# Blind Compensation of \(I/q\) Impairments in Wireless Transceivers
## 1 Introduction  
Modern communication networks suffer from various imperfections that significantly degrade their performance. Transmitters (Tx) and receivers (Rx) employ local oscillators to up- and down-convert the in-phase (\(I\)) and quadrature-phase (\(Q\)) signals to the carrier frequency and baseband, respectively. However, due to the non-idealities of the local oscillators and mixers, there is an imbalance in the gain and phase of the up/down-converted \(I/Q\) signals. \(I/Q\) imbalance results in mirror frequency imaging [1]. This problem is quite prevalent in direct-conversion radio architecture where the image falls within the band of interest after down-conversion. The effects of \(I/Q\) imbalance can be mitigated using signal processing techniques, instead of adding hardware components to the transceiver topology or making changes in the transceiver topology. Various methods have been proposed to mitigate the effects of modulator and/or demodulator imperfections [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. These include both frequency-independent (e.g., [4; 5; 6; 7; 10; 11; 15]) and frequency-dependent models (e.g., [8]). The method proposed in [1] employs two solutions to mitigate the effects of \(I/Q\) mismatch in OFDM receivers. The first method is based on least squares, while the second is an adaptive least mean squares based method using pilot tones. However, the effect of transmitter imbalances has not been considered. Similarly, the methods suggested in [2; 3] aim to mitigate the joint effect of receiver \(I/Q\) imbalance in the presence of phase noise and carrier frequency offset, respectively. The imbalance parametersare estimated by minimizing the mean squared error. A maximum likelihood based estimation for transceiver's \(I/Q\) imbalance is presented in [4] in the presence of Additive White Gaussian Noise (AWGN) channel. Many of these methods, however, require long training sequences to estimate the model coefficients. These techniques also require the disruption of signal transmission as the imbalanced signals need to be time and phase aligned with the training/input signal for accurate modeling. Blind methods are attractive in cases where long training sequences are not available or need to be avoided. Blind and semi-blind methods have been proposed in literature [5; 6; 7; 8; 9; 11] with the aim of mitigating these imperfections. A gradient-based adaption algorithm based on image suppression ratio has been suggested in [7] for low-IF transmitters. The method proposed in [8] uses second-order statistics, i.e., the circularity property of the signals, to eliminate the effect of \(I/Q\) imbalance in quadrature radio receivers. Similarly, the method proposed in [10] provides a maximum likelihood estimation of receiver \(I/Q\) imbalance parameters. A higher order statistics based method for blind transmitter \(I/Q\) imbalance calibration has been proposed in [11]. The authors make use of a diode detector in the feedback path for detecting the instantaneous envelope of the transmitted signal, for its simplicity. As pointed out by the authors, demodulator or mixer based circuits can also be used in the feedback path. However, a limitation of these methods is that they either consider the effect of the modulator or the demodulator \(I/Q\) imbalance, i.e., if the effect of the modulator imbalance is considered, the demodulator is considered as ideal and vice versa. Similarly, the authors in [12] investigated the effects of Tx \(I/Q\) imbalance in direct-conversion transmitters while considering a super-heterodyne receiver. This, however, eliminates the direct-conversion architecture's advantage of low complexity and cost efficiency. Hence, the methods that can compensate for both the Tx and Rx impairments without the need for a change in the architecture are attractive for practical systems. Methods that consider this combined effect have been presented in [13; 14].