We study the artificial neural network (ANN) and feed forward equalizer jointly to mitigate the inter-symbol interference (ISI) and the nonlinear distortions in the optical single-side band and direct-detection (DD) system. The optical SSB signal is generated by a relatively low-cost dual-drive Mach-Zehnder modulator (DDMZM). The two driving signals are a pair of Hilbert signals with Nyquist pulse-shaped four-level pulse amplitude modulation (NPAM-4). In this paper, we experimentally compare the performance of feedforward equalizer (FFE) with BP (back propagation) network combined with FFE. The experimental results show that the performance of BP network with FFE equalizer outperforms FFE equalizer, because it can efficiently mitigate the nonlinear distortions. We have experimentally demonstrated that a 40 Gb/s NPAM-4 signal transmit over 80 km standard single mode fiber (SSMF) without any inline dispersion compensation.