Optimal Discrete to Continuous Transfer for Band Limited Inputs

Central to the discrete to continuous (d2C) time transfer of state space models is the selection of an intersample parameterization of the input. At present the zero-order hold is widely used but the premise is unnecessarily crude in the typical situation where the input is not constant within the sampling intervals. The paper shows that the optimal d2C transfer for models obtained from low pass filtered observations is realized by treating the input as a sampled-modulated train of Dirac impulses, designated here as the band limited hold (BLH). The foregoing result rests primarily on the fact that the spectrum of the residual between the BLH reconstruction and the true input is zero in the first Nyquist band. The merit of the BLH is illustrated by contrasting its residue predictions with those obtained using the zero-order hold, the noncausal first-order hold, and a reconstruction based on a half time step forward shift of the zero-order hold premise. It is also shown that accuracy of the direct transmission matrix can be improved, over that realized from the d2C transfer, by computing it from constraints that connect it to the state space triplet $\left\{A_c{B}_c{C}_c\right\}$ .