the proposed procedures in terms of number of symbols for training, coordination data size and number of floating point operations per second (flops).

The numerical results corroborate the GA-based procedures in achieving high performance, specifically in crowded XL-MIMO applications. Additionally, the decentralized GA version offers a good trade-off between performance, number of operations and coordination data size, outperforming the centralized procedures by adopting proper settings.

The rest of the paper is organized as follows. In Section II is described the system model, including the distributed subarrays processing at the BS. Next, in Section III are described the centralized and distributed GA-based optimization procedures for joint AS and PA in XL-MIMO systems, while Section IV discusses two feasible AS procedures adopted as a result of decoupling the joint AS and PA optimization problem. Section V examines the complexity of the proposed algorithms. Extensive numerical results are discussed in Section VI. Final comments and conclusions are provided in Section VII.

Boldface small $\mathbf{a}$ and capital $\mathbf{A}$ letters represent respectively vectors and matrices. Capital calligraphic letters $\mathcal{A}$ represent finite sets, and $|\mathcal{A}|$ denotes the cardinality of the set $\mathcal{A}$. $\mathbf{I}_{n}$ denotes the identity matrix of size $n$. $\{\cdot\}^{T}$ and $\{\cdot\}^{H}$ denote respectively the transpose and the conjugate transpose operators. $\textrm{diag}(\cdot)$, $\textrm{tr}(\cdot)$ and $\textrm{det}(\cdot)$ denote respectively the diagonal matrix, trace and determinant operators. $\left\lceil\cdot\right\rceil$ denotes the greatest integer operator. $\binom{n}{k}$ denotes the binomial coefficient. $\mathcal{CN}(\mu,\sigma^{2})$ is a circularly symmetric complex Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$. $\mathbb{E}[\cdot]$ denotes the expectation operator.

In the XL-MIMO scenario, spatial non-stationarities arise due to the large array physical dimensions and number of antenna elements. Such non-stationarities are addressed in the adopted channel model as the variation of the mean received power along the array, as in [22, 17]. The path-loss coefficient associated to the BS antenna $m$ and the user $k$ is defined as

where $q_{0}$ is the path-loss attenuation at a reference distance, $d_{m,k}$ is the distance between the antenna $m$ and the user $k$ and $\kappa$ is the path-loss exponent.

Let $\mathbf{R}_{k}\in\mathbb{C}^{M\times M}$, $\mathbf{R}_{k}=\textrm{diag}([\beta_{1,k}\;\cdots\;\beta_{M,k}]^{T})$ be the matrix with the long-term fading coefficients of the user $k$. The channel vector of the user $k$ is defined as

where $\mathbf{h}_{k}^{\prime}\in\mathbb{C}^{M\times 1}$, $\mathbf{h}_{k}^{\prime}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{I}_{M}\right)$ is the short-term fading vector. From the users channel vectors, the channel matrix $\mathbf{H}\in\mathbb{C}^{M\times K}$ is defined as

considering $\mathbf{\underline{h}}_{m}\in\mathbb{C}^{1\times K}$ as the channel vector with the coefficients associated to the antenna $m$.

During the DL, the BS activates a group of antennas represented by the set $\mathcal{S}\subseteq\left\{1,\dots,M\right\}$ such that $|\mathcal{S}|\leq N$. A partition of the set $\mathcal{S}$, i.e. $\{\mathcal{S}_{b}\}$, $\forall b=1,\dots,B$, contains the index of the selected antennas in the subarray $b$. This set is defined such that $|\mathcal{S}_{b}|\leq N_{b}\;\forall b$, meeting the adopted subarray structure. The equivalent channel matrix of the active antennas is defined as a row-wise submatrix of $\mathbf{H}$, $\mathbf{H}_{\mathcal{S}}{\in\mathbb{C}^{|\mathcal{S}|\times K}}$. Similarly, the matrix $\mathbf{H}_{\mathcal{S}_{b}}{\in\mathbb{C}^{|\mathcal{S}_{b}|\times K}}$ contains only the channel vectors related to the active antennas in the subarray $b$.

Let $D_{m}\in\left\{0,1\right\},\;\forall m=1,\dots,M$ be an indicator equal to 1 if the antenna $m$ is active during the DL and 0 otherwise. These indicators form the diagonal matrix $\mathbf{D}=\textrm{diag}([D_{1}\;\cdots\;D_{M}]^{T})$. During the precoding and SE computations, it is required to calculate the matrix product $\mathbf{H}_{\mathcal{S}}^{H}\mathbf{H}_{\mathcal{S}}$ of the active antennas channel matrix. Intended to enable this computation by the distributed signal processing architecture, the Gramian matrix is defined as in the following.

