# Blind Compensation of \(I/q\) Impairments in Wireless Transceivers
## 2 Model Formulation and Mitigation Methodology
### Closed-form PDF of the Imbalanced Signal  
The following assumptions were made before deriving the PDF of the demodulated signal:  
\(A1\): The \(I\) and \(Q\) components are jointly Gaussian and independent, i.e., (\(E[IQ]=E[I]E[Q]\)), where \(E[.]\) is the expectation operation.  
\(A2\): The input \(I\) and \(Q\) signals have similar statistics, i.e., both components have zero means (\(u_{I}=u_{Q}=0\)) and same variances (\(\sigma_{I}^{2}=\sigma_{Q}^{2}=\sigma^{2}\)).  
These assumptions result in the covariance matrix of \(\mathbf{x}\) to be diagonal i.e., \(\mathbf{C_{x}}=\sigma^{2}\mathbf{I}\), where \(\mathbf{I}\) represents identity matrix and are valid for practical communication signals without making significant changes in system specifications. The assumption of \(I\) and \(Q\) signals being independent is considered in many works (e.g., reference [8] in this work and references [20; 21; 22; 23; 24; 25] found in [11]). Orthogonal frequency division multiplexing (OFDM) signals have been shown to exhibit Gaussian characteristics [19]. The authors in [11] also assume similar statistics of \(I\) and \(Q\) signals.  
For \(N\) samples of the received signal, the \(N\)-point PDF of \(\mathbf{y}\) can be written as [10]:  
\[f_{y}(\mathbf{y};\mathbf{\Gamma})=\prod_{i=1}^{N}\frac{1}{2\pi\sqrt{\left| \mathbf{C_{y}}(\mathbf{\Gamma})\right|}}\exp(-\frac{1}{2}\mathbf{y}_{i}^{ \mathrm{T}}\mathbf{C_{y}^{-1}}(\mathbf{\Gamma})\mathbf{y}_{i}) \tag{16}\]  
where \(\mathbf{C_{y}}\) denotes the covariance matrix of \(\mathbf{y}\) and \(G=\left[\alpha_{1}\ \beta_{1}\ \alpha_{2}\ \beta_{2}\right]^{\mathrm{T}}\) is the vector of unknown imbalance parameters defined in Equations (12)-(15). Using the expressions for the covariance matrix of \(\mathbf{y}\) i.e., \(\mathbf{C_{y}}\) and its determinant, the joint PDF of \(I_{y}\) and \(Q_{y}\) takes the following form (see Appendix A for derivation):  
\[f_{y}(\mathbf{y};\mathbf{\Gamma})=\frac{1}{\left(2\pi\sigma^{2}|\gamma|\right) ^{N}}e^{-(\frac{1}{2\pi\sigma^{2}}\sum\limits_{i=1}^{N}p(i))} \tag{17}\]  
where \(\sigma\) represents the standard deviation of the \(I\) and \(Q\) signals, \(\left|\,\gamma\,\right|\) is the determinant of \(\mathbf{A}\). An auxiliary set of variables has been used for mathematical simplicity, which are defined as:  
\[p(i)=p_{1}^{2}(i)+p_{2}^{2}(i) \tag{18}\]  
\[p_{1}(i)=\beta_{2}I_{y}(i)-\beta_{1}Q_{y}(i) \tag{19}\]  
\[p_{2}(i)=-\alpha_{2}I_{y}(i)+\alpha_{1}Q_{y}(i) \tag{20}\]  
The analytical closed-form expression for the PDF of the imbalanced signal obtained using Equation (17) is herein referred to as the 'derived' PDF. It should be noted here that the noise has not been included in the analysis i.e., \(\mathbf{y}_{R}=\mathbf{y}\) in Equation (10). A maximum likelihood estimation of the imbalance parameters is obtained using this PDF. In the presence of noise, however, the PDF of the imbalanced signal \(\mathbf{y}_{R}\) is given below, where * represents convolution of the two PDFs [20]:  
\[f_{y_{R}}(\mathbf{y}_{R};\mathbf{\Gamma})=f_{y}(\mathbf{y};\mathbf{\Gamma})*f _{\overline{n}}(\mathbf{\widetilde{n}};\mathbf{\Gamma}) \tag{21}\]