A new method is proposed for estimating the time-varying velocity of a moving target with a low-cost laser sensor using optical feedback interferometry. A new algorithm is proposed to track velocity variations from real-time analysis of the output signal of a self-mixing laser diode. This signal is strongly corrupted by a multiplicative noise caused by the speckle effect, which occurs very often with noncooperative targets used in many industrial applications. The proposed signal processing method is based on a second order adaptive linear predictor filter, which enables us to track the digital instantaneous Doppler frequency, which is proportional to the velocity. A model of the laser diode output signal is proposed, and it is shown that the sensor and its associated algorithm have a global first-order lowpass transfer function with a cutoff frequency expressed as a function of the speckle perturbations, the signal to noise ratio and the mean Doppler frequency. Numerical as well as experimental results illustrate the properties of this sensor.