# Low-Complexity Online Calibration for Large-Scale Multi-Beam Antennas
## 3 Low-Complexity Online Calibration Method
### Calibration Accuracy Analysis  
To investigate the calibration performance of the proposed method, we derive closed-form calibration accuracy expressions with respect to noise. As shown in Figure 4, two types of noise are involved [22]: \(\mathbf{n}\) and \(\mathbf{x}\). The noise \(\mathbf{n}\) in Equation (2) is the channel noise \"before\" correlation, The noise \(x_{k}=\mathbf{c}_{k}^{\rm T}\mathbf{n}\) in Equation (4) and Equation (9) is the noise \"after\" correlation. As mentioned before, \(x_{k}\) follows complex Gaussian distribution \(\mathcal{CN}(0,\sigma^{2})\). Then we can define \(\gamma_{k}\) as the ratio of calibrated signal power to noise \"after\" correlation:  
\[\gamma_{k}=\frac{a_{k}^{2}}{\sigma^{2}}\quad\text{and}\quad[\gamma_{k}]_{\rm dB }=10\text{log}_{10}\Bigg{(}\frac{a_{k}^{2}}{\sigma^{2}}\Bigg{)}. \tag{18}\]  
Correspondingly, the ratio of calibrated signal power to noise \"before\" correlation can be defined as  
\[\delta_{k}=\frac{\gamma_{k}}{L^{2}}\quad\text{and}\quad[\delta_{k}]_{\rm dB}= [\gamma_{k}]_{\rm dB}-20\text{log}_{10}(L). \tag{19}\]  
Figure 5: The elements of the matrix \(M^{-1}\) versus \(K\). (**a**) The off-diagonal element of \(M^{-1}\). (**b**) The diagonal element of \(M^{-1}\). The difference between \(\gamma_{k}\) and \(\delta_{k}\) is the code length \(L\), i.e., the signal-to-noise ratio can be improved by \(20\text{log}_{10}(L)\) through spread spectrum technology. In this section, we derive the calibration accuracy at \(\gamma_{k}\) and the relationship with \(\delta_{k}\) can be easily obtained. The performance of calibration accuracy can be evaluated by the root mean square error (RMSE) of the relative amplitude in decibel (dB) and phase in degree (deg) as follows:  
\[A_{k,1}^{\text{RMSE}} =\sqrt{\text{E}\Bigg{\{}\Bigg{[}10\log_{10}\bigg{(}\frac{\hat{a}_ {k}}{\hat{a}_{1}}\bigg{)}^{2}-10\log_{10}\bigg{(}\frac{a_{k}}{a_{1}}\bigg{)}^{2 }\Bigg{]}^{2}\Bigg{\}}},\quad k=2,\cdots,K \tag{20}\] \[\phi_{k,1}^{\text{RMSE}} =\frac{180}{\pi}\sqrt{\text{E}\Big{\{}[(\widehat{\varphi}_{k}- \widehat{\varphi}_{1})-(\varphi_{k}-\varphi_{1})]^{2}\Big{\}}},\quad k=2, \cdots,K \tag{21}\]  
where \(\{a_{k}\}_{k=1}^{K}\) and \(\{\varphi_{k}\}_{k=1}^{K}\) are true amplitude and phase coefficients, respectively. We still assume that the first antenna is chosen as the reference antenna, the same as (7) and (8). The key to obtain closed-form expressions of (20) and (21) is to estimate \(\widehat{a}_{k}\) in (16) and \(\widehat{\varphi}_{k}\) in (17). From (16) and (17), we can express the following estimated complex weight coefficients:  
\[\widehat{\omega}_{k}=\omega_{k}+y_{k},\quad k=1,2,\cdots,K \tag{22}\]  
where  
\[y_{k}=x_{k}+\frac{1}{\beta}\bigg{(}u_{k}^{2}x_{k}+\sum_{i=1,i\
eq k}^{K}u_{i}u _{k}x_{i}-u_{k}x_{0}\bigg{)}.\quad k=1,2,\cdots,K \tag{23}\]  
As shown in Appendix A, the expectation of \(y_{k}\) is zero and its variance of the real part and imaginary part can be approximately derived as \(\sigma^{2}/2\) for a large-scale antenna system \(L>>K\). The approximate derivation results are closer to theoretical values when L is larger and K is smaller. The RMSE of the relative amplitude and phase can be further approximated as follows (see the detailed proof in Appendix B):  
\[A_{k,1}^{\text{RMSE}} \approx\frac{10}{\ln(10)}\sqrt{\frac{2}{\gamma_{k}}\bigg{(}\frac {1}{2\gamma_{k}}+1\bigg{)}+\frac{2}{\gamma_{1}}\bigg{(}\frac{1}{2\gamma_{1}}+ 1\bigg{)}+\bigg{(}\frac{1}{\gamma_{k}}-\frac{1}{\gamma_{1}}\bigg{)}^{2}}, \tag{24}\] \[\phi_{k,1}^{\text{RMSE}} \approx\frac{180}{\pi}\sqrt{\frac{1}{2\gamma_{k}}+\frac{1}{2 \gamma_{1}}}. \tag{25}\]  
From (24) and (25), it is clear that the calibration accuracy of the proposed method depends solely on the signal-to-noise ratio as long as \(L>>K\) (i.e., \(L>2K\)) and the strong interference from communication signal has been eliminated efficiently. The closed-form expressions of \(A_{k,1}^{\text{RMSE}}\) and \(\phi_{k,1}^{\text{RMSE}}\) are gradually invalidated along with the increase of \(K\).