Preconditioning Filter Bank Decomposition Using Structured Normalized Tight Frames

Martin Ehler

doi:10.1155/2015/861563

Abstract

We turn a given filter bank into a filtering scheme that provides perfect reconstruction, synthesis is the adjoint of the analysis part (so-called unitary filter banks), all filters have equal norm, and the essential features of the original filter bank are preserved. Unitary filter banks providing perfect reconstruction are induced by tight generalized frames, which enable signal decomposition using a set of linear operators. If, in addition, frame elements have equal norm, then the signal energy is spread through the various filter bank channels in some uniform fashion, which is often more suitable for further signal processing. We start with a given generalized frame whose elements allow for fast matrix vector multiplication, as, for instance, convolution operators, and compute a normalized tight frame, for which signal analysis and synthesis still preserve those fast algorithmic schemes.

Highlights

Detailed data are acquired in all sorts of measurements nowadays, so that fast algorithms are an important factor for successful signal processing
The concept of generalized frames has a long tradition in signal processing and many unitary filter bank schemes with the perfect reconstruction property are induced by tight generalized frames
If the frame is tight and its elements have unit norm, it resembles the concept of an orthonormal basis, with the add-on of useful redundancy, and frame coefficients measure the signal energy in a uniform fashion

Summary

Introduction

Detailed data are acquired in all sorts of measurements nowadays, so that fast algorithms are an important factor for successful signal processing. In the present paper we start with a generalized frame, whose elements allow for fast matrix vector multiplications (e.g., convolution operators) and construct a unit norm tight generalized frame that induces a filter bank scheme preserving those fast algorithms. The latter is related to the so-called Paulsen problem for frames, where one is given a unit norm frame and one asks for the closest tight frame with unit norm and for an algorithm to find it.

Generalized Frames

Constructing Unit Norm Tight g-Frames That Preserve Fast Algorithms

Examples of Random Matrices Satisfying the Assumptions for Convergence

Closeness to the Original g-Frame

Some Concluding Remarks