# Calibration for Sample-And-Hold Mismatches in \(M\)-Channel TIADCs Based on Statistics
## 5 Simulation and Comparison
### Effectiveness  
We evaluate the effectiveness of the proposed approach by simulating two cases.  
In the first case, an 8-channel TIADC is simulated. The parameters are set as Tables 1 and 2. The output power spectra are demonstrated in Figure 8. The higher vertical lines up to 0 dB stand for the desirable input signals, while the lower vertical lines are the bandwidth mismatch induced errors which should be suppressed below the noise floor. The signal-to-noise-and-distortion (SINAD) is  
Figure 7: The calculation of the cost function in Stage 1. (\(\arcsec\))\({}^{2\prime}\) is the square operation, and \(\arcsec\)(\(\cdot\))\({}^{\prime\prime}\) is the averaging operation).  
62.15 dB and the LSSR is 59.07 dBc before calibration, while they are enhanced to 74.95 dB and 77.53 dBc after Stage 1 calibration, and then to 75.78 dB and 91.39 dBc after Stage 2 calibration. The identified coefficients in Stage 2 calibration plotted in Figure 9 and those in Table 2 are close but not identical, because we use \(y[n]\) and \(z[n]\) to approximate the error rather than \(\tilde{y}[n]\) in (8) and (9). Therefore, the algorithm proposed in this paper can accurately identify the mismatch coefficients without much prior information of the input signal other than its WSS and modulo 8 quasi-stationary properties.  
In the second case, the performance enhancement for different mismatch coefficients in a 2-channel TIADC is illustrated. Parameters are set as Table 3. The performance before and after calibration are depicted in Figure 10, showing that our algorithm works well under circumstances of different mismatch levels. Theoretically, larger promotion should have been achieved at smaller mismatch circumstance where the error in the compensation resulted from approximating \(y\) or \(z\) to \(\tilde{y}\) is smaller  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency Range** & **Signal Type** & **Batch Size** & **Differentiator Order** \\ \hline \(0\sim 0.4/s\) & 10-tone sinusoid & \(N=512\) & \(N_{d}=20\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\beta_{1}\) & \(\beta_{2}\) & \(\beta_{3}\) & \(\beta_{4}\) & \(\beta_{5}\) & \(\beta_{6}\) & \(\beta_{7}\) \\ \hline
0.01 & \(-\)0.005 & 0.012 & \(-\)0.014 & 0.02 & \(-\)0.008 & 0.008 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mismatch coefficients.  
Figure 8: Output power spectra of the 8-channel TIADC (**a**) without calibration, (**b**) after Stage 1 calibration, and (**c**) after Stage 2 calibration.  
Figure 9: Identified coefficients for the 8-channel TIADC in Stage 2 calibration.  
(ref. (8) and (9)). Whereas actually in Figure 10, the SINAD and the LSSR initially increase with the decreasing mismatch and then saturate at certain values. This is because the quantization bits of the TIADC limit the further improvement. By increasing the quantization bits, a higher promotion can be achieved, and the SINAD and LSSR will saturate at larger values.  
In this subsection, the calibration method performs well for an 8-channel TIADC and a 2-channel TIADC with different mismatch coefficients.