For maximum-likelihood (ML) estimation to jointly track carrier frequency offset (CFO) and channel impulse response (CIR) in orthogonal frequency division multiplexing (OFDM) systems, we present a finite high order approximation method utilizing the MATLAB ‘roots’ command on the log-likelihood function derived from the OFDM received signal, coupled with an adaptive iteration algorithm. The tracking performance of this high order approximation algorithm is found to be excellent, and as expected, the algorithm outperforms the other existing first order approximation algorithms.