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|<math>\mathfrak{F}^{-1}[X(j\omega + omega_0)] = \frac{1}{2\pi} \int_{-\infty}^{\infty}X(j(\omega +\omega_0))e^{j\omega t} d\omega </math><br/> | |<math>\mathfrak{F}^{-1}[X(j\omega + omega_0)] = \frac{1}{2\pi} \int_{-\infty}^{\infty}X(j(\omega +\omega_0))e^{j\omega t} d\omega </math><br/> | ||
<math>=\frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega ')e^{j\omega (\omega ' + \omega_0)} d\omega </math><br/> | <math>=\frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega ')e^{j\omega (\omega ' + \omega_0)} d\omega </math><br/> | ||
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+ | <math>=e^{j\omega_0 t}\frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega ')e^{j\omega '} d\omega' </math><br/> | ||
+ | <math>=x(t)e^{j\omega_0 t} | ||
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Revision as of 22:07, 22 April 2018
Table of CT Fourier Series Coefficients and Properties
Fourier series Coefficients
Function | Fourier Series | Coefficients |
---|---|---|
Properties of CT Fourier systems
Property Name | Property | Proof |
---|---|---|
Linearity | $ \mathfrak{F}(c_1g(t) + c_2h(t) = c_1G(f) + c_2H(f) $ | $ \mathfrak{F}(c_1g(t) + c_2h(t) = \int_{-\infty}^\infty c_1g(t) dt + \int_{-\infty}^\infty c_2h(t) dt $ $ =c_1\int_{-\infty}^\infty g(t)e^{i2\pi ft} dt + c_2 \int_{-\infty}^\infty g(t)e^{i2\pi ft} dt $ |
Time Shifting | $ \mathfrak{F}(g(t - a)) = e^{-i2\pi fa}*G(f) $ | $ \mathfrak{F}(g(t - a)) = \int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt $ $ =\int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du $ |
Time Scaling | $ \mathfrak{F}(g(ct)) = \frac{G(\frac{f}{c})}{|c|} $ |
$ \mathfrak{F}(g(ct)) = \int_{-\infty}^\infty g(ct)e^{-i2\pi ft}dt $ subtitute : u = ct, du = cdt |
Frequency Shifting | $ \mathfrak{F}^{-1}[X(j\omega + omega_0)] = x(t)e^{-j\omega_0t} $ | $ \mathfrak{F}^{-1}[X(j\omega + omega_0)] = \frac{1}{2\pi} \int_{-\infty}^{\infty}X(j(\omega +\omega_0))e^{j\omega t} d\omega $ $ =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega ')e^{j\omega (\omega ' + \omega_0)} d\omega $ $ =e^{j\omega_0 t}\frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega ')e^{j\omega '} d\omega' $ $ =x(t)e^{j\omega_0 t} |- | | | |- } $ |