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Dragon Notes

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  UNDER CONSTRUCTION

Fourier Transform Properties


Property\(x(t)\)\(\bn{X}(\omega)\)
\([1]\ \t{Linearity}\) \(\ds K_1x_1(t)+K_2x_2(t)\) \(\ds K_1\bn{X}_1(\omega)+K_2\bn{X}_2(\omega)\)
\([2]\ \t{Time scaling}\) \(\ds x(at)\) \(\ds \frac{1}{\abs{a}}\bn{X}\Frac{\omega}{a}\)
\([3]\ \t{Time shift}\) \(\ds x(t-t_0)\) \(\ds e^{-j\omega t_0}\bn{X}(\omega)\)
\([4]\ \t{Frequency shift}\) \(\ds e^{j\omega_0 t}x(t)\) \(\ds \bn{X}(\omega - \omega_0)\)
\([5]\ \t{Time 1st derivative}\) \(\ds x'=\frac{dx}{dt}\) \(\ds j\omega \bn{X}(\omega)\)
\([6]\ \t{Time \(n\)th derivative}\) \(\ds \frac{d^nx}{dt^n}\) \(\ds (j\omega)^n \bn{X}(\omega)\)
\([7]\ \t{Time integral}\) \(\ds \s{}\int_{-\infty}^{t}\ns{}x(\tau)\ d\tau\)\(\ds \frac{\bn{X}(\omega)}{j\omega}+\pi\bn{X}(0)\delta(\omega),\ \bn{X}(0)=\s{}\pnint \ns{}x(t)\hspace{3px}dt\)
\([8]\ \t{Frequency derivative}\) \(\ds t^n x(t)\) \(\ds (j)^n\frac{d^n\bn{X}(\omega)}{d\omega^n}\)
\([9]\ \t{Modulation}\) \(\ds x(t)\Cos{\omega_0 t}\) \(\ds \sfrac{1}{2}[\bn{X}(\omega-\omega_0)+\bn{X}(\omega+\omega_0)]\)
\([10]\ \t{Convolution in \(t\)}\) \(\ds x_1(t)*x_2(t)\) \(\ds \bn{X}_1(\omega)\bn{X}_2(\omega)\)
\([11]\ \t{Convolution in \(\omega\)}\)\(\ds x_1(t)\ x_2(t)\) \(\ds \frac{1}{2\pi}\bn{X}_1(\omega)*\bn{X}_2(\omega)\)
\([12]\ \t{Conjugate Symmetry}\) \(\ds \bn{X}(-\omega)=\bn{X}^*(\omega)\)




Dragon Notes,   Est. 2018     About

By OverLordGoldDragon