# Low-Complexity Online Calibration for Large-Scale Multi-Beam Antennas
## 2 System Model and Traffic Impact Analysis  
### System Model  
An online calibration system equipped with \(K\) antennas is depicted in Figure 1. A set of code-division calibration signals is generated in parallel by the calibration controller and injected after the beamforming network (BFN). Then \(K\) transmitter and receiver (TR) chains produce amplitude and phase errors on calibration signals due to temperature drift and other factors of electronic devices. Each TR chain contains the upconverter (UC), downconverter (DC), power amplifier (PA), and low noise amplifier (LNA). Temperature drift and other characteristics in-orbit after satellite launch are time-varying, causing amplitude and phase errors fluctuation during calibration window. To ensure effective calibration results, calibration must be completed quickly in a short time. Referring to the calibration in TerraSAR-X, the typical calibration time required for real-time estimation and compensation is about 1 second [16]. Otherwise, calibration results are lagging. During the short calibration time, the assumption of constant amplitude and phase errors is reasonable and the calibration result is valuable. One communication signal was equivalent to a random spread spectrum modulated signal during the calibration time window [17]. Considering multiple communication signals, we perform the following conversion. We can define the equivalent communication signal as follows:  
\[\mathbf{d^{\prime}} =\omega_{01}\mathbf{d}_{1}+\omega_{02}\mathbf{d}_{2}+\cdots+\omega_{0M}\bm {d}_{M},\] \[\mathbf{d} =\frac{1}{\omega_{0}}\mathbf{d^{\prime}}, \tag{1}\]  
where \(M\) is the number of codes for communication, \(\{\mathbf{d}_{m}\}_{m=1}^{M}\) is the \(m\)-th communication signal and \(\omega_{0m}\) is its weights. The coefficient \(\omega_{0}=\sqrt{{\mathbf{d^{\prime}}}^{\intercal}\mathbf{d^{\prime}}}\) represents the equivalent superimposed complex gain of communication signal and \(\mathbf{d}\in\mathbb{C}^{L\times 1}\) represents the normalized equivalent binary communication data. The received signal involves the normal communication signal and calibration signals with amplitude/phase errors as follows:  
\[\mathbf{r}=\underbrace{\omega_{1}\mathbf{c}_{1}+\omega_{2}\mathbf{c}_{2}+\cdots+\omega_{ K}\mathbf{c}_{K}}_{\text{Calibration signals}}+\underbrace{\omega_{0}\mathbf{d}}_{\text{Traffic signal}}+\underbrace{\mathbf{n}}_{\text{noise}} \tag{2}\]  
where \(\mathbf{r}\in\mathbb{C}^{L\times 1}\), \(\{\omega_{k}\}_{k=1}^{K}\) represents the complex weights [18] which consist of amplitude \(a_{k}\) and phase \(\varphi_{k}\), \(\mathbf{c}_{k}\in\mathbb{R}^{L\times 1}\) represents the normalized spread spectrum code sequence assigned to the \(k\)-th transmitting antenna, and \(\mathbf{n}\in\mathbb{C}^{L\times 1}\) denotes the complex noise vector, in which the elements are independent distrbuted (i.i.d.) complex Gaussian random variables [19]. The three addition items in (2) are calibration signals, the communication traffic signal, and additive noise, respectively. The code-division calibration receiver can achieve despreading with local spread spectrum code sequences. The \(k\)-th peak value of correlation peak which contains the amplitude and phase characteristics of the corresponding transmitted antenna can be given by  
\[p_{k}=c_{k}^{\intercal}\mathbf{r}. \tag{3}\]  
Figure 1: Online calibration model for multi-beam antennas.