## 4 Test Verification
### Simulation Result  
The signal-to-noise distortion ratio (SNDR) is a metric that describes the signal performance. Additionally, SNDR can be calculated by the following:  
\[\rm{SNDR}=10\rm{lg}\bigg{(}\frac{P_{\rm Signal}}{P_{\rm Noise}+P_{\rm Distortion }}\bigg{)} \tag{34}\]where \(P_{\text{Signal}}\) is the power of the ideal signal, \(P_{\text{Noise}}\) is the power of the noise signal, and \(P_{\text{Distortion}}\) is the spurious power of the output signal. In the TI-ADCs system, the presence of mismatches leads to an increase in \(P_{\text{Noise}}\). When and \(P_{\text{Noise}}\) are fixed, the smaller the calibration error, the higher the SNDR. Therefore, the SNDR is applied to evaluate the effect of error calibration. The calibration of offset is simpler than other mismatches. Hence, the offset is set to 0 in the simulation tests. The 12-bit 8 GS/s 8-channel TI-ADCs simulation system is established according to Equation (3) with MATLAB. Additionally, the mismatches of the simulation model are shown in Table 1. Moreover, the parameters of the bandpass fractional filters are shown in Table 2. Additionally, the number of taps of \(h_{\Delta g_{i}\Delta\tau_{i}\Delta\omega_{i}}[k]\) is 73. In the 12-bit 8 GS/s 8-channel TI-ADCs simulation system, the SNDR of the input signal before sampling is 60 dB because of the Gaussian white noise in each channel. Figure 6 shows the values of SNDR and spurious-free dynamic range (SFDR) for sinusoidal signals with the different frequencies of the first Nyquist bandwidth. The frequency range is limited within 0.05-0.42 GHz. Additionally, the frequency spacing between two single-tone sinusoidal signals is 0.01 GHz. Where \(f_{\text{c}}\) is the sampling frequency of sub-ADC. In Figure 6a, the SNDR is stable above 65 dB in the test frequency range. Additionally, the SFDR is stable above 80 dB in Figure 6b. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sub-ADC Number** & **Offset** & **Gain** & **Time Skew** & **Bandwidth** \\ \hline
1 & 0 & 1 & 0 & \(4\times f_{s}\) \\
2 & 0 & 1 & 0 & \(1\times f_{s}\) \\
3 & 0 & 1 & 0 & \(2\times f_{s}\) \\
4 & 0 & 1 & 0 & \(3\times f_{s}\) \\
5 & 0 & 1 & 0 & \(1\times f_{s}\) \\
6 & 0 & 1 & 0 & \(2\times f_{s}\) \\
7 & 0 & 1 & 0 & \(3\times f_{s}\) \\
8 & 0 & 1 & 0 & \(3\times f_{s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The mismatches of the TI-ADCs simulation system. Figure 5: Bandpass fractional delay filter. The mismatches shown in Table 3 are set to verify that the proposed methods are still effective when there are gain, time skew, and bandwidth mismatch at the same time. The frequency range is limited within 0.05-0.42 GHz. Where \(f_{\rm c}\) is the frequency of sub-ADC, and \(f_{\rm s}\) is the frequency of the TI-ADCs system. Additionally, Figure 6 shows the SNDR and SFDR before and after calibration. It can be seen from Figure 7 that the proposed calibration method still has a good effect in the cases of gain, time skew, and bandwidth mismatch. The SNDR before calibration does not exceed 20 dB. Additionally, the SNDR is above 65 dB after calibration. Therefore, the value of SNDR is improved by at least 45 dB after calibration. Moreover, the value of SFDR is above 80 dB after calibration. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sub-ADC Number** & **Offset** & **Gain** & **Time Skew** & **Bandwidth** \\ \hline
1 & 0.000 & 1.000 & 0.000 & 4.000 \(\times f_{\rm s}\) \\
2 & 0.000 & 1.400 & \(\pm\)0.500 & 1.000 \(\times f_{\rm s}\) \\
3 & 0.000 & 1.300 & \(\pm\)\(\frac{500}{2}\) & 2.000 \(\times f_{\rm s}\) \\
4 & 0.000 & 1.200 & \(\pm\)\(\frac{500}{100}\) & 3.000 \(\times f_{\rm s}\) \\
5 & 0.000 & 1.100 & \(\pm\)\(\frac{400}{400}\) & 1.000 \(\times f_{\rm s}\) \\
6 & 0.000 & 1.200 & \(\pm\)\(\frac{500}{300}\) & 2.000 \(\times f_{\rm s}\) \\
7 & 0.000 & 1.300 & \(\pm\)\(\frac{500}{100}\) & 3.000 \(\times f_{\rm s}\) \\
8 & 0.000 & 1.100 & \(\pm\)\(\frac{500}{f_{\rm s}}\) & 3.000 \(\times f_{\rm s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The mismatches of the TI-ADCs system.