# Blind Compensation of \(I/q\) Impairments in Wireless Transceivers
## Appendix C  
Here we provide the Cramer-Rao lower bound (CRLB) for the proposed estimator using the Fisher information matrix. The variance of the proposed estimator is given by:  
\[var(\hat{\mathbf{\Gamma}})\geq\left[\mathbf{I}(\mathbf{\Gamma})\right]_{mn}{}^ {-1} \tag{102}\]  
where \(\left[\mathbf{I}(\mathbf{\Gamma})\right]_{mn}\) is the Fisher information matrix. From Equation (100), the log of the likelihood function can be obtained as:  
\[ln(f_{y}(\mathbf{y};\mathbf{\Gamma}))=-Nln(2\pi\sigma^{2})-Nln(\gamma)-\frac{ 1}{2\sigma^{2}}{\sum_{i}}\mathbf{y}_{i}^{\mathrm{T}}(\mathbf{A}\mathbf{A}^{ \mathrm{T}})^{-1}\mathbf{y}_{i} \tag{103}\]  
The Hessian of this log likelihood function is given by:  
\[\left[\mathbf{H}(\Gamma)\right]_{mn}=\frac{\partial^{2}ln(f_{y}(\mathbf{y}; \Gamma))}{\partial\Gamma_{m}\partial\Gamma_{n}}=\left[\mathbf{H}_{1}\right]_ {mn}+\left[\mathbf{H}_{2}\right]_{mn} \tag{104}\]  
where \(\mathbf{H}_{1}\) and \(\mathbf{H}_{2}\) are the Hessian matrices of the second and third term, respectively. Also:  
\[\left[\mathbf{I}(\mathbf{\Gamma})\right]_{mn}=-E(\left[\mathbf{H}(\mathbf{ \Gamma})\right]_{mn}) \tag{105}\]  
\[E(\left[\mathbf{H}_{1}\right]_{mn})=E(\frac{N}{\gamma^{2}}\frac{\partial \gamma}{\partial\mathbf{\Gamma}}\frac{\partial\gamma^{\mathrm{T}}}{\partial \mathbf{\Gamma}})=\frac{N}{\gamma^{2}}\left(\begin{array}{cccc}\beta_{2}{}^ {2}&-\alpha_{2}\beta_{2}&-\beta_{1}\beta_{2}&\alpha_{1}\beta_{2}\\ -\alpha_{2}\beta_{2}&\alpha_{2}^{2}&\alpha_{2}\beta_{1}&-\alpha_{1}\alpha_{2} \\ -\beta_{1}\beta_{2}&\alpha_{2}\beta_{1}&\beta_{1}{}^{2}&-\alpha_{1}\beta_{1}\\ \alpha_{1}\beta_{2}&-\alpha_{1}\alpha_{2}&-\alpha_{1}\beta_{1}&\alpha_{1}^{2} \end{array}\right) \tag{106}\]  
Using the following statistics of the received signal:  
\[E[Q_{y}^{2}]=(\alpha_{2}^{2}+\beta_{2}^{2})\sigma^{2} \tag{107}\]  
\[E[I_{y}^{2}]=(\alpha_{1}^{2}+\beta_{1}^{2})\sigma^{2} \tag{108}\]  
\[E[I_{y}Q_{y}]=(\alpha_{1}\alpha_{2}+\beta_{1}\beta_{2})\sigma^{2} \tag{109}\]  
The expected value of the second Hessian matrix can be written as:  
\[E(\left[\mathbf{H}_{2}\right]_{mn})=-\frac{N}{\gamma^{2}}\left(\begin{array}[] {cccc}\alpha_{2}^{2}+\beta_{2}^{2}&0&-(\alpha_{1}\alpha_{2}+\beta_{1}\beta_{ 2})&0\\ 0&\alpha_{2}^{2}+\beta_{2}^{2}&0&-(\alpha_{1}\alpha_{2}+\beta_{1}\beta_{2})\\ -(\alpha_{1}\alpha_{2}+\beta_{1}\beta_{2})&0&\alpha_{1}^{2}+\beta_{1}^{2} \end{array}\right) \tag{110}\]  
\[\left[\mathbf{I}(\mathbf{\Gamma})\right]_{mn}=\frac{N}{\gamma^{2}}\left( \begin{array}{cccc}\alpha_{2}^{2}&\alpha_{2}\beta_{2}&-\alpha_{1}\alpha_{2} &-\alpha_{1}\beta_{2}\\ \alpha_{2}\beta_{2}&\beta_{2}^{2}&-\alpha_{2}\beta_{1}&-\beta_{1}\beta_{2}\\ -\alpha_{1}\alpha_{2}&-\alpha_{2}\beta_{1}&\alpha_{1}^{2}&\alpha_{1}\beta_{1} \\ -\alpha_{1}\beta_{2}&-\beta_{1}\beta_{2}&\alpha_{1}\beta_{1}&\beta_{1}^{2} \end{array}\right)=\frac{N}{\gamma^{2}}\mathbf{\Lambda} \tag{111}\]Having gained the expression for the Fisher information matrix using Equation (A25), the variance of the proposed estimator can be calculated by Equation (A16) as:  
\[var(\hat{\mathbf{\Gamma}})\geq\frac{\gamma^{2}}{N\boldsymbol{\Lambda}}\] (A26)