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.

The primary use of Fourier transformation is to extract the frequency spectrum of a given waveform with respect to an independent variable (space or time). This appendix is a primer for Chapters 12 and 13.

A.1 Fourier Series

According to the Fourier theorem, any periodic waveform can be expressed as a summation of a series of simple harmonic terms. The notation used in this context are (1) analysis: the decomposition of a parent function [f(x)] into a collection of harmonic components (sine and cosine waveforms); and (2) synthesis: the reconstitutions of f(x) from these harmonic components. The basic form of the Fourier theorem says that if a function f(x) is periodic over the interval x0xx0 + l0, is absolutely integrable, and has a finite number of discontinuities, it can be represented by the following infinite sum:


In this polar form of the Fourier series, the fundamental mode (lowest-frequency waveform) has an angular frequency ω0 = (2π/x0) with amplitude A1. The n harmonics have frequencies nω0 and amplitudes An; the constant A0 represents the d.c. term, and ωn is the radial frequency in rad/cycle. The independent variable on the x axis is the spatial distance (mm) in image processingz and time (sec) in temporal signal analysis. One of the widely referred-to examples of the Fourier series is the reconstruction of an asymmetrical square wave (pulse train) from a collection of harmonics:


Online access to SPIE eBooks is limited to subscribing institutions.

Back to Top