There is disclosed a predictive filter for predicting the future position of a look direction of a rotating directional antenna mounted on a vessel. Prediction of a look direction of the antenna at any instant is made from input of ship's compass data including roll data, pitch data and heading data, as well as data describing a rotational position of the antenna, output from an antenna synchro device. Smoothing of the digital roll, pitch and heading data and antenna position data is followed by application of a moving time window, filter which removes any offset from the data within of the moving time window to produce data having a zero mean. The zero mean data is input into an auto-regression algorithm, the output of which is combined with mean roll, pitch, heading, position and antenna velocity data to obtain predicted positions of the antenna relative to the vessel, and a predicted motion of the vessel. The combined vessel motion prediction and antenna position and velocity prediction can be used to establish a predictive future look direction of the antenna for a period of between 0.1 and 5 seconds into the future.