This letter addresses the problem of causal velocity estimation of motion signals exhibiting sudden velocity changes. It is explicitly assumed that only sampled quantized position data are available, and that no dynamic system model is at hand, to allow for an effective solution with minimal a priori information. Our solution consists of a finite-response filter with adaptive window length and a velocity jump detection algorithm. A velocity jump is detected when the difference between sampled position data and predicted position is larger than a parametric threshold. The specific value for the threshold is obtained via an analytical formula that depends on sampling time, maximum allowed variation in acceleration (away from velocity jumps), and encoder resolution. The developed method is first demonstrated on simulated quantized data corresponding to a bouncing motion and its performance compared to that of other existing non-model-based methods. Its effectiveness is furthermore illustrated by means of trajectory tracking results on an experimental setup experiencing impacts.