# Convolution in Spatial Domain Is Multiplication in Frequency Domain

I have to prove convolution in spatial domain is equivalent to multiplication in frequency domain using two matrices.

$$x(m, n) = \begin{bmatrix} 1 && 2 \\ 3 && 4 \end{bmatrix}$$

$$h(m, n) = \begin{bmatrix} 1 && 0 \\ 0 && 0 \end{bmatrix}$$

When I used matrix method, I got the following result

$$\begin{bmatrix} 1 && 2 && 0 \\ 3 && 4 && 0 \\ 0 && 0 && 0 \end{bmatrix}$$

But when I converted both to DFT using the kernel

$$\begin{bmatrix} 1 && 1 \\ 1 && -1 \end{bmatrix}$$

and multiply the result, I get

$$\begin{bmatrix} 1 && 2 \\ 3 && 4 \end{bmatrix}$$

They are not equal. The dimensions are not matching. The first one is $$3\times 3$$ while the second one is $$2\times 2$$. Am I doing it correct? Is there something I am missing?