# Blind Compensation of \(I/q\) Impairments in Wireless Transceivers
## Appendix A  
To derive the closed form expression for the PDF of the output signal \(\mathbf{y}\), we start with the following PDF provided in Equation (16):  
\[f_{y}(\mathbf{y};\mathbf{\Gamma})=\prod_{i=1}^{N}\frac{1}{2\pi\sqrt{\left| \mathbf{C_{y}}(\mathbf{\Gamma})\right|}}\times\exp(-\frac{1}{2}\mathbf{y}_{i} ^{\mathrm{T}}\mathbf{C_{y}^{-1}}(\mathbf{\Gamma})\mathbf{y}_{i}) \tag{16}\]  
The covariance matrix of \(\mathbf{y}\) is given by:  
\[\mathbf{C_{y}}(\mathbf{\Gamma})=E(\mathbf{y}\mathbf{y}^{\mathrm{T}})=E( \mathbf{A}\mathbf{x}\mathbf{x}^{\mathrm{T}}\mathbf{A}^{\mathrm{T}})=\mathbf{A} \mathbf{C_{x}}\mathbf{A}^{\mathrm{T}} \tag{17}\]  
where \(\mathbf{C_{x}}\) is the covariance matrix of \(\mathbf{x}\). Replacing the expression of \(\mathbf{C_{x}}=\sigma^{2}\mathbf{I_{D}}\) (using assumptions \(A1\) and \(A2\)) in the above equation, we have  
\[\mathbf{C_{y}}(\mathbf{\Gamma})=\sigma^{2}\mathbf{A}\mathbf{A}^{\mathrm{T}} \tag{18}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**EVM** & **EVM** & **EVM** & **EVM** & **EVM** & **NMSE (dB)** & **NMSE** & **NMSE** & **NMSE** & **NMSE** \\
**Uncompensated Signal** & **(This Work)** & **(15)** & **(No imb.)** & **(Sim.)** & **Uncompensated Signal** & **(This Work)** & **(15)** & **(No imb.)** & **(Sim.)** \\ \hline
7.1651 & 0.7809 & 0.986 & 0.59 & 0.6162 & \(-\)22.89 & \(-\)41.77 & \(-\)39.87 & \(-\)43.96 & \(-\)41.87 \\ \hline \hline \end{tabular}
\end{table}
Table 4: NMSE and EVM for 1 MHz 16 QAM signal using experimental data and comparison with simulations.  
The determinant of \(\mathbf{C_{y}}\) is givssssen by:  
\[|\mathbf{C_{y}}(\mathbf{\Gamma})|=\left(\sigma^{2}\right)^{R}|\mathbf{A}\mathbf{ A}^{\mathsf{T}}| \tag{10}\]  
where \(R\) denotes the size of the matrix which is 2 in our case. Furthermore,  
\[\left|\mathbf{C_{y}}(\mathbf{\Gamma})\right|_{1/2}=\sigma^{2}\gamma;\gamma= \alpha_{1}\beta_{2}-\alpha_{2}\beta_{1} \tag{11}\]  
The inverse of \(\mathbf{C_{y}}\) is given by:  
\[\mathbf{C_{y}^{-1}}(\mathbf{\Gamma})=\frac{1}{\sigma^{2}}(\mathbf{A}^{ \mathsf{T}})^{-1}(\mathbf{A})^{-1} \tag{12}\]  
where  
\[(\mathbf{A}^{\mathsf{T}})^{-1}=\frac{1}{\gamma}\Bigg{(}\begin{array}{cc} \beta_{2}&-\alpha_{2}\\ -\beta_{1}&\alpha_{1}\end{array}\Bigg{)};\mathbf{A}^{-1}=\frac{1}{\gamma} \Bigg{(}\begin{array}{cc}\beta_{2}&-\beta_{1}\\ -\alpha_{2}&\alpha_{1}\end{array}\Bigg{)} \tag{13}\]  
Hence,  
\[\mathbf{C_{y}^{-1}}(\mathbf{\Gamma})=\frac{1}{\sigma^{2}\gamma^{2}}\Bigg{(} \begin{array}{cc}\beta_{2}{}^{2}+\alpha_{2}{}^{2}&-\beta_{1}\beta_{2}- \alpha_{1}\alpha_{2}\\ -\beta_{1}\beta_{2}-\alpha_{1}\alpha_{2}&\beta_{1}{}^{2}+\alpha_{1}{}^{2} \end{array}\Bigg{)} \tag{14}\]  
Putting the expressions for \(\mathbf{C_{y}}^{-1}\) and \(\mid\mathbf{C_{y}}\mid^{1/2}\) in Equation (10), we obtain the closed form expression for the demodulated imbalanced signal as:  
\[f_{y}(\mathbf{y};\mathbf{\Gamma})=\frac{1}{\left(2\pi\sigma^{2}|\gamma|\right) ^{N}}e^{-\left(\frac{1}{2\pi\sigma^{2}\gamma}\sum\limits_{i=1}^{N}\left(\beta_ {2}I_{y}(i)-\beta_{1}Q_{y}(i)\right)^{2}+\left(\alpha_{1}Q_{y}(i)-\alpha_{2}I_ {y}(i)\right)^{2}\right)} \tag{15}\]