On stability properties of three- and higher dimensional linear shift-invariant digital filters

This paper presents a detailed discussion on the stability properties of 3- and higher dimensional linear digital filters, and the differences between 2- and higher dimensional cases. In particular, it is shown that straightforward extension of Goodman's 2-D bounded-input-bounded-output (BIBO) stability theorem [1] to 3- and higher dimensions would be too restrictive. Necessary conditions for stability are presented along with results, showing that these conditions are the same for any case where the number of dimensions is higher than two. Two examples of 3-D digital filters illustrating these results are given. Sufficient conditions for the stability of
n-D (n > 2)
filters are also presented.