## 4 Conclusions  
In this brief, we propose an all-digital background calibration technique for timing mismatches in TIADC. The basic theory of spurious signals caused by timing mismatches in the frequency domain is illustrated. Based on the above theory, we propose a novel compensation algorithm to eliminate the effect of timing mismatch. Moreover, we utilized the conjugate property between spurious signals to reduce the cost of filter and estimation parameters. A coarse-fine correction architecture was adopted to eliminate higher-order error terms. We also proposed a novel feedforward estimation method to accurately estimate the time-skew errors. We utilized the correlation between adjacent channels to construct an underdetermined equation and solved this using singular value decomposition to obtain time-skew errors. Compared with other feedback architecture, our estimation method can exhibit rapid convergence and is immune to potential stability issues. The proposed calibration technique is applicable to TIADCs with any number of channels, and does not necessitate the use of pilot input signals or supplementary auxiliary ADC channels. The simulation results show the effectiveness of the proposed method in both single-tone and multi-tone input conditions. It also brings about the best improvements in SNFR and SFDR, of more than 27 dB and 49 dB, respectively, for a four-channel TIADC. Finally, the ASIC design flow was adopted to evaluate the hardware cost of the proposed calibration structure.  
However, our proposed algorithm is limited by the Nyquist sampling theorem and is not effective for inputs with arbitrary Nyquist bands. Besides, the proposed feedforward estimation structure may result in inaccurate calculations when significant timing mismatches are present. Hence, further research will focus on breaking through the limitations of our calibration on arbitrary input signals through the Nyquist zone by the addition of Hilbert filters. We will also consider the use of a hybrid structure of feedforward and feedback to improve the accuracy of the calibration algorithm while maintaining a fast convergence speed.  
Conceptualization, W.X.; validation, W.X. and Z.Z.; methodology, investigation, and simulation, W.X.; writing, original draft preparation, W.X.; writing, review and editing, W.X., L.L. and Y.D.; funding acquisition, Y.D. All authors have read and agreed to the published version of the manuscript.  
This research was supported in part by the Research Foundation of Strategic Priority Research Program of the Chinese Academy of Sciences (grant number: XDA18030100).  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Characteristics** & [18] & [19] & [20] & [21] & **This Work** \\ \hline Background & Yes & Yes & Yes & Yes & Yes \\ Resolution & 10 & 11 & - & 14 & 14 \\ Channels & 2\&8 & 16 & 4\&16 & 4\&16 & 4\&8 \\ Arbitrary channel calibration & No & Yes & Yes & Yes & Yes \\ Matrix operation & Yes & No & No & No & Yes \\ Estimation parameters num. & M & 2M\(-\)2 & 2M\(-\)2 & 2M\(-\)2 & M \\ Convergence time (samples) & 50 k & 1200 k & 60 k & 20 k & 6 k \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the state-of-the-art calibration methods.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Module Name** & **Area [\(\mu\)m\({}^{2}\)]** & **Area \%** & **Power [mW]** & **Power [\%]** \\ \hline Correction & 27,344 & 77.2 & 17.9 & 84 \\ Estimation & 8076 & 22.8 & 3.4 & 16 \\ \hline Total & 35,420 & 100 & 21.3 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The synthesized results of power and area for the proposed calibration algorithm.