# 11. Define Fourier Transform and its inverse.

## 11. Define Fourier Transform and its inverse.

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)│ejØ(u)

where
│F(u)│ = [R2(u) + I2(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)|  = [R2(u, v) + I2(u, v )]1/2

Ø(u, v) = tan-1[ I(u, v)/R(u, v) ]