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! Coefficients | ! Coefficients | ||
|- | |- | ||
| − | | | + | |
| − | | | + | |<math>sin(\omega_0t) </math> |
| − | | | + | |<math>\frac{1}{2j}e^{j\omega_0 t} - \frac{1}{2j}e^{-j\omega_0 t} </math> |
| − | |} | + | |<math>a_1 = \frac{1}{2j} a_{-1} = -\frac{1}{2j} and a_k = 0 for k not 1,-1 </math> |
| + | |- | ||
| + | |||
| + | |<math>cos(\omega_0t) </math> | ||
| + | |<math>\frac{1}{2}e^{j\omega_0 t} + \frac{1}{2}e^{-j\omega_0 t} </math> | ||
| + | |<math