Fourier transforms

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The Fourier Transform represents a function s \left( t \right) as a "linear combination" of complex sinusoids at different frequencies \omega\,. Fourier proposed that a function may be written in terms of a sum of complex sine and cosine functions with weighted amplitudes.

In Euler notation the complex exponential may be represented as:

e^{j\omega t}\, = cos(\omega t) + j sin(\omega t)

Thus, the definition of a Fourier transform is usually represented in complex exponential notation.


s \left( t \right) = \frac{1}{2\pi} \int\limits_{-\infty}^\infty S\left( \omega\right) e^{j\omega t}\,d\omega.

The function  S \left( \omega \right) is the Fourier transform of s \left( t \right). This is often denoted with the operator \mathcal{F}, in the case above,  S \left( \omega \right) = \mathcal{F} \left(s ( t) \right)

The function  s \left( t \right) must satisfy the Dirichlet conditions in order for  s \left( t \right) to have a valid Fourier transform.

Forward Fourier Transform(FT)/Anaysis Equation



S \left( \omega \right) = \int\limits_{-\infty}^\infty s\left( t\right) e^{-j\omega t}\,dt.


Inverse Fourier Transform(IFT)/Synthesis Equation


s \left( t \right) = \frac{1}{2\pi} \int\limits_{-\infty}^\infty S\left( \omega\right) e^{j\omega t}\,d\omega.


Relation to the Laplace Transform [edit]

In fact, the Fourier Transform can be viewed as a special case of the bilateral Laplace Transform. If the complex Laplace variable s were written as  s = \sigma + j  \omega \,, then the Fourier transform is just the bilateral Laplace transform evaluated at \sigma = 0 \,. This justification is not mathematically rigorous, but for most applications in engineering the correspondence holds.

Properties [edit]

× Time Function Fourier Transform Property
1 z(t)=x(t) \pm \ y(t) Z(\omega)=X(\omega) \pm \ Y(\omega) Linearity
2 Z(t) 2\pi z(-\omega) Duality
3 c\, x(t), c = constant c\, X(\omega) Scalar Multiplication
4 \frac {dx(t)}{dt} j \omega\,X(\omega) Differentiation in time domain
5 \int\limits_{-x}^{t} x(\tau)d \tau \frac {X(\omega)}{j \omega}, if \int\limits_{-\infty}^{\infty} x(t)\,dt = 0 Integration in Time domain
6 t\,x(t) j\,\frac {dX(\omega)}{d \omega} Differentiation in Frequency Domain
7 x(-\,t) X(-\,\omega) Time reversal
8 x(a\,t) \frac{1}{\left | a \right |}X\left( \frac {\omega}{a} \right ) Time Scaling
9 x(t\,-\,a) e^{-\,j \omega\,a}\,X(\omega) Time shifting
10 x(t) \cos {\omega_0\,t} \frac{1}{2}\left [ X(\omega\,+\,\omega_0)\,+\,X(\omega\,-\,\omega_0) \right ] Modulation
11 x(t) \sin {\omega_0\,t} \frac{1}{2j}\left [ X(\omega\,-\,\omega_0)\,-\,X(\omega\,+\,\omega_0) \right ] Modulation
12 e^{-\,a\,t}x(t) X(\omega\,+\,a) Frequency shifting
13 x_1(t)\times\,x_2(t) \frac{1}{2 \pi}\int\limits_{- \pi}^{\pi}X_1(\lambda)\,X_2(\omega\,-\lambda)\,d\lambda Convolution