Joint Antenna Selection and Multicast Precoding in Spatial Modulation Systems

Abstract

In this paper, the downlink of the multicast based spatial modulation systems is investigated. Specifically, physical layer multicasting is introduced to increase the number of access users and to improve the communication rate of the spatial modulation system in which only single radio frequency chain is activated in each transmission. To minimize the bit error rate (BER) of the multicast based spatial modulation system, a joint optimizing algorithm of antenna selection and multicast precoding is proposed. Firstly, the joint optimization is transformed into a mixed-integer non-linear program based on single-stage reformulation. Then, a novel iterative algorithm based on the idea of branch and bound is proposed to obtain the quasioptimal solution. Furthermore, in order to balance the performance and time complexity, a low-complexity deflation algorithm based on the successive convex approximation is proposed which can obtain a sub-optimal solution. Finally, numerical results are showed that the convergence of our proposed iterative algorithm is between 10 and 15 iterations and the signal-to-noise-ratio (SNR) of the iterative algorithm is 1-2dB lower than the exhaustive search based algorithm under the same BER accuracy conditions.

1. Introduction

Spatial modulation is an energy efficient transmission scheme, in which only one antenna is activated during each transmission. Hence only one radio frequency (RF) chain is needed in the spatial modulation systems (SMSs). In SMSs, the input signals are transmitted at both the transmit antenna (TA) domain and amplitude-phase modulation (APM) domain [1]. Although SMSs utilize the spatial dimension to transmit information bits, they lack transmit diversity due to limited number of active transmit antennas. Therefore, the achievable transmission rate is imposed a few limitations at the APM domain. On the other hand, physical layer multicasting is a spectral efficient transmission scheme, in which a common message is multicasted to a couple of users [2]. The integration of multicast and spatial modulation is highly desired. The multicast based SMSs can achieve good balance between the energy efficiency and spectral efficiency.

When channel state information is available at the base station, transmit antenna selection and transmit precoding can be employed to enhance the performance of SMSs. Specifically, the diversity order can be obtained by employing transmit antenna selection [3], [4]. Moreover, for SMSs, when the total number of transmit antennas is not the power of 2 in practice, the antenna selection process is also needed [4]. On the other hand, the precoding can be optimized to shape the received APM signal constellation and boost the bit error rate (BER) performance [5]–[7]. Note that in SMSs, only one antenna is activated in each transmission period and the precoding matrix should be diagonal [7].

In the references of multicast spatial modulation, reference [9] firstly analyzes the BER performance of the multicast SMSs and provide a closed-form asymptotic BER upper by exploiting the system statistics. Finally, reference [9] studies the impact of the receiver number on BER in the multicast spatial modulation. Furthermore, reference [10] proposes two heuristic transmit mode selection algorithms, including tree pruning technique and approximating the signal constellation. Then reference [10] also design a precoder approach to improve the system. Similarly, reference [8] considers both antenna selection and the precoding coefficients, and gives the optimization problem. However, the optimization problem was not solved directly and reference [8] uses exhaustive search algorithm and convex optimization tools in turn to solve antenna selection and precoding design problems.

In this paper, antenna selection and multicast precoding in multicast based SMSs are design jointly. To facilitate the joint optimization of antenna selection and precoding, a single-stage reformulation is proposed to reconstruct the problem of joint optimization into a mix-integer non-linear program. Secondly, a branch and bound based algorithm is designed to obtain the quasi-optimal solution, whose convergence behavior and performance superiority are verified by the numerical results. Specifically, when the number of iterations reaches 10 to 15, the proposed branch and bound based algorithm converges with the accuracy of 0.0001, which is defined as the gap between the obtained upper bounds and lower bounds. This shows that the proposed algorithm converges fast, thus proving the effectiveness of joint antenna selection and multicast precoding in SMS. Moreover, a low-complexity deflation algorithm based on successive convex approximation (SCA) is proposed to obtain a suboptimal solution, which can achieve BER performance close to the exhaustive search based algorithm. To the best of our knowledge, this is the first work studying the single-stage reformulation for jointly solving the antenna selection and precoding problems. Meanwhile, under the same BER condition, the SNR corresponding to the deflation algorithm achieve 1dB lower than the result of the exhaustive search based algorithm.

