# Accurate Phase Calibration of Multistatic Imaging System for Medical and Industrial Applications
## 2 Phase Calibration Method
### Apply Phase Calibration  
The round-trip phase is converted to a one-way phase for phase calibration. When converting the phase, it is necessary to consider the phase fringes. When the electrical length is longer than the wavelength, the measured phase has advanced by more than 360 degrees. When determining the one-way phase, it is necessary to have an absolute phase. To find the absolute phase, the phase wraps \(p\), which is the number of times the phase has cycled \(2\pi\) radians at the lower frequency limit, is required. There are two ways to obtain the phase wraps at lower frequency. The first method is to adjust the lower frequency limit to \(p=0\). If the wavelength is sufficiently long compared to the assumed electrical length, there is no need to consider phase fringes. The second method is to calculate \(p\). If the assumed electrical length is known, it is possible to evaluate the absolute phase from the relative phase. The electrical length can be obtained from the peak of the reflection coefficient in the time domain, but attention must be paid to conditions such as the frequency to be used for the inverse Fourier transform, as it cannot consider dispersion. The absolute phase \(\phi_{\text{tx}}(\omega,n_{\text{tx}})\) is calculated using \(p\):  
\[\phi_{\text{tx}}(\omega,n_{\text{tx}})=\arg\bigl{(}S^{\prime}_{11}(\omega,n_{ \text{tx}})\bigr{)}+2\pi p \tag{6}\]  
The round-trip absolute phase \(\phi^{\prime}_{\text{tx}}\) is converted to the one-way phase for phase calibration. Dividing the unwrapped phase value by two yields the one-way phase. \(\phi^{\prime}_{\text{tx}}(\omega,n_{\text{tx}})\) represents the calibration data. \(\phi^{\prime}_{\text{rx}}(\omega,n_{\text{tx}})\) is also calculated in the same way. \[\phi^{\prime}_{\text{tx}}(\omega,\,n_{\text{tx}})=\frac{\phi_{\text{tx}}(\omega,n_{\text{tx}})}{2} \tag{7}\]Finally, calibration data are applied to the measured data using De-Embedding [22; 23]. The phase shift is eliminated using the difference in electrical lengths determined from the transmission coefficients measured between the calibration planes. The circuit between the calibration planes in Figure 1 is connected in series, with three circuits as shown in Figure 3.  
\(S(\omega,\ n_{\text{tx}},\ n_{\text{rx}})\) represents the data between the measurement reference planes of the Tx and Rx antennas. \(S_{\text{tx}}(\omega,n_{\text{tx}})\) is the S-parameter between the calibration plane and the measurement reference plane of the Tx antenna side, and \(S_{\text{rx}}(\omega,\ n_{\text{tx}})\) is the S-parameter between the calibration plane and the measurement reference plane of the Rx antenna side. Assuming that these two circuits are lossless, they are represented by the following S-parameters:  
\[S_{\text{tx}}(\omega,n_{\text{tx}})= \begin{bmatrix}0&e^{i\phi^{\prime}_{\text{ rx}}(\omega,n_{\text{tx}})}\\ e^{i\phi^{\prime}_{\text{rx}}(\omega,n_{\text{rx}})}&0\end{bmatrix} ,S_{\text{ rx}}(\omega,n_{\text{tx}})= \begin{bmatrix}0&e^{i\phi^{\prime}_{\text{rx}}(\omega,n_{ \text{rx}})}\\ e^{i\phi^{\prime}_{\text{rx}}(\omega,n_{\text{rx}})}&0\end{bmatrix}  \tag{8}\]  
These circuits are expressed using T-parameters, as follows:  
\[T_{\text{meas}}(\omega,n_{\text{tx}},\ n_{\text{rx}})\ =\ T_{\text{tx}}(\omega,\ n_{\text{tx}})T( \omega,\ n_{\text{tx}},\ n_{\text{rx}})\ T_{\text{rx}}(\omega,\ n_{\text{tx}}) \tag{9}\]  
S-parameters and T-parameters are transformed by the following equations:  
\[[S]=\frac{1}{T_{22}} \begin{bmatrix}T_{12}&T_{11}T_{22}\ -\ T_{12}T_{21}\\ 1&-T_{21}\end{bmatrix}  \tag{10}\]  
\[[T]=\frac{1}{S_{21}} \begin{bmatrix}S_{12}S_{21}-S_{11}S_{22}&S_{11}\\ -S_{22}&1\end{bmatrix}  \tag{11}\]  
Multiply the transmission coefficient \(T_{\text{meas}}(\omega,n_{\text{tx}},n_{\text{rx}})\) between the calibration planes by the inverse of \(T_{\text{tx}}(\omega,n_{\text{tx}})\) and \(T_{\text{rx}}(\omega,n_{\text{rx}})\) to obtain the transmission coefficient \(T(\omega,n_{\text{tx}},n_{\text{rx}})\) between the measurement reference planes.