Fourier Transform

Last Updated: November 12, 2020

Definition - What does Fourier Transform mean?

The Fourier transform is a mathematical function that takes a time-based pattern as input and determines the overall cycle offset, rotation speed and strength for every possible cycle in the given pattern. The Fourier transform is applied to waveforms which are basically a function of time, space or some other variable. The Fourier transform decomposes a waveform into a sinusoid and thus provides another way to represent a waveform.

Techopedia explains Fourier Transform



The Fourier transform of a function f(x) is given by:

equations for the Fourier transform of a function f(x)

Where F(k) can be obtained using inverse Fourier transform.

Some of the properties of Fourier transform include:

  • It is a linear transform – If g(t) and h(t) are two Fourier transforms given by G(f) and H(f) respectively, then the Fourier transform of the linear combination of g and t can be easily calculated.
  • Time shift property – The Fourier transform of g(t–a) where a is a real number that shifts the original function has the same amount of shift in the magnitude of the spectrum.
  • Modulation property – A function is modulated by another function when it is multiplied in time.
  • Parseval’s theorem – Fourier transform is unitary, i.e., the sum of square of a function g(t) equals the sum of the square of its Fourier transform, G(f).
  • Duality – If g(t) has the Fourier transform G(f), then the Fourier transform of G(t) is g(-f).

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