Balanced Frames: A Useful Tool in Signal Processing with Good Properties

Abstract

So far there has not been paid attention to frames that are balanced, i.e. those frames which sum is zero. In this paper we consider balanced frames, and in particular balanced unit norm tight frames, in finite dimensional Hilbert spaces. Here we discover various advantages of balanced unit norm tight frames in signal processing. They give an exact reconstruction in the presence of systematic errors in the transmitted coefficients, and are optimal when these coefficients are corrupted with noises that can have non-zero mean. Moreover, using balanced frames we can know that the transmitted coefficients were perturbed, and we also have an indication of the source of the error. We analyze several properties of these types of frames. We define an equivalence relation in the set of the dual frames of a balanced frame, and use it to show that we can obtain all the duals from the balanced ones. We study the problem of finding the nearest balanced frame to a given frame, characterizing completely its existence and giving its expression. We introduce and study a concept of complement for balanced frames. Finally, we present many examples and methods for constructing balanced unit norm tight frames.