## 1 Introduction  
The well-known increase in information rate is a particular concern for inter-data center communication due to chromatic dispersion (CD). Increasing the rate of information reduces the optimal reachable transmission [1]. Using an O-band for standard single-mode fibers, where the CD can almost be neglected, is one way to avoid needing to mitigate it. Nonetheless, transmission in the O-band has high attenuation, which reduces the length of the transmission link [2]. Alternatively, solutions in the C-band, which address the intersymbol interference induced by dispersion, can also be considered [3; 4; 5]. Coherent and direct detection (DD) systems are both possible technologies for C-band short-reach communication [3; 6]. A coherent receiver has access to the amplitude and phase of the signal. Adding it together with digital signal processing (DSP), it brings with it huge potential to mitigate transmission impairments. However, inter-data center applications require cheaper and low-power transponders, which can be a challenge in these systems. DD-receivers are the current alternative, but CD cannot be fully compensated in the digital domain as only the power is detected in the receiver. Different options for inter- and intra-data center interconnection, as well as for mobile fronthaul application for operation on the C-band have already been discussed in recent years [4; 5]. A few of them operate on PAM-4 modulation and different options for DSP-based receiver equalization, such as Volterra and maximum likelihood sequence estimation (MLSE) [7; 8; 9] and DSP-based pre-distortion, such as Tomlinson Harashima precoding and the kramers-kronig receiver [10; 11]. Works with an on-off keying (OOK) modulation format have also been presented in the literature [12]. Recently, further signal processing technologies for this application range have been proposed, such as DSP-implemented neural networks [13; 14] or silicon photonic-based opto-electronic reservoir computers with analog electronic processing [15]. Alternatively, optical dispersion compensation modules as fiber Bragg grating and dispersion compensation fibers are more commonly used to compensate for the CD in the optical domain. However, they are not easy to tune and need to be designed for a specific link. The latter also has issues with high attenuation and a large footprint. Here, we propose a tuneable optoelectronic solution to mitigate the CD in DD systems. The processing structure we are investigating is located close to the approach using a simplified optical reservoir processor, discussed in [15], together with an NN digital processor [14]. This work is an extension of our preliminary results of [16]. Here, we improve the digital equalization by using a neural network equalizer and propose a simplification of the optical structure to reduce footprint. The optical module slices the signal into narrow frequency sub-bands and delays them accordingly to the CD, before recombining everything back into a single signal. Increasing the number of sub-bands improves performance, but increases complexity. Alternatively, we show that the performance can be improved using the optical component with fewer sub-bands if a further stage of equalization in the digital domain is also applied. Furthermore, a complexity reduction is demonstrated by using half of the proposed optical module structure together with an NN equalizer. This approach slightly reduces the gain, but can be attractive for inter-data center communication due to the reduced footprint. Finally, the number of neurons required for the NN equalizer is analyzed. The paper is organized as follows. Section 2 presents the system description for short-reach communication and simulation setup. Sections 2.1 and 2.2 describe the hybrid signal process module. The former is the optical structure used to mitigate CD, and the latter describes the NN equalization and training process. Section 3 shows the simulation results starting with the complete hybrid structure and ending with a simplified version of it.