## 2 The Proposed Timing Mismatch Calibration Technique
### Timing Mismatch Compensation  
From Equation (6), we know that the spurious signals \(Q_{i}(j\omega)\) are associated with \(X(j\omega)\). Theoretically, if we estimate the \(\gamma_{k}\) accurately, the product of modulation coefficient and frequency shift in ideal input signal \(C_{i}\cdot X\left(j\left(\omega-\frac{2\pi i}{M}\right)\right)\) approximately equals \(Q_{i}(j\omega)\), as shown in Figure 3b,c. Thus, the corrected outputs \(\hat{Y}(j\omega)\) can be written as  
\[\hat{Y}(j\omega)=Y(j\omega)-\sum_{i=1}^{M-1}Q_{i}(j\omega). \tag{7}\]  
Figure 3: (**a**) The TIADC output spectrum with timing mismatch. (**b**) The frequency shift in \(X(j\omega)\). (**c**) The amplitude modulation of the frequency-shifting parameters. (**d**) The spectrum of corrected signal \(\hat{Y}(j\omega)\). Figure 2: The block diagram of sampling in an M-channel TIADC. The spectrum of corrected signal \(\hat{Y}(j\omega)\) is presented in Figure 3d. However, the IDFTs of Equations (4)-(7) are complex-valued signals and we cannot directly handle them in the time domain. To solve the above issue, many reports [19; 20; 21] exploit the additional filters to structure the imaginary components, but this approach has undoubtedly led to an increase in computational complexity and hardware resource consumption. Hence, we propose a novel compensation algorithm to optimize the calculation process and circumvent the calculation from the imaginary components. Note that the time-skew errors are usually much less than the sampling period and, for the sake of description, we ignore the presence of higher-order error terms and use the first-order Taylor series to rewrite \(Q_{i}(j\omega)\), expressed as follows:  
\[Q_{i}(j\omega) \approx \frac{1}{M}\sum_{k=0}^{M-1}e^{-j2\pi ik/M}\bigg{(}1+j\bigg{(} \omega-\frac{2\pi i}{M}\bigg{)}\cdot\gamma_{k}\bigg{)}X\bigg{(}j\bigg{(}\omega- \frac{2\pi i}{M}\bigg{)}\bigg{)} \tag{8}\] \[= \frac{1}{M}\sum_{k=0}^{M-1}e^{-j2\pi ik/M}\gamma_{k}\cdot j\bigg{(} \omega-\frac{2\pi i}{M}\bigg{)}X\bigg{(}j\bigg{(}\omega-\frac{2\pi i}{M}\bigg{)} \bigg{)}\]  
where \(j\bigg{(}\omega-\frac{2\pi i}{M}\bigg{)}\) is the frequency response of the derivative filter. Similarly, \(Q_{M-i}(j\omega)\) can be written as  
\[Q_{M-i}(j\omega) = \frac{1}{M}\sum_{k=0}^{M-1}e^{-j2\pi(M-i)k/M}\gamma_{k}\cdot j \bigg{(}\omega-\frac{2\pi(M-i)}{M}\bigg{)}X\bigg{(}j\bigg{(}\omega-\frac{2\pi (M-i)}{M}\bigg{)}\bigg{)} \tag{9}\] \[= \frac{1}{M}\sum_{k=0}^{M-1}e^{j2\pi ik/M}\gamma_{k}\cdot j\bigg{(} \omega+\frac{2\pi i}{M}\bigg{)}X\bigg{(}j\bigg{(}\omega+\frac{2\pi i}{M} \bigg{)}\bigg{)}.\]  
Thus, the IDFT of \(Q_{i}(j\omega)\) can be written as  
\[Q_{i}[n] = \frac{1}{M}\sum_{k=0}^{M-1}e^{-j2\pi ik/M}\gamma_{k}e^{j2\pi in/M }\cdot(h_{d}[n]*x[n]) \tag{10}\] \[= \frac{1}{M}\sum_{k=0}^{M-1}\gamma_{k}\bigg{(}cos\bigg{(}\frac{2 \pi ik}{M}\bigg{)}cos\bigg{(}\frac{2\pi in}{M}\bigg{)}+sin\bigg{(}\frac{2\pi ik }{M}\bigg{)}sin\bigg{(}\frac{2\pi in}{M}\bigg{)}\] \[+j\bigg{(}sin\bigg{(}\frac{2\pi in}{M}\bigg{)}cos\bigg{(}\frac{2 \pi ik}{M}\bigg{)}-sin\bigg{(}\frac{2\pi ik}{M}\bigg{)}cos\bigg{(}\frac{2\pi in }{M}\bigg{)}\bigg{)}\bigg{)}\cdot x^{{}^{\prime}}[n]\]  
where \(h_{d}[n]\) denotes the unit sample response of the derivative filter and \(x^{{}^{\prime}}[n]\) denotes the derivative of ideal input signal \(x[n]\). Note that when we calculate the IDFT the \(Q_{M-i}(j\omega)\), the terms of the frequency shift parameter in \(Q_{M-i}[n]\) conjugate with the frequency shift parameter in \(Q_{i}[n]\). Hence, the IDFT of \(Q_{M-i}(j\omega)\) can be written as  
Figure 3:0 (11) Figure 3:1 Figure 3:2 Figure 3:3  
with  
Figure 3:4  
where \(e_{i}[n]\) denotes the reconstruction conjugate error pair and \(w_{i}[n]\) denotes the corresponding coefficient. \(w_{i}[n]\) is a function of period M and, with a specific number of channels, M, the coefficient in front of the time error can be calculated in advance, which can save hardware resource consumption.