Multidimensional signal processing: Difference between revisions

Revision as of 15:13, 28 October 2014

Revision as of 15:14, 28 October 2014

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In [[signal processing]], '''multidimensional signal processing''' covers all signal processing done using [[multidimensional sampling]]. While multidimensional signal processing is a subset of signal processing, it is unique in the sense that it deals specifically with data that can only be adequately detailed using more than one dimension. Examples of this are [[image processing]] and multi-sensor radar detection.

Multidimensional signals are part of [[multidimensional systems]], and as such are generally more complex than classical, single dimension signal processing. Processing in mmulti-Ddimension (multim-dimensionD) requires more complex algorithms to handle calculations such as the [[Fast Fourier Transform]] due to more degrees of freedom<ref name="dudmer83">D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, First Edition, pp. 2, 1983.</ref>. In some cases, m-D signals and systems can be simplified into single dimension signal processing methods, utilizing assumptions such as symmetry.

Typically, multidimensional signal processing is directly associated with digital signal processing because its complexity warrants the use of computer modelling and computation<ref name="dudmer83"/>.

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== Fourier Analysis ==

{{main| Fourier Analysis| Multidimensional Transform| Fast Fourier Transform}}

A multidimensional signal can be represented in terms of sinusoidal components. This is typically done with a type of [[Fourier transform]]. The Mm-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>.