Notations: We employ uppercase boldface letters for matrices and lowercase boldface for vectors. E(⦁), Tr(⦁), ║⦁║_p, (⦁)^T, (⦁)^H and Re(⦁) return the expectation, trace, pl -norm, transpose, conjugate transpose and the real part of the input respectively.

2. System Description

As shown in Fig. 1, we consider the downlink of a multicasting SMS where a base station (BS) multicasts a common message to K users simultaneously using both the TA domain and APM domain signal transmission [1]. The BS is equipped with N_tot antennas whereas each user equipment (UE) has N_R antennas. For brevity, we denote K = {1, 2,...,K} and N_tot = {1, 2,..., N_tot} as the user set and transmit antenna set, respectively.

Fig. 1. Downlink of a multicasting SMS.

2.1 Signal Model

In each transmission, the BS selects NSM antennas from Ntot to perform spatial modulation, where NSM is power of 2 [3]. There are total CNSMNtot possible combinations of antenna selection. I is denoted as the set of selected NSM antennas with |I| = NSM, and L as the set of enumerations of all possible antenna selections with |I| = CNSMNtot. Let Hk ∈ CNRxNtot be the channel matrix from the BS to the k -th user. Then after antenna selection, the channel of user k used for spatial modulation becomes Hk,I ∈ CNSMxNtot where I ∈ L. In each transmission, the first log2 NSM bits of the transmitted signal are mapped to one point in the spatial constellation set defined as

S = {e₁, ..., e_i, ..., e_NSM}       (1)

where e_i ∈ C^N_SMx1 is an all-zero vector except for the i-th entry, which is 1. The selection of e_i indicates that the i-th antenna among all the selected N_SM antennas is activated. Next, the last log₂ M bits are transmitted from the activated antenna by M-QAM or M-PSK signal constellation defined by

M = {s₁, ..., s_m, ..., s_M}       (2)

where s_m ∈ C with E{|s_m|²} = 1

The spectral efficiency in the described system can be computed as log₂ N_SM M bits per channel use. The transmitted spatial modulation constellation SM with cardinality N_SMM is the Cartesian product of S and M which is given by

SM = {e₁s₁, ..., e_is_m, ..., e_NSMs_M}       (3)

Next, denote the transmitted spatial modulation symbol as x_l = e_is_m ∈ C^N_SMx1 with x_l ∈ SM. Then x_l is precoded by a diagonal matrix W = diag(w) ∈ C^N_SMxN_SM with w = [w_I1, ...,W_ISM]^T ∈ C^N_SMx1 given I. Then the received signal at user k given antenna selection I ∈ L can be written as

y_k = H_k,IWx_l + n_k = h_k,IiW_IiS_m + n_k       (4)

Where, I_i is the i-th element in I and h_k,Ii is the i-th column in H_k,I which also is the I_i-th column in the total channel matrix H_k. And w_Ii is the precoding coefficient for antenna I_i in N_tot. Moreover, n_k ~ CN(0,σ²I_NR) denotes the additive noise at user k.

At the receiver side, user k decodes the TA domain signal (i.e., index of the activated antenna) and APM domain signal jointly with maximum likelihood (ML) detector as follows

\(\begin{align}\hat{x}_{l}^{\mathrm{ML}}=\arg \min _{\hat{x}_{l} \in \mathcal{S} M}\left\|\mathbf{y}_{k}-\mathbf{H}_{k, I} \mathbf{W} \hat{x}_{l}\right\|_{2}^{2}, \forall k \in \mathcal{K}\end{align}\)       (5)

The error performance of the ML detector at user k can be approximated by the sum of the pairwise error probability (PEP), which is given by [5]

