The Discrete Fourier Transform (DF T) is recognized as one of the key tools in digital data processing, such as data compression, dominant frequency identification, and audio and image processing.
This research aims to provide a theoretical analysis of the eigenstructure of DF T and its related families, with the goal of introducing a new eigenbasis that enhances the efficiency of these transforms in
signal processing. To achieve this, a novel and effective transformation system has been introduced,
establishing a deep connection between the eigenstructure of the DF T and the eigenstructure of certain
symmetric normalized discrete trigonometric transforms. This shift in approach led to the analysis of the
eigenstructure of normalized discrete trigonometric transforms, through which a class of symmetric matrices that are square roots of the identity matrix, including symmetric normalized discrete trigonometric
transforms, was introduced. Finally, utilizing advanced linear algebra techniques, a specific theoretical
framework for extracting eigenvectors and performing spectral decomposition for this class was presented. Another significant achievement of this research is the introduction of an eigenbasis converging
to Hermite-Gaussian functions, which play a key role in the spectral analysis of the continuous Fourier
transform and have diverse applications in physics and electrical engineering. Although direct sampling
from these functions cannot form a complete orthonormal basis for the DF T, this research, through
innovative methods and precise design, introduces an eigenbasis for the centered discrete Fourier transform (CDF T) that, while simple, closely approximates Hermite-Gaussian functions. The results of the
study indicate that leveraging the eigenbasis of DT T can improve the spectral analysis of DF T and
have widespread applications in various fields. In conclusion, suggestions for expanding this research
emphasize the importance of further exploring the