Multipath interference (MPI) noise suppression is crucial for high-speed 4-level pulse amplitude modulation (PAM4) intensity modulation direct detection (IMDD) transmission systems. In this paper, two algorithms are proposed to eliminate the MPI noise, and these algorithms can be realized in the receiver digital signal processing (DSP) module without changing the existing IMDD system architecture. The first algorithm (A1) suppresses the MPI noise by removing the fluctuation of MPI-impaired PAM4 signals. The second algorithm (A2) can estimate the MPI noise which is intensity-dependent, and then the estimated MPI noise is subtracted from the impaired signal. The complexity of the proposed algorithms is lower than that of the traditional high-pass filter (HPF) scheme. Simulation results indicate that the proposed schemes can effectively suppress the MPI noise and improve the transmission performance under different signal-interference ratios (SIR), different laser linewidth, and random optical delay of the reflected signal. Besides, the cascade of A1 and A2 can further improve the transmission performance, and it has a higher tolerance for laser linewidth compared to the traditional HPF scheme. With the help of the proposed scheme, 56 Gb/s PAM4 signals are experimentally transmitted over 15.5 km standard single-mode fiber under the SIR of 18 dB and the laser linewidth of 100 kHz, and the measured bit error rate (BER) can be below the 7% overhead hard-decision forward error correction limit. When the laser linewidth is increased to 4 MHz, the traditional HPF scheme becomes less effective and may degrade the transmission performance, but the proposed schemes are still effective and the SIR tolerance can be improved by about 3 dB. 100 Gb/s PAM4 under optical back-to-back transmission with 24 dB SIR can be achieved, and the BER performance can be increased from 4.63e-3 to 2.06e-3 with the help of the proposed schemes.