\(\begin{align}P_{k}^{\mathrm{Errror}}(I, \mathbf{w}) \leq \sum_{l=1}^{N_{\mathrm{SM}} M} \sum_{t=1, t \neq l}^{N_{\mathrm{sM}} M} Q\left(\sqrt{\frac{1}{2 \sigma^{2}} d_{k, l t}(I, \mathbf{w})}\right)\end{align}\)       (6)

where, \(\begin{align}Q(x)=\frac{1}{\sqrt{2 \pi}} \int_{x}^{\infty} e^{-\frac{y^{2}}{2}} d y\end{align}\) denotes the Gaussian tail probability [1], and \(\begin{align}d_{k, l t}(I, \mathbf{w})=\left\|\mathbf{H}_{k, I} \mathbf{W}\left(x_{l}-x_{t}\right)\right\|_{2}^{2}\end{align}\) is the squared Euclidian distance between two different spatial modulation symbols x_l ∈ SM and x_t ∈ SM observed at receiver k. Similar to [1] and [5], we employ the nearest neighbor approximation for PEP in (6) and we have

\(\begin{align}\tilde{p}_{k}^{\mathrm{e}}(I, \mathbf{w})=\lambda Q\left(\sqrt{\frac{1}{2 \sigma^{2}} d_{\min }^{k}(I, \mathbf{w})}\right)\end{align}\)       (7)

where, λ is the number of neighboring constellation points [1] and

\(\begin{align}\begin{array}{l}d_{\text {min }}^{k}(I, \mathbf{w})=\min _{\substack{x_{l}, x_{x} \in \mathcal{S} \\ x_{l} \neq x_{i}}}^{k}\left\|\mathbf{H}_{k, I} \mathbf{W}\left(x_{l}-x_{t}\right)\right\|_{2}^{2} \\ =\min _{\substack{i, j \in \mathcal{N}_{\mathrm{S}}, i \leq j \\ S_{m}, S_{n}, \mathcal{M} \\(i, m) \neq(j, n)}}\left\|\mathbf{h}_{k, I_{i}} \mathrm{~W}_{I_{i}} S_{m}-\mathbf{h}_{k, I_{j}} \mathrm{~W}_{I_{j}} S_{n}\right\|_{2}^{2} \\\end{array}\end{align}\)       (8)

In multicast system, the system performance is dominated by the worst user. Similar to (9), the BER performance of the considered multicasting SMS is defined by the worst BER among all the users, and can be written by

\(\begin{align}\tilde{P}^{\mathrm{e}}(I, \mathbf{w})=\max _{k \in \mathcal{K}} \tilde{P}_{k}^{\mathrm{e}}(I, \mathbf{w})\end{align}\)       (9)

2.2 Problem Formulation

In this paper, the BER of the multicasting SMS by jointly optimizing the transmit antenna selection and multicast precoding subject is minimized to the power constraint at the BS, which can be formulated as

\(\begin{align}\begin{array}{c}\min _{I, \mathbf{w}} \max _{k \in \mathcal{K}} \tilde{P}_{k}^{\mathrm{e}}(I, \mathbf{w}) \end{array} \end{align}\)       (10a)

s.t. ||w||²₂ ≤ P_T       (10b)

where, P_T is the total power constraint at the BS.

Since \(\begin{align}\tilde{P}_{k}^{\mathrm{e}}(I, \mathbf{w})\end{align}\) in (7) is a monotonically decreasing function of d^k_min(I,w), the objective (10a) is equivalent to max_I,w min_k∈Kd^k_min(I,w). Hence, problem (10) can be equivalently written as

\(\begin{align}\max _{I, \mathbf{W}} \min _{k \in \mathcal{K}} \min _{\substack{i, j \in \mathcal{N}_{\mathrm{SM}}, i \leq j \\ S_{m}, S_{n} \in \mathcal{M} \\(i, m) \neq(j, n)}}\left\|\mathbf{h}_{k, I_i} \text {w}_{I_i} S_{m}-\mathbf{h}_{k, I_j} \text {w}_{I_j} s_{n}\right\|_{2}^{2}\end{align}\)       (11)

s.t. ||w||²₂ ≤ P_T       (11b)

