L² Sampled signal reconstruction with causality constraints - Part I: Setup and solutions

This paper studies the problem of reconstructing an analog signal from its sampled measurements, in which the sampler (acquisition device) is given and the reconstructor (interpolator/hold) is the design parameter. We formulate this problem as an L² (Wiener/Kalman filtering like) optimization problem and place the main emphasis on a systematic incorporation of causality constraints into the design procedure. Specifically, the optimization problem is solved under the constraint that the interpolation kernel is l-causal for a given I ∈ N, i.e., that its impulse response is zero in the time interval ( -∞, -Ih), where h is the sampling period. We present a closed-form state-space solution of the problem, whose computational complexity does not depend on I and which can be efficiently calculated and implemented.