## 2 Discrete-Time Volterra Series Model for TIADC
### Frequency Domain Analysis of the Discrete-Time Volterra Series Model for a Four-Channel TIADC  
In this sub-section, the discrete-time Volterra series model of the four-channel TIADC with nonlinear order 2 and memory order 3 proposed above is analyzed in the frequency domain. At the same time, these frequency-domain analyses can be easily extended to M-channel TIADC systems with nonlinear order P and memory effect K.  
The sinusoidal wave fitting method is adopted in this study. For the sake of calculation, suppose the input is a cosine signal; Fourier transform is applied to Equation (3) in Section 2.2 to obtain the frequency domain expression of the time-domain signal:  
\[ \begin{split} Y(w)&=\sum\limits_{m=1}^{4}\left[F_{m,0} (w)+F_{m,1}(w)+F_{m,2}(w)\right]*S_{m}(w)\\ &=\sum\limits_{m=1}^{4}\left\{\begin{array}{c}\left\{\begin{array} []{c}2\pi h_{m,0}\delta(w)\\ +\sum\limits_{k_{1}=0}^{2}\pi h_{m,1}[\delta(w+\Omega_{0})+\delta(w-\Omega_{0} )]e^{-j\pi k_{1}T_{S}}\\ +\sum\limits_{k_{1}=0}^{2}\sum\limits_{k_{2}=0}^{2}\pi^{2}h_{m,1}[\delta(w+ \Omega_{0})+\delta(w-\Omega_{0})]^{2}e^{-j\pi(k_{1}+k_{2})T_{S}}\\ \ast\Omega_{S}\sum\limits_{k=-\infty}^{\infty}\delta\Big{(}w-\frac{k\Omega_{S} }{4}\Big{)}e^{-j\pi T_{S}\frac{\pi}{M}}\end{array}\right\}\end{split} \right. \end{split} \tag{7}\]  
where \(\Omega_{S}=\frac{2\pi}{T_{S}}\), \(\Omega_{0}\) is the input signal frequency and \(F_{m,p}(w)\) is the Fourier transform of the \(p\)th Volterra series of the \(m\)th ADC. In order to facilitate the understanding of the influence of error signals on the signal spectrum, the frequency domain signals are decomposed into three parts: \(p=0\), 1 and 2. The expression of the memory nonlinear mismatch error in frequency domain is:\[Y_{p}(w)=\sum_{m=1}^{4}\left[F_{m,p}(w)*S_{m}(w)\right] \tag{8}\]  
According to the property that convolution satisfies the distributive law:  
\[Y(w)=\sum_{p=0}^{2}Y_{p}(w) \tag{9}\]  
When \(p=0\):  
\[Y_{0}(w) =\sum_{m=1}^{4}\left[F_{m,0}(w)*S_{m}(w)\right] \tag{10}\] \[=\sum_{m=1}^{4}\left[2\pi h_{m,0}\delta(w)*\Omega_{5}\sum_{k=- \infty}^{\infty}\delta(w-\frac{k\Omega_{5}}{4})e^{-jwT_{5}\frac{m}{4}}\right]\] \[=\sum_{m=1}^{4}\left[2\pi\Omega_{5}h_{m,0}\sum_{k=-\infty}^{ \infty}\delta(w-\frac{k\Omega_{5}}{4})e^{-jwT_{5}\frac{m}{4}}\right]\]  
It can be seen from Equation (10) that the mismatch error spectrum generated by the 0 th term of the Volterra series appears at \(\frac{k\Omega_{5}}{4}\) (\(k=1,2,3,\,\dots\) ). When \(p=1\):  
\[Y_{1}(w) =\sum_{m=1}^{4}F_{m,1}(w)*S_{m}(w) \tag{11}\] \[=\sum_{m=1}^{4}\left\{\sum_{k_{1}=0}^{2}\pi h_{m,1}[\delta(w+ \Omega_{0})+\delta(w-\Omega_{0})]e^{-jwk_{1}T_{5}}*\Omega_{5}\sum_{k=-\infty}^ {\infty}\delta(w-\frac{k\Omega_{5}}{4})e^{-jwT_{5}\frac{m}{4}}\right\}\] \[=\sum_{m=1}^{4}\left\{\sum_{k_{1}=0}^{2}\pi\Omega_{5}h_{m,1}[\sum_ {k=-\infty}^{\infty}\delta(w-\frac{k\Omega_{5}}{4}+\Omega_{0})e^{-jwT_{5}\frac {m}{4}}+\sum_{k=-\infty}^{\infty}\delta(w-\frac{k\Omega_{5}}{4}-\Omega_{0})e^ {-jwT_{5}\frac{m}{4}}]e^{-jwk_{1}T_{5}}\right\}\]  
It can be seen from Equation (11) that the mismatch error spectrum generated by the first term of the Volterra series appears at \(\frac{k\Omega_{5}}{4}\pm\Omega_{0}\) (\(k=1,2,3,\,\dots\) ). When \(p=2\):  
\[Y_{2}(w) =\sum_{m=1}^{4}F_{m,2}(w)*S_{m}(w) \tag{12}\] \[=\sum_{m=1}^{4}\left\{\sum_{k_{1}=0}^{2}\sum_{k_{2}=0}^{2}\sum_{ k_{2}=0}^{2}\frac{\pi}{2}h_{m,2}[\delta(w+\Omega_{0})+\delta(w-\Omega_{0})]e^{-jwk_{1}T _{5}}*\delta(w+\Omega_{0})+\delta(w-\Omega_{0})]e^{-jwk_{2}T_{5}}*\Omega_{5} \sum_{k=-\infty}^{\infty}\delta(w-\frac{k\Omega_{5}}{M})e^{-jwT_{5}\frac{m}{4 }}\right\}\]  
It can be seen from Equation (12) that the spectrum of mismatch errors generated by the second term of the Volterra series appears at \(\frac{k\Omega_{5}}{4}\pm 2\Omega_{0}\) (\(k=1,2,3,\,\dots\) ). From the above analysis, it can be seen that the frequency spectrum produced by the high-order memory nonlinear mismatch error is greater, but its coefficient decreases with the increase in order.