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Practice Problem on Discrete-space Fourier transform computation
Compute the discrete-space Fourier transform of the following signal:
$ f[m,n]= \left( u[n]-u[n-3] \right) \left( u[m+1]-u[m-2] \right) $
(Write enough intermediate steps to fully justify your answer.)
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Answer 1
$ \begin{align} F [u,v] &= \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} f[m,n]e^{-j(mu + nv)}\\ &= \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \left( u[n]-u[n-3] \right) \left( u[m+1]-u[m-2] \right)e^{-j(mu + nv)}\\ &= \sum_{m=-\infty}^{\infty} \left( u[m+1]-u[m-2] \right) e^{-j(mu)} \sum_{n=-\infty}^{\infty} \left( u[n]-u[n-3] \right)e^{-j(nv)}\\ &= (e^{jmu} + 1 + e^{-jmu})\cdot(1 + e^{-jnv} + e^{-2jnv})\\ \end{align} $
Answer 2
Write it here.
