The Fourier Transform

Table 2.1: Fourier transform pairs

Function	Transform
$e^{\pi t^2}$	$e^{\pi \nu^2}$
$1$	$\delta(\nu)$
$\cos(\pi t)$	${1 \over 2}\delta(\nu - {1 \over 2}) + {1 \over 2}\delta(\nu + {1 \over 2})$
$\sin(\pi t)$	${i \over 2}\delta(\nu - {1 \over 2}) - {i \over 2}\delta(\nu + {1 \over 2})$
$rect(t)$	$sinc(\nu)$

The Fourier transform $U(\nu)$ of a function $u(t)$ is defined as

$$U(\nu) = \int_{-\infty}^{\infty} u(t)e^{-i2\pi\nu t} dt$$

and can be shown to exist for any function $u(t)$ for which

$$\int_{-\infty}^{\infty} \vert u(t)\vert dt < \infty$$

The Fourier transform is invertible, i.e. given $U(\nu)$, $u(t)$ can be obtained using the inverse Fourier transform, viz.

$$u(t) = \int_{-\infty}^{\infty} U(\nu)e^{i2\pi\nu t} d\nu$$

Some important properties of the Fourier transform are listed below (where by convention capitalized functions refer to the Fourier transform)

1. Linearity
   
   $$\mathcal{F}\{au(t) + bv(t)\} = aU(\nu) + b V(\nu)$$
   
   
   where $a, b$ are arbitrary complex constants.

2. Similarity
   
   $$\mathcal{F}\{u(at)\} = {1\over a}U({\nu\over a})$$
   
   
   where $a$ is an arbitrary real constant.

3. Shift
   
   $$\mathcal{F}\{u(t-a)\} = e^{-i2\pi a}U({\nu})$$
   
   
   where $a$ is an arbitrary real constant.

4. Parseval's Theorem
   
   $$\int_{-\infty}^{\infty} \vert u(t)\vert^2 dt = \int_{-\infty}^{\infty} \vert U(\nu)\vert^2 d\nu$$
   
   

5. Convolution Theorem
   
   $$\mathcal{F}{\int_{-\infty}^{\infty}} u(t)v(t-\tau) dt = U(\nu) V(\nu)$$
   
   

6. Autocorrelation Theorem
   
   $$\mathcal{F}{\int_{-\infty}^{\infty}} u(t)u(t+\tau) dt = \vert U(\nu)\vert^2$$
   
   

Some commonly used Fourier transform pairs are: