Methods for Recovering a Random Waveform from a Finite Number of Samples

The reconstruction of a random waveform from finite data presents an interesting statistical problem, especially if practical constraints are imposed on the method of reconstruction. Consideration is given to the merits of Lagrange polynomial interpolation; RC filtering; linear, least-squares time-varying interpolation; and linear, least-squares, time-invariant interpolation. Numerical results for random waveforms having various correlation functions are presented. From these results a quantitative comparison of the merits of each of the interpolation procedures can be made. From a theoretical point of view one set of mathematical results is new, namely, those results associated with the optimum, linear, time-invariant interpolation of a finite number of samples. The analysis of this problem is complicated by the combination of the constraints of time invariance (i.e., the same interpolatory function is used for each sample) and a finite number of samples. Removal of either constraint makes the problem simpler and leads one to known results.