# Accurate Phase Calibration of Multistatic Imaging System for Medical and Industrial Applications
## 2 Phase Calibration Method
### Time Domain Gating  
The measured reflection signals are transformed into the time domain to extract only the reflection signals at the phase origin. This technique is called Time Domain Gating [21]. Initially, the discrete inverse Fourier transform is used to convert the frequency domain reflection signals \(S_{11}(\omega,n_{\text{tx}})\) into the time domain signals \(s_{11}(t,n_{\text{tx}})\). \[s_{11}(t[m_{\text{t}}],n_{\text{tx}})=\frac{1}{M}\sum_{m_{\text{f}}=0}^{M-1}S_{ 11}(\omega[m_{\text{f}}],n_{\text{tx}})\text{e}^{j\frac{2\pi}{M}m_{\text{f}}m_ {\text{f}}} \tag{1}\]  
where \(M\) is the number of measurement data in the frequency domain; \(m_{\text{f}}\) is the discrete frequency index \(m_{\text{f}}=0,\,1,\,2,\,\ldots,\,M-1\); \(m_{\text{t}}\) is the discrete time index \(m_{t}=0,1,2,\ldots,M-1\); \(t\) is the time axis in the time domain \(t[m_{\text{t}}]=\Delta t\cdot m_{\text{t}}\); and \(\Delta t\) is the time resolution of the time axis. \[\Delta t=\frac{2\pi}{\omega_{\text{stop}}-\omega_{\text{start}}} \tag{2}\]  
where \(\omega_{\text{start}}\) is the lowest angular frequency \(\omega_{\text{start}}=\omega[0]\) and \(\omega_{\text{stop}}\) is the highest angular frequency \(\omega_{\text{stop}}=\omega[M-1]\). To improve the accuracy of phase calibration, it is necessary to improve the time resolution \(\Delta t\). The reflection coefficients of the Rx \(s_{22}(t,n_{\text{rx}})\) are also calculated the same way. The time resolution \(\Delta t\) depends on the bandwidth in the frequency domain \(\omega_{\text{stop}}-\omega_{\text{start}}\). Since it is difficult to ensure sufficient bandwidth in the experiments, the bandwidth was ensured by padding the data. In this case, a method of adding zeros to the measured frequency domain data to widen the bandwidth is used. Figure 2: Flow chart of the phase calibration. The phase calibration is performed in three steps. Next, the reflection signals at the measurement reference plane \(s^{\prime}{}_{11}(t,n_{\text{tx}})\) and \(s^{\prime}_{22}(t,n_{\text{rx}})\) are extracted from the time domain signals. The signals in the time domain were extracted by applying a gate function. \[s^{\prime}{}_{11}(t[m_{\text{t}}],n_{\text{tx}})=s_{11}(t[m_{\text{t}}],n_{\text {tx}})\cdot\text{Window}(m_{\text{t}}) \tag{3}\]  
The Kaiser-Bessel function was used for gate processing. The Kaiser-Bessel function is represented by the following equation:  
\[\text{Window}(m_{\text{t}})=\left\{ \begin{array}{ll}\text{I}_{0}\left( \beta\sqrt{1-\frac{4m_{\text{t}}^{2}}{\left(m_{\text{t,span}}-1\right)^{2}}} \right)\\ \text{I}_{0}(\beta)\\ 0\end{array} &\text{if}\,-\frac{m_{\text{t,span}}-1}{2}+m_{\text{t,peak}}\leq m _{\text{t}}\leq\frac{m_{\text{t,span}}-1}{2}+m_{\text{t,peak}}\\ 0&\text{otherwise}\end{array}\right. \tag{4}\]  
where \(m_{\text{t,peak}}\) is a time index of the center of the peak, and \(m_{\text{t,span}}\) is the index number of the peak width. \(\text{I}_{0}\) is the modified zeroth-order Bessel function. \(\beta\) is the shape parameter of the function. When \(\beta=0\), the function is rectangular. As \(\beta\) increases, the function's half-width decreases. \(\beta=6\), like the Han function, which is commonly used. Finally, the discrete Fourier transform was used to convert the time domain signal into the frequency domain:  
\[S^{\prime}{}_{11}\left(\omega^{\prime}[m_{\text{f}}^{\prime}],n_{\text{tx}} \right)=\sum_{m_{\text{t}}=0}^{M-1}s^{\prime}{}_{11}(t[m_{\text{t}}],n_{\text{ tx}})e^{-j\frac{2\pi}{M}m_{\text{t}}m_{\text{f}}^{\prime}} \tag{5}\]  
where \(m_{\text{f}}^{\prime}\) is a new discrete frequency index \(m_{\text{f}}^{\prime}=0,1,2,\ldots,M-1\), and \(\omega^{\prime}\) is the new frequency axis in the frequency domain \(\omega^{\prime}[m_{\text{f}}^{\prime}]=\Delta\omega^{\prime}.m_{\text{f}}^{ \prime}+\omega_{\text{start}}\). \(\Delta\omega^{\prime}\) is the frequency resolution of the time axis \(\Delta\omega^{\prime}=\frac{2\pi}{\Delta t\cdot(M-1)}\).