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Compute the two-dimensional DFT [4x4] for the following 4x4 image $ \begin{matrix} 0.5 & 0.5 & 0.5 & 0.5 \\ 0.5 & 0.5 & 0.5 & 0.5\\ 0.5 & 0.5 & 0.5 & 0.5\\ 0.5 & 0.5 & 0.5 & 0.5 \end{matrix} $

I know that DFT is separable by dimensions – one can calculate 4 vertical transforms first, then 4 horizontal ones

For each row we get [2 0 0 0] in the first and third row and zero elsewhere.

For each column we get [1 0 1 0] in the first and third column and zero elsewhere.

How from this two one-dimensional dfts obtain two-dimensional one?

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No you are not doing the separation correctly :

The horizontal 1D-DFT of the rows of input will be:

$ H_1 = \begin{matrix} 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{matrix} $

and the vertical 1D-DFT of the columns of $H_1$ will be:

$ H_2 = \begin{matrix} 8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{matrix} $

which is equivalent to the 2D-DFT of the original input.

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