3. Single-stage Reformulation

In the formulation (11), the antenna selection variable I is a set of antenna indices, which cannot be designed jointly with precoding coefficients w unless enumerating all possible antenna combinations as in formulation (8). In what follows, we integrate the antenna selection and multicast precoding into an equivalent single-stage reformulation of (11), by which we can leverage the tools of optimization to design them jointly. To this end, we introduce a binary vector b = [b₁,...,b_Ntot] to denote the antenna selection, i.e., b_i =1 if antenna i is selected; otherwise, b_i = 0. Since there are N_SM antennas selected, we have \(\begin{align}\sum_{i=1}^{N_{\mathrm{tot}}} b_{i}=N_{\mathrm{SM}}\end{align}\). Note that there is an one-to-one mapping between b and I. Next, using the idea of Big-M formulation (11), formulation (11) is reformulated as follows

\(\begin{align}\max _{I, \mathbf{W}} \min _{k \in \mathcal{K}} \min _{\substack{i, j \in \mathcal{N}_{\mathrm{SM}}, i \leq j \\ s_{m}, S_{m} \in \mathcal{M} \\(i, m) \neq(j, n)}}\left\|\mathbf{h}_{k, i} v_{i} S_{m}-\mathbf{h}_{k, j} v_{j} s_{n}\right\|_{2}^{2}+\Phi_{i j}^{k}\end{align}\)       (12a)

s.t. b_i∈{0,1}, ∀i∈N_tot       (12b)

\(\begin{align}\sum_{i=1}^{N_{\text {tot }}} b_{i}=N_{S M}\end{align}\)       (12c)

|v_i|² ≤ b_iP_T, ∀i ∈ N_tot       (12d)

||v||²₂≤P_T       (12e)

where Φ^k_ij = (1 - b_i)Ω_k + (1 - b_j)Ω_k. Constant Ω_k has a sufficiently large value which should be chosen such that when b_i = 0 or b_j = 0 the term ||h_k,iv_is_m - h_k,jv_js_m||²₂ + Φ^k_ij is inactive in the second min function of (12a) for every feasible channel realization, precoding coefficients, and APM symbols. A simple choice could be \(\begin{align}\Omega_{k}=\max _{i \in \mathcal{N}_{\text {tot }}} 2 P_{T}\left\|\mathbf{h}_{k, i}\right\|_{2}^{2}\end{align}\). The variable v_i is the precoding coefficient. Besides, v = [v₁,...,v_Ntot]^T ∈ C^N_totx1. Since there are N_SM antennas selected from N_tot, we have ||v||₀ = N_SM which is guaranteed by constraints (12c) and (12d). We note that the indices i, j in (12) take values from 1 to N_tot, which is different from i, j in (11) taking values from 1 to N_SM.

Proposition 1: Problems (11) and (12) are equivalent.

Proof: First, based on the definition of binary variable b_i =1 and constraint (12c), we can see that there is an one-to-one mapping between and I ∈ L in (11) and b in (12). Also, constraints (12c) and (12d) ensure ||v||₀ = N_SM and only the selected antennas have non-zero precoding coefficients. Hence, there is an one-to-one mapping between w in (11) and v in (12).

On the other hand, for the objective in (12a), we denote

\(\begin{align}\tilde{d}_{\min }^{k}(\mathbf{b}, \mathbf{v})=\min _{\substack{i, j \in \mathcal{N}_{\mathrm{SM}}, i \leq j \\ s_{m}, s_{n} \in \mathcal{N} \\(i, m) \neq(j, n)}}\left\|\mathbf{h}_{k, i} v_{i} s_{m}-\mathbf{h}_{k, j} v_{j} s_{n}\right\|_{2}^{2}+\Phi_{i j}^{k}\end{align}\)       (13)

