## 1 Introduction
### Related Work  
The main features of OTSs are high capacity and long-haul transmission due to huge amount of available bandwidth and extremely low propagation loss. To ensure transmission integrity, a large volume of literature exists on treatment of channel effects. Some advanced contributions and their limitations are presented as follow. Digital signal processing techniques for compensating inter-and intra-channel fiber non-linearities have been investigated in [11], and input parameters such as launch power and fiber length are used to carry out simulation work. Self phase modulation (SPM) non-linear issue is investigated in [12] where conversion between different formats, such as 8-quadrature amplitude modulation (8-QAM) to quadrature phase shift keying (QPSK) and on off keying (OOK), is performed through optical methods to adapt to the channel conditions. Different values of optical signal-to-noise ratio (OSNR), optical power, and chromatic dispersion have been used to conduct analysis on the performance of the proposed system, however, the format conversion to a lower order results in decreased data rates. In [13], the authors discuss intermodel non-linear issues in few-mode fiber using 16-QAM up to 36 km transmission and report mitigation of the XPM non-linear effect. High cost components are used in [14], such as indium phosphate and silicon photonics to achieve high bandwidth in the presence of fiber non-linearities. The voltra-assisted method is explained in [15], in which the authors overcome the limitations of optical phase conjugation (OPC) technique. In [16], the authors mitigate nonlinearity for 92-channel 22.5 band rate, using 16-QAM modulation format. Similarly, SPM and XPM nonlinear parameters are discussed in [17], implementing 16-QAM and 64-QAM super channels and advanced equalization procedures. In [18], linearly swept optical source methodology is developed for mitigating high order FWM effects. The impact of nonlinearity inside optical fiber is investigated by spectral efficient frequency division multiplexing (SEFDM) technique and the work is compared with orthogonal-FDM (O-FDM). In [19], nonlinear mitigation technique is reported for multi subcarrier PM-16-QAM using symbol rate optimization in 31 channel WDM system. In [20], the authors have explored nonlinear compensation mechanism with the help of multi-channel digital back propagation in Nyquist spaced optical communication system. However, the effect of RIrNLs still needs to be investigated, in terms of analytical characterization, to understand their effect to achieve an optimum mitigation solution, which is the main focus of our paper. Moreover, modern systems require multiplexing of several channels for which different parameters for a WDM based system need to be analyzed in this context. Therefore, we present a detailed analysis of RIrNLs and their treatment by varying input power, fiber length, channel spacing, and the number of channels. We present a WDM system with a 100 Gb/s aggregate data rate, which can support transmission distances up to 500 km,with 32 number of channels. The performance analysis has been conducted for a range of non-linear distortions. To the best of our knowledge, such performance has only been possible with the mathematically proposed mitigation technique for second and third order non-linearities.