AN EXTENDED FORM OF SUB-BAND INTERPOLATION OF MULTI-DIMENSIONAL DISCRETE SIGNALS AND ITS APPLICATIONS

Abstract

In this paper, we establish an extended form of the optimum sub-band interpolation for a family of n-dimensional discrete signals. We assume that the Fourier spectrums of these discrete signals have weighted L2 norms smaller than a given positive number. It is assumed that the sample points of these discrete signals are identical with the whole vertices of an n-dimensional rectangular lattice in Rn. Among these sample points, certain subsets are used for the interpolation. Selecting appropriate subsets of the sample points, we can realize a wide variety of periodic arrangements of sample points for interpolation such as hexagonal and octagonal lattices or a set of sample points used in interlaces scanning of digital television. The proposed method minimizes the measure of error which is equal to the envelope of the approximation errors with respect to the discrete signals. In the following discussion, we assume initially that the corresponding approximation formula uses an infinite number of interpolation functions having limited supports and functional forms different from each other. However, it should be noted that the resultant optimum interpolation functions are expressed as the parallel shifts of the impulse responses of the finite number of n-dimensional FIR filters. Equivalent analog approximation formula corresponding to the proposed discrete approximation, is derived and interesting reciprocal relation in the approximation, is also discussed. A necessary and sufficient condition for the convergence of the corresponding analog approximation formula to the original band limited signal, is presented. An equivalent expression of the analog approximation formula in the frequency domain, is derived in relation to the convergence condition.