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 x₀ ≤ x ≤ x₀ + l₀, is absolutely integrable, and has a finite number of discontinuities, it can be represented by the following infinite sum:

(A.1a)

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 A_n; the constant A₀ represents the d.c. term, and ω_n 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:

(A.1b)