## 4 Simulation Results
### Computational Complexity  
The complexity of the proposed method is analyzed in this section. The major advantage of our method over the approach proposed in [25] is that our method requires much less hardware resources and avoids the generation of sinusoidal modulation sequences \(\frac{2}{M}cos(\frac{2\pi}{M}nk)\) and \(-\frac{2}{M}sin(\frac{2\pi}{M}nk)\). A brief comparison between this work and the BWE method proposed in [25] is shown in Table 1. One drawback of our method is that the number of channels is limited to the power of two, i.e., \(M=1,2,4,8,...\), which is less general than the BWE method, as it can deal with the nonlinear mismatch problem in TIADCs with any integer number of channels. However, this drawback has little influence on the application of the proposed approach because the number of channels in a TIADC is usually the power of two in practice. The bandwidth efficiency property is maintained in the proposed method because the input-free band is utilized at the overall sampling frequency \(F_{\mathrm{s}}\). The modulation sequences in our method only consist of \(1\) and \(-1\), which are easy to process compared with the cumbersome modulation sequences \(\frac{2}{M}cos(\frac{2\pi}{M}nk)\) and \(-\frac{2}{M}sin(\frac{2\pi}{M}nk)\) adopted in [25]. Firstly, no specific circuit is required to generate modulation sequences composed of \(1\) and \(-1\). By comparison, generating accurate sinusoidal signals requires a large amount of hardware resources. Secondly, no multiplier is required to implement the modulation by Hadamard sequences because it can be done by just changing the sign of the modulated signal. This means the multipliers used to multiply the modulation sequences are not required in our method. Apart from the resources to generate the modulation sequences, the number of multipliers consumed in the BWE method is \((2M+1)(L-1)+\frac{N_{\mathrm{up}}}{2}+1\), while our method only requires \((M+1)(L-1)+\frac{N_{\mathrm{up}}}{2}+1\) multipliers. Figure 13 shows the number of multipliers consumed by the BWE method and our approach, as well as the ratio of the consumed multipliers. The order of the high-pass filter is \(44\); thus, \(22\) multipliers are needed exploiting the symmetric property of the FIR filter. As the figure shows, the difference between the multiplier consumed by the two methods increases with the increase of the number of channels and polynomial orders. The number of multipliers is the same for the two calibration methods in the case of single ADC and 2-TIADC, where \(27\) and \(29\) multipliers are consumed for \(L=3\) and \(31\) and \(35\) multipliers for \(L=5\). The difference of multiplier number and the complexity \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **This Work** & **BWE [25]** \\ \hline Number of channels & \(M=1,2,4,8,...\) & \(M=1,2,3,...\) \\ Bandwidth efficiency & \(\eta\) & \(\eta\) \\ Elements of modulation sequences & \(\{1,-1\}\) & \(\{\frac{2}{M}cos(\frac{2\pi}{M}nk),\)\(-\frac{2}{M}sin(\frac{2\pi}{M}nk)\}\) \\ Complexity (multipliers/sample) & \((M+1)(L-1)+\frac{N_{\mathrm{up}}}{2}+1\) & \((2M+1)(L-1)+\frac{N_{\mathrm{up}}}{2}+1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison. BWE, bandwidth efficient. Figure 12: The spectra of 4-TIADC output (**a**) before and (**b**) after calibration for bandpass multi-tone sinusoidal signal input. The mean square deviation (MSD) of the estimated parameters is adopted as the figure of merit for a clear view of the performance. The MSD is given by [25]:  
\[MSD_{l}[n]=10\log 10(\sum_{k=0}^{M-1}|c_{k,l}-\hat{c}_{k,l}[n]|^{2}). \tag{31}\]  
Figure 16 shows the MSD curves for one-stage calibration and two-stage calibration. It is inferred from the figure that the accuracy of nonlinear parameters' estimation is greatly improved exploiting the two-stage calibration structure. Figure 17 shows the SNDR of the TIADC output with and without calibration versus the standard deviation of nonlinearity mismatches.