# Fourier Transform

## 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 is a mathematical function that decomposes a waveform, which is a function of time, into the frequencies that make it up. The result produced by the Fourier transform is a complex valued function of frequency. The absolute value of the Fourier transform represents the frequency value present in the original function and its complex argument represents the phase offset of the basic sinusoidal in that frequency.

The Fourier transform is also called a generalization of the Fourier series. This term can also be applied to both the frequency domain representation and the mathematical function used. The Fourier transform helps in extending the Fourier series to non-periodic functions, which allows viewing any function as a sum of simple sinusoids.

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

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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