## 2 The Proposed Timing Mismatch Calibration Technique
### Timing Mismatch Estimation  
In our previous work [22], the product's average adjacent channels can be written as follows:  
\[R_{k,k+1} =E(y_{k}[n]\cdot y_{k+1}[n])\] \[=R_{x}(T_{s}+\gamma_{k+1}-\gamma_{k}) \tag{26}\]  
where \(R_{x}\) denotes the autocorrelation function, \(y_{k}[n]\) denotes the output of \(k\)-th channel, and \(E(\cdot)\) denotes the mean value. The approximation of Equation (26) can be written as  
\[A_{k}=R_{k,k+1}\approx R_{x}(T_{s})+(\gamma_{k+1}-\gamma_{k})\cdot R_{x}^{{}^{ \prime}}(T_{s}), \tag{27}\]  
where \(R_{x}^{{}^{\prime}}(T_{s})\) is the derivative of the autocorrelation of the input signal. To solve the potential stability issues in the feedback architecture [18; 20; 23; 24; 25; 26], we propose a novel feedforward estimation algorithm to directly calculate the value of timing mismatches. Firstly, to eliminate \(R_{x}(T_{s})\), the timing-skew error function can be written as  
\[B_{k} =A_{k+2}-A_{k}\] \[=(\gamma_{k}-\gamma_{k+1}-\gamma_{k+2}+\gamma_{k+3})\cdot R_{x}^{ {}^{\prime}}(T_{s}),\;\;k=0,1,\cdots,M-3. \tag{28}\]  
Figure 4: The structure of the proposed compensation technique. (**a**) The structure of the coarse correction. (**b**) The structure of the fine correction. For convenience of expression, we define the \(\gamma_{M}=\gamma_{0}\). Note that \(B_{i}\) is the difference between odd or even terms. To increase the integrity of the error function, we add an additional error function, expressed as follows:  
\[B_{M-2} =A_{2}-A_{1}\] \[=(\gamma_{1}-2\gamma_{2}+\gamma_{3})\cdot R^{{}^{\prime}}_{x}(T_{s }). \tag{29}\]  
From Equations (28) and (29), we can obtain the following linear equations:  
\[ \begin{cases}B_{0}=(\gamma_{0}-\gamma_{1}-\gamma_{2}+\gamma_{3})\cdot R^{{}^{ \prime}}_{x}(T_{s})\\ B_{1}=(\gamma_{1}-\gamma_{2}-\gamma_{3}+\gamma_{4})\cdot R^{{}^{\prime}}_{x}(T _{s})\\ \vdots\\ B_{k}=(\gamma_{k}-\gamma_{k+1}-\gamma_{k+2}+\gamma_{k+3})\cdot R^{{}^{\prime} }_{x}(T_{s})\\ \vdots\\ B_{M-3}=(\gamma_{M-3}-\gamma_{M-2}-\gamma_{M-1}+\gamma_{0})\cdot R^{{}^{ \prime}}_{x}(T_{s})\\ B_{M-2}=(\gamma_{1}-2\gamma_{2}+\gamma_{3})\cdot R^{{}^{\prime}}_{x}(T_{s}). \end{cases}  \tag{30}\]  
The linear equation system of Equation (30) can be written in matrix form as follows:  
\[\mathbf{C_{T}}\times\mathbf{\gamma_{T}}=\mathbf{B_{T}}\cdot\frac{1}{R^{{}^{\prime}}_{x}(T _{s})} \tag{31}\]  
where \(\mathbf{\gamma_{T}}\) is a column vector and its elements are each sub-ADC's time-skew error; \(\mathbf{B_{T}}\) is also a column vector and its elements are each time-skew error function, which can be expressed as:  
\[\mathbf{\gamma_{T}}=\big{[}\gamma_{0},\gamma_{1},\cdots,\gamma_{M-1}\big{]}^{T}, \tag{32}\]  
\[\mathbf{B_{T}}=[B_{0},B_{1},\cdots,B_{M-2}]^{T} \tag{33}\]  
and  
\[\mathbf{C}= \begin{bmatrix}1&-1&-1&1&0&\cdots&0\\ 0&1&-1&-1&1&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1&0&0&0&0&\cdots&-1\\ 0&1&-2&1&0&\cdots&0\end{bmatrix} _{(M-1)\times M} \tag{34}\]  
is a constant circular \((M-1)\times M\) matrix. The sum of each row is zero. Note that the rank of \(\mathbf{C_{T}}\) is \(M-1\), indicating that we cannot directly find its inverse matrix. In addition, the number of unknown parameters \((\gamma_{k},k=0,\cdots,M-1)\) is higher than the error equations. Therefore, the linear equation system is under-determined and finding the unique minimal norm solution of this system becomes a key challenge. Although the inverse matrix of \(\mathbf{C_{T}}\) does not exist, we can use the Moore-Penrose generalized inverse matrix (M-P inverse matrix) to replace it. Firstly, the singular value decomposition of \(\mathbf{C_{T}}\) can be written as  
\[\mathbf{C_{T}}=\mathbf{U}\mathbf{S}\mathbf{V}, \tag{35}\]  
where \(\mathbf{U}\) is a \((M-1)\times(M-1)\) matrix whose column vectors are left singular vectors; \(\mathbf{S}\) denotes the diagonal matrix whose elements are singular values; \(\mathbf{V}\) is a \(M\times M\) matrix whose column vectors are right singular vectors. \(\mathbf{U}\) and \(\mathbf{V}\) are all orthogonal matrices.