(New page: DT Fourier Transform Properties * DT Fourier Transform Multiplication x[n]y[n]\longleftrightarrow \frac{1}{2\pi} \int_{2\pi} X(e^{j\theta})Y(e^{j(\omega-\theta)})d\theta * DT Four...) |
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| − | + | CT Fourier Transform Properties | |
| − | * | + | * CT_Fourier_Int/Diff\; \; \; (1)\frac{dx(t)}{dt} \rightarrow j\omega \Chi (\omega)\; \; \; \; \; \; (2) \int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{1}{j\omega}\Chi (\omega) + \pi \Chi (0) \delta (\omega) |
| − | * | + | |
| − | * | + | * CT Time and Frequency Scaling : x(at) \leftarrow \rightarrow \frac{1}{|a|}X(\frac{j\omega }{a})\, |
| − | * | + | |
| + | * CT Differentiation in Frequencyx(t)\rightarrow j\frac{d}{d\omega}X(j\omega) | ||
| + | |||
| + | * CT Convolution: F(x_1(t)*x_2(t)) = X_1(\omega)X_2(\omega) \! | ||
| + | |||
| + | * CT Frequency Shifting : F(e^{jw0t}x(t)) = X(j(w - w0)) \! | ||
| + | |||
| + | F(x(t)y(t))=\frac{1}{2\pi}X(j\omega)*Y(j\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\theta)Y(j(\omega-\theta))d\theta | ||
| + | |||
| + | * CT Time Reversal(-t) \leftarrow \rightarrow X(-j\omega )\, | ||
| + | |||
| + | * CT Multiplication Property Mimis VersionF(x_1(t)x_2(t)) = \frac {1} {2\pi} X_1(\omega)*X_2(\omega) | ||
| + | |||
| + | * CT Duality Property : F(x(t)) = X(w) = 2\pi x(-w) \! | ||
| + | |||
| + | F(x(t)) = X(w) = 2\pi x(-w) \! | ||
| + | |||
| + | * CT Conjugate Symmetry ==Conjugate Symmetry== | ||
| + | |||
| + | if | ||
| + | |||
| + | \ F(x(t)) = X(w) | ||
| + | |||
| + | then, | ||
| + | |||
| + | \ F(x(t)^*) = X^*(-w) | ||
Latest revision as of 04:40, 23 July 2009
CT Fourier Transform Properties
* CT_Fourier_Int/Diff\; \; \; (1)\frac{dx(t)}{dt} \rightarrow j\omega \Chi (\omega)\; \; \; \; \; \; (2) \int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{1}{j\omega}\Chi (\omega) + \pi \Chi (0) \delta (\omega)
* CT Time and Frequency Scaling : x(at) \leftarrow \rightarrow \frac{1}{|a|}X(\frac{j\omega }{a})\,
* CT Differentiation in Frequencyx(t)\rightarrow j\frac{d}{d\omega}X(j\omega)
* CT Convolution: F(x_1(t)*x_2(t)) = X_1(\omega)X_2(\omega) \!
* CT Frequency Shifting : F(e^{jw0t}x(t)) = X(j(w - w0)) \!
F(x(t)y(t))=\frac{1}{2\pi}X(j\omega)*Y(j\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\theta)Y(j(\omega-\theta))d\theta
* CT Time Reversal(-t) \leftarrow \rightarrow X(-j\omega )\,
* CT Multiplication Property Mimis VersionF(x_1(t)x_2(t)) = \frac {1} {2\pi} X_1(\omega)*X_2(\omega)
* CT Duality Property : F(x(t)) = X(w) = 2\pi x(-w) \!
F(x(t)) = X(w) = 2\pi x(-w) \!
* CT Conjugate Symmetry ==Conjugate Symmetry==
if
\ F(x(t)) = X(w)
then,
\ F(x(t)^*) = X^*(-w)
