Multidimensional transform: Difference between revisions

Revision as of 03:06, 15 November 2016

Revision as of 21:31, 15 November 2016

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====[[Multidimensional modulation|Modulation]]====

if <math>x(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1,...\ldots,\omega_M)</math>, then<br />

: <math>e^{j(\theta_1 n_1 +,...,\cdots+ \theta_M n_M)} x(n_1 - a_1,...\ldots,n_M - a_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1 - \theta_1,...\ldots,\omega_M - \theta_M)</math>

====Multiplication====

if <math>x_1(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X_1(\omega_1,...\ldots,\omega_M)</math>, and <math>x_2(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X_2 (\omega_1,...\ldots,\omega_M)</math><br />

then,<br />

{{NumBlk|:|<math> x_1(n_1,...,n_M) x_2(n_1,...,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} \frac{1}{(2\pi)^M} \int\limits_{-\pi}^{\pi} ... \int\limits_{-\pi}^{\pi}X_1(\omega_1 - \theta_1,...,\omega_M - \theta_M) X_2(\theta_1,...,\theta_M)d\theta_1...d\theta_M</math>|{{EquationRef|MD Convolution in Frequency Domain}}}}

: {{NumBlk|:|<math> x_1(n_1,...\ldots,n_M) x_2(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} \frac{1}{(2\pi)^M} \int \limits_{-\pi}^{\pi} ...\cdots \int\limits_{-\pi}^{\pi} X_1(\omega_1 - \theta_1,...\ldots,\omega_M - \theta_M) X_2 (\theta_1,... \ldots, \theta_M) \, d\theta_1... \cdots d\theta_M</math>|{{EquationRef|MD Convolution in Frequency Domain}}}}

or,<br />

: {{NumBlk|:|<math> x_1(n_1,...\ldots,n_M) x_2(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} \frac{1}{(2\pi)^M} \int\limits_{-\pi}^{\pi} ...\cdots \int\limits_{-\pi}^{\pi} X_2(\omega_1 - \theta_1,...\ldots,\omega_M - \theta_M) X_1(\theta_1,...\ldots,\theta_M) \, d\theta_1...\cdots d\theta_M</math>|{{EquationRef|MD Convolution in Frequency Domain}}}}

====Differentiation====

ifIf <math>x(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1,...\ldots,\omega_M)</math>, then

: <math>-jn_1x(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} \frac{\delta}{(\delta\omega_1)} X(\omega_1,...\ldots,\omega_M), </math>,<br />

: <math>-jn_2x(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} \frac{\delta}{(\delta\omega_2)} X(\omega_1,...\ldots,\omega_M), </math>,<br />

: <math>(-j)^M(n_1n_2...\cdots n_M)x(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} \frac{(\delta)^M}{(\delta\omega_1\delta\omega_2...\cdots\delta\omega_M)} X(\omega_1,...\ldots,\omega_M),</math>,

====Transposition====

ifIf <math>x(n_1,...\ldots,n_M) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X(\omega_1,...\ldots,\omega_M)</math>, then

: <math>x(n_M,...\ldots,n_1) \overset{\underset{\mathrm{FT}}{}}{\longleftrightarrow} X (\omega_M,...\ldots,\omega_1)</math>

====Reflection====

ifIf <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(\pm \omega_1,...\ldots,\pm \omega_M)</math>

====Complex conjugation====

ifIf <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^{*}(-\omega_1,...\ldots,-\omega_M)</math>

====Parseval's theorem (MD)====

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* [[Multidimensional discrete convolution]]

* [[2D Z-transform]]

* [[Multidimensional Empiricalempirical Modemode Decompositiondecomposition]]

* [[Multidimensional Signalsignal Reconstructionreconstruction]]

== References ==