# What is a translation property in DFT

Hi someone please explain me the translation property of $\textrm{DFT}$. I am not able to understand it neither from Gonzalez nor from internet. I have done extensive study on this but not able to get it. Really Frustrated!!

I don't understand how shifting is done? What is the difference between $f(x-x_0, y-y_0)$ and $f(x+x_0, y+y_0)$ and what are their corresponding $\textrm{DFT}$'s?

The idea is the same as the 1D case. Start with the regular old continuous time fourier transform definition (the integral) and calculate the Fourier transform of $e^{j \omega_0 t} f(t)$. Similarly, calculate the inverse Fourier transform (using the integral) for $e^{j \alpha \omega} F(\omega)$.

Alternatively, do this with the discrete time fourier transform (so the transform is a sum and the inverse transform is an integral).

• Just try calculating it yourself. Sep 25, 2014 at 15:15

I think all the question is trying to show is that a rotation or phase shift in one domain causes a shift or translation in the other domain. Multiplying by exp(jx) or exp(-jx) is nothing but a phase shift by angle x. I know these are correct properties but I don't know if that answers your question.

-K

Shifting is done by moving the origin point of an image. If $f(x,y)$ is your original image, then $f(x+x_0,y+y_0)$ is your spatially shifted image where point of image originally located at $(0,0)$ is now located at $(x_0,y_0)$.