On the Papoulis Sampling Theorem: Some General Conditions

Abstract

Some general conditions for multichannel sampling are established for wide sense stationary sequences with spectral density. First, necessary and sufficient conditions are given for these processes so that they are linearly determined by the samples obtained from a multichannel sampling scheme. Some results are studied for stationary sequences, and then applied to the problem of sampling, not necessarily band limited, wide sense stationary processes. Conditions are also given for the existence of a frame sequence of the samples. In the case of frames, the condition that the spectral measure is absolutely continuous is proved to be necessary.

Optimal Spline Generators for Derivative Sampling

“…Ideally, we are looking for a scheme where H is the identity filter, which translates into generators being interpolatory. Specifically, a pair of generators φ int = (φ int,1 , φ int,2 ) is said to be interpolatory if, they satisfy (7) with a[k] = f [k] for all k ∈ Z. To construct such generators, we simply need to apply the digital filter H T (the inverse z-transform of Φ −T ) on the original generators φ, which results in…”