Appendix C Discrete Fourier Transform

Abstract

The discrete Fourier transform (DFT) represents the Fourier transformation of a sequence of discrete signal values. The continuous-Fourier-transform pair is derived through closed-form analytical integration of the continuous parent function g(x) with respect to the continuous independent variables (space in image processing); the DFT is the weighted sum of the sampled values g(kΔx). This appendix extends some of the core results of Appendix A. In deriving the DFT, the sampling process and supporting functions are chosen to closely approximate the continuous Fourier transform. The sequence of deriving the frequency spectrum of a given continuous signal is illustrated in Fig. C.1; spatial waveforms are shown on the top rows, and their frequency-domain counterparts are shown along the bottom rows. The analytical steps of deriving the DFT from the continuous FT are summarized below (all transformations are given in terms of the spatial frequency u). (a) For the parent function g(x) to meet the Fourier transformation condition, the continuous Fourier transform G(u); -umax ≤ u ≤ +umax is derived. (b) With the bandlimited assumption, g(x) is impulse sampled at twice its bandlimiting frequency umax using the spatial sampling function (C.1a) and its Fourier transform (C.1b) or (C.1c)