An upper bound for the system performance considering the active antennas in the set $\mathcal{S}$, namely the DL sum-capacity, is calculated by [14]:

where $\sigma_{z}^{2}$ is the additive noise power, while $\mathbf{P}=\textrm{diag}\left([p_{1}\;\cdots\;p_{K}]\right)$ denotes the matrix with the allocated power for each user. The powers $p_{k},\;\forall k=1,\dots,K$ are defined in order to meet the total power constraint $\sum_{k=1}^{K}p_{k}=P_{\max}$. The DL sum-capacity is achieved by the dirty paper coding (DPC) precoder, which has prohibitive high-complexity for practical implementations.

The data signal transmitted by the BS is defined as $\mathbf{x}\in\mathbb{C}^{|\mathcal{S}|\times 1}$,

where $\mathbf{F}{\in\mathbb{C}^{|\mathcal{S}|\times K}}$ denotes the ZF precoding matrix, calculated by

$\mathbf{s}=[s_{1}\;\cdots\;s_{K}]^{T}$ denotes the vector of modulated data symbols such that $\mathbb{E}\left[\|s_{k}\|_{2}^{2}\right]=1,\;\forall k=1,\dots,K$ and $\mathbb{E}\left[s_{k}^{*}s_{k^{\prime}}\right]=0,\;\forall k\neq k^{\prime}$. The allocated powers in (7) are calculated in order to meet the following power constraint

Therefore, the entries of $\mathbf{P}$ depend on the active antennas set $\mathcal{S}$ and the PA policy.

The signal received by the users in the DL is defined as $\mathbf{y}\in\mathbb{C}^{K\times 1}$,

where $\mathbf{z}\in\mathbb{C}^{K\times 1}$, $\mathbf{z}\sim\mathcal{CN}\left(\mathbf{0},\sigma_{z}^{2}\mathbf{I}_{K}\right)$ denotes the additive noise vector.

Given the ZF precoding design, the system SE is calculated by

which is equivalent to the SE of $K$ independent Gaussian channels with received signal-to-noise ratio (SNR) equal to $p_{k}/\sigma_{z}^{2}\;\forall k$.

The OPA policy is the one that solves the problem of maximizing the system SE at (11), subjected to the maximum power constraint in (9):

The optimization problem in (12) is equivalent to the well-known PA problem on independent Gaussian channels. It has an analytical closed-form solution derived by the Lagrange multipliers method (water filling solution). The optimal power distribution is calculated by [32]:

where $(x)^{+}=\textrm{max}(x,0)$ and $\mu$ is a constant calculated by

If $p_{k}=0$ for some user $k$, the PA problem including this user is not feasible. For this reason, the $k$-th user is deactivated and the power distribution is recalculated considering only the group of the remaining active users. This process must be repeated until a group of users which results in a feasible solution is found.

One metaheuristic procedure adopted to solve many different combinatorial problems in wireless communications is the GA. This technique implements different search phases to efficiently explore the feasible set and exploit the good candidates properties in order to find promising regions in the feasible sub-spaces. Differently from exact optimization methods, evolutionary metaheuristics do not require convex objective functions or constraints. In addition, the execution complexity can be fitted to the available computational burden by adjusting the input parameters and number of iterations. Despite the advantages, the GA, as well as other metaheuristics, does not ensure finding the optimal solution.

As the GA is a procedure inspired by principles of genetics and natural selection, it inherited several terms from biology. To simplify understanding, Table I contains a glossary of some common GA terms adopted throughout this work. In the following, the implemented GA procedures, phases and variables deployed to solve the problem (15) are briefly described.

The implemented GA contains the following phases: a) elitism, b) tournament selection, c) crossover and d) mutation. These phases require the definition of the parameters: population size $N_{p}$, number of individuals for elitism $N_{e}$, number of tournaments $N_{s}$, crossover probability $p_{c}$ and mutation probability $p_{m}$. Each procedure is summarized in the sequel.

Algorithm 2 summarizes the implemented procedure, named genetic algorithm for resource allocation (GA-RA). The set $\mathcal{P}_{0}$ denotes the initial population, $\mathcal{P}_{t}$ the population of the generation $t$, $\mathcal{P}_{s}$ the winners of the tournament selection and $\mathcal{P}_{\rm temp}$ a temporary set for the elitism phase.

The proposed GA-RA procedure requires the entire channel matrix $\mathbf{H}$ knowledge at the CPU to compute the individuals score values. Such requirement is unfeasible in the XL-MIMO scenario due to the high bandwidth to transfer all the channel coefficients associated to thousands of antennas to the CPU. For this reason, one solution that does not depend on the knowledge of full CSI at the CPU is preferable.