Then to prove that (11) and (12) are equivalent, we only need to prove \(\begin{align}\tilde{d}_{\text {min }}^{k}(\mathbf{b}, \mathbf{v})=\tilde{d}_{\text {min }}^{k}(I, \mathbf{w})\end{align}\) for any given antenna selection, precoding coefficients and user index. Suppose I = {I₁,...,I_NSM} are the selected N_SM antennas and the corresponding precoding coefficients are w = {w_I1,..., w_INSM}. Let tot I^c = N_tot\I, i.e., the relative complement of set I with respect to set N_tot, denote the set of unused antennas. Hence we have b_i =1 and v_i = w_i if i ∈ I; otherwise, if i ∈ I^c, we have b_i = 0 and v_i = 0. Then it follows that \(\begin{align}\tilde{d}_{\text {min }}^{k}(\mathbf{b}, \mathbf{v})=\min \left\{\tilde{d}_{\min }^{k, I}(\mathbf{b}, \mathbf{v}), \tilde{d}_{\text {min }}^{k, I^{c}}(\mathbf{b}, \mathbf{v})\right\}\end{align}\), where

\(\begin{align}\tilde{d}_{\min }^{k, I}(\mathbf{b}, \mathbf{v})=\min _{\substack{i, j \in I, i \leq j \\ s_{m}, s_{n} \in \mathcal{M} \\(i, m) \neq(j, n)}}\left\|\mathbf{h}_{k, i} v_{i} s_{m}-\mathbf{h}_{k, j} v_{j} s_{n}\right\|_{2}^{2}\end{align}\)       (14)

\(\begin{align}\tilde{d}_{\min }^{k, I^{c}}(\mathbf{b}, \mathbf{v})=\min _{\substack{i \in I^{c} \| j \in I^{c}, i \leq j, j \\ s_{m}, s_{n} \in \mathcal{M} \\(i, m) \neq(j, n)}}\left\|\mathbf{h}_{k, i} v_{i} s_{m}-\mathbf{h}_{k, j} v_{j} s_{n}\right\|_{2}^{2}+\Phi_{i j}^{k}\end{align}\)       (15)

where i ∈ I^C || j ∈ I^C means that at least one element in {i, j} belongs to I^c, i.e., at least one element in {b_i, b_j} is zero.

Note that in (14), i and j are the indices of selected transmit antennas and b_i = b_j = 1. Hence, (1 - b_i)Ω_k + (1 - b_j)Ω_k = 0.

Recalling (8), we can obtain that \(\begin{align}\tilde{d}_{\text {min }}^{k, I}(\mathbf{b}, \mathbf{v})=\tilde{d}_{\text {min }}^{k}(\mathbf{b}, \mathbf{v})\end{align}\). Since \(\begin{align}\Omega_{k}=\max _{i \in \mathcal{N}_{\text {tot }}} 2 P_{T}\left\|\mathbf{h}_{k, i}\right\|_{2}^{2}\end{align}\), we have \(\begin{align}\tilde{d}_{\text {min }}^{k, I}(\mathbf{b}, \mathbf{v})<\tilde{d}_{\min }^{k, I^{c}}(\mathbf{b}, \mathbf{v})\end{align}\) which results in \(\begin{align}\tilde{d}_{\min }^{k}(\mathbf{b}, \mathbf{v})=\tilde{d}_{\min }^{k, I}(\mathbf{b}, \mathbf{v})=\tilde{d}_{\min }^{k}(I, \mathbf{w})\end{align}\). Hence, (11) and (12) are equivalent.

4. Joint Antenna Selection and Multicast Precoding Design

The single-stage reformulation in problem (12) is a mixed-integer non-linear program (MINLP), which is NP-hard to solve. Besides, the objective in (12a) is non-smooth and non-convex. To proceed, we first show that problem (12) can be further reformulated into a mixed-integer quadratically constrained quadratic programming (MI-QCQP) problem.

