| Line 40: | Line 40: | ||
|<math>\int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt </math><br/> | |<math>\int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt </math><br/> | ||
<math>\int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du </math><br/> | <math>\int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du </math><br/> | ||
| − | <math>\int_{-\infty}^\infty g(u)e^{-i2\pi | + | <math>e^{-i2\pi fa}\int_{-\infty}^\infty g(u)e^{-i2\pi fu}du </math><br/> |
<math>e^{-i2\pi fa}\int_{-\infty}^\infty g(u)e^{-i2\pi fu}du <\math><br/> | <math>e^{-i2\pi fa}\int_{-\infty}^\infty g(u)e^{-i2\pi fu}du <\math><br/> | ||
<math>e^{-i2\pi fa}G(f)<\math><br/> | <math>e^{-i2\pi fa}G(f)<\math><br/> | ||
Revision as of 20:37, 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) $ | $ \int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt $ $ \int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du $ |
