# Calibration for Sample-And-Hold Mismatches in \(M\)-Channel TIADCs Based on Statistics
## 3 Mismatch Compensation  
Since the compensation structure is utilized in the identification, the compensation is firstly considered. When the mismatch coefficient \(\hat{\beta}_{m}\) is identified (which is described in detail in Section 4), the error \(e_{m}[Mn]\) in (7) can be generated in theory. In practice, however, no prior knowledge about the desirable signal \(\tilde{y}\left[n\right]\) (defined in Figure 5) is available. One widely acceptable solution is to approximate it by \(y\left[n\right]\)[6,7,22,28]. The Richardson iteration structure illuminated in Figure 6 is introduced for high precision compensation [40]. In Stage 1, we try to generate the 1 st-order term of Taylor series in (7) using the output signal \(y\left[n\right]\) and then eliminate it from the original output signal. Then the compensated down-sampled signal of Channel \(m\) is  
\[ \begin{split} p_{m}\left[Mn\right]&=v_{m}\left[Mn \right]+\frac{\hat{\beta}_{m1}}{\Omega_{c}}y_{m}^{{}^{\prime}}\left[Mn\right] \\ &=\tilde{v}_{m}\left[Mn\right]+\frac{\hat{\beta}_{m1}-\hat{\beta }_{m}}{\Omega_{c}}\tilde{y}_{m}^{{}^{\prime}}\left[Mn\right]+\frac{\beta_{m} \left(1+\beta_{m}-\hat{\beta}_{m1}\right)}{\Omega_{c}^{2}}\tilde{y}_{m}^{{}^{ \prime\prime}}\left[Mn\right]+o\left(\tilde{y}_{m}^{{}^{\prime\prime}}\left[Mn \right]\right),\end{split}  \tag{8}\]  
where \(\hat{\beta}_{m1}\) is the identified coefficient for \(\beta_{m}\) in Stage 1, and \(o\left(\tilde{y}_{m}^{{}^{\prime\prime}}\left[Mn\right]\right)\) is higher order differential of \(\tilde{y}_{m}\left[Mn\right]\) which is too weak to consider. Comparing (8) with (7), \(p_{m}\left[Mn\right]\) is closer to \(\tilde{v}_{m}\left[Mn\right]\) than \(v_{m}\left[Mn\right]\), where the detailed derivation is given in [22,40]. Such being the case, we can use \(p_{m}\left[Mn\right]\) instead of \(v_{m}\left[Mn\right]\) for further approximation to \(\tilde{v}_{m}\left[Mn\right]\) in the error cancellation of Stage 2. Figure 5: (**a**) \(\tilde{v}_{m}\left[Mn\right]\) is obtained through a TIADC without mismatches. (**b**) \(\tilde{y}_{m}^{{}^{\prime}}\left[Mn\right]\) and \(\tilde{y}_{m}^{{}^{\prime\prime}}\left[Mn\right]\) are the subsequences of signal \(\tilde{y}\)’s first differential and second differential, respectively. The \(j\omega\) is a first-order differentiator, and \(\left(j\omega\right)^{2}\) is a second-order differentiator. In Stage 2, we try to approximate the 1 st- and 2 nd-order terms of Taylor series in (7). Note that the generated error should be subtracted from the original signal \(v_{m}\left[Mn\right]\) to further suppress the distortion. Then the compensated down-sampled signal of Channel \(m\) is denoted as  
\[ \begin{split}& q_{m}\left[Mn\right]=v_{m}\left[Mn\right]+\frac{ \hat{\beta}_{m2}}{\Omega_{c}}z_{m}^{{}^{\prime}}\left[Mn\right]-\frac{\hat{ \beta}_{m2}\left(1+\hat{\beta}_{m2}\right)}{\Omega_{c}^{2}}z_{m}^{{}^{\prime \prime}}\left[Mn\right]\\ &=\tilde{v}_{m}\left[Mn\right]+\frac{\hat{\beta}_{m2}-\hat{\beta }_{m}}{\Omega_{c}}\hat{y}_{m}^{{}^{\prime}}\left[Mn\right]+\frac{\left(1+\beta _{m}-\hat{\beta}_{m1}\right)\left(\beta_{m}-\hat{\beta}_{m2}\right)-\hat{\beta }_{m2}^{2}}{\Omega_{c}^{2}}\hat{y}_{m}^{{}^{\prime\prime}}\left[Mn\right]+o \left(\hat{y}_{m}^{{}^{\prime\prime}}\left[Mn\right]\right),\end{split}  \tag{9}\]  
where \(\hat{\beta}_{m2}\) is the identified coefficient for \(\beta_{m}\) in Stage 2. The error is further suppressed in Stage 2 compared with Stage 1. More stages are needed if the mismatch is larger or higher precision is required. For instance, in Stage 3, the 1 st-, 2 nd-, and 3 rd-order terms of the Taylor's series can be approximated using \(u[n]\), and are then subtracted from \(y[n]\) to get a signal closer to the expected one. **4. Coefficient Identification**  
The coefficient identification is based on the statistical properties of the input signals. Most real-life signals are WSS and modulo \(M\) quasi-stationary, whose definitions are given below [3,5]. **Definition 1**.