$L^2$ Sampled Signal Reconstruction With Causality Constraints—Part II: Theory

This paper provides the theoretic foundation for the design of L² optimal reconstructors (also known as interpolators/holds) with a prescribed degree of causality. A compact frequency-domain solution is derived that mimics known interpolation techniques for ordinary transfer functions. In parallel, an extensive state space solution is documented. It complements the frequency-domain solution in that it constructively proves the various claims, and it also makes the solution concrete. The state space solution requires the solution of one Riccati and one Lyapunov matrix equation.