# Low-Complexity Online Calibration for Large-Scale Multi-Beam Antennas
## Appendix A Expectation and Variance for \(y_{k}\)
### Expression of \(A_{k,1}^{\text{RMSE}}\)  
The RMSE of the relative amplitude in (20) can be rewritten as  
\[A_{k,1}^{\text{RMSE}}=\sqrt{\left\{\mathrm{E}\bigg{[}10\log_{10}\bigg{(}\frac {\widehat{a}_{k}/a_{k}}{\widehat{a}_{1}/a_{1}}\bigg{)}^{2}\bigg{]}\right\}^{ 2}+\mathrm{V}\bigg{[}10\log_{10}\bigg{(}\frac{\widehat{a}_{k}/a_{k}}{\widehat {a}_{1}/a_{1}}\bigg{)}^{2}\bigg{]}}. \tag{13}\]  
The derivation problem of \(A_{k,1}^{\text{RMSE}}\) can be transformed into estimating the statistical characteristics of \(\widehat{a}_{k}/a_{k}\). Since the estimated amplitude and phase must achieve high accuracy in the calibration mission, we can reasonably assume that \(\widehat{a}_{k}/a_{k}\to 1\) and \(\widehat{a}_{1}/a_{1}\to 1\)[26]. Then we can obtain the following approximation results:  
\[10\log_{10}\biggl{(}\frac{\widehat{a}_{k}/a_{k}}{\widehat{a}_{1}/ a_{1}}\biggr{)}^{2} = \frac{10}{\ln(10)}\Bigl{\{}\ln\Bigl{[}(\widehat{a}_{k}/a_{k})^{2} \Bigr{]}-\ln\Bigl{[}(\widehat{a}_{1}/a_{1})^{2}\Bigr{]}\Bigr{\}} \tag{10}\] \[\approx \frac{10}{\ln(10)}\Bigl{[}(\widehat{a}_{k}/a_{k})^{2}-(\widehat{ a}_{1}/a_{1})^{2}\Bigr{]}.\]  
Substituting (22) into \(\widehat{a}_{k}=|\widehat{c}_{k}|\) and separating the real part from the imaginary part, \((\widehat{a}_{k}/a_{k})^{2}\) can be given by  
\[\biggl{(}\frac{\widehat{a}_{k}}{a_{k}}\biggr{)}^{2} = \biggl{(}\frac{|\omega_{k}+y_{k}|}{a_{k}}\biggr{)}^{2} \tag{11}\] \[= \biggl{(}\cos\varphi_{k}+\frac{y_{k}}{a_{k}}\biggr{)}^{2}+ \biggl{(}\sin\varphi_{k}+\frac{y_{kQ}}{a_{k}}\biggr{)}^{2}.\]  
According to (10), we can obtain that  
\[\cos\varphi_{k}+\frac{y_{kI}}{a_{k}}\stackrel{{\text{ Approx.}}}{{\sim}}\mathcal{N}\biggl{(}\cos\varphi_{k},\frac{\sigma^{2}}{2a_{k}^{2}} \biggr{)}, \tag{12}\] \[\sin\varphi_{k}+\frac{y_{kQ}}{a_{k}}\stackrel{{\text{ Approx.}}}{{\sim}}\mathcal{N}\biggl{(}\sin\varphi_{k},\frac{\sigma^{2}}{2a_{k}^{2}} \biggr{)}. \tag{13}\]  
Equation (11) is converted into the sum of squares of two non-zero mean Gaussian random variables. Then we convert Equation (11) to the Gaussian distribution with variance 1 and obtain a random variable \(Y\) with non-central chi-square distribution as follows:  
\[Y=\biggl{(}\frac{2a_{k}^{2}}{\sigma^{2}}\biggr{)}\biggl{(}\frac{\widehat{a}_{ k}}{a_{k}}\biggr{)}^{2}. \tag{14}\]  
where \(Y\stackrel{{\text{Approx.}}}{{\sim}}\chi^{2}(2,2a_{k}^{2}/\sigma^ {2})\). The expectation and variance of \(Y\) can be given by  
\[\text{E}(Y) \approx 2+\frac{2a_{k}^{2}}{\sigma^{2}}, \tag{15}\] \[\text{V}(Y) \approx 4+\frac{8a_{k}^{2}}{\sigma^{2}}. \tag{16}\]  
Considering the coefficient \((2a_{k}^{2})/\sigma^{2}\) in \(Y\), the expectation and variance of \((\widehat{a}_{k}/a_{k})^{2}\) can be approximated as  
\[\text{E}\Bigl{[}(\widehat{a}_{k}/a_{k})^{2}\Bigr{]} \approx 1+\frac{\sigma^{2}}{a_{k}^{2}}, \tag{17}\] \[\text{V}\Bigl{[}(\widehat{a}_{k}/a_{k})^{2}\Bigr{]} \approx \frac{2\sigma^{2}}{a_{k}^{2}}\biggl{(}\frac{\sigma^{2}}{2a_{k}^{2} }+1\biggr{)}. \tag{18}\]  
Similarly, \(\text{E}\bigl{[}(\widehat{a}_{1}/a_{1})^{2}\bigr{]}\) and \(\text{V}\bigl{[}(\widehat{a}_{1}/a_{1})^{2}\bigr{]}\) can be obtained. Further, we substitude the expectation and variance of (10) into (11) and we can obtain the RMSE of the relative amplitude (in dB) as follows:  
\[A_{k,1}^{\text{RMSE}}\approx\frac{10}{\ln(10)}\sqrt{\frac{2\sigma^{2}}{a_{k}^{ 2}}\Biggl{(}\frac{\sigma^{2}}{2a_{k}^{2}}+1\Biggr{)}+\frac{2\sigma^{2}}{a_{1}^ {2}}\Biggl{(}\frac{\sigma^{2}}{2a_{1}^{2}}+1\Biggr{)}+\Biggl{(}\frac{\sigma^{2} }{a_{k}^{2}}-\frac{\sigma^{2}}{a_{1}^{2}}\[PERSON]{)}^{2}}. \tag{19}\]