Fourier Transform represents a function ${\displaystyle s\left(t\right)}$ as a "linear combination" of complex sinusoids at different frequencies ${\displaystyle \omega \,}$. 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:

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

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

The Fourier transform of s(t) is defined by ${\displaystyle S\left(\omega \right)=\int \limits _{-\infty }^{\infty }s\left(t\right)e^{-j\omega t}\,dt.}$

Under appropriate conditions original function can be recovered by:

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

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

The function ${\displaystyle s\left(t\right)}$ must satisfy the Dirichlet conditions in order for ${\displaystyle s\left(t\right)}$ for the integral defining Fourier transform to converge.

Forward Fourier Transform(FT)/Anaysis Equation

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

Inverse Fourier Transform(IFT)/Synthesis Equation

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

Relation to the Laplace Transform

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 ${\displaystyle s=\sigma +j\omega \,}$, then the Fourier transform is just the bilateral Laplace transform evaluated at ${\displaystyle \sigma =0\,}$. This justification is not mathematically rigorous, but for most applications in engineering the correspondence holds.

Properties

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