# Blind Compensation of \(I/q\) Impairments in Wireless Transceivers
## 2 Model Formulation and Mitigation Methodology  
### 1/Q Impairment Model Formulation  
Figure 1 shows a simplified block diagram of a transceiver system with modulator and demodulator imperfections, along with the post-compensation block. In a typical transmitter, the baseband \(I\) and \(Q\) signals pass through the modulator and are up-converted to the desired radio frequency (RF). Ideally, the local oscillator should produce a 90 degrees phase difference between the \(I\) and the \(Q\) branches, while maintaining equal gain between the two branches. However, this difference is not exactly 90 degrees, due to the non-idealities of various components used for up- and down-conversion, resulting in phase imbalance. Similarly, a gain mismatch is introduced between the \(I\) and the \(Q\) branches of the modulator/-demodulator, which also affects the quality of the signal. The combined effect of the gain and phase imbalances is referred to as \(I/Q\) imbalance. The RF output of the modulator can be represented as: \(y_{RF}(t)=Re\Big{\{}(I_{T}(t)+jQ_{T}(t))e^{2\pi ft}\Big{\}}\), where \(f\) is the carrier frequency and \(t\) is the time. The baseband equivalent of the output \(I\) and \(Q\) of the modulator can be expressed as [11]:  
\[I_{T}(t)=g_{I}\cos(\theta)I(t)+g_{Q}\sin(\theta)Q(t) \tag{1}\]  
\[Q_{T}(t)=g_{I}\sin(\theta)I(t)+g_{Q}\cos(\theta)Q(t) \tag{2}\]  
where \(g_{I}\) and \(g_{Q}\) are the transmitter's gain imbalance parameters in the \(I\) and the \(Q\) paths, respectively. The phase imbalance i.e., \(\phi\) is split equally between the two branches i.e., \(\theta=\phi/2\). The transmitted signal after passing through an AWGN channel, in matrix form, can be written as:  
\[\mathbf{y}_{T}=\varsigma\mathbf{T}\mathbf{x}+\mathbf{n} \tag{3}\]  
where  
\[\varsigma=\sqrt{\frac{g_{I}^{2}+g_{Q}^{2}}{2}};\mathbf{T}=\sqrt{\frac{2}{(1+g ^{2})}}\left( \begin{array}{cc}g\cos(\theta)&\sin(\theta)\\ g\sin(\theta)&\cos(\theta)\end{array} \right) \tag{4}\]  
Figure 1: Block diagram of transceiver system with modulator’s and demodulator’s imperfections. Here \(g=g_{I}/g_{Q_{I}}\,\mathbf{y}_{T}=\left[I_{T}(t)\ Q_{T}(t)\right]^{\mathrm{T}}\) is the imbalanced signal at the output of the modulator, while \(\mathbf{x}=\left[I(t)\ Q(t)\right]^{\mathrm{T}}\) is the input signal to the modulator. \(\mathbf{T}\) is the matrix of modulator's imbalance parameters, \(\mathbf{n}=\left[I_{I}(t)\ n_{Q}(t)\right]^{\mathrm{T}}\) is the noise vector, and \(n_{I}(t)\) and \(n_{Q}(t)\) are the noise components in the \(I\) and the \(Q\) branches respectively. This signal is then demodulated at the receiver, which also introduces an \(I/Q\) imbalance resulting in the following expressions for the received signals:  
\[I_{R}(t)=g_{II}\cos(\theta^{\prime})I_{T}(t)+g_{QQ}\sin(\theta^{\prime})Q_{T}(t )+\widetilde{n}_{I}(t) \tag{5}\]  
\[Q_{R}(t)=g_{II}\sin(\theta^{\prime})I_{T}(t)+g_{QQ}\cos(\theta^{\prime})Q_{T}(t )+\widetilde{n}_{Q}(t) \tag{6}\]  
where \(g_{II}\) and \(g_{QQ}\) are the receiver's gain imbalance parameters in the \(I\) and \(Q\) components, respectively. Similar to the modulator, \(\theta^{\prime}\) is the phase imbalance in each branch of the receiver. Defining \(g^{\prime}=g_{II}/g_{QQ}\), the output of the demodulator can be written as:  
\[\mathbf{y}_{R}=\zeta^{\prime}\mathbf{R}\mathbf{y}_{T}=\zeta\mathbf{c}^{\prime }\mathbf{RTx}+\widetilde{\mathbf{n}} \tag{7}\]  
where \(\mathbf{y}_{R}=\left[I_{R}(t)\ Q_{R}(t)\right]^{\mathrm{T}}\) is the signal at the output of the demodulator, \(\mathbf{R}\) is the matrix of the receiver's imbalance parameters and  
\[\zeta^{\prime}=\sqrt{\frac{g_{II}^{2}+g_{QQ}^{2}}{2}};\ \mathbf{R}=\sqrt{\frac{2}{ \left(1+{g^{\prime}}^{2}\right)}}\left( \begin{array}{cc}g^{\prime}\cos( \theta^{\prime})&\sin(\theta^{\prime})\\ g^{\prime}\sin(\theta^{\prime})&\cos(\theta^{\prime})\end{array}