A single sideband direct detection digital filtered multiple access-passive optical network (DFMA-PON) system is proposed to suppress the frequency selective fading effect caused by the fiber dispersion. Moreover, in order to eliminate the signal-signal beat interference (SSBI) caused by the square-law characteristic during the direct detection, a Kramers-Kronig (KK) algorithm suitable for the DFMA-PON system is designed at the receiver. The KK algorithm recovers the original signal by reconstructing the phase information from the amplitude information of the received signal, so as to eliminate the influence of SSBI on the received signal and improve the sensitivity of the receiver. The simulation results demonstrate that the performance of single sideband transmission system is significantly improved compared with that of double sideband transmission system without additional dispersion compensation algorithms. The receiver sensitivity can be increased by about 5.0 dB at most after 25 km standard single-mode fiber transmission compared with the single sideband system without KK algorithm at the receiver.