## 2 Theory
### Error Correction for DUT Measurement  
According to the above analysis, a two-port DUT measurement will result in a matrix Equation similar to (19). Therefore, the T-matrix of the DUT can be expressed as follows:  
\[[T_{12}^{\rm D}]=\left[\begin{array}{cc}0&1\\ 1&0\end{array}\right][T_{10}]\left[\begin{array}{cc}a_{\rm m11}^{\rm D}&a_{ \rm m12}^{\rm D}\\ b_{\rm m11}^{\rm D}&b_{\rm m12}^{\rm D}\end{array}\right]\left[\begin{array}{ cc}a_{\rm m21}^{\rm D}&a_{\rm m22}^{\rm D}\\ b_{\rm m21}^{\rm D}&b_{\rm m22}^{\rm D}\end{array}\right]^{-1}[T_{23}]^{-1} \tag{23}\]  
where \(a_{\rm mij}^{\rm D}\) and \(b_{\rm mij}^{\rm D}\) (i, j = 1, 2) are the measured wave parameters of the DUT. After substitution of (6) and (7) in (23), an Equation similar to (20) can be obtained for the corrected \([T_{12}^{\rm D}]\) as shown below.  
\[[T_{12}^{\rm D}]=z\left[\begin{array}{cc}\Gamma_{\rm R11}&\Gamma_{\rm R21}x \\ 1&x\end{array}\right][W_{\rm D}]\left[\begin{array}{cc}1&y\\ \Gamma_{\rm R12}&\Gamma_{\rm R22}y\end{array}\right]^{-1} \tag{24}\]  
The matrix \([W_{\rm D}]\) with the same form of (21) is obtained from measurement. From the above results, it is easy to find out that the elements of \([T_{10}]\) and \([T_{23}]\) must be known except a common factor \(a_{11}^{\rm R11}\). A knowledge of \(a_{11}^{\rm R11}\) is not required for the de-embedding or calibration task, therefore one may arbitrarily set it to any nonzero value. In this work, we set  
\[a_{11}^{\rm R11}=\sqrt{\frac{a_{\rm m11}^{\rm R11}b_{\rm m11}^{\rm R21}-a_{\rm m 11}^{\rm R21}b_{\rm m11}^{\rm R11}}{(\Gamma_{\rm R21}-\Gamma_{\rm R11})x}}. \tag{25}\]  
Equation (25) can make the determinant of \([T_{10}]\) equal to 1, which does not mean that the real error box is reciprocal. Substituting (25) into (6) and (7) can conclude  
\[[T_{10}]=\sqrt{\frac{a_{\rm m11}^{\rm R11}b_{\rm m11}^{\rm R21}-a_{\rm m11}^{ \rm R21}b_{\rm m11}^{\rm R11}}{(\Gamma_{\rm R21}-\Gamma_{\rm R11})x}}\left[ \begin{array}{cc}1&x\\ \Gamma_{\rm R11}&\Gamma_{\rm R21}x\end{array}\right]\left[\begin{array}{ cc}a_{\rm m11}^{\rm R11}&a_{\rm m11}^{\rm R21}\\ b_{\rm m11}^{\rm R11}&b_{\rm m11}^{\rm R21}\end{array}\right]^{-1} \tag{26}\]  
\[[T_{23}]=\frac{1}{z}\sqrt{\frac{a_{\rm m11}^{\rm R21}-a_{\rm m11}^{\rm R21}b_{ \rm m11}^{\rm R11}}{(\Gamma_{\rm R21}-\Gamma_{\rm R11})x}}\left[\begin{array}[ ]{cc}1&y\\ \Gamma_{\rm R12}&\Gamma_{\rm R22}y\end{array}\right]\left[\begin{array}{ cc}a_{\rm m22}^{\rm R22}&a_{\rm m22}^{\rm R22}\\ b_{\rm m22}^{\rm R22}&b_{\rm m22}^{\rm R22}\end{array}\right]^{-1}. \tag{27}\]  
To determine \([T_{10}]\) and \([T_{23}]\), besides the definition and measurement of standards R\({}_{\rm kl}\) (k, l = 1, 2), only the newly introduced \(x\), \(y\), and \(z\) are required to be solved. Once the unknowns in (26) and (27) are all determined, the T-matrix of the DUT can be corrected by (23), and then the following Equation can be used for the S-matrix transformation:  
\[[S]=\left(\left[\begin{array}{cc}1&0\\ 0&0\end{array}\right]-[T_{12}^{\rm D}]\left[\begin{array}{cc}0&0\\ 0&1\end{array}\right]\right)^{-1}\left([T_{12}^{\rm D}]\left[\begin{array}{ cc}0&1\\ 0&0\end{array}\right]-\left[\begin{array}{cc}0&0\\ 1&0\end{array}\right]\right) \tag{28}\]  
where \([S]\) is defined as the S-matrix of the DUT.  
The T-matrix representation of error boxes can be also converted into the cascade matrix form defined in [6] by using the following Equations:  
\[[R_{\text{left}}]=\left[\begin{array}{cc}0&1\\ 1&0\end{array}\right][T_{10}]^{-1}\left[\begin{array}{cc}0&1\\ 1&0\end{array}\right] \tag{29}\]  
\[\left[R_{\text{right}}]=[T_{23}] \tag{30}\]  
where \([R_{\text{left}}]\) and \([R_{\text{right}}]\) are the cascade matrix of the left error box and the reverse cascade matrix of the right error box, respectively.