# 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
capacity improves accordingly. For instance, at CDF = 1 (representing the maximum achievable capacity), the I-BAB algorithm achieves a peak channel capacity of 23.5 bps/Hz at \(N_{r}\) = 8, increasing to 25.7 bps/Hz at \(N_{r}\) = 256, reflecting a gain of 2.2 bps/Hz. It can also be noted that both the BAB and I-BAB algorithms exhibit progressively tighter convergence toward the theoretical upper bound of the full antenna array configuration. A detailed analysis reveals that within the antenna configuration range of \(N_{r}\) = 8 to 256, the I-BAB algorithm achieves a channel capacity remarkably close to that of the BAB algorithm, while the greedy algorithm also performs comparably to the BAB. For instance, when the CDF = 0.1 with \(N_{r}\) = 8, the channel capacity obtained through the BAB algorithm reaches approximately 17.23 bps/Hz, while a greedy search yields 17.17 bps/Hz and I-BAB achieves 17.09 bps/Hz. This represents a difference of 0.14 bps/Hz compared to the BAB algorithm and 0.08 bps/Hz relative to the greedy search. Notably, at CDF = 1, all three algorithms demonstrate an equivalent maximum channel capacity of 23.5 bps/Hz. Subsequently, we configured \(N_{t}\) = 4 with \(N_{r}\) ranging from 128 to 256 and performed simulations of the channel capacity CDF under different algorithms by setting \(d\) to 8 and 16, respectively. The resultant CDF curves are illustrated in Figures 3 and 4. As illustrated in Figure 2, the channel capacity of various algorithms demonstrates an increasing trend with \(N_{r}\) increasing from 8 to 256 while maintaining constant \(N_{t}\) and \(d\) values. This observation confirms that the channel capacity escalates with the \(N_{r}\) growth when \(d\) is configured as 8 and 16, respectively. To specifically compare the channel capacity discrepancies

Figure 2: CDF of the simulated capacity bounds and capacities with I-BAB search, BAB search and greedy search with \(N_{t}\) = 4, \(d\) = 4 and \(\rho\) = 15 dB. (**a**) \(N_{r}\) = 8; (**b**) \(N_{r}\) = 32; (**c**) \(N_{r}\) = 128; (**d**) \(N_{r}\) = 256.

induced by different antenna selection algorithms under distinct \(d\) configurations, the simulation results for \(N_{r}=8\) and \(N_{r}=32\) are intentionally omitted in Figures 3 and 4, thereby focusing analytical attention on the comparative performance evaluation between different \(d\) scenarios. From Figures 3 and 4, we can see that three antenna selection algorithms exhibit minimal disparities in channel capacity under identical operational conditions. Notably, when \(d=8\), the maximum channel capacities achievable by all three algorithms at CDF = 1 exceed those observed at \(d=4\), irrespective of whether \(N_{r}=128\) or \(N_{r}=256\). Furthermore, at \(d=16\) with CDF = 1, all antenna selection strategies demonstrate higher maximum channel capacities compared to both \(d=4\) and \(d=8\) configurations. Specifically, when \(d=8\) and \(N_{r}=128\), the peak channel capacity reaches 28.64 bps/Hz, exceeding the \(d=4\) case by 3.39 bps/Hz. With \(N_{r}=128\) fixed, increasing \(d\) to 16 yields a maximum capacity of 31.49 bps/Hz, demonstrating a 2.85 bps/Hz improvement over the \(d=8\) configuration. These results corroborate the theory that channel capacity scales with the antenna quantity in multiple-input multiple-output systems, while confirming the broad applicability of the I-BAB algorithm across varying \(d\) values. However, the convergence behavior between the three antenna algorithms and the simulated bound exhibits degradation with an increasing

Figure 4: CDF of the simulated capacity bounds and capacities with I-BAB search, BAB search and greedy search with \(N_{l}=4\), \(d=16\) and \(\rho=15\) dB. (**a**) \(N_{r}=128\); (**b**) \(N_{r}=256\). Figure 3: CDF of the simulated capacity bounds and capacities with I-BAB search, BAB search and greedy search with \(N_{l}=4\), \(d=8\) and \(\rho=15\) dB. (**a**) \(N_{r}=128\); (**b**) \(N_{r}=256\). \(d\), a phenomenon that merits further investigation in subsequent research to optimize algorithmic performance.