14.State and prove the translation property of 2D-DFT

14.State and prove the translation property of 2D-DFT

   December 05, 2013       legend

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The translation properties of the Fourier transform pair are

and

Where the double arrow indicates the correspondence between a function and its Fourier
Transform,
Equation (1) shows that multiplying f(x, y) by the indicated exponential term and taking
the transform of the product results in a shift of the origin of the frequency plane to the point (u_o,
v_o).

Consider the equation (1) with u_o = v_o = N/2 or
                                              exp[j2Π(u_ox + v_oy)/N] = e^{jΠ(x + y)}
                                                                                   = (-1)^{(x + y)}
and
                                          f(x, y)(-1)^x+y = F(u – N/2, v – N/2)

Thus the origin of the Fourier transform of f(x, y) can be moved to the center of its
corresponding N x N frequency square simply by multiplying f(x, y) by (-1)^x+y . In the one
variable case this shift reduces to multiplication of f(x) by the term (-1)^x. Note from equation (2)
that a shift in f(x, y) does not affect the magnitude of its Fourier transform as,

 

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