# Blind Compensation of \(I/q\) Impairments in Wireless Transceivers
## 3 Simulation Results  
### Normalized Mean Squared Error and Image Suppression Evaluation  
The figures of merit to assess the accuracy of the proposed method include the normalized mean squared error (NMSE), image suppression (IMGsup) and the bit error rate (BER). NMSE is an effective figure of merit for in-band performance analysis and is given by [26]:  
\[\mathrm{NMSE(dB)}=10\log_{10}(\frac{\sum\limits_{l=1}^{L}\left|y_{act}(l)-y_{ est}(l)\right|^{2}}{\sum\limits_{l=1}^{L}\left|y_{act}(l)\right|^{2}}) \tag{29}\]  
where \(y_{\mathrm{act}}\) is the desired/measured output, \(y_{\mathrm{est}}\) is the output estimated by the proposed methodology and \(L\) is the length of data used for evaluation. Image suppression [7] (reciprocal of image rejection ratio) is the ratio of the image power to the desired signal power and measures the effectiveness of the algorithm to mitigate the \(I/Q\) imbalance. \[\mathrm{IMG}_{\mathrm{sup}}(f)=10\log_{10}(\frac{P_{\mathrm{image}}(f)}{P_{ \mathrm{signal}}(f)}) \tag{30}\]  
where \(P_{\mathrm{image}}\) is the power in the image band and \(P_{\mathrm{signal}}\) is the power in the signal band and \(f\) denotes the frequency. Figure 5 shows the NMSE performance of the proposed methodology for various signals. The signals used for the evaluation of the proposed methodology were a 20 MHz four-carrier Wideband Code Division Multiple Access (WCDMA 1111) signal, a 9 MHz Long Term Evolution (LTE) signal and 20 MHz WCDMA 1101 signal. It can be seen in the figure, that with the proposed mitigation approach, the NMSE is considerably reduced as compared to the uncompensated case. As mentioned in [27], \(I/Q\) imbalance results in mirror frequency imaging, i.e., if the baseband/low-IF signal is up-converted to RF at a carrier frequency of \(w_{\rm c}\), a mirror image of this signal is created at \(-w_{\rm c}\) which when down-converted to baseband/low-IF results in cross talk between the signals, distorting the desired signal. For signals symmetric around the DC such as the WCDMA 1111 signal, the image lies exactly on top of the desired signal when down-converted. Hence, it is difficult to observe the image caused by the \(I/Q\) imbalance using the power spectral density. This can be seen in Figure 6a. However, as shown in Figure 6b, for signals not symmetric about DC, it is possible to observe the power in the image band using the spectrum of the signal. For this reason, three cases of the WCDMA 1101 signals (namely S1, S2 and S3 as shown in Table 2) with different center frequencies (\(f_{\rm c}\)) are used to measure the image suppression performance of the proposed technique as shown in Figure 7. These signals are centered on different frequencies, i.e., \(f_{\rm c}\) = 10 MHz, 5 MHz and around DC, respectively, and are used to evaluate the performance of the proposed methodology by observing the suppression in the image band. For WCDMA 1101 signal S1, the image lies out of band when down-converted, however, for WCDMA 1101 signals S2 and S3, the image band falls into the band of interest, degrading the signal quality. Figure 5: Normalized mean squared error (NMSE) vs. SNR using the proposed mitigation approach for \(g=g^{\prime}=0.92\) dB and \(\theta=\theta^{\prime}=-6\) degrees. Figure 6: Mirror frequency Imaging (MFI) due to \(I/Q\) imbalance. (**a**) MFI for symmetric 1111 signal for which the power in the image band cannot be seen using PSD; (**b**) MFI for asymmetric signals and the power in the image band is visible. The spectrum of compensated signal achieved by the proposed algorithm for these three WCDMA 1101 signals can also be seen in Figure 7 for \(g\) = 0.92 dB, \(g^{\prime}\) = 0.5 dB and \(\theta\) = \(-6\) degrees, \(\theta^{\prime}\) = \(-2\) degrees.