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.

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