# Fourier transform

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.

The Fourier transform of s(t) is defined by $S\left(\omega \right)=\int \limits _{-\infty }^{\infty }s\left(t\right)e^{-j\omega t}\,dt.$ Under appropriate conditions original function can be recovered by:

$s\left(t\right)={\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }S\left(\omega \right)e^{j\omega t}\,d\omega .$ 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 $s\left(t\right)$ for the integral defining Fourier transform to converge.

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

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

× 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