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====Differentiation====
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=== MD FFT ===
A [[fast Fourier transform]] (FFT) is an algorithm to compute the discrete Fourier transform (DFT) and its inverse. An FFT computes the DFT and produces exactly the same result as evaluating the DFT definition directly; the only difference is that an FFT is much faster. (In the presence of [[round-off error]], many FFT algorithms are also much more accurate than evaluating the DFT definition directly).There are many different FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory. See more in [[Fast Fourier transform|FFT]].
=== MD DFT ===
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== Multidimensional Laplace transform ==
The multidimensional [[Laplace transform]] is useful for the solution of boundary value problems. Boundary value problems in two or more variables characterized by partial differential equations can be solved by a direct use of the Laplace transform.<ref name=":0">{{Cite journal|title = Theorems on multidimensional laplace transform for solution of boundary value problems|journal = Computers & Mathematics with Applications|date = 1989-01-01|pages = 1033–1056|volume = 18|issue = 12|doi = 10.1016/0898-1221(89)90031-X|first1 = Joyati|last1 = Debnath|first2 = R. S.|last2 = Dahiya|doi-access = free}}</ref> The Laplace transform for an M-dimensional case is defined<ref name=":0"/> as
<math> F(s_1,s_2,\ldots,s_n) = \int_{0}^{\infty} \cdots \int_{0}^{\infty}
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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 ==
Source:<ref>{{Cite web|url = http://dsp-book.narod.ru/HFTSP/8579ch08.pdf|title = Narod Book}}</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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<math> F(z_1,z_2)= \sum_{n_1=-\infty}^{\infty} \sum_{n_2=-\infty}^{\infty} f(n_1,n_2) z_1^{-n_1} z_2^{-n_2} </math>
The Fourier transform is a special case of the Z transform evaluated along the [[unit circle]] (in 1D) and unit bi-circle (in 2D). i.e. at
<math display="inline"> z=e^{iw} </math> where z and w are vectors.
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If a sequence has a support as shown in Figure 1.1a, then its ROC is shown in Figure 1.1b. This follows that |''F''(''z''<sub>1</sub>,''z''<sub>2</sub>)| < '''∞''' .
<math>(z_{01},z_{02})</math> lies in the ROC, then all points<math>(z_1,z_2)</math>that satisfy |z1|≥|z01| and |z2|≥|z02| lie in the ROC.
Therefore, for figure 1.1a and 1.1b, the ROC would be
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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 ===
[[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.
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