Systematic DFT Frames: Principle, Eigenvalues Structure, and Applications

TL;DR

This paper investigates systematic DFT frames, analyzing their eigenvalues and tightness conditions, with applications in distributed source coding, and provides bounds on eigenvalues of DFT subframes for improved signal reconstruction.

Contribution

It introduces systematic DFT frames, studies their eigenvalues and tightness conditions, and derives bounds on eigenvalues of DFT subframes for enhanced understanding and application.

Findings

Systematic DFT frames are not necessarily tight.

Conditions for tightness of systematic DFT frames are established.

Bounds on eigenvalues of DFT subframes are derived.

Abstract

Motivated by a host of recent applications requiring some amount of redundancy, frames are becoming a standard tool in the signal processing toolbox. In this paper, we study a specific class of frames, known as discrete Fourier transform (DFT) codes, and introduce the notion of systematic frames for this class. This is encouraged by a new application of frames, namely, distributed source coding that uses DFT codes for compression. Studying their extreme eigenvalues, we show that, unlike DFT frames, systematic DFT frames are not necessarily tight. Then, we come up with conditions for which these frames can be tight. In either case, the best and worst systematic frames are established in the minimum mean-squared reconstruction error sense. Eigenvalues of DFT frames and their subframes play a pivotal role in this work. Particularly, we derive some bounds on the extreme eigenvalues DFT subframes which are used to prove most of the results; these bounds are valuable independently.