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

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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

_{o}x + v

_{o}y)/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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