# Low-Complexity Online Calibration for Large-Scale Multi-Beam Antennas
## Appendix A Expectation and Variance for \(y_{k}\)  
There are two random variables in \(y_{k}\): \(u_{k}\) and \(x_{k}\). From the above analysis, the variable \(u_{k}\) follows Gaussian distribution \(\mathcal{N}(0,1/L)\) and the variable \(x_{k}\) follows complex Gaussian distribution \(\mathcal{CN}(0,\sigma^{2})\). Because the square of the standard normal distribution follows the chi-square distribution and \(\sqrt{L}u_{k}\sim\mathcal{N}(0,1)\), then \(Lu_{k}^{2}\sim\chi^{2}(1)\). Then the expectation and  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **Beam 1** & **Beam 2** & **Beam 3** & **Beam 4** & **Beam 5** & **Beam 6** & **Beam 7** & **Beam 8** \\ \hline True values & 0 dB & 0 dB & 0 dB & \(-5\) dB & \(-10\) dB & \(-15\) dB & \(-20\) dB & \(-25\) dB \\ Means & / & \(-0.0251\) dB & \(-0.00261\) dB & \(-5.0204\) dB & \(-10.0204\) dB & \(-14.9986\) dB & \(-19.9986\) dB & \(-24.9899\) dB \\ RMSEs & / & 0.0021 dB & 0.0021 dB & 0.0030 dB & 0.0036 dB & 0.0073 dB & 0.0137 dB & 0.0236 dB \\ \hline \end{tabular}
\end{table}
Table 2: Relative amplitude statistical results.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **Beam 1** & **Beam 2** & **Beam 3** & **Beam 4** & **Beam 5** & **Beam 6** & **Beam 7** & **Beam 8** \\ \hline True values & 0 deg & 90 deg & 45 deg & 30 deg & 60 deg & 170 deg & 135 deg & 160 deg \\ Means & / & 90.181 deg & 45.034 deg & 29.992 deg & 60.066 deg & 170.014 deg & 135.148 deg & 160.086 deg \\ RMSEs & / & 0.0093 deg & 0.0089 deg & 0.0194 deg & 0.0292 deg & 0.0527 deg & 0.0641 deg & 0.1497 deg \\ \hline \end{tabular}
\end{table}
Table 3: Relative phase statistical results.  
variance of \(u_{k}^{2}\) can be derived as \(\mathrm{E}(u_{k}^{2})=1/L\) and \(\mathrm{V}(u_{k}^{2})=2/L^{2}\), respectively. According to the central limit theorem, the distribution of \(\sum_{k=1}^{K}u_{k}^{2}\) can be approximated as follows:  
\[\sum_{k=1}^{K}u_{k}^{2}\sim\mathcal{N}(K/L,2K/L^{2}). \tag{10}\]  
In deriving (10), we have assumed that \(K>>1\) which is reasonable for a large-scale antenna system. The Monte Carlo simulation and the above approximation results of probability density function (PDF) [24] for \(\sum_{k=1}^{K}u_{k}^{2}\) are compared in Figure 10. It can be seen that the approximation results are quite accurate with simulation results.  
The third item of (23) involves the product of two Gaussian random variables (\(u_{i}u_{k},i\
eq k\)) [25]. The expectation and variance can be computed as \(\mathrm{E}(u_{i}u_{k})=0\) and \(\mathrm{V}(u_{i}u_{k})=1/L^{2}\), respectively. When \(L\) is very large (i.e., \(L>>1\)), \(\mathrm{V}(u_{i}u_{k})\) is close to zero and we can neglect statistical characteristics of this item. The last item of (23) also involves the product of two independent Gaussian random variables (\(u_{k}x_{0}\)). Its expectation and variance are \(\mathrm{E}(u_{k}x_{0})=0\) and \(\mathrm{V}(u_{k}x_{0})=\sigma^{2}/(2L^{2})\), respectively. Note that \(\beta=1-\sum_{k=1}^{K}u_{k}^{2}\), the coefficient \(\beta\) in (23) follows the distribution \(\mathcal{N}(1-K/L,2K/L^{2})\). Thus, the expectation of \(y_{k}\) is zero and the variance can be expressed as  
\[\mathrm{V}(y_{kI})=\mathrm{V}(y_{k\mathrm{Q}})\approx\frac{\sigma^{2}}{2} \bigg{[}1+\frac{3L-1}{(L-K)^{2}}\bigg{]}, \tag{11}\]  
where \(y_{kI}\) and \(y_{k\mathrm{Q}}\) are the real part and imaginary part of \(y_{k}\), respectively. For a large-scale antenna system and \(L>>K\), (11) can be simplified as follows:  
\[\mathrm{V}(y_{kI})=\mathrm{V}(y_{k\mathrm{Q}})\approx\frac{\sigma^{2}}{2}. \tag{12}\]  
Appendix B Closed-Form Expressions of \(A_{k,1}^{\text{RMSE}}\) and \(\phi_{k,1}^{\text{RMSE}}\)