## 4 Test Verification
### A Fixed Fractional Delay Filter  
The proposed method adopts the fractional delay filters to estimate the gain, time skew, and bandwidth mismatch in each channel. The fractional delay filters are used in each sub-ADC. Therefore, the input signal is limited to the first Nyquist bandwidth of the sub-ADC. To make the proposed algorithm work for the high Nyquist bandwidth of the sub-ADC in the TI-ADCs system, the fractional delay filters should be effective in high sub-ADC Nyquist bandwidth. When the bandpass (BP) signals are in the \(N_{\text{BP}}\)-th Nyquist bandwidth, the frequency can be expressed as follows:  
\[(N_{\text{BP}}-1)\times\frac{f_{\text{s}}}{2}<f_{\text{L}}<|f|<f_{\text{H}}<N _{\text{BP}}\times\frac{f_{\text{s}}}{2}. \tag{22}\]  
Here \(N_{\text{BP}}\geq 1,f_{\text{L}}\), and \(f_{\text{H}}\) are the low and high cutoff frequencies of the signal, respectively. Additionally, \(f_{\text{s}}\) is the sampling frequency of sub-ADC. The BP signal is constructed by the sum of two complex signals of \(x^{+}(t)\) and \(x^{-}(t)\). Therefore, the frequency response of the complex signals can be respectively represented as \(X^{+}(f)\) and \(X^{-}(f)\). According to the band-pass sampling theory, the original spectrum of the band-pass signal can be folded back to the baseband in the under-sampled TI-ADC converter whose sampling frequency conforms to the Nyquist criterion. In the odd-order Nyquist band, the baseband spectrum of the under-sampling signal has the same shape as the original spectrum. Additionally, the baseband spectrum of even order NB input signal is reversed. In the odd-order Nyquist band, \(X(f)\) can be represented as follows:  
\[ \begin{array}{ll}X(f)&=X^{+}(f)+X^{-}(f)\\ &=T_{\text{s}}Y^{+}(f-n_{\text{n}}f_{\text{s}})+T_{\text{s}}Y^{-}(f+n_{\text{ n}}f_{\text{s}}).\end{array}  \tag{23}\]  
Here, \(Y^{+}(f)\) and \(Y^{-}(f)\) is the output of \(X^{+}(f)\) and \(X^{-}(f)\), respectively. Furthermore, \(n_{\text{n}}\) can be obtained as follows:  
\[n_{\text{n}}=\left\lfloor\frac{N_{\text{BP}}}{2}\right\rfloor. \tag{24}\]  
where the input signal is the \(N_{\text{BP}}\)-th Nyquist bandwidth and \(\lfloor N_{\text{BP}}/2\rfloor\) is the result of rounding down \(N_{\text{BP}}/2\). Therefore, the inverse fast Fourier transform (IFFT) of \(X(f)\) can be expressed as  
\[x(t)=T_{\text{s}}y^{+}(t)e^{j2\pi n_{\text{n}}f_{\text{s}}t}+T_{\text{s}}y^{- }(t)e^{-j2\pi n_{\text{n}}f_{\text{s}}t}. \tag{25}\]  
Figure 4: Proposed structure of correction of the offset, gain, time skew, and bandwidth mismatch. According to the Euler formula, \(x(t)\) can be rewritten as follows:  
\[ \begin{array}{ll}x(t)&=T_{\rm s}\cos(2\pi n_{\rm n}f_{\rm s}t)(y^{+}(t)+y^{-}(t) )\\ &+jT_{\rm s}\sin(2\pi n_{\rm n}f_{\rm s}t)(y^{+}(t)-y^{-}(t)).\end{array}  \tag{26}\]  
In the time domain, \(y^{+}(t)\) and \(y^{-}(t)\) can be obtained as follows:  
\[ \begin{array}{ll}y^{+}(t)=\frac{y(t)+jy_{\rm n}(t)}{2},\\ y^{-}(t)=\frac{y(t)-jy_{\rm n}(t)}{2}.\end{array}  \tag{27}\]  
According to Equation (27), the reference of the \(i\)-th channel can be represented as follows:  
\[ \begin{array}{ll}x_{0}\Big{(}t+\frac{i}{Mf_{\rm s}}\Big{)}&=T_{\rm s}\cos \Big{(}2\pi n_{\rm n}f_{\rm s}\Big{(}t+\frac{i}{Mf_{\rm s}}\Big{)}\Big{)} \Big{(}y_{0}\Big{(}t+\frac{i}{Mf_{\rm s}}\Big{)}\Big{)}\\ &+jT_{\rm s}\sin\Big{(}2\pi n_{\rm n}f_{\rm s}\Big{(}t+\frac{i}{Mf_{\rm s}} \Big{)}\Big{)}\Big{(}jy_{\rm h0}\Big{(}t+\frac{i}{Mf_{\rm s}}\Big{)}\Big{)}. \end{array}  \tag{28}\]  
Therefore, the DFT of Equation (28) can be expressed as follows:  
\[ \begin{array}{ll}x_{0}\Big{[}n+\frac{i}{Mf_{\rm s}}\Big{]}&=\cos\Big{[}2 \pi n_{\rm n}\Big{(}n+\frac{i}{Mf_{\rm s}}\Big{)}\Big{]}\Big{(}y_{0}\Big{[}n +\frac{i}{Mf_{\rm s}}\Big{]}\Big{)}\\ &-\sin\Big{[}2\pi n_{\rm n}\Big{(}n+\frac{i}{M}\Big{)}\Big{]}\Big{(}y_{\rm h0 }\Big{[}n+\frac{i}{Mf_{\rm s}}\Big{]}\Big{)},\end{array}