Solution to Q4 of Week 13 Quiz Pool
a. y[m,n] = h[m,n] ** x[m,n]
Using definition of convolution,
$ \begin{align} y[m,n] &= \sum_{k=-1}^{1} \sum_{l=-1}^{1} h[k,l] x[m-k,n-l] \\ \end{align} $
Expanding,
y[m,n] = h[-1,-1] x[m+1,n+1] + h[-1,0] x[m+1,n] + h[-1,1] x[m+1,n-1] + h[0,-1] x[m,n+1] + h[0,0] x[m,n] + h[0,1] x[m,n-1] + h[1,-1] x[m-1,n+1] + h[1,0] x[m-1,n] + h[1,1] x[m-1,n-1]
Sub values of h[m,n] from table, zero terms go away,
$ \begin{align} y[m,n] &= h[-1,-1] x[m+1,n+1] + h[0,0] x[m,n] + h[1,-1] x[m-1,n+1] + h[1,0] x[m-1,n] + h[1,1] x[m-1,n-1] \\ &= h[-1,-1] x[m+1,n+1] + h[0,0] x[m,n] + h[1,-1] x[m-1,n+1] + h[1,0] x[m-1,n] + h[1,1] x[m-1,n-1] \end{align} $
to be continued...
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