One solution to avoid the requirement of full knowledge of the $\mathbf{H}$ matrix consists of performing local AS at each subarray, considering fixed the AS indices in the other subarrays. The contribution of these fixed AS indices can be calculated previously by the CPU and transmitted to the RPUs with reduced bandwidth and processing power resources. Therefore, each subarray can selects its antennas using the GA. The proposed quasi-distributed genetic algorithm for resource allocation (DGA-RA) implements this concept and is presented in the following.

Analyzing the fitness function of the GA-RA procedure in (11), one can observe that it depends on the inverse of the array Gramian matrix, $\mathbf{G}_{\mathcal{S}}^{-1}=(\mathbf{H}_{\mathcal{S}}^{H}\mathbf{H}_{\mathcal{S}})^{-1}$. The computation of $\mathbf{G}_{\mathcal{S}}^{-1}$ can be done from the subarrays Gramian matrices by

Therefore, the CPU can compute the inverse of the array Gramian matrix to calculate the GA-RA fitness function only with the subarrays Gramian matrices calculated locally at the RPUs. Each subarray Gramian matrix has $K^{2}$ entries, while the channel matrix has $MK$. Therefore, calulating the contribution of the selected antennas at the CPU using the Gramian matrix strategy requires less bandwidth than by using the centralized strategy if $BK^{2}<MK$ holds.

Based on (17), the DGA-RA procedure operates as follows. Initially, each subarray selects an active antennas set based on a simple criterion, such as the norm-based antenna selection (N-AS) described in the subsection IV-B. Then, the subarrays compute their Gramian matrices based on the selected set and transmit them to the CPU. At the CPU, the array Gramian matrix is computed by (17) and transmitted back to the subarrays. Afterwards, every subarray performs local antenna selection by a GA implementation, considering that the other subarrays are fixed. To evaluate the fitness function in eq. (11), the subarrays compute the array Gramian inverse matrix adopting the SMW formula for matrix inversion, as follows.

After performing local AS, each subarray transmits their achieved SE values to the CPU. The CPU updates the AS indices of the subarray that has achieved the maximum SE values at the iteration $n$. Then, the CPU requests the subarray Gramian matrix of the updated subarray, and recalculates the inverse of the array Gramian matrix, $(\mathbf{G}_{\mathcal{S}}^{(n)})^{-1}$. The process can be executed iteratively following the scheme depicted in Fig. 3.

The GA implemented in the DGA-RA procedure is similar to that one described in the Algorithm 2, except for some details at the optimization variables encoding and the crossover phase. About the individual encoding, the optimization variables at each subarray are reduced from $M$ to $M_{b}$, since local AS is performed at each RPU. In addition, as the optimization variables consider only one subarray at each RPU, the individuals have two chromosomes: one represented by the first $M_{b}/2$ genes, and another composed by the remaining genes.

Due to this new chromosome definition, one further procedure after the crossover phase is required to preserve the solution’s feasibility. The chosen method is to deactivate antennas of individuals with more than $N_{b}$ antennas in a random fashion until they become feasible.

Firstly, we analize equal power allocation (EPA) strategy, i.e. $\mathbf{P}=\frac{P_{\max}}{K}\mathbf{I}_{K}$, intended to obtain a manageable optimization problem. The problem of selecting the set of active antennas in order to maximize the DL sum-capacity with the constraints of maximum number of RF transceivers and subarray connections is formulated as [14]:

Despite the concavity of the objective function in (22a) [13], the problem (22) is not convex due to the binary constraint in (22c). Hence, we define a convex relaxation of (22) by taking the variables $D_{m}$ in the range $(0,1)$. This new problem, which can be solved with convex optimization tools, has the constraint (22c) replaced by

Notice that the solution of the convex relaxation results in non-binary values for the active antenna indicators $D_{m}$, which is outside the original problem domain.

One method for performing the antenna selection by solving the convex relaxation is to activate the $N_{b}$ antennas with the highest $D_{m}$ values at each subarray. This procedure is named in this work as SCMAX-AS, and is followed by the OPA policy in eq. (13). This AS procedure gives near-optimal results, except for $N\ll M$ [14]. Therefore, in a XL-MIMO system where the number of available RF transceivers is much less than the array antennas, the achieved system SE with the SCMAX-AS algorithm will be sub-optimal.

The N-AS procedure focus on selecting the subset of $N_{b}$ antennas with the highest channel vector norm values [7]. We adopt this method to initiate the population of the GA-based procedures due to its low computational cost. The N-AS method solves the optimization problem formulated as

where the objective function consists of the sum of the squared norms of the channel vectors associated to the selected antennas.

The problem (24) can be solved quickly by selecting the $N_{b}$ antennas with the highest channel vector norms at each subarray. After selection, the PA is performed by the OPA policy in (13).

