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In this thesis we consider the adaptive filtering problem with manifold-valued measurements. Using motivations from radiological wireless systems, special consideration is given to measurements on Stiefel and Grassmann manifolds. We complement it with a mathematical background to differential geometry and, specifically, to manifold-valued statistics.

The adaptive filter with Stiefel and Grassmann manifold-valued measurements is considered. This is done by patching together the extended Kalman filter introduced in Paper III, with a method of system identification using homogeneous space-valued autoregressive processes introduced in Paper I. Thus, turning the filter in Paper III into an adaptive filter. In Paper II we show that inference of the variance parameter is possible when observing projected normal random variables for the special case of the 2-sphere. In paper IV we show that the Filter given in Paper III converges when a non-moving particle is observed.