## 1 Introduction
Therefore, optimal joint BF and phase adjustment does not lend itself to an easy solution, and some suboptimal and practical approaches to this problem are needed. In this paper, we propose two AP BF schemes and two IRS phase adjustment schemes, from which four (2 \(\times\) 2) combinations of alternating schemes are presented and their performances are compared in various parameter settings. A SDR-based scheme is proposed for the AP BF, though it is already known to be the optimal one for the MC channel without IRS. Additionally, least squares (LS)-based algorithms that try to equalize user channel strengths are proposed for the AP BF and the IRS phase alignment, respectively. These schemes require a less-demanding computational load compared to the SDR scheme when the number of reflecting elements and/or the number of AP antennas becomes large. Finally, a grouping algorithm of IRS elements is proposed for the IRS phase alignment, where each IRS reflecting element is associated to a specific user to strengthen the paths toward the user. A similar idea is applied to the multiuser channel with an IRS support in [19]. Although the SDR scheme is optimal, it requires high computational load when the dimension of the signal vector increases. On the other hand, the other three are sub-optimal with reasonable computation, and thus are good candidates for practical implementation. This paper is organized as follows. In Section 2, the system model of the proposed multi-antenna MC channel with an IRS is presented. Alternation-based beamforming designs are presented in Section 3. After presenting the corroborating numerical results in Section 4, Section 5 concludes this paper. _Notations_: The notations \(\mathbf{A}^{H}\), \(\mathbf{A}^{\intercal}\), \(\mathbf{A}^{T}\), \(\mathbf{A}^{*}\), and \(\mathbf{Tr}[\mathbf{A}]\) are the Hermitian transpose, the pseudo inverse, the transpose, the conjugate and the trace of a matrix \(\mathbf{A}\), respectively. \(|a|\) and \(\|\mathbf{a}\|\) denote the absolute value of \(a\) and the norm of \(\mathbf{a}\), respectively. The notation \(\mathbf{1}_{I}\) denotes the all-one vector with \(J\) elements and \(\phi(\mathbf{a})\) generates a vector whose \(n\)th element is the phase of the \(n\)th element of \(\mathbf{a}\) with unit magnitude. \(\mathbf{A}_{i,j}\) represents the \((i,j)\)th element of \(\mathbf{A}\); \(\mathbf{a}(i)\) represents the \(i\)-th element of vector \(\mathbf{a}\); \(\text{diag}[\mathbf{A}]\) returns a vector whose elements are the diagonal elements of the matrix \(\mathbf{A}\) and \(\text{diag}[\mathbf{a}]\) returns a diagonal matrix with the elements of the vector \(\mathbf{a}\) as its diagonal entries. \(\mathbf{a}\sim\mathcal{CN}(\mathbf{0},\mathbf{C})\) denotes a complex white Gaussian random vector \(\mathbf{a}\) with zero mean vector \(\mathbf{0}\) and the covariance matrix \(\mathbf{C}\). \(\mathbb{E}[a]\) takes the expectation of \(a\) and \(\angle(\mathbf{a})\) returns the phase vector of a complex vector \(\mathbf{a}\). The function \(\lambda_{max}(\mathbf{A})\) returns the maximum singular value of the matrix \(\mathbf{A}\). The operator \([a,b]^{+}\) returns the \(\max\{a,b\}\).