11. Define Fourier Transform and its inverse.

11. Define Fourier Transform and its inverse.

   May 02, 2013       legend

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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)│ = [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) ]



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