## 4 Test Verification
### Simulation Result
Figure 6: Performances of bandwidth mismatch calibration with the proposed method. (**a**) The SNDR of TI-ADCs with different frequencies; (**b**) the SFDR of TI-ADC with different frequencies. Figure 7: Performances of the proposed method. (**a**) The SNDR of TI-ADCs with different frequencies; (**b**) the SFDR of TI-ADC with different frequencies. is improved by 92.81 dB. Figure 10 shows that the proposed calibration algorithm is still effective for multi-tone bandpass signals with high sub-ADC Nyquist bandwidth. There are matrix inversions in the LS algorithm. Additionally, the implementation of matrix inversion in FPGA is complex. Therefore, Equation (19) is rewritten as follows:  
\[\left(\mathbf{y}_{i\text{cal}\text{ }\text{ }\text{o}}^{\text{T}}\mathbf{y}_{i \text{cal}\text{ }\text{o}}\right)h_{\Delta g_{i},\Delta\tau_{i},\Delta\omega_{i}}[k]=\mathbf{y }_{i\text{cal}\text{ }\text{cal}\text{ }\text{o}}^{\text{T}}\mathbf{y}_{ni}. \tag{35}\]  
Hence, a system of linear equations can be constructed as follows:  
\[\mathbf{A}_{N\times N}h_{\Delta g_{i},\Delta\tau_{i},\Delta\omega_{i}}[k]= \mathbf{b}_{N\times 1}. \tag{36}\]  
According to [PERSON]'s rule [29], the solution of \(h_{\Delta g_{i},\Delta\tau_{i},\Delta\omega_{i}}[k]\) requires about \(N! (N-1)(N+1)\) times of addition and \(N! (N+1)N\) times of multiplication. In this way, the matrix inversion can be avoided when solving \(h_{\Delta g_{i},\Delta\tau_{i},\Delta\omega_{i}}[k]\). Additionally, the solution of \(\mathbf{A}_{N\times N}\) requires about \(N(L_{s}-1)N\) times of addition and \(NL_{s}N\) times of multiplication. Similarly, the solution of \(\mathbf{b}_{N\times 1}\) requires about \(N(L_{s}-1)\) times of addition and \(NL_{s}\) times of multiplication. Therefore, the estimation in the \(i\)-th channel of gain, time skew, and bandwidth mismatch requires about \((N+1)(N(L_{s}-1)+(N-1)N! )+(M-1)(L-1)L_{s}\) times of addition and \(N(N+1)(L_{s}+N! )+(M-1)LL_{s}\) times of multiplication. Where \(L\) is the number of bandpass fractional delay filter taps. The comparison results of the proposed technique with the prior state-of-the-art are summarized in Table 4. The bandwidth mismatch can be calibrated with the proposed technique. In the reference [13], the calibration of time skew is based on the correlation between channels. However, Equation (36) of reference [13] is established under the necessary condition that the time skew of the adjacent channels is much smaller than the sampling period of the sub-ADC. To summarize, the calibration techniques of reference [13] must meet the necessary condition that the time skew between adjacent channels or compared with the reference channel is much smaller than the sampling period of the sub-ADC. Similarly, there are restrictions on the size of the time skew in the references [30,31]. References [16,27] and this work approach enables a wide range of time skew mismatch compensation. Therefore, references [16,27], and this work have a good effect on the improvement of SNDR and SFDR. Work [13,16,27], and this work needs 1045.5 K, 277.5 K, 720 K, and 128.1 K multiplication modules to estimate time skew, respectively. Additionally, the multiplication modules occupy most of the computing resources in the techniques of time skew estimation or correction. Therefore, it can be considered that the proposed estimation technique in this work has the lowest power consumption. Work [13], work [16], work [27], work [30], work [31], and this work needs 31, 33, 12, 25, 21, and 5 multiplication modules to correct time skew respectively. Therefore, the power consumption of the proposed technique  
Figure 10: Performances of mismatches calibration for a multi-tone signal. (**a**) Before calibration; (**b**) after calibration. of correction time skew is far less than techniques [13; 16; 27; 30; 31]. ENOB represents the final performance of the TI-ADC system after calibration. Compared with work [16; 27], and [30], the proposed technique has the best performance in ENOB. The quantization error is ignored in the work [31].