# Finding the equivalent filter H(u,v) in the frequency domain of a 3x3 spatial mask

I'm trying to find the equivalent frequency domain filter, $$H(u,v)$$, of a 3x3 spatial mask that averages all neighbours of a point $$(x,y)$$ in said 3x3 neighbourhood excluding the point itself. So far, I have been able to calculate:

$$g(x,y) = 1/8 \cdot [f(x+1,y) + f(x+1,y+1) + f(x+1,y-1).... f(x-1,y-1)]$$ -(no $$f(x,y)$$)- but here is where I get stuck.

I've read that I can use linearity and translation properties to get $$H(u,v)$$ after performing DFT on $$g(x,y)$$ but for me that is currently a 'draw the rest of the owl' situation.

How do I get $$H(u,v)$$ from $$g(x,y)$$ and what form does it take?
I.e., what will the DFT of $$f(x+1,y-1)$$ look like and how do I then get $$H(u,v)$$ from that?

Hint: $$g(x,y)$$ averages all pixel values around $$f(x,y)$$, at 8-neighbour connectivity distance 1, except $$g(x,y)$$. The classical uniform $$3\times 3$$ average is:
$$m(x,y) = \frac{1}{9}\sum_{i\in[-1,0,1]}\sum_{j\in[-1,0,1]}f(x-i,y-j)\,.$$
Thus: $$9m(x,y) = 8g(x,y)+f(x,y)\,.$$
Knowing the Fourier transform of the uniform $$3\times 3$$ mask, and that of the identity, and using linear superposition, you should be on track.