# 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
In the low-SNR regime (SNR \(<5\) dB), the capacity grows linearly, while transitioning to the high-SNR regime (SNR \(>5\) dB) results in decelerated growth, demonstrating logarithmic characteristics. This SNR-dependent behavior aligns with fundamental information-theoretic principles governing additive white Gaussian noise channels. It can also be observed that across different SNR levels, massive MIMO systems employing antenna selection algorithms still exhibit significantly higher system capacity compared to \(Nt\times Nt\) small-scale MIMO configurations. Furthermore, we compare the channel capacity performance of three antenna selection algorithms under varying SNR levels. As illustrated in Figure 6, when the SNR increases from 0 to 50 dB, the channel capacity curves of the I-BAB algorithm remain closely aligned with those of BAB and greedy search algorithms. When the SNR \(\leq 15\)dB, the three curves exhibit near-overlapping characteristics, indicating negligible discrepancies between I-BAB and BAB algorithms. Remarkably, during the SNR increase from 0 to 10 dB, the I-BAB algorithm achieves a marginally higher channel capacity than the BAB. For instance, at SNR = 5 dB, I-BAB yields 15.72 bps/Hz versus the BAB's 15.52 bps/Hz, showing a slight increase of 0.2 bps/Hz. However, when the SNR exceeds 15 dB, the I-BAB's capacity begins to underperform the BAB. At SNR = 25 dB, the I-BAB delivers 24.09 bps/Hz compared to the BAB's 24.41 bps/Hz, with a differential of 0.32 bps/Hz. Specifically, under a SNR of 40 dB, the I-BAB algorithm exhibits a 0.51 bps/Hz deficit in channel capacity compared to the BAB algorithm. This performance gap widens to 0.58 bps/Hz when the SNR increases to 50 dB. Empirical observations demonstrate that for SNR levels exceeding 15 dB, the discrepancy in channel capacity between I-BAB and BAB implementations progressively expands with ascending SNR values, albeit at a slow rate of increment. The performance gap remains modest, but the relative inferiority of the I-BAB suggests potential areas for algorithmic refinement. Therefore, enhancing the performance of the I-BAB under SNR \(>15\) dB conditions warrants further investigation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name of Algorithm** & **Value of d** & **Execution Time in Seconds** & **Total Time** \\ \hline \multirow{3}{*}{BAB} & d = 2 & 0.001967s & \multirow{3}{*}{0.172804 s} \\  & d = 4 & 0.004900s & \\ \cline{1-1}  & d = 8 & 0.165937s & \\ \hline I-BAB & d = [2, 4, 8] & 0.030619s & 0.030619 s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Execution time of I-BAB and BAB when \(N_{r}=16\). All results mentioned above are derived from simulations under ideal channel conditions. To validate the practical performance of the I-BAB algorithm, we constructed a 4 \(\times\) 8 MIMO testbed using USRP N310 (Figure 7), which constitutes a MIMO system with \(N_{r}\) = 8 and \(N_{t}\) = 4. The SNR range was configured from 5 to 25 dB to emulate both indoor office environments (with strong multipath propagation) and outdoor open scenarios (dominated by line-of-sight transmission). Channel data acquisition was implemented through Orthogonal Frequency Division Multiplexing (OFDM) pilot signals with a typical 20 MHz bandwidth. Experimental measurements are systematically documented in Tables 4 and 5. Channel capacity represents the theoretical upper bound derived from idealized simulations, whereas the practical throughput denotes the empirically measured achievable rate under real-world channel conditions. As indicated by simulation data in Figure 2, the corresponding channel capacities under this configuration are 23.5 bps/Hz for the BAB and 23.37 bps/Hz for the I-BAB. In Table 4, the aggregate capacity is calculated as the bandwidth multiplied by channel capacity. Under ideal conditions, the actual throughput should closely approach this theoretical maximum.