A promotion for odd symmetric discrete Fourier transform

TL;DR

This paper advocates for using odd-length symmetric discrete Fourier transforms (SDFT) over the traditional even-length, highlighting their superior properties and correcting misconceptions about their symmetry and application.

Contribution

The study demonstrates the unique nature of SDFT, corrects its symmetry properties for even points, and advocates for odd-length SDFT in applications.

Findings

Odd SDFT has better FT properties than ODFT.

The time-domain of even-point SDFT is not symmetric to zero.

Reasons for favoring odd SDFT are provided.

Abstract

DFT is the numerical implementation of Fourier transform (FT), and it has many forms. Ordinary DFT (ODFT) and symmetric DFT (SDFT) are the two main forms of DFT. The most widely used DFT is ODFT, and the phase spectrum of this form is widely used in engineering applications. However, it is found ODFT has the problem of phase aliasing. Moreover, ODFT does not have many FT properties, such as symmetry, integration, and interpolation. When compared with ODFT, SDFT has more FT properties. Theoretically, the more properties a transformation has, the wider its application range. Hence, SDFT is more suitable as the discrete form of FT. In order to promote SDFT, the unique nature of SDFT is demonstrated. The time-domain of even-point SDFT is not symmetric to zero, and the author corrects it in this study. The author raises a new issue that should the signal length be odd or even when performing SDFT. The answer is odd. However, scientists and engineers are accustomed to using even-numbered sequences. At the end of this study, the reasons why the author advocates odd SDFT are given. Besides, even sampling function, discrete frequency Fourier transform, and the Gibbs phenomenon of the SDFT are introduced.