U-invariant sampling: Extrapolation and causal interpolation from generalized samples

Causal processing of a signal's samples is crucial in on-line applications such as audio rate conversion, compression, tracking and more. This paper addresses the problems of predicting future samples and causally interpolating deterministic signals. We treat a rich variety of sampling mechanisms encountered in practice, namely in which each sampling function is obtained by applying a unitary operator on its predecessor. Examples include pointwise sampling at the output of an antialiasing filter and magnetic resonance imaging (MRI), which correspond respectively to the translation and modulation operators. From an abstract Hilbert-space viewpoint, such sequences of functions were studied extensively in the context of stationary random processes. We thus utilize powerful tools from this discipline, although our problems are deterministic by nature. In particular, we provide necessary and sufficient conditions on the sampling mechanism such that perfect prediction is possible. For cases where perfect prediction is impossible, we derive the predictor minimizing the prediction error. We also derive a causal interpolation method that best approximates the commonly used noncausal solution. Finally, we study when causal processing of the samples of a signal can be performed in a stable manner.