## 2 Methods and Simulations
### Distortion of Misaligned Synthetic Signal  
Due to the limited adjustment range of the TDL, the larger the adjustment range, the higher the price. In order to reduce costs and ensure that EADC can collect the peak value of the optical signal pulse, an adjustment range of 1-2 times \(t_{s}\) can be selected. But when the delay of each channel exceeds 2 \(t_{s}\), it will be impossible to maintain real-time synchronization between each channel. At this point, the signal we obtain will be distorted and significantly different from the original signal. Taking the PADC system with \(N=4\) and \(f_{s}=5.2\) GHz as an example, the total sampling frequency \(F_{S}=Nf_{s}=20.8\) GHz. It is assumed that the sampled RF signal is a normalized sine signal with a frequency of \(f_{0}=100\) MHz, i.e., \(\mathrm{V}=\sin\left(2\pi f_{0}t\right)\). Further assuming that the first channel signal is synchronized with the original signal, the delays of channels 2-4 relative to the 1 st channel is \(\Delta t_{2}=3t_{s}\), \(\Delta t_{3}=-8t_{s}\), and \(\Delta t_{4}=5t_{s}\), respectively, where the positive sign represents leading, and the negative value represents lagging. The signals of each channel can be represented as follows:  
\[\left\{ \begin{array}{l}V_{1}=\sin(2\pi\ f_{0}t_{1})\\ V_{2}=\sin[2\pi\ f_{0}(t_{2}+\Delta t_{2})]=\sin[2\pi\ f_{0}(t_{2}+3t_{s})] \\ V_{3}=\sin[2\pi\ f_{0}(t_{3}+\Delta t_{3})]=\sin[2\pi\ f_{0}(t_{3}-8t_{s})]\\ V_{4}=\sin[2\pi\ f_{0}(t_{4}+\Delta t_{4})]=\sin[2\pi\ f_{0}(t_{4}+5t_{s})] \end{array} \right. \tag{1}\]  
Figure 1: N-channel, time–wavelength interleaved PADC system sketch. EDFA, erbium-doped fiber amplifier; MZM, Mach–Zehnder modulator; WDM, wavelength division multiplexer; TDL, tunable delay line; TFOA, tunable fiber optic attenuator; PD, photoelectric detector; LPF, low-pass filter. in which \(t_{1}\)-\(t_{4}\) are the time series in the original signal when the 4 channel signals are synchronized, which is a time vector. Taking \(t_{1}\) as an example, \(t_{1}=[t_{11},\,t_{12},\,\dots\,t_{1i},\,\dots,\,t_{1M}]\), where \(M\) is the number of samples. The following is the relationship between them:  
\[t_{2i}-t_{1i}=t_{3i}-t_{2i}=t_{4i}-t_{3i}=t_{s}/N=1/(4f_{s}) \tag{2}\]  
The RF signals collected by the four channels of the EADC are shown in Figure 2a, and the actual original signal should overlap with the signal from channel 1. The signals of channels 1-4 are represented in red, green, blue, and purple, respectively, with solid dots representing the quantified values collected by the EADC. It is obvious that the signals of channels 2 and 4 are ahead of channel 1, while channel 3 lags behind channel 1. The signal directly synthesized according to the timing of the signals collected by the EADC is shown in Figure 2b, which shows significant distortion compared to the original signal. The detailed synthesis process is shown in Figure 2c. The solid circles in each channel represent the points collected by the EADC, and the dashed lines represent the fitted signals. The black solid line represents the signal directly synthesized by the four channels, which can be obtained by connecting them in chronological order. In order to further illustrate the difference between the original signal and the directly synthesized signal, the effective number of bits (ENOB) and spectral information are  
Figure 2: Four-channel signals collected by EADC and the directly synthesized signal. (**a**) 4-channel signals with different delays; (**b**) directly synthesized signal; (**c**) signal synthesis process. calculated from the simulation, assuming that the original signal in the simulation is ideal and only constrained by the numerical accuracy of double-precision, floating-point calculations. The ENOB of the original signal can exceed 44 bits, while the ENOB of the directly synthesized signal is only 0.2918 bits. It can be found that there is only 100 MHz of spectrum information in the original signal from Figure 3a.