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Linearly frequency modulation (LFM) pulse train and linearly stepped frequency (LSF) pulse train are mostly used in radar systems. However, the estimation performance of target motion parameters may be affected by the high recurrent lobe levels and the range–Doppler coupling phenomenon appearing in ambiguity function (AF). In multi-target scenarios, the estimation performance becomes even worse. The Costas frequency-modulation coded (CFMC) LFM pulse train has an ideal thumbtack-shaped AF, thus it can provide motion parameter estimation with high precision. However, the estimation of target motion parameter for the coherent CFMC LFM pulse train has not been investigated in depth. In this paper, we first analyze the properties of the AF of the CFMC LFM pulse train. Based on its convexity and narrow mainlobe width, a fast method to implement the maximum likelihood estimator (MLE) is proposed to estimate the motion parameters. To reduce the computation complexity, the Chirp-Z Transform (CZT) is introduced. Then, the Cramer–Rao lower bounds (CRLBs) on range, velocity and acceleration for frequency-modulation coded (FMC) pulse train are derived. It is shown that the CRLBs are relevant to the frequency coding patterns. Finally, Monte Carlo simulations are performed to verify the performance of the MLE. The results show that the performance of our proposed method can achieve the CRLBs when the signal-noise ratio (SNR) is higher than the threshold SNR. [ABSTRACT FROM AUTHOR]