page_title: Multidimensional discrete convolution
section_title: Pictorial Method of Operation
hierarchical_section_title: Multidimensional discrete convolution / Overlap and Add / Pictorial Method of Operation
caption_attribution_description: English: decomposed input
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: In order to visualize the overlap-add method more clearly, the following illustrations examine the method graphically. Assume that the input  has a square region support of length N in both vertical and horizontal directions as shown in the figure below. It is then broken up into four smaller segments in such a way that it is now composed of four smaller squares. Each block of the aggregate signal has dimensions   . 
  Then, each component is convolved with the impulse response of the filter. Note that an advantage for an implementation such as this can be visualized here since each of these convolutions can be parallelized on a computer, as long as the computer has sufficient memory and resources to store and compute simultaneously.
  In the figure below, the first graph on the left represents the convolution corresponding to the component of the input  with the corresponding impulse response . To the right of that, the input  is then convolved with the impulse response .
  The same process is done for the other two inputs respectively, and they are accumulated together in order to form the convolution. This is depicted to the left.
  Assume that the filter impulse response   has a region of support  of  in both dimensions. This entails that each convolution convolves signals with dimensions    in both  and  directions, which leads to overlap (highlighted in blue) since the length of each individual convolution is equivalent to:
      = 
  in both directions. The lighter blue portion correlates to the overlap between two adjacent convolutions, whereas the darker blue portion correlates to overlap between all four convolutions. All of these overlap portions are added together in addition to the convolutions in order to form the combined convolution .
caption: Decomposed Input Signal
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