Access to eBooks is limited to institutions that have purchased or currently subscribe to the SPIE eBooks program. eBooks are not available via an individual subscription. SPIE books (print and digital) may be purchased individually on SPIE.Org.

Contact your librarian to recommend SPIE eBooks for your organization.
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); -umaxu ≤ +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)

Online access to SPIE eBooks is limited to subscribing institutions.
CHAPTER 3
10 PAGES


SHARE
Back to Top