Fourier unit pulse

Plot of the real and imaginary parts of the Fourier transform of the pulse function with and without a phase shift.

A Fourier transform is a linear transformation that decomposes a function into the inputs from its constituent frequencies, or, informally, gives the amount of each frequency that composes a signal. The Fourier transform of a function is complex, with the magnitude representing the amount of a given frequency and the argument representing the phase shift from a sine wave of that frequency. It can be thought of as an extension of the Fourier series, and can be used for non-periodic functions.

An intuitive way to think about the Fourier transform is by thinking as a non-periodic function as a periodic function of infinite wavelength. As such, it will have a Fourier series, generally with an infinite number of terms. Since the wavelength is infinite, the frequency of the sine and cosine waves do not have to be integer multiples of the original frequency, and so the frequencies that compose the signal can be continuous, rather than discrete. As such, the Fourier transform gives the amplitude and phase shift of the wave at each frequency. Because of this, the more concentrated the signal, the more spread out the Fourier transform, and vice versa.

The Fourier transform of $ g(t) $ is equal to

$ G(f)=\mathcal{F}\{g(t)\}=\int\limits_{-\infty}^\infty e^{-2\pi ift}g(t)dt $

This is sometimes represented using angular frequency, rather than frequency, as

$ G(\omega)=\int\limits_{-\infty}^\infty e^{-\omega ift}g(t)dt $

The inverse Fourier transform, which takes a function of frequency and transforms it into a function of time, is equal to

$ g(t)=\mathcal{F}^{-1}\{G(f)\}=\int\limits_{-\infty}^\infty e^{2\pi ift}G(f)df $

See also
