# Blind Compensation of \(I/q\) Impairments in Wireless Transceivers
## 2 Model Formulation and Mitigation Methodology
### Accuracy of Derived PDF  
The accuracy of the derived PDF can be measured by various figures of merit, including the Kullback-Leibler (KL) divergence, which is the measure of similarity or dissimilarity between two PDFs. If we consider two discrete PDFs, \(f(x_{k})\) and \(g(x_{k})\), using \(K\) independent and identically distributed samples \(\{x_{k}\}_{k=1}^{K}\), KL divergence can then be defined as [21]:  
\[D_{\mathrm{KL}}(f||g)=\sum\limits_{k=1}^{K}f(x_{k})log\left(\frac{f(x_{k})}{g( x_{k})}\right) \tag{22}\]A higher value of KL divergence indicates dissimilarity between the compared PDFs, while, a smaller value indicates similarity or closeness of the compared PDFs. Another figure of merit for comparing two PDFs is the Hellinger distance. The square of the Hellinger distance is defined as [22]:  
\[D_{\mathrm{H}}^{2}(f||g)=\frac{1}{2}\underset{k=1}{\overset{K}{\sum}}\bigg{(} \sqrt{f(x_{k})}-\sqrt{g(x_{k})}\bigg{)}^{2} \tag{23}\]  
The first step is to analyze the effect of the \(I/Q\) imbalances on the PDF of the signal. Kernel Density Estimation (KDE) is a well-established method for PDF estimation and hence has been used here for comparison. Table 1 compares the input signal's PDF (\(f_{\mathrm{in,KDE}}\)) and the imbalanced signal's PDF (\(g_{\mathrm{KDE}}\)) using the nonparametric KDE method for the metrics provided in Equations (22) and (23) for the noiseless case. These values of the KL divergence and Hellinger distance show the divergence in the PDFs of the modulator's input signal and the demodulator's output signal. This is due to the presence of the \(I/Q\) imbalance in the system, which changes the PDF of the input signal. The next step is to determine the accuracy of the derived analytical expression in Equation (17). For this purpose, the derived PDF of the imbalanced signal (\(f_{y}(\mathbf{y};G)\) was compared to the KDE-based estimate of the output i.e., \(g_{\mathrm{KDE}}\). In this case, Table 1 shows that the values of both the KL divergence and the Hellinger square distance are quite small for both Wideband Code Division Multiple Access (WCDMA) and Long-Term Evolution (LTE) signals, resulting in the conclusion that the derived PDF is very similar to the desired PDF. This is also evident from Figures 2 and 3, which show the effect of \(I/Q\) imbalance on the PDF contours and the accuracy of the derived PDF of the output signal of the demodulator, respectively. It can be concluded from Table 1 and Figure 3 that there is an excellent correspondence between the derived and KDE-based PDFs of the imbalanced signals. It should be noted that the analysis provided here is for the PDF obtained in Equation (17) for the noiseless case i.e., derived PDF. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Signal**} & \multicolumn{2}{c}{\(D_{\mathrm{KL}}\)} & \multicolumn{2}{c}{\(D_{\mathrm{H}}^{2}\)} \\ \cline{2-5}  & \(f_{\mathrm{in,KDE}}\parallel g_{\mathrm{KDE}}\) & \(f_{y}(\mathbf{y};\mathbf{\Gamma})\parallel g_{\mathrm{KDE}}\) & \(f_{\mathrm{in,KDE}}\parallel g_{\mathrm{KDE}}\) & \(f_{y}(\mathbf{y};\mathbf{\Gamma})\parallel g_{\mathrm{KDE}}\) \\ \hline WCDMA 1111 & \(1.146\times 10^{-1}\) & \(2.2\times 10^{-3}\) & \(2.38\times 10^{-2}\) & \(4.75\times 10^{-4}\) \\ LTE 101 & \(1.368\times 10^{-1}\) & \(2\times 10^{-3}\) & \(2.8\times 10^{-2}\) & \(4.97\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of probability density functions (PDFs) for noiseless case and \(g=g^{\prime}=0.92\) dB and \(\theta=\theta^{\prime}=-6\) degrees. Figure 2: Contour plot for the PDF of Wideband Code Division Multiple Access (WCDMA) 1111 (**a**) input and (**b**) imbalanced output \(I/Q\) signals using the Kernel Density Estimation (KDE) method. A deviation from the input signal’s PDF can be seen due to \(I/Q\) imbalance. Figure 4 shows the comparative analysis for the noisy case. It can be seen that the KL divergence between the derived PDF and KDE based estimate of \(\mathbf{y}_{R}\) is higher at lower Signal to Noise Ratios (SNRs).