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A multidimensional signal can be represented in terms of sinusoidal components. This is typically done with a type of [[Fourier transform]]. The M-D [[Fourier transform]] transforms a signal from a [[time ___domain]] representation to a [[frequency ___domain]] representation of the signal. In the case of digital processing, a [[discrete time Fourier transform]] is utilized to transform a sampled time ___domain representation into a frequency ___domain representation:

:<math> X(k_1,k_2,\dots,k_m) = \sum_{n_1=-\infty}^\infty \sum_{n_2=-\infty}^\infty \cdots \sum_{n_m=-\infty}^\infty x(n_1,n_2,\dots,n_m) e^{-j k_1 n_1} e^{-j k_2 n_2} \cdots e^{-j k_m n_m}</math>

where ''X'' stands for the multidimensional discrete Fourier transform, ''x'' stands for the sampled time ___domain signal, ''m'' stands for the number of dimensions in the system, ''n'' are time samples and ''k'' are frequency samples<ref name="dudmer83_2">D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, First Edition, pp. 61,112, 1983.</ref>.

Computational complexity is usually the main concern when implementing any Fourier transform. For multidimensional signals, the complexity can be reduced by a number of different methods. The computation may be simplified if there is [[independence]] between [[variables]] of the multidimensional signal<ref name="dudmer83_2"/>. In general, [[Fast Fourier Transforms]] (FFTs), utilize efficiencies of the system to reduce the number of computations by a substantial factor. While there are a number of different implementations of this [[algorithm]] for m-D signals, two often used variations are the vector-radix FFT and the row-column FFT.

== Filtering ==

{{main|Filter (signal processing)}}

[[File:2-D filter frequency response and 1-D filter prototype frequency response.gif|thumb|500px|right|A 2-D filter (left) defined by its 1-D prototype function (right) and a mapping function.]]

<gallery>

Sample 1_D.bmp|In the above, a 1-D prototype, band-pass filter is defined.

Sample 2-D.jpg|The above is a 2-D filter, defined by a 1-D prototype and a mapping function.

</gallery>

Filtering is an important part of any signal processing application. Similar to typical single dimension signal processing applications, there are varying degrees of complexity within filter design for a given system. M-D systems utilize [[digital filters]] in many different applications. The actual implementation of these m-D filters can pose a design problem depending on whether the multidimensional polynomial is factorable<ref name="dudmer83_2"/>. Typically, a [[prototype]] filter is designed in a single dimension and that filter is [[extrapolate|extrapolated]] to m-D using a [[map (mathematics)|mapping function]]<ref name="dudmer83_2"/>. Both [[FIR]] and [[IIR]] filters can be utilized in this manner, depending on the application.

== Applicable Fields ==

* [[Image processing]]

* [[Towed array sonar]]

* [[X-ray computed tomography]]

== References ==

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{{Reflist}}

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