This paper presents a low sampling rate digital pre-distortion technique based on an improved Chebyshev polynomial for the non-linear distortion problem of amplifiers in 5G broadband communication systems. An improved Chebyshev polynomial is used to construct the behavioural model of the broadband amplifier, and an undersampling technique is used to sample the output signal of the amplifier, reduce the sampling rate, and extract the pre-distortion parameters from the sampled signal through an indirect learning structure to finally correct the non-linearity of the amplifier system. This technique is able to improve the linearity and efficiency of the power amplifier and provides better flexibility. Experimental results show that by constructing the behav-ioural model of the amplifier using memory polynomials (MP), generalised polynomials (GMP) and modified Chebyshev polynomials respectively, the adjacent channel power ratio of the obtained system can be improved by more than 13.87dB, 17.6dB and 19.98dB respectively compared to the output signal of the amplifier without digital pre-distortion. The Chebyshev polynomial improves the neighbourhood channel power ratio by 6.11dB and 2.38dB compared to the memory polynomial and generalised polynomial respectively, while the normalised mean square error is effectively improved and enhanced. This shows that the improved Chebyshev pre-distortion can guarantee the performance of the system and improve the non-linearity better.