# Low-Complexity Online Calibration for Large-Scale Multi-Beam Antennas
## 3 Low-Complexity Online Calibration Method
### Online Calibration Method Based on Decorrelation Algorithm  
Let \(\mathbf{M}\in\mathbb{R}^{K\times K}\) be the correlation matrix of code sequences for calibration. According to (4), the vector form of \((K+1)\) correlation peak values can be obtained as  
\[\left[ \begin{array}{c}p_{1}\\ p_{2}\\ \vdots\\ p_{K}\\ p_{0}\end{array} \right] = \left[ \begin{array}{c}\mathbf{c}_{1}^{\mathsf{T}}\\ \mathbf{c}_{2}^{\mathsf{T}}\\ \vdots\\ \mathbf{c}_{k}^{\mathsf{T}}\\ \mathbf{d}^{\mathsf{T}}\end{array} \right](\omega_{1}\mathbf{c}_{1}+\omega_{2}\mathbf{c}_{2 }+\cdots+\omega_{K}\mathbf{c}_{K}+\omega_{0}\mathbf{d}+\mathbf{n}) \tag{9}\] \[= \mathbf{R}\left[ \begin{array}{c}\mathbf{\omega}\\ \omega_{0}\end{array} \right]+\left[ \begin{array}{c}\mathbf{x}\\ x_{0}\end{array} \right],\]  
Figure 3: Amplitude and phase mismatches. (**a**) Amplitude mismatches without the traffic interference. (**b**) Amplitude mismatches in the presence of traffic. (**c**) Phase mismatches without the traffic interference. (**d**) Phase mismatches in the presence of traffic. where \(\mathbf{\omega}\in\mathbb{C}^{K\times 1}\), \(\mathbf{x}=[\mathbf{c}_{1}^{\mathsf{T}}\mathbf{n},\mathbf{c}_{2}^{\mathsf{T}}\mathbf{n},\cdots,\mathbf{ c}_{K}^{\mathsf{T}}\mathbf{n}]^{\mathsf{T}}\in\mathbb{C}^{K\times 1}\), and \(x_{0}=d^{\mathsf{T}}\mathbf{n}\). We define \(\mathbf{u}=[d^{\mathsf{T}}\mathbf{c}_{1},d^{\mathsf{T}}\mathbf{c}_{2},\cdots,d^{\mathsf{ T}}\mathbf{c}_{K}]\in\mathbb{C}^{1\times K}\) and the matrix \(\mathbf{R}\) in (9) is  
\[\mathbf{R}=\left[ \begin{array}{cc}\mathbf{M}&\mathbf{u}^{\mathsf{T}}\\ \mathbf{u}&1\end{array} \right]. \tag{10}\]  
As shown in Figure 3, the calibration process is seriously disturbed by the high-power communication signal. In order to eliminate the strong interference from communication signal, a decorrelation algorithm is introduced by multiplying the inverse matrix of the correlation matrix with the correlation peak values. The structure of calibration receiver based on decorrelation is shown in Figure 4.  
The inverse of the correlation matrix \(\mathbf{R}\) can be expressed as [20]  
\[\mathbf{R}^{-1}=\frac{1}{\beta}\left[ \begin{array}{cc}\beta M^{-1}+M^{-1}\mathbf{u} ^{\mathsf{T}}\mathbf{u}M^{-1}&-M^{-1}\mathbf{u}^{\mathsf{T}}\\ -\mathbf{u}\mathbf{M}^{-1}&1\end{array} \right], \tag{11}\]  
where \(\beta=1-\mathbf{u}\mathbf{M}^{-1}\mathbf{u}^{\mathsf{T}}\). A code-division calibration method proposed in [21] can take advantage of the binary autocorrelation property of m-sequence. The elements on the diagonal of the matrix \(\mathbf{M}\) are all 1 and other elements are \(-1/L\). Therefore, its inverse matrix has particularity, that is, the elements on the diagonal are \(\frac{L(L-K+2)}{(L+1)(L-K+1)}\), and other elements are \(\frac{L}{(L+1)(L-K+1)}\). The relationship between the elements of the matrix \(\mathbf{M}^{-1}\) and \(K\) is shown in Figure 5. When \(L>2K\), the diagonal and off-diagonal elements of the matrix \(\mathbf{M}^{-1}\) can be approximately 1 and 0, respectively. Then the matrix \(\mathbf{M}^{-1}\) can be approximately obtained by \(\mathbf{I}_{K}\). Then (11) can be rewritten as  
\[\mathbf{R}^{-1}=\frac{1}{\beta}\left[ \begin{array}{cc}\beta\mathbf{I}_{K}+\mathbf{u}^{ \mathsf{T}}\mathbf{u}&-\mathbf{u}^{\mathsf{T}}\\ -\mathbf{u}&1\end{array} \right], \tag{12}\]  
where \(\beta=1-\mathbf{u}\mathbf{u}^{\mathsf{T}}=1-\sum_{k=1}^{K}u_{k}^{2}\). Multiplying (9) by (11) and taking out the first \(K\) elements, we can obtain \(K\) estimated complex coefficients as follows:  
\[\widehat{\mathbf{\omega}}=\mathbf{\omega}+\mathbf{M}^{-1}\mathbf{x}+\frac{1}{\beta}\mathbf{M}^{-1 }\mathbf{u}\mathbf{u}\mathbf{M}^{-1}\mathbf{x}-\frac{1}{\beta}\mathbf{M}^{-1}\mathbf{u}^{\mathsf{T}}x_ {0}. \tag{13}\]  
As previously analyzed, \(\mathbf{M}^{-1}\) can approximate the identity matrix when \(L>2K\) is satisfied.