Chris' Page for HW 4 (Fourier Transforms)

The Fourier Transform is a process or formula that converts a signal from one domain to another. Often it is used to go between the time domain and the frequency domain.

Developed by Frenchman, Jean Baptiste Joseph Fourier (1768 - 1830), the Fourier Transform stems from the more general Fourier Analysis, which is the representation of a function with sine and cosine terms. Unlike the Fourier Series the Fourier Transform is capable of representing aperiodic signals.

Mathematical Description

The Fourier Transform is detonated by;

$X(f)=\int _{-\infty }^{\infty }x(t)\ e^{-j\omega t}\,dt,$

The Inverse Fourier Transform is;

$x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )\ e^{j\omega t}\,d\omega ,$

Relation to Laplace Transform

Unless otherwise noted, a Laplace Transform is defined by the unilateral or one-sided integral

${\mathcal {L}}\left\{f(t)\right\}=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt.$

The Laplace Transform can be applied from $-\infty$ to $\infty$, this is known as the Bilateral Laplace Transform and is denoted by

${\mathcal {L}}\left\{f(t)\right\}=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.$

Setting $s=j\omega$ ($\sigma =0$) gives the equation

${\mathcal {L}}\left\{f(t)\right\}=\int _{-\infty }^{\infty }e^{-j\omega t}f(t)\,dt.$

which is identical to the Fourier Transform. The same relationship exists between the Inverse Laplace and the Inverse Fourier transforms.