# Fourier transforms

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

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