Approximating Multi-Dimensional and Multiband Signals

TL;DR

This paper develops low-dimensional dictionaries for representing multi-dimensional, multiband signals, using spectral analysis and DPSS approximation, applicable to high-dimensional tensor data.

Contribution

It introduces a novel spectral analysis approach to construct optimal dictionaries for multi-dimensional multiband signals, extending to high-dimensional cases.

Findings

Optimal dictionaries have size proportional to sampling count and subband volume.

DPSS can approximate the optimal dictionaries efficiently.

The methods are primarily demonstrated in 2D but extend to higher dimensions.

Abstract

We study the problem of representing a discrete tensor that comes from finite uniform samplings of a multi-dimensional and multiband analog signal. Particularly, we consider two typical cases in which the shape of the subbands is cubic or parallelepipedic. For the cubic case, by examining the spectrum of its corresponding time- and band-limited operators, we obtain a low-dimensional optimal dictionary to represent the original tensor. We further prove that the optimal dictionary can be approximated by the famous \ac{dpss} with certain modulation, leading to an efficient constructing method. For the parallelepipedic case, we show that there also exists a low-dimensional dictionary to represent the original tensor. We present rigorous proof that the numbers of atoms in both dictionaries are approximately equal to the dot of the total number of samplings and the total volume of the subbands. Our derivations are mainly focused on the \ac{2d} scenarios but can be naturally extended to high dimensions.