# Calibration for Sample-And-Hold Mismatches in \(M\)-Channel TIADCs Based on Statistics
## 5 Simulation and Comparison
### Comparisons  
In this section, we choose the IFB-based method [23; 35] and the I/Q-based method [24; 25] for comparison which are better than other methods both in bandwidth efficiency and complexity. A 2-channel TIADC is used for the following two cases. In the first case, the LSSR improvements of the three methods are compared. The parameters are set as Tables 4-7, and the differentiators used in the IFB method and our method are identical. The output power spectra are demonstrated in Figure 13. The SINAD is 59.39 dB and the LSSR is 58.00 dBc before calibration, and they are slightly enhanced to 62.89 dB and 61.77 dBc using the IFB-based method, and they are enhanced to 75.08 dB and 84.95 dBc using the I/Q-based method, and they are enhanced to 77.34 dB and 93.05 dBc employing the method proposed in this paper. For the IFB-based method, an input-free band is created in the frequency spectrum by oversampling where the mismatch-induced error exists without input signal. By exerting a high-pass filter whose passband coincides with the IFB, the mismatch coefficients can be identified. However, in this case, the error spectrum does not appear in the IFB and the identification fails (ref. Figure 13b). For some other kinds of narrow-band signals, the IFB-based method cannot also work. For the I/Q-based method, the TIADC's output signal is converted to a complex signal with frequency shift, which is similar to the homodyne receiver's output signal with I/Q mismatch, and the I/Q mismatch calibration technique is used to calibrate the TIADC's mismatch. The compensation filter's coefficients are determined by restoring the complex signal's circularity. However, in this case where the signals are multi-tone sinusoids, there is no remarkable difference between the circularity of the signal without mismatch and that of the signal with mismatch, and therefore the calibration cannot suppress the error to the noise floor. For some other types of signals mainly composed of sinusoids, the I/Q-based method also cannot work. So compared with the other two methods, our method has more extensive applicability for different types of signals. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Frequency Range** & **Signal Type** & **Mismatch Coefficient** \\ \hline
0.26 \(\sim\) 0.38\(f_{s}\) & 10-tone sinusoids & \(\beta=0.02\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Common simulation parameters for LSSR improvement comparison. Figure 12: The (**a**) SINAD and (**b**) LSSR after two stages of calibration for different batch sizes. In the second case, the resource consumptions are compared among the three methods when the same error attenuation is achieved after calibration. The multiplications used in one loop of calibration is chosen as the indicator. The input signal is set as Table 8 so that all the methods can work effectively. \(\beta=0.08\). The variables \(N_{d}\), \(N_{hp}\), \(N_{cp}\), \(N_{hb}\) and \(N\) are defined in Tables 1, 5 and 6. The resource consumption for the I/Q-based method is calculated as follows. \(\langle 1\rangle\) It needs \(0.5N_{hb}\) multiplications to pass the output signal through the Hilbert filter because the filter's taps are anti-symmetric with a null center tap. \(\langle 2\rangle\) It needs \(4(N_{cp}+1)\) multiplications to compensate the signal, since both the signal and the filter taps are complex, and multiplying two complex numbers actually \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Hilbert Filter Order** & **Hilbert Filter PB** & **Compensation Filter Order** \\ \hline \(N_{hb}=10\) & \(0.075f_{s}\sim 0.425f_{s}\) & \(N_{cp}=2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Simulation parameters for the I/Q method. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Signal Type** & **Carrier Frequency** & **Duration/Sample** \\ \hline
16-QAM & \(0.05f_{s}\) & \(10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Input signal for resource consumption comparison.