In the following, we analyze the procedures in terms of training symbols for CSI acquisition. The length of the mutually orthogonal pilot signals used to estimate the channel vectors at the BS depends on: a) the number of users; b) the number of available RF transceivers; c) the number of antennas at the BS.

The number of symbols to acquire the entire channel matrix, required in all the presented procedures except in the N-AS, is $K\left\lceil\frac{M}{N}\right\rceil$. Particularly, the N-AS algorithm requires only the knowledge of the channel vector norms for selection. For this reason, the N-AS can be implemented without explicit channel estimation, supported by physical power-meters [21]. With this implementation, the N-AS requires a total of $2K$ symbols to operate. From this total, $K$ symbols are required to estimate the norms of the channel vectors, and the remaining $K$ symbols are used to estimate the channel vectors associated to the selected antennas.

The coordination data is defined as the data originated at the RPUs that is required at the CPU during the RA procedures. Determining the coordination data size is crucial since it can grow tremendously in the XL-MIMO scenario. In practical implementations, techniques as data compression helps alleviating the high interconnection bandwidth associated to the coordination data. However, such kind of consideration and optimization are out of the scope of this work.

Table II contains the coordination data size associated to the considered RA procedures, detailing the type of required data in each one. The GA-RA and SCMAX-AS procedures require the entire channel matrix at the CPU, while the DGA-RA one relies on the subarrays Gramian matrices. On the other hand, the N-AS procedure does not require any CSI knowledge at the CPU for antenna selection purpose, being the most appealing technique in terms of the coordination data size.

The third complexity metric is the number of flops executed by each procedure. The complexity analyses for the N-AS and the GA-based AS algorithms are as follows. The SCMAX-AS procedure is not considered due to the high complexity associated with computing the number of executed operations by the convex optimization solver.

As the OPA policy involves simple computations, the complexity of the fitness function is reduced to the inversion of the array Gramian matrix. The flops to compute the array Gramian matrix inverse is derived in Appendix B. From this result, the total flops for the GA-RA algorithm is

The numerical results in the following cover the computational complexity of the proposed procedures. In Fig. 7 the coordination data size of the centralized procedures (GA-RA and SCMAX-AS) and the DGA-RA one versus the number of users is illustrated. The curves are evaluated by the expressions in Table II. The result considers $M\in\{512,2048\}$ and,for the DGA-RA procedure, $N_{\rm it}=16$ and $B\in\{2,4,8\}$. Comparing the RA approaches when the number of users is low, the quasi-distributed one get lower coordination data sizes than the centralized procedures. For higher numbers of users, the coordination data size associated to DGA-RA acquires larger values than the obtained by the centralized procedures. This point of inversion of behavior depends on the numbers of antennas, subarrays and iterations w.r.t. the DGA-RA procedure. It is worth mentioning that the coordination data size grows quadratically with $K$ for the DGA-RA procedure, while it grows linearly with $K$ for the centralized RA procedure.

Fig. 7 depicts the coordination data size of the centralized procedures and the DGA-RA one versus the number of antennas in the BS. The results consider $K=50$ and, for the DGA-RA method, $N_{\rm it}\in\{5,16\}$ and $B\in\{2,4,8\}$. The coordination data size grows linearly with $M$ in the centralized procedures, while for the DGA-RA procedure, it does not depend on $M$. In fact, this is the primary aim for choosing a distributed RA technique in XL-MIMO, in which the BS is equipped with an asymptotically high number of antennas.

The next results are related to the complexity in terms of flops. Fig. 8 illustrates the number of flops per processing unit of the GA-based procedures versus the number of available RF transceivers. The curves are evaluated by the eqs. (26) and (27). Such results consider $K=50$ and, for the DGA-RA procedure, $B=8$ and $N_{\rm it}\in\{1,5,16\}$. For low numbers of RF transceivers, the flops’ values for the DGA-RA procedure are lower than the GA-RA algorithm. Again, after a point of inversion of behavior, the flops’ values for GA-RA get lower than the ones for the quasi-distributed procedure. This point of changing of behavior decreases as $N_{\rm it}$ increases.

The curves with the number of flops per processing unit of the GA-based procedures versus the number of users are depicted in Fig. 8. This result considers $N=256$ and, for the DGA-RA procedure, $B=8$ and $N_{\rm it}=\{1,5,16\}$. For low numbers of users, the flops’ values of the GA-RA procedure are lower than the ones get for the DGA-RA. However, this behavior inverts quickly, and the gap between the flops’ values for both centralized and distributed procedures becomes constant. This constant behavior for large $K$ is due to the fact that both eqs. (26) and (27) grow asymptotically with $K^{3}$.