Fourier Transform, Formula, Properties
Table of Contents
A mathematical operation known as the Fourier transform uses a time-based pattern as input to calculate the strength, rotation speed, and overall cycle offset for each potential cycle. Waveforms, which are fundamentally a function of time, space, or another variable, are subjected to the Fourier transform. It offers yet another way to represent a waveform by breaking it down into a sinusoid.
It is a mathematical operation that separates the frequencies that make up a waveform, which is a function of time. And The Fourier transform’s output is a complex valued function of frequency. The Fourier transform’s complex argument,. Which represents the phase offset of the fundamental sinusoidal in that frequency, represents the Fourier transform’s absolute value,. Which represents the frequency value present in the original function.
A generalization of the Fourier series is another name for the Fourier transforms. This phrase can be used to describe both the mathematical operation used and the representation in the frequency domain. So Any function can be seen as a collection of straight forward sinusoids thanks to the Fourier transform’s assistance in extending the Fourier series to non-periodic functions.
Fourier Transform Formula
The Fourier transforms of a function f(x) is given by:
Where F(k) can be obtained using inverse Fourier transforms.
Properties of Fourier Transform
It is a linear transform :–
It is simple to calculate the Fourier transform of the linear combination of g and t if g(t) and h(t) are two Fourier transforms provided by G(f) and H(f), respectively.
Time shift property :–
The magnitude of the spectrum is shifted by the same amount in the original function’s Fourier transform, g(t-a), where an is a real number.
Modulation property :–
A function is modulated by another function when it is multiplied in time.
Parseval’s theorem :–
Since the Fourier transforms is unitary,. The sum of a function’s square, g(t), and its Fourier transforms, G(f), are equal.
If g(t) has the Fourier transforms G(f), then the Fourier transforms of G(t) is g(-f).