4.1 Equivalent formulation

Proposition 2: The MINLP problem in (12) is equivalent to the following MI-QCQP problem

\(\begin{align}\max _{\mathbf{b}, \mathbf{v}} \min _{k \in \mathcal{K}} \min _{\substack{i, j \in \mathcal{N}_{\text {tot }} \\ i \leq j \\ s_{m}, s_{n} \in \mathcal{M} \\(i, m) \neq(j, n)}} \mathbf{v}^{H} \mathbf{R}_{i m, j n}^{k} \mathbf{v}+\Phi_{i j}^{k}\end{align}\)       (16a)

s.t. b_i ∈ {0,1}, ∀i ∈ N_tot       (16b)

\(\begin{align}\sum_{i=1}^{N} b_{i}=N_{S M}\end{align}\)       (16c)

|v_i|² ≤ b_iP_T, ∀i ∈ N_tot       (16d)

||v||²₂ ≤ P_T       (16e)

where, R^k_im,jn = H^H_kH_koΘ^T_im,jn is positive semidefinite matrix in which the matrix Θ_im,jn = (e_is_m - e_js_n)(e_is_m - e_js_n)^H. The operator o denotes the Hadamard product. Moreover, e_i ∈ C^N_totx1 is an all-zero vector except for the i-th entry, which is 1.

Proof: In order to see the equivalence between the objective functions in (16a) and (12a), we denote the distance as d^k_im,jn = ||h_k,iv_is_m - h_k,jv_js_n||². Next, introduce a diagonal matrix V = diag(v) ∈ C^N_totxN_tot where diag(v) represents the diagonal matrix which its main diagonal elements are the components of the vector v. Then we have

\(\begin{align}\begin{aligned} d_{i m, j n}^{k} & =\left\|\mathbf{H}_{k} \mathbf{V}\left(\mathbf{e}_{i} s_{m}-\mathbf{e}_{j} s_{n}\right)\right\|^{2} \\ & =\operatorname{Tr}\left\{\mathbf{V}^{H} \mathbf{H}_{k}^{H} \mathbf{H}_{k} \mathbf{V}\left(\mathbf{e}_{i} s_{m}-\mathbf{e}_{j} s_{n}\right)\left(\mathbf{e}_{i} s_{m}-\mathbf{e}_{j} s_{n}\right)^{H}\right\} \\ & =\operatorname{Tr}\left\{\mathbf{V}^{H} \mathbf{H}_{k}^{H} \mathbf{H}_{k} \mathbf{V} \boldsymbol{\Theta}_{i m, j n}\right\} \\ & \stackrel{\text { (a) }}{=} \mathbf{v}^{H}\left(\mathbf{H}_{k}^{H} \mathbf{H}_{k} \circ \boldsymbol{\Theta}_{i m, j n}^{T}\right) \mathbf{v} \\ & =\mathbf{v}^{H} \mathbf{R}_{i m, j n}^{k} \mathbf{V}\end{aligned}\end{align}\)      (17)

where, (a) follows from Eq. (1.10.6) in (12). Hence, the equivalence of problems (16) and (12) has been proved.

Next, the MI-QCQP problem (16) can be further converted (16) to the following equivalent problem

\(\begin{align}\max _{\mathbf{b}, \mathbf{v}, t} t\end{align}\)       (18a)

s.t. v^HR^k_im,jnv + Φ^k_ij ≥ t, ∀k ∈ K, i, j ∈N, i ≤ j       (18b)

\(\begin{align}\begin{array}{r}s_{m}, s_{n} \in \mathcal{M},(i, m) \neq(j, n) \\ (16 \mathrm{~b})-(16 \mathrm{e})\end{array}\end{align}\)       (18c)

4.2 Quasi-Optimal Solution Based on Branch and Bound

In this subsection, we design a quasi-optimal algorithm for solving (18) by employing the branch and bound framework, in which branching and bounding are main tasks. In general, the branching step divides the feasible region into subsets and constructs sub-problems with those subsets. The bounding step finds the upper and lower bounds for those sub-problems within the corresponding subset. Specifically, we denote Q = − {b, v|(16b) - (16e)} as the feasible region. Then we branch the feasible region Q according to integer variables b , and divide the original problem into sub-problems accordingly.

