###### Abstract  
In order to meet the increasing requirements of speed and distance, an advanced digital signal processing (DSP) algorithm is preferred without changing the system structure in intensity modulation and the direct detection (IM/DD) system. As the transmission distance increases, the power fading induced by dispersion must be mitigated. In addition, linear and nonlinear inter symbol interference (ISI) introduced by bandwidth limitation and device imperfections becomes an obstacle to achieving higher capacity. The Gerchberg-Saxton (GS) algorithm was recently used to compensate for dispersion. In this paper, GS-based pre- and post-compensation schemes in the IM/DD system with nonlinearity were investigated. We investigated and compared the performance of the GS-based pre- and post-compensation algorithm in a 28 GB and four-level pulse amplitude modulation (PAM-4) transmission over 40 km standard single-mode fiber (SSMF). The bit error rate (BER) achieved a threshold of \(3.8\times 10^{-3}\) using look-up-table (LUT), FFE, and the GS-based pre-compensation algorithm without iterations. Turning to the GS-based post-compensation scheme, 80 iterations are needed. However, the demand for FFE is reduced. The algorithm selection depends on the tolerance of the transmitter or receiver complexity in specific scenarios. The joint LUT and GS-based pre-compensation algorithm may be a preferable approach in scenarios where a low-complexity receiver is desired. intensity modulation and direct detection (IM/DD); chromatic dispersion (CD); Gerchberg-Saxton algorithm; look-up-table (LUT) +
Footnote †: journal: _Photonics_
the iterative GS algorithm was deduced in [3], which is also known as GS algorithm based on small signal analysis [4]. In [3], a 28 GBaud PAM-4 signal over 40 km standard single-mode fiber (SSMF) was realized through GS-based pre-electrical dispersion compensation (EDC) at the transmitter and FFE at the receiver. The iterative GS algorithm mitigated the effect of CD, and the FFE with 25 taps compensated for the residual ISI. After that, a series of modified GS algorithms were proposed. The phase at the transmitter and the amplitude at the receiver are regarded as the degrees of freedom, and two error-reversing factors were introduced [5; 6]. Afterward, [PERSON] and [PERSON] et al. summarized the theory of a GS-based finite impulse response (FIR) filter for electronic dispersion compensation and correlative experimental demonstration [7; 8; 9]. In this process, researchers are pursuing better performance and faster convergence of the GS algorithm. However, tens of iterations are needed for the modified GS and GS-based FIR filter. In terms of ease of implementation, the non-iterative version is preferred. As for CD post-compensation, the GS algorithm is firstly applied for optical field recovery. Beyond that, data-aided iterative algorithm (DIA) and decision-directed DIA (DD-DIA) were proposed to increase convergence speeds and obtain a better global optimum [10]. The error-controlled iterative algorithm (ECIA) was then proposed to digitally linearize the IM/DD transmission systems. A minimum number of iterations required is reduced in comparison with DD-DIA [11]. Furthermore, the multi-constraint iterative algorithm (MCIA) was proposed to accelerate the convergence speed and realize nonlinear-equalization-free detection [12; 13]. Furthermore, the adaptive hybrid MCIA improves the sensitivity and the convergence performance [14]. Considering the tradeoff between performance and complexity, DIA was selected as a representative of the GS-based post-compensation algorithm in this paper. The CD compensation scheme based on GS at the transceiver plays a great role in alleviating power fading caused by CD. However, the current research is confined to the transmitter or the receiver. The performance comparison of the GS algorithm between the transmitter and the receiver has not been investigated. Moreover, investigation of the GS-based CD compensation scheme under nonlinearity is absent.