A(x+M,y+N) = A(x,y)

in which case the results are always the same as the FFT method of convolution since it assumes periodicty in its calculation. This would also mean that if you are defining linear convolultion by not assuming periodicity, and merely padding to the side with zeroes, that for periodicity not to be a factor you would require, if A=p by q and B=q by p, that p+q-2 < max(M,N)

Or am I misunderstanding the question?