对离散Fourier变换公式四种形式的讨论

Four Forms of Discrete Fourier Transformtion Formulae

分析了离散Fourier变换公式中求和符号前的常系数选取规律,提出了离散Fourier变换公式的分类概念,将离散Fourier变换公式分为通用型、缩频扩谱型、缩谱扩频型和频谱均衡型等4种类型;讨论了4种不同类型的离散Fourier变换公式的内在联系和一致性,分析了各种形式的离散Fourier变换公式所强调的侧重面,并讨论了它们的适用范围.主要结论为:离散Fourier正、逆变换公式中求和符号前常系数之乘积等于信号采样数的倒数;按此规律调整常系数,可以有重点地对信号开展频-谱分析;缩频扩谱型、缩谱扩频型离散Fourier变换公式分别适合于弱幅信号、低频信号的处理.

On account of random choices of instant coefficients, there are various forms of discrete Fourier transformation formulae. Based on the principle of Fourier Transformation, the choice rules of instant coefficients of discrete Fourier transformation formulae were summarized, and a classified conception about the formulae was put forward. There are four forms of discrete Fourier transformation formulae： they are General Transformation Formula （GTF）; Compressing Frequency- enlarging Spectrum Transformation Formula （CFESTF;Compressing Spectrum - enlarging Frequency Transformation Formula （CSEFTF） and Equilibrium Transformation Formula （ETF）. We discussed the relation and uniformity among the 4 forms of formulae, analyzed the emphasized aspect about each form, and discussed their suitable fields. The results indicate that the product of two instant coefficients in the formulae of discrete Fourier and its inverse Transformations was equal to the reciprocal of the sampling number of observation signal;that CFESTF can be properly used in the processing of the weak amplitude signal or high frequency signal;and that CSEFTF can be properly used in the processing of the low frequency signal. These conclusions will help us to choose some reasonable type from discrete Fourier and its inverse transformations.