# Blind Compensation of \(I/q\) Impairments in Wireless Transceivers
## 2 Model Formulation and Mitigation Methodology
### Parameter Estimation  
Once the PDF of the imbalanced signals is obtained and verified, the next step is the estimation of imbalance parameters. These modeling parameters are obtained by maximizing the likelihood function or the derived PDF. Hence, the imbalance parameters are obtained by solving the system of following equations (see Appendix B for derivation):  
\[\frac{\beta_{2}}{2}(\mathbf{1}^{\mathrm{T}}.\mathbf{p})-\gamma( \mathbf{p}_{2}^{\mathrm{T}}.\mathbf{Q}_{\mathbf{y}})=0 \tag{24}\] \[\frac{\alpha_{2}}{2}(\mathbf{1}^{\mathrm{T}}.\mathbf{p})-\gamma( \mathbf{p}_{1}^{\mathrm{T}}.\mathbf{Q}_{\mathbf{y}})=0 \tag{25}\]  
Figure 4: Kullback-Leibler (KL) divergence between KDE based estimate of \(\mathbf{y}_{R}\) and derived PDF for WCDMA 1111 signal. The dotted line shows the KL divergence for the noiseless case while the solid line includes the effect of noise.  
Figure 3: Contour plot for the PDF of imbalanced (\(g=0.92\) dB, \(g^{\prime}=0.5\) dB and \(\theta=-6\) degrees, \(\theta^{\prime}=-2\) degrees) signal using the KDE based method and the derived expression (17) for noiseless case. The derived PDF follows the PDF of the imbalanced signal obtained using the KDE method.  
\[\frac{\alpha_{1}}{2}(\mathbf{1}^{\mathrm{T}}.\mathbf{p})-\gamma(\mathbf{p}_{1}^{ \mathrm{T}}.\mathbf{I}_{\mathbf{y}}) = 0 \tag{26}\]  
\[\frac{\beta_{1}}{2}(\mathbf{1}^{\mathrm{T}}.\mathbf{p})-\gamma(\mathbf{p}_{2}^{ \mathrm{T}}.\mathbf{I}_{\mathbf{y}}) = 0 \tag{27}\]  
where \(\mathbf{1}\) is the unit vector containing all ones i.e., \(\mathbf{1}_{\mathrm{N}\times 1}=[1\ 1\ \ldots\ 1]^{\mathrm{T}}\); \(\mathbf{p}\), \(\mathbf{p}_{1}\) and \(\mathbf{p}_{2}\) are the vector representations of \(p(i)\), \(p_{1}(i)\) and \(p_{2}(i)\) from Equations (18)-(20), respectively for \(i=1\), \(\ldots\)\(N\); and \(\mathbf{I}_{\mathbf{y}}\) and \(\mathbf{Q}_{\mathbf{y}}\) are the vector representations of \(I_{y}(i)\) and \(Q_{y}(i)\).  
By solving this set of four equations, the unknown imbalance parameters, i.e., \(\alpha_{1}\ \beta_{1}\ \alpha_{2}\) and \(\beta_{2}\) can be estimated. After an estimate is obtained, the estimated \(I\) and \(Q\) signals can finally be obtained with the following expression:  
\[\left(\begin{array}{c}\wedge\\ I(i)\\ \wedge\\ Q(i)\end{array}\right)=\left(\begin{array}{cc}\wedge&\wedge\\ \alpha_{1}&\wedge\\ \alpha_{2}&\wedge\\ \end{array}\right)^{-1}\left(\begin{array}{c}I_{R}(i)\\ Q_{R}(i)\end{array}\right) \tag{28}\]  
It should be noted that we are able to estimate the final matrix \(\mathbf{A}\) and not the transmitter and receiver imbalance matrices \(\mathbf{T}\) and \(\mathbf{R}\) individually. The Cramer-Rao Lower Bound (CRLB) for the proposed estimator is provided in Appendix C.  
Solution of nonlinear equations: Here we discuss possible solutions to Equations (24)-(27) for the estimation of the imbalance parameters. Nonlinear multivariable optimization techniques, such as the simulated annealing algorithm [23] or the nonlinear system solver-based Levenberg-Marquardt method [24], can be used to solve these nonlinear system of equations. The simulated annealing algorithm relies on minimizing the system's energy by lowering the temperature of the system until the convergence criterion is met. This technique is quite effective for non-convex systems, as it has the ability to avoid being stuck in the local minima. The complexity of the proposed methodology relies on the optimization technique required to estimate the imbalance parameters using Equations (24)-(27). Simulated Annealing algorithm has been used in this work to estimate the unknown parameters. A detailed analysis on the complexity of the simulated annealing algorithm has been provided in [25].