Multidimensional discrete convolution

In signal processing, multidimensional discrete convolution refers to the mathematical operation between two functions f and g on an n-dimensional lattice that produces a third function, also of n-dimensions. Multidimensional discrete convolution is the discrete analog of the multidimensional convolution of functions on Euclidean space. It is also a special case of convolution on groups when the group is the group of n-tuples of integers. Similar to the one-dimensional case, an asterisk is used to represent the convolution operation. The number of dimensions in the given operation is reflected in the number of asterisks. For example, an M-dimensional convolution would be written with M asterisks. The following represents a M-dimensional convolution of discrete signals: For discrete-valued signals, this convolution can be directly computed via the following: The resulting output region of support of a discrete multidimensional convolution will be determined based on the size and regions of support of the two input signals. Listed are several properties of the two-dimensional convolution operator. Note that these can also be extended for signals of -dimensions. Commutative Property: Associate Property: Distributive Property: These properties are seen in use in the figure below. Given some input that goes into a filter with impulse response and then another filter with impulse response , the output is given by . Assume that the output of the first filter is given by , this means that: Further, that intermediate function is then convolved with the impulse response of the second filter, and thus the output can be represented by: Using the associative property, this can be rewritten as follows: meaning that the equivalent impulse response for a cascaded system is given by: A similar analysis can be done on a set of parallel systems illustrated below. In this case, it is clear that: Using the distributive law, it is demonstrated that: This means that in the case of a parallel system, the equivalent impulse response is provided by: The equivalent impulse responses in both cascaded systems and parallel systems can be generalized to systems with -number of filters.