# Improved Branch-and-Bound Antenna Selection Algorithm for Massive MIMO
## 3 Efficient Antenna Selection via Enhanced Branch-and-Bound Optimization
### Branch-and-Bound Antenna Selection
3:for\(i=1\) to \(|K|\)do  Calculate \(\Delta_{k,\ i},\ k\in K\) endfor  Sort \(\Delta_{k,\ i}\) in descending order to get an ordered index vector \(J\). for\(



Therefore, we raise two critical questions regarding the existing limitations of the BAB algorithm: (1) Given that the BAB algorithm still searches for certain unnecessary nodes, can we develop a mechanism to prune more redundant nodes effectively? (2) When parameter \(d\) extends from a single value to multiple values within a range, is it feasible to obtain optimal subsets corresponding to different \(d\) values through a single algorithm execution? ### Improved Branch-and-Bound Algorithm

In response to the aforementioned two issues, we propose an Improved BAB algorithm (I-BAB). Although the antenna channel capacity increases with the quantity of antennas, practical constraints including hardware costs and computational space limitations necessitate the selection of an optimal subset of antennas to satisfy actual communication requirements. As proposed in [21], the essence of antenna selection lies in constructing an appropriate search tree and executing a tree traversal to identify suitable tree nodes. Through the derivation presented in Section 3.2, we establish a monotonic criterion: if antenna subset \(H_{1}\subseteq H_{2}\), the channel capacity \(C_{1}\leq C_{2}\). The monotonicity remains valid under non-ideal practical conditions, with the rigorous proof provided in Appendix A. The I-BAB algorithm improves the search process by initiating from the complete set of candidate antennas and iteratively generating child nodes through the progressive elimination of antennas exhibiting minimal contributions to channel capacity. As demonstrated in Section 3.2, we can analogously derive the channel capacity decrement value \(\Delta\) resulting from the removal of a specific antenna. Let \(\mathbf{H}_{l}\) denote the matrix obtained by removing \(l\) rows from the channel matrix \(\mathbf{H}\) through a \(l\)-step selection process, where the channel capacity \(C_{l}=\log_{2}\det\left(\mathbf{I}_{N_{l}}+\frac{\rho}{N_{l}}\mathbf{H}_{l}^{ \mathrm{H}}\mathbf{H}_{l}\right)\), \(l=1\), \(2\), \(\cdots\), \(N_{r}-d\), \(C_{0}=\log_{2}\det\left(\mathbf{I}_{N_{l}}+\frac{\rho}{N_{l}}\mathbf{H}^{ \mathrm{H}}\mathbf{H}\right)\). Assuming at the \((l+1)\)th selection step, we select and remove the \(k_{l+1}\)th row from \(\mathbf{H}_{l}\); \(\mathbf{h}_{k_{l+1}}\) denotes \(k_{l+1}\)th row in matrix \(\mathbf{H}_{l}\), resulting in a new antenna subset \(\mathbf{H}_{l+1}=\mathbf{H}_{l}\backslash\left\{\mathbf{h}_{k_{l+1}}\right\}\). Consequently, the channel capacity \(C_{l+1}\) is expressed as:

\[ \begin{array}{ll}C_{l+1}&=\log_{2}\det\left(\mathbf{I}_{N_{l}}+\frac{\rho}{ N_{l}}\mathbf{H}_{l+1}^{\mathrm{H}}\mathbf{H}_{l+1}\right)\\ &=\log_{2}\det\left(\mathbf{I}_{N_{l}}+\frac{\rho}{N_{l}}\mathbf{H}_{l}^{ \mathrm{H}}\mathbf{H}_{l}-\frac{\rho}{N_{l}}\mathbf{h}_{k_{l+1}}\mathbf{h}_{ k_{l+1}}^{\mathrm{H}}\right)\\ &=C_{l}-\log_{2}\det\left(\mathbf{I}_{N_{l}}+\frac{\rho}{N_{l}}\left(\mathbf{I }_{N_{l}}+\frac{\rho}{N_{l}}\mathbf{H}_{l}^{\mathrm{H}}\mathbf{H}_{l}\right)^ {-1}\mathbf{h}_{k_{l+1}}\mathbf{h}_{k_{l+1}}^{\mathrm{H}}\right)\\ &\stackrel{{(2)}}{{=}}\ C_{l}-\underbrace{\log_{2}\left(1+\frac{ \rho}{N_{l}}\mathbf{h}_{k_{l+1}}^{\mathrm{H}}\mathbf{G}_{l}\mathbf{h}_{k_{l+1} }\right)}_{\Delta_{k_{l+1},l}},\end{array}  \tag{15}\]

where step \"(2)\" establishes the Sylvester determinant identity, \(\mathbf{G}_{l}=\left(\mathbf{I}_{N_{l}}+\frac{\rho}{N_{l}}\mathbf{H}_{l}^{ \mathrm{H}}\mathbf{H}_{l}\right)^{-1}\). Similarly, referring to Section 3.2, the loss reduction \(\Delta_{k_{l+1},l}\) by eliminating the \(k_{l+1}\)th antenna can be calculated as:

\[\Delta_{k_{l+1},\ l+1}=\log_{2}\left(1+\frac{\rho}{N_{l}}\alpha_{k_{l+1},\ l+1} \right), \tag{16}\]

where \(\alpha_{k_{l+1},\ l+1}=\mathbf{h}_{k_{l+1}}^{\mathrm{H}}\mathbf{G}_{l+1} \mathbf{h}_{k_{l+1}}=\alpha_{k_{l+1},\ l}-\left|\mathbf{h}_{k_{l+1}}^{\mathrm{ H}}\mathbf{g}_{l+1}\right|^{2}\).