page_title: Multidimensional discrete convolution
section_title: Problem Statement & Basics
hierarchical_section_title: Multidimensional discrete convolution / Definition / Problem Statement & Basics
caption_attribution_description: English: Simple Visualization of two-dimensional Convolution
context_page_description: 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.
context_section_description: 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.
caption: Visualization of Convolution between two Simple Two-Dimensional Signals
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