# Calibration for Sample-And-Hold Mismatches in \(M\)-Channel TIADCs Based on Statistics
## 3 Mismatch Compensation
: _Wide-Sense Stationary_  
A discrete-time signal \(u[k]\) is said to be WSS if its 1 st and 2 nd moments are time-invariant. That is  
\[m_{u}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}E\left(u\left[k\right]\right), \tag{10}\]  
Figure 6: (**a**) The compensation structure using Richardson iteration. (**b**) The details of block \(H_{Am}\). (**c**) The details of block \(H_{Bm}\). (**d**) The details of block \(H_{Cm}\). (**e**) The details of block \(H_{Dm}\). (**f**) The details of block \(\Sigma_{m}\), where \(\gamma\) represents for \(z\) or \(u\) in (**a**). and  
\[R_{u}\left[n\right]=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}E\left(u \left[k+n\right]u\left[k\right]\right), \tag{11}\]  
where \(E(\cdot)\) is the expectation. **Definition 2**. : _Modulo M Quasi-Stationary_  
Assume  
\[\tilde{f}_{u_{1},u_{2},\cdots}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{t=1} ^{N}f\left(u_{i_{1}}\left[t\right],u_{i_{2}}\left[t\right],\cdots\right),\ i_{1},i_{2},\cdots=0,1,\cdots,M-1 \tag{12}\]  
exists for a function \(f(\cdot,\cdot,\cdots)\). Then \(u\) is modulo \(M\) quasi-stationary with respect to \(f\) if  
\[\tilde{f}_{i_{1},i_{2},\cdots}=\tilde{f}_{\left(\left(i_{1}+l\right)\text{ mod }M,\ \left(i_{2}+l\right)\text{ mod }M,\cdots\right.},\ l\in Z. \tag{13}\]  
(\"mod\" is the remainder operator, and \"Z\" is the set of integers.) The modulo \(M\) quasi-stationary property guarantees that the input signal manifests the same statistical properties for all channels in the time-interleaved system. Assume \(x[n]\) is WSS and modulo \(M\) quasi-stationary with respect to the function \(f(x_{i},x_{i-1})=(x_{i}-x_{i-1})^{2}\). Then the down-sampled signal for Channel \(m\) is denoted as  
\[v_{m}\left[Mn\right]=A_{m}x\left(MnT_{s}+t_{m}\right),\quad m=0,1,2,\cdots,M-1, \tag{14}\]  
where  
\[A_{m}=\frac{1}{\sqrt{1+\left(\frac{\left(1+\beta_{m}\right)\omega}{\Omega_{c }}\right)^{2}}} \tag{15}\]  
is the magnitude response of the filter \(H_{m}(e^{i\omega})\), and  
\[t_{m}=-\frac{1}{\omega}\arctan\left(\frac{\left(1+\beta_{m}\right)\omega}{ \Omega_{c}}\right),\ s.t.\ t_{m}\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \tag{16}\]  
is the phase response of the filter \(H_{m}(e^{i\omega})\). Then the mean squared difference between the down-sampled signals of adjacent channels is  
\[ \begin{split} D_{v_{m},v_{m-1+\left|\lfloor\frac{m-1}{M}\right \rfloor\right|M}}&=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1 }^{N}\left(v_{m}\left[Mn\right]-v_{m-1+\left|\lfloor\frac{m-1}{M}\right\rfloor \right|M}\left[M\left(n+\lfloor\frac{m-1}{M}\rfloor\right)\right]\right)^{2} \\ &=\left(A_{m}+A_{m-1+\left|\lfloor\frac{m-1}{M}\rfloor\right| \cdot M}\right)\sigma^{2}-2A_{m}A_{m-1+\left|\lfloor\frac{m-1}{M}\rfloor \right|\cdot M}R_{x}\left(T_{s}+t_{m}-t_{m-1+\left|\lfloor\frac{m-1}{M} \right\rfloor\right|M}\right),\\ &\quad m=0,1,\cdots,M-1,\end{split}  \tag{17}\]  
where \(\left\lfloor\cdot\right\rfloor\) is the floor operator, and \(\left|\cdot\right|\) is the absolution operator. In (17), \(\sigma^{2}\) stands for the variance of the signal \(x[n]\)  
\[\sigma^{2}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N}x^{2}\left[n\right], \tag{18}\]  
and \(R_{x}(\cdot)\) means the autocorrelation of the signal \(x[n]\)  
\[R_{x}\left(\tau\right)=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N}x \left[n\right]x\left[n+\tau\right]. \tag{19}\]When \(m>1\), (17) becomes  
\[D_{v_{m},v_{m-1}}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\left(v_{m} \left[Mn\right]-v_{m-1}\left[Mn\right]\right)^{2},\quad m=1,\cdots,M-1. \tag{20}\]  
When \(m=0\), (17) becomes  
\[D_{v_{0},v_{M-1}}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\left(v_{0 }\left[Mn\right]-v_{M-1}\left[M\left(n-1\right)\right]\right)^{2}. \tag{21}\]  
The value of (17) varies with the bandwidth mismatch \(\beta_{m}\). LMS algorithm is adopted to estimate \(\beta_{m}\).