Eigenvalues of Symmetric Non-normalized Discrete Trigonometric Transforms

TL;DR

This paper provides a comprehensive spectrum characterization of symmetric discrete trigonometric transforms, deriving explicit eigenvalues and multiplicities for several types, and introduces new formulas for matrix trace and squares.

Contribution

It offers the first explicit analytic expressions for eigenvalues and multiplicities of three specific DTTs, enhancing understanding of their spectral properties.

Findings

Explicit eigenvalues for DCT$_{(1)}$, DCT$_{(5)}$, and DST$_{(8)}$

New formulas for the trace and square of DTT matrices

Complete spectrum characterization for eight DTT types

Abstract

A comprehensive approach to the spectrum characterization (derivation of eigenvalues and the corresponding multiplicities) for non-normalized, symmetric discrete trigonometric transforms (DTT) is presented in the paper. Eight types of the DTT are analyzed. New explicit analytic expressions for the eigenvalues, together with their multiplicities, for the cases of three DTT (DCT$_{(1)}$, DCT$_{(5)}$, and DST$_{(8)}$), are the main contribution of this paper. Moreover, the presented theory is supplemented by new, original derivations for the closed-form expressions of the square and the trace of analyzed DTT matrices.