Line 45: | Line 45: | ||
|Time Scaling | |Time Scaling | ||
− | |<math>\mathfrak{F}(g(ct)) = \frac{G(\frac{f}{c}}{|c|} </math><br/> | + | |<math>\mathfrak{F}(g(ct)) = \frac{G(\frac{f}{c})}{|c|} </math><br/> |
|<math>\mathfrak{F}(g(ct)) = \int_{-\infty}^\infty g(ct)e^{-i2\pi ft}dt </math><br/> | |<math>\mathfrak{F}(g(ct)) = \int_{-\infty}^\infty g(ct)e^{-i2\pi ft}dt </math><br/> | ||
+ | subtitute : u = ct, du = cdt <br/> | ||
+ | <math> \mathfrak{F}(g(ct)) = \int_{-c\infty}^{c\infty} \frac{g(u)}{c}e^{-i2\pi f\frac{u}{c}}du </math><br/> | ||
+ | if c is greater than 0: then no signs change. | ||
+ | if c is less than 0: the integration must be flipped as well as the negative from the c so you still get the same equation. therefore the absolute value of c is obtained. | ||
|- | |- | ||
} | } |
Revision as of 21:49, 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 |