Cross Complex Conjugate Symmetry of the Coefficients of Discrete Fourier Transforms of Real Finite Discrete One-Dimensional and Two-Dimensional Signals

Across diverse scientific disciplines, the methods and algorithms of digital Fourier processing stand as indispensable tools for tackling a spectrum of real-world challenges. At the heart of this signal processing prowess lies the one-dimensional, two-dimensional, or, more generally, multidimensional discrete Fourier transform, the foundation upon which complex and real discrete finite signals are analyzed and manipulated. Yet, practical application of these transforms reveals a duality: while they offer undeniable advantages, they also harbor inherent limitations, manifesting as well-documented negative effects. This paper introduces two novel discrete Fourier transforms, conceived as generalizations of their classical counterparts, engineered to mitigate or substantially diminish the troublesome effects that plague traditional approaches. Recognizing the ubiquity of real-valued signals, we delve into the cross complex conjugate symmetry properties of the Fourier coefficients derived from these innovative transforms. Furthermore, we develop methods and algorithms designed to amplify the efficiency and effectiveness of these new discrete Fourier transforms, paving the way for more robust and insightful signal analysis.