Overview

The Fourier transform is the mathematical procedure that breaks up a function into the frequencies that compose it, as a prism breaks up light into colors. It transforms a function f that depends on time into a new function, , or " f hat," which depends on frequency. This new function is called the Fourier transform of the original function (or, when the original function is periodic, its Fourier series). For functions or signals that vary with time music, for example, or the fluctuations of the stock market frequency is most often measured in hertz, or cycles per second, illustrated in Figure 1.

Figure 1: The frequency of sin 2 ?x is 1 cycle per second (1 hertz). The frequency of sin 2 ?2 x is 2 cycles per second (2 hertz).

Functions can also vary with space. The Fourier transform of a fingerprint might have important values near the "frequency" of 15 ridges per centimeter. (Strictly speaking, frequency is the inverse of time, so for a function that depends on space one often says wave number, the inverse of space.)

A function and its Fourier transform are two faces of the same information. The function displays the time information and hides the information about frequencies. The function corresponding to a musical recording shows how the air pressure (produced by sound waves) changes with time, but it doesn't tells us what frequencies what notes make up the music. The Fourier transform displays information about...