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In this paper, we propose a learning-based moving-object tracking method utilizing our newly developed LiDAR sensor, Frequency Modulated Continuous Wave (FMCW) LiDAR. Compared with most existing commercial LiDAR sensors, our FMCW LiDAR can provide additional Doppler velocity information to each 3D point of the point clouds. Benefiting from this, we can generate instance labels as ground truth in a semi-automatic manner. Given the labels, we propose a contrastive learning framework, which pulls together the features from the same instance in embedding space and pushes apart the features from different instances, to improve the tracking quality. Extensive experiments are conducted on our recorded driving data, and the results show that our method outperforms the baseline methods by a large margin.