User:Tjhuston225/Multidimensional Signal Processing: Difference between revisions

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:<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, 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. In general, [[Fast Fourier TransformTransforms]] (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 ==