Unification analysis of the four major classifications of Fourier transform

In response to the problems of complex branches of existing Fourier theory and insufficient comprehensive and unified research and analysis methods, four types of Fourier transform formulas were analysed and demonstrated for uniformity. Firstly, through rigorous mathematical formula derivation, the decomposition and synthesis processes of four main Fourier transforms, namely Fourier Series, FT, DFT, and DTFT, were reorganized and revised, and the complete equivalence of these four transformations was demonstrated from a quantitative perspective. Secondly, based on the unity analysis of the four Fourier transforms, the application characteristics of the four Fourier transforms were revealed from both theoretical analysis and engineering practice, and application suggestions were provided. Finally, the correctness of the proposed theory was verified through simulation testing, indicating that the Fourier unified analysis method proposed in this paper can effectively solve Fourier related quantization research and provide important guidance for downsampling and signal recovery.