On the Consistency of L-Optimal Sampled Signal Reconstructors

The problem of restoring an analog signal from its sampled measurements is called the signal reconstruction problem. A reconstructor is said to be consistent if the resampling of the reconstructed signal by the acquisition system would produce exactly the same measurements. The consistency requirement is frequently used in signal processing applications as the design criterion for signal reconstruction. System-theoretic reconstruction, in which the analog reconstruction error is minimized, is a promising alternative to consistency-based approaches. The primary objective of this paper is to investigate, what are conditions under which consistency might be a byproduct of the system-theoretic design that uses the L criterion. By analyzing the L reconstruction in the lifted frequency domain, we show that non-causal solutions are always consistent. When causality constraints are imposed, the situation is more complicated. We prove that optimal relaxedly causal reconstructors are consistent either if the acquisition device is a zero-order generalized sampler or if the measured signal is the ideally sampled state vector of the antialiasing filter. In other cases consistency can no longer be guaranteed as we demonstrate by a numerical example.