## 2 System Description and Simulation Setup
### Optical Pre-Processing  
The optical module to mitigate the CD is based on the spectrum decomposition (SD) and spectral composition (SC) blocks [17]. SD is a technique in which the signal is sliced into narrow frequency sub-bands. The basic building-block for this process is a phase-tunable Mach-Zehnder delay interferometer (MZDI). Figure 3a is an example of a two-stage SD (four sub-bands). The orange dashed lines highlight the MZDI. The bandwidth of the sub-bands and the relative position of each other depends on the delay (\(\Delta t_{i}\)) and phase shift (\(\Delta\phi_{i}\)) of the MZDI. \(i\) is the number of stages. Equations (1) and (2) describe it.  
\[\Delta t_{i}=\frac{1}{2\cdot d_{i}} \tag{1}\]  
\[\Delta\phi_{i}=\pi+k2\pi+\omega_{i}\Delta t_{i},\ k\ \in\ \mathbb{Z} \tag{2}\]  
in which \(\omega\) is the angular frequency where the signal's power is maximized in the output upper arm of the MZDI and minimized for the other. The inverse delay, \(d\), determines the distance in frequency where the maximum and minimum of the signal's power is alternated in each output arm of the MZDI. For example, considering \(\omega_{1}=2\cdot\pi\cdot 12\cdot 10^{9}\) rad/s and \(d_{1}=8\) GHz, the signal's power is maximized at 12 GHz and alternates between minimum and maximum power every 8 GHz for the output upper arm, and the opposite for the lower arm. For a two-stage SD, we considered in the first stage \(d_{1}=8\) GHz and \(\omega_{1}=2\cdot\pi\cdot 12\cdot 10^{9}\) rad/s. For the second stage we considered, \(d_{2}=16\) GHz and \(\omega_{2}=2\cdot\pi\cdot 12\cdot 10^{9}\) rad/s.  
SC applies the inverse transfer function of the SD, and it is highlighted in blue dashed lines in Figure 3b. Perfect reconstruction of the signal is obtained if the SD is applied, followed directly by the SC. By knowing the transfer function of CD, a time delay and phase shift in each of the sub-bands of the SD can approach the inverse of the CD transfer function. The approximation increases in accuracy for increasingly narrower spectral slices. The values of the delay and phase depend on the number of sub-bands and the CD. The SD and SC structures, together with the delay and phase to mitigate the CD in each sub-band, is called the n-stage SD/SC Rx (Figure 4a) in this work.  
Figure 4 shows the optical pre-processing modules. Figure 4a shows an n-stage SD/SC Rx, which is composed of \(2\times(2^{n}-1)\) MZDI modules and creates \(2^{n}\) sub-bands. Figure 4b shows a simplified version of the optical module. We are proposing to remove the SC part in the two-stage SD/SC structure. In this configuration, the optical module is acting as an optical pre-processing structure to assist the equalization process in the digital domain. The phase shift is not considered because the signal is directly connected to the PD. The different signals in the electrical domain need to be recombined and the equalizer can perform this operation implicitly. An addition of three PDs is necessary for receiving all four optical outputs of a two-stage SD. The performance of both optical modules is analyzed in the results section.  
Figure 3: Optical structure to slice the signal into frequency sub-bands (**a**) and reconstruct it (**b**).