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Oppenheim-Willsky

"In Chapter 3, we developed a representation of periodic singals as linear combinations of complex exponentials.. Whereas for periodic signals the complex exponential building blocks are harmonically related, for aperiodic singals they are infinitesimally close in frequency, and the resulting spectrum of coefficients in this representation is called the Fourier Transform.... In particular, Fourier reasoned that an aperiodic signal can be viewed as a periodic singal with an infinite period." An example of choosing the decision (this is a link)!!##!!

Wikipedia [1]

"Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components. A useful analogy is the relationship between a set of notes in musical notation (the frequency components) and the sound of the musical chord represented by these notes (the function/signal itself).
Using physical terminology, the Fourier transform of a signal x(t) can be thought of as a representation of a signal in the "frequency domain"; i.e. how much each frequency contributes to the signal. This is similar to the basic idea of the various other Fourier transforms including the Fourier series of a periodic function."

My Definition

When thinking about the Fourier Transform, I like to think about the analogy of a musical chord given by Wikipedia. A chord of music can be represented as our function x(t). Taking the Fourier transform of x(t) breaks the chord down into the frequency components. The inverse of the Fourier Transform then "sums" the frequency components to form the function x(t) or the musical chord. The difference between the Fourier Series and Fourier Transform is that a Fourier Series can only be applied to periodic functions which can be broken into a finite number of frequency components. The Fourier Transform applies to aperiodic functions and breaks the function into as infinite number of infinitesimally close frequency components using the integral.

The Human Perspective

The cool thing is that humans (and other animals) do Fourier Transforms all the time, without having to deal with the underlying mathematics. When you hear sounds, you hear tones (which is just the "frequency" abstracted out of the to-and-fro motions of your ear-drum). When you see, you see colors, which are frequency components in the light spectrum. How our sensory organs work as "analog computers" is itself a fascinating topic to pursue.

[1] Wikipedia - Fourier Transform

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva