## 2 Methods and Simulations
### Measurement and Correction of Misaligned Signals  
As assumed in Section 2.1, the original RF signal to be tested is kept in phase synchronization with the signal collected in channel 1. A triangular interpolation can be performed on the 1 st channel's signal to achieve the N-fold expansion, i.e.,  
\[\left\{ \begin{array}{l}V_{1}=\sin(2\pi\ f_{0}t_{1})\\ V_{2}^{\prime}=\sin(2\pi\ f_{0}t_{2}^{\prime})\\ V_{3}^{\prime}=\sin(2\pi\ f_{0}t_{3}^{\prime})\\ V_{4}^{\prime}=\sin(2\pi\ f_{0}t_{4}^{\prime})\end{array} \right. \tag{3}\]  
in which the following relationship between \(t_{1}\), \(t_{2}\), \(t_{3}\), and \(t_{4}\)' exists:  
\[t_{2i}^{\prime}-t_{1i}=t_{3i}^{\prime}-t_{2i}^{\prime}=t_{4i}^{\prime}-t_{3i} ^{\prime}=1/(Nf_{s}) \tag{4}\]  
Subsequently, the signals collected by channels 2-4 of the EADC are shifted separately. Taking channel 2 as an example, assuming that after moving \(k_{2}\) sampling periods, the measured values align closely with the corresponding interpolation, and the root mean square error between them will be minimized, that is, the MRMSE will be found. If \(k_{2}>0\), it indicates that channel 2 is leading. In order to maintain synchronization with channel 1, an increase in the delay is required, which means that channel 2 needs to move forward along the timeline, artificially increasing the delay amount, corresponding to the \(V_{2}\) sequence moving \(k_{2}\) positions to the right as a whole. Then, by removing the last \(k_{2}\) terms from \(V_{2}\) and subtracting the last (\(M-k_{2}\)) terms of \(V_{2}\)', each term is squared, the average is calculated, and is then taken to the square root to obtain the MRMSE, i.e.,  
\[ \begin{array}{ll}\text{MRMSE}_{2}&=\sqrt{\left[\left(V_{21}-V_{2(k_{2}+1)}^ {\prime}\right)^{2}+\left(V_{22}-V_{2(k_{2}+2)}^{\prime}\right)^{2}+\ldots+ \left(V_{2(M-k_{2})}-V_{2M}^{\prime}\right)^{2}\right]/\left(M-k_{2}\right)} \\ &=\sqrt{\sum\limits_{i=1}^{M-k_{2}}\left(V_{2i}-V_{2(k_{2}+i)}^{\prime} \right)^{2}/\left(M-k_{2}\right)}\end{array}  \tag{5}\]  
Figure 3: Spectrum information of original signal and directly synthesized signal. (**a**) Original signal; (**b**) directly synthesized signal. Assuming that \(Z\) is the smallest integer in the common multiple of \(f_{3}\!/\!f_{0}\), where \(f_{0}\)' is the frequency of the RF signal to be tested within the Nyquist bandwidth at an \(f\)s sampling rate, two \(k_{2}\) values can always be found between \((-Z,Z)\), resulting in the MRMSE\({}_{2}\). One value is positive, indicating that it needs to be shifted forward, and the other is negative, meaning a backward shift. The shift process is illustrated in Figure 4, where the red line represents the signal fitted to channel 1, the red solid points represent the values collected by the EADC of channel 1, the black solid points represent the (\(N-1\)) times values that obtained through the trigonometric interpolation on channel 1 (excluding the values of channel 1), and the green line and solid points represent the fitted signal and the values collected by the EADC of channel 2, respectively. When shifted 3 positions forward, \(V_{2}\) is essentially aligned with the corresponding interpolation points of channel 1. At this moment, green dashed lines and circles are used to represent the signal and collected values of channel 2 after the shift, allowing for the calculation of the MRMSE. Similarly, when shifted backward by (\(f_{3}\!/\!f_{0}\)' \(-3\)) = 49 positions, i.e., \(k_{2}=-49\), channel 2 also basically coincides with the interpolation points. Since the delay amount of the PADC system remains constant regardless of the frequency of the signal being tested, substituting a new RF signal will result in two additional \(k_{2}\) values, with the unchanged \(k_{2}\) representing the actual delay amount. Similarly, channels 3 and 4 can also achieve their corresponding MRMSEs through appropriate shifts. Assuming \(k_{3}<0\) and \(k_{4}>0\), then  
\[