The Schur decomposition of discrete Sine and Cosine transformations of type IV, By Taher Yazdanpanah

Research

Title	The Schur decomposition of discrete Sine and Cosine transformations of type IV
Type	Article
Keywords	Discrete trigonometric transforms Discrete cosine transforms Discrete sine transforms eigenvalues eigenvectors
Journal	DIGITAL SIGNAL PROCESSING
DOI	https://doi.org/10.1016/j.dsp.2023.104260
Researchers	Ali Bagheri-Bardi (First researcher) , Milos Dakovic (Second researcher) , Taher Yazdanpanah (Third researcher) , Ljubiša Stanković (Fifth researcher)

Abstract

Eigendecomposition of discrete sine and cosine transformations of type IV is the main objective of this study. A closed form formula for eigenvectors is derived using the property that a square of the transform matrix is proportional to the identity matrix. Odd and even transform sizes are considered separately. In both cases, it has been demonstrated that the eigenvectors can be obtained by a straightforward modification of the columns of the transformation matrix. Comparing this approach with the related works, it provides a simple explanation of the eigendecomposition with a reliable numerical stability. Additionally, these eigenvectors can be used to obtain the eigendecomposition of the counterpart offset discrete Fourier transform.