## 11. Define Fourier Transform and its inverse.

>Table of contents

Let f(x) be a continuous function of a real variable x. The Fourier transform of f(x) is

defined by the equation

Where j = √-1

Given F(u), f(x) can be obtained by using the inverse Fourier transform

The Fourier transform exists if f(x) is continuous and integrable and F(u) is integrable.

The Fourier transform of a real function, is generally complex,

F(u) = R(u) + jI(u)

Where R(u) and I(u) are the real and imaginary components of F(u). F(u) can be expressed in

exponential form as

F(u) = │F(u)│e

^{jØ(u)}

where

│F(u)│ = [R

^{2}(u) + I

^{2}(u)]

^{1/2}

and

Ø (u, v) = tan

^{-1}[ I (u, v)/R (u, v) ]

The magnitude function |F (u)| is called the Fourier Spectrum of f(x) and Φ(u) its phase angle.

The variable u appearing in the Fourier transform is called the frequency variable.

Fig 1 A simple function and its Fourier spectrum

The Fourier transform can be easily extended to a function f(x, y) of two variables. If f(x, y) is

continuous and integrable and F(u,v) is integrable, following Fourier transform pair exists

and

Where u, v are the frequency variables

The Fourier spectrum, phase, are

|F(u, v)| = [R

^{2}(u, v) + I

^{2}(u, v )]

^{1/2}

Ø(u, v) = tan

^{-1}[ I(u, v)/R(u, v) ]

>Table of contents

## 0 comments :

## Post a Comment