Branching Step: Pick any index i , and we can divide the origin set Q into two subsets Q₁ = − {b, v|b_i = 0, (16b) - (16e)} and Q₂ = {b, v|b_i = 1, (16b) - (16e)}. Then the bounding steps described below are performed to obtain the upper and lower bounds of the original problem in these subsets respectively. Next, the subset with respect to the current overall upper bound is chosen to be further branched.

Bounding Step: In this step, we calculate the upper and lower bounds for the objective of (18) in different subsets. For convenience, we introduce ξ = [ξ₁,...,ξ_Ntot] ∈ R^N_totx1 as the state vector for the integer variables b and the corresponding subset Q_ξ in the branching step, the elements of which take values from {0,1, -1}. Specifically, ξ_i =1 or 0 indicates that b_i =1 or 0. If ξ_i = −1, b_i has not been branched yet. Denote A_ξ = {i ∈ N_tot|ξ_i = 0or1}. Then subset Q_ξ which is divided by A_ξ can be written as Q_ξ = {b, v|b_i = ξ_i(i ∈ A_ξ), (16b) - (16e)}.

Next, to obtain the upper bound in subset Q_ξ, we relax the undetermined binary variables b_i(i∈N_tot\A_ξ) into 0 ≤ b_i ≤ 1, and then solve problem (18) using the SCA framework [13], which convexity the nonconvex constraint (18b) and iteratively approximate the nonconvex problem to convex ones. Specifically, at SCA iteration l, the left-hand-side of (18b) can be lower bounded by its first-order Taylor series expansion around v^l, and (18b) can approximated by

2Re(V^l,HR^k_im,jnv) - V^l,HR^k_im,jnv^l + Φ^k_i,j≥ t       (19)

Then the approximated convex problem at SCA iteration l can be written as

\(\begin{align}\max _{\mathbf{b}, \mathbf{v}, t} t\end{align}\)       (20a)

s.t. b_i = ξ_i, i ∈ A_ξ       (20b)

0 ≤ b_i≤ 1, i ∈N_tot\A_ξ       (20c)

(16c) - (16e), (19)       (20d)

which is convex and can be solved by modern solvers. The SCA algorithm is summarized for obtaining v giving ξ in Algorithm 1. The value of t at convergence is the upper bound.

Algorithm 1 SCA Algorithm for Obtaining (v,t) v t Given ξ

Function [U*,b*] = UB(ξ) is defined to represent the processing for computing the upper bound, in which U* is the obtained upper bound and b* = [b*₁,...,b*_Ntot] is not integral. In order to get the lower bound for the objective of (18) in Q_ξ, round b*_i(i ∈ A_ξ) to 0 or 1, and denote the obtained integer variables as be \(\begin{align}\tilde{\mathbf{b}}\end{align}\). Then similar to Algorithm 1, we apply SCA algorithm to solve (18) given be \(\begin{align}\tilde{\mathbf{b}}\end{align}\), and the obtained lower bound is denoted as L* = LB(ξ, \(\begin{align}\tilde{\mathbf{b}}\end{align}\)). If it is infeasible, the lower bound in subset Q_ξ is L* = ∞.

The branching and bounding steps are performed iteratively until the overall upper bound and overall lower bound converge. The details of the proposed branch and bound algorithm for solving problem (18) is summarized in Algorithm 2.

Algorithm 2 Algorithm Based on Branch and Bound

4.3 Low-complexity Suboptimal Deflation Algorithm Based on SCA

In this subsection, we propose a low-complexity algorithm for solving (18) in which the integer variables b are treated by the deflation process [14]. At first, all N_tot antennas are assumed to be selected. Next, we drop some poor communication link one by one until there remains NSM antennas, i.e., ║b║₀ = N_SM. In each step of the deflation process, after the remaining set of antennas is determined, i.e., giving b , the precoding vector v is obtained by solving (18) using the SCA framework similar to Algorithm 1.

