Multidimensional signal processing: Difference between revisions

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Latest revision as of 19:51, 15 August 2020

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In [[signal processing]], '''multidimensional signal processing''' covers all signal processing done using multidimensional signals and systems. 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. In m-D digital signal processing, useful data is sampled in more than one dimension. Examples of this are [[image processing]] and multi-sensor radar detection. Both of these examples use multiple sensors to sample signals and form images based on the manipulation of these multiple signals.

Processing in multi-dimension (m-D) requires more complex algorithms, compared to the 1-D case, to handle calculations such as the [[Fastfast Fourier Transformtransform]] 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, if the considered systems are separable.

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"/> A multidimensional signal is similar to a single dimensional signal as far as manipulations that can be performed, such as [[Sampling (signal processing)|sampling]], [[Fourier analysis]], and [[Filter (signal processing)|filtering]]. The actual computations of these manipulations grow with the number of dimensions.

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== Fourier Analysis ==

{{main| Fourier Analysisanalysis| Multidimensional Transformtransform| Fast Fourier Transformtransform}}

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 signal ___domain representation to a [[frequency ___domain]] representation of the signal. In the case of digital processing, a discrete Fourier Transform (DFT) is utilized to transform a sampled signal ___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 2 \pi k_1 n_1} e^{-j 2 \pi k_2 n_2} \cdots e^{-j 2 \pi k_m n_m}</math>

where ''X'' stands for the multidimensional discrete Fourier transform, ''x'' stands for the sampled time/space ___domain signal, ''m'' stands for the number of dimensions in the system, ''n'' are sample indices 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 [[variable (mathematics)|variables]] of the multidimensional signal.<ref name="dudmer83_2"/> In general, [[Fastfast Fourier Transformstransform]]s (FFTs), 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 ==