Multidimensional transform: Difference between revisions

Revision as of 22:08, 19 September 2020

Revision as of 18:50, 11 December 2020

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where F stands for the s-___domain representation of the signal f(t).

A special case (along 2 dimensions) of the multi-dimensional Laplace transform of function f(x,y) is defined<ref>{{Cite book|title = Operational Calculus in two Variables and its Application (1st English edition) - translated by D.M.G. Wishart (Calcul opérationnel)|last = |first = |publisher = |year = |isbn = |___location = |pages = }}</ref> as

<math display="block">F(s_1,s_2)= \int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\ f(x,y) e^{-s_1x-s_2y}\, dxdy</math>

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<math> F(s_1,s_2) </math> is called the image of <math> f(x,y) </math> and <math> f(x,y) </math> is known as the original of <math> F(s_1,s_2) </math>.{{cn|date=November 2019}} This special case can be used to solve the [[Telegrapher's equations]].{{cn|date=November 2019}}}

== Multidimensional Z transform<ref>{{Cite web|url = http://dsp-book.narod.ru/HFTSP/8579ch08.pdf|title = Narod Book|date = |accessdate = |website = |publisher = |last = |first = }}</ref> ==

The multidimensional Z transform is used to map the discrete time ___domain multidimensional signal to the Z ___domain. This can be used to check the stability of filters. The equation of the multidimensional Z transform is given by

[[File:Figure 1.1a depicting region of support.png|thumb|209x209px|Figure 1.1a]]

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Laplace transforms are used to solve partial differential equations. The general theory for obtaining solutions in this technique is developed by theorems on Laplace transform in n dimensions.<ref name=":0" />

The multidimensional Z transform can also be used to solve partial differential equations.<ref>{{Cite journal|url = http://dml.cz/bitstream/handle/10338.dmlcz/124265/Kybernetika_24-1988-7_1.pdf|title = Kybernetika|last = Gregor|first = Jiří |date= 1998 |journal= Kybernetika |volume=24 |access-date = }}</ref>

=== Image processing for arts surface analysis by FFT ===

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museums without affecting their daily use. But this method doesn’t allow a quantitative measure of the corrosion rate.

=== Application to weakly nonlinear circuit simulation<ref>{{Cite book|chapter-url = http://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&queryText=Weakly%20Nonlinear%20Circuit%20Analysis%20Based%20on%20Fast%20Multidimensional%20Inverse%20Laplace%20Transform|chapter = Weakly Nonlinear Circuit Analysis Based on Fast Multidimensional Inverse Laplace Transform|last = Wang|first = Tingting|date = 2012|pages = 547–552|doi = 10.1109/ASPDAC.2012.6165013|pmid = |access-date = |isbn = 978-1-4673-0772-7|title = 17th Asia and South Pacific Design Automation Conference}}</ref> ===

[[File:A weakly circuit.PNG|thumb|330x330px|An example of a weakly nonlinear circuit]]

The inverse multidimensional Laplace transform can be applied to simulate nonlinear circuits. This is done so by formulating a circuit as a state-space and expanding the Inverse Laplace Transform based on [[Laguerre function]] expansion.