Then the approximated convex problem at SCA iteration l can be formulated as

\(\begin{align}\max _{\mathbf{v}, t} t\end{align}\)       (21a)

s.t. (16c) - (16e), (19)       (21b)

The overall procedure of deflation SCA algorithm for solving problem (18) is summarized in Algorithm 3.

Algorithm 3 Deflation Algorithm Based on SCA

4.4 Complexity Analysis

In this section, the complexity of the proposed algorithms is discussed. Firstly, in Algorithm 3, the number of the subsets are needed to be considered increases exponentially with the problem dimension. Problem (20) approximately consist of 2N_tot + 1 variables and 1 second-order cone constraint of dimension N_tot . Hence, In the worst case, we need to traverse all 2N_tot − 2 subsets and the worst-case complexity is given by O(w^N_tot+1N²_tot). Secondly, the iteration times in Algorithm 3 is N_tot - N_SM, and the constraints in problem (21) consist of N_tot +1 variables and 1 second-order cone constraint of dimension N_tot approximately. Thus, the complexity of Algorithm 3 can be given by O(N³_tot).

5. Numerical results

In this section, we present numerical results to evaluate the performance of the proposed algorithms, where K = 2, N_R = 2, N_SM = 2, ε = 10⁻⁴ and P_T = N_SM Watt. Moreover, QPSK constellations are employed to modulate the source information bits.

Firstly, we investigate the convergence behaviors of the branch and bound based algorithm in Algorithm 2 and the SCA algorithm in Algorithm 3 as depicted in Fig. 2 and Fig. 3, respectively. The parameters are set as N_tot = 6 and SNR = 3dB. The results in Fig. 2 corroborate the fact that the proposed branch and bound algorithm converges to its optimal value exactly. Note that in the first three iterations, the value of the lower bound is ∞, so it is not drawn in Fig. 2. Then, as the number of iterations increases, the upper bound decreases and the lower bound increases until convergence. Moreover, as shown in Fig. 3, we can see that the SCA algorithm converges fast toward the optimal value illustrated by the dotted line therein.

Next, we investigate the BER comparison versus the signal-to-noise-ratio (SNR) in Fig. 4 and Fig. 5 with N_tot = 4 or 6. As shown in Fig. 4 and Fig. 5, the branch and bound algorithm can achieve the best BER performance among all the algorithms since it can get the optimal solution, and the superiority becomes more pronounced as N_tot increases. Moreover, the performance of the deflation SCA algorithm is close to the exhaustive search based algorithm while avoiding enumerating all the antenna combinations. Furthermore, the BER performance of these algorithms with N_tot = 6 in Fig. 4 is better than that with N_tot = 4 in Fig. 5.

Fig. 2. Convergence behavior of Algorithm 2 giving N_tot = 6, SNR = 3dB with QPSK.

Fig. 3. Convergence behavior of SCA in Algorithm 3 giving N_tot = 6 , SNR = 3dB with QPSK.

Fig. 4. BER for SMS giving N_SM = 2 with QPSK, N_tot = 4 .

Fig. 5. BER for SMS giving N_SM = 2 with QPSK, N_tot = 4 .

6. Conclusion

In this paper, the transmit antenna selection and multicast precoding problem has been treated in multicast based SMSs. To this end, an equivalent single-stage reformulation of the original problem has been proposed to leverage the tools of optimization to design antenna selection and precoding coefficients jointly and simultaneously. Then, two algorithms to solve the reformulated mixed-integer non-linear problem have been proposed, where a branch and bound based iterative algorithm that can achieve quasi-optimal solution, and a deflation SCA algorithm that achieves elegant performance with low-complexity. The outcomes of this study highlight the significance of our approaches in improving the performance of transmit antenna selection and multicast precoding in SMSs. Numerical results demonstrate the convergence and efficiency of the proposed schemes, the convergence of our proposed iterative algorithm is between 10 and 15 iterations. Moreover, when the iterative algorithm and the exhaustive search based algorithm take the same BER accuracy conditions, the signal-to-noise-ratio (SNR) of the iterative algorithm is 1-2dB lower than the exhaustive search based algorithm.