# 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
Finally, it is crucial to address the fact that while the I-BAB algorithm progressively eliminates antennas based on their contribution values to system capacity, this elimination sequence inherently relies on the assumption of a strong correlation between the greedy backward path and the globally optimal forward selection. Therefore, a rigorous validation of this assumption is imperative to ensure the rationality and correctness of the reverse elimination mechanism in the I-BAB algorithm. The mathematical proof is provided in Appendix B. ## 4 Results and Discussion

In this section, we evaluate the performance of the I-BAB algorithm based on two key metrics: channel capacity and computational complexity. For this experiment, we assume that channel matrix \(\mathbf{H}\) is perfectly estimated at the receiver and conduct extensive simulations to approximate the upper bound of system capacity. Even when \(N_{r}\) increases to 256, 10,000 trials are sufficient to achieve accurate results. In each experiment, we randomly generate an independent and identically distributed (i.i.d.) channel matrix \(\mathbf{H}\) and compute the capacity upper bound using Equation (9). We fix \(N_{t}=4\) and gradually increase \(N_{r}\) from 8 to 256 while keeping the number of selected antennas \(d\) constant at 4. To comprehensively evaluate the performance, we compare the I-BAB algorithm against both the conventional BAB algorithm and the greedy selection algorithm in our channel capacity analysis. As a benchmark, we simulate the channel capacity of a small-scale MIMO system with \(N_{t}\) antenna elements at both the transmitter and receiver, emphasizing the advantages of large-scale MIMO systems in enhancing channel capacity. Additionally, we use the channel capacity derived from Equation (9) as a reference for comparison. Due to the reliance of CNN-based training models on exhaustive search algorithms for antenna classification and labeling, their application remains challenging in MIMO systems with \(N_{r}>16\). Therefore, they are not included as reference algorithms in our comparative analysis. Similarly, the self-supervised MCTS algorithm struggles to scale effectively to large-scale MIMO systems with \(N_{r}>64\). Its computational efficiency heavily depends on GPU acceleration, yet the acceleration performance varies significantly across different GPU configurations. Specifically, due to discrepancies in GPU specifications between our experimental setup and that in [15], we have excluded direct comparisons between the I-BAB algorithm and the self-supervised MCTS approach. Figure 2 shows the Cumulative Distribution Function (CDF) curves of channel capacity after antenna selection with the I-BAB algorithm. The horizontal coordinate in Figure 2 represents the channel capacity and the vertical coordinate represents the cumulative distribution function. Here, \(d\) is set to an exact value rather than a range, making it easier to compare with other antenna selection methods. Overall, MIMO systems with antenna selection algorithms demonstrate significantly higher system capacity than small-scale \(N_{t}\times N_{t}\) MIMO systems. As the number of \(N_{r}\) increases from 8 to 256, the system channel \begin{table}
\begin{tabular}{l l l} \hline \hline
**BAB** & **Improvement** & **I-BAB** \\ \hline \multirow{2}{*}{Generates all possible child nodes} & Introduction of child node & “Full generation” \(\rightarrow\) “Selective \\  & generation Formula (Equation (18)) & generation” \\ \hline Natural-order stacking may lead to & Child nodes are sorted by & \\ inefficient deep searches in & \(\Delta i\) (ascending) and reverse- & “Blind search” \(\rightarrow\) “Guided search”. \\ suboptimal branches. & pushed into the stack. & \\ \hline \multirow{2}{*}{Requires separate executions for each \(d\).} & A single execution covers & “Single-objective” \(\rightarrow\) \\  & \(d\in\) [MIN, MAX]. & “Multi-objective” \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Interrelationships between the BAB and the I-BAB algorithms.