Content deleted Content added
→Properties of Fourier transform: missing subscripts |
Link suggestions feature: 3 links added. |
||
| (14 intermediate revisions by 13 users not shown) | |||
Line 1:
{{Short description|Mathematical analysis of frequency content of signals}}
In [[mathematical analysis]] and applications, '''multidimensional transforms''' are used to analyze the frequency content of signals in a ___domain of two or more dimensions.
Line 28 ⟶ 29:
if <math>x(n_1,...,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1,...,\omega_M)</math>, then<br />
<math>x(n_1 - a_1,...,n_M - a_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} e^{-
====[[Multidimensional modulation|Modulation]]====
Line 34 ⟶ 35:
if <math>x(n_1,\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1,\ldots,\omega_M)</math>, then
: <math>e^{
====Multiplication====
Line 42 ⟶ 43:
then,
or,
====Differentiation====
Line 51 ⟶ 52:
If <math>x(n_1,\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1,\ldots,\omega_M)</math>, then
: <math>-
: <math>-
: <math>(-
====Transposition====
Line 71 ⟶ 72:
If <math>x(n_1,\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1,\ldots,\omega_M)</math>, then
: <math>x^{*}(\pm n_1,\ldots,\pm n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X^{*}(
====Parseval's theorem (MD)====
Line 83 ⟶ 84:
<math>\sum_{n_1=-\infty}^\infty ... \sum_{n_M =-\infty}^\infty |x_1 (n_1,...,n_M)|^2 {=} \frac{1}{(2\pi)^M} \int\limits_{-\pi}^{\pi} ... \int\limits_{-\pi}^{\pi}|X_1(\omega_1,...,\omega_M)|^2 d\omega_1...d\omega_M</math>
A special case of the [[Parseval's theorem]] is when the two multi-dimensional signals are the same. In this case, the theorem portrays the energy conservation of the signal and the term in the summation or integral is the energy-density of the signal.
====Separability====
▲One property is the separability property. A signal or system is said to be separable if it can be expressed as a product of 1-D functions with different independent variables. This phenomenon allows computing the FT transform as a product of 1-D FTs instead of multi-dimensional FT.
if <math>x(n_1,...,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1,...,\omega_M)</math>, <math>a(n_1) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} A(\omega_1)</math>,
Line 98:
=== 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 ===
Line 123:
== 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|
<math> F(s_1,s_2,\ldots,s_n) = \int_{0}^{\infty} \cdots \int_{0}^{\infty}
Line 136:
<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]]
Line 147 ⟶ 149:
<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^{
=== Region of convergence ===
Line 159 ⟶ 161:
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
Line 177 ⟶ 179:
[[File:Dctjpeg.png|thumb|250px|Two-dimensional DCT frequencies from the [[JPEG#Discrete cosine transform|JPEG DCT]]]]
The DCT is used in [[JPEG]] image compression, [[MJPEG]], [[MPEG]], [[DV (video format)|DV]], [[Daala]], and [[Theora]] [[video compression]]. There, the two-dimensional DCT-II of ''N''x''N'' blocks are computed and the results are [[Quantization (signal processing)|quantized]] and [[Entropy encoding|entropy coded]]. In this case, ''N'' is typically 8 and the DCT-II formula is applied to each row and column of the block. The result is an 8x8 transform coefficient array in which the: (0,0) element (top-left) is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies, as shown in the picture on the right.
In image processing, one can also analyze and describe unconventional cryptographic methods based on 2D DCTs, for inserting non-visible binary watermarks into the 2D image plane,<ref>Peter KULLAI, Pavol SABAKAI, JozefHUSKAI. Simple Possibilities of 2D DCT Application in Digital
Line 221 ⟶ 223:
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 ===
Source:<ref>{{Cite book|chapter-url = [[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.
The
It is observed that a high accuracy and significant speedup can be achieved for simulating large nonlinear circuits using multidimensional Laplace transforms.
| |||