Frame Analysis of Irregular Periodic Sampling of Signals and Their Derivatives

Given a bandlimited signal, we consider the sampling of the signal and some of its derivatives in a periodic manner. The mathematical concept of frames is utilized in the analysis of the properties of the sequence of sampling functions. The frame operator of this sequence is expressed as a matrix-valued function multiplying a vector-valued function. An important property of this matrix is that the maximum and minimum eigenvalues are equal (in some sense) to the upper and lower frame bounds. We present a method for finding the dual frame and, thereby, a method for reconstructing the signal from its samples. Using the matrix approach we prove that the sequence of sampling functions is always complete in the cases of critical sampling and oversampling. A sufficient condition for the sequence of sampling functions to constitute a frame is derived. We show that if no sampling of the signal itself is involved, the sampling is not stable and cannot be stabilized by oversampling. Examples are considered, and the frame bounds in the case of sampling of the signal and its first derivative are calculated explicitly. Finally, the matrix approach can be similarly applied to other problems of signal representation.