Systematic DFT Frames: Principle and Eigenvalues Structure

TL;DR

This paper investigates systematic DFT frames, analyzing their eigenvalues and tightness conditions, with implications for distributed source coding and signal reconstruction accuracy.

Contribution

It introduces systematic DFT frames, explores their eigenvalue structure, and establishes conditions for tightness, advancing frame design for signal processing applications.

Findings

Systematic DFT frames are not necessarily tight.

Conditions for systematic DFT frames to be tight are identified.

Eigenvalues of DFT frames are crucial for understanding their properties.

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 application of systematic DFT codes in distributed source coding using DFT codes, a new application for frames. 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 from reconstruction error point of view. Eigenvalues of DFT frames, and their subframes, play a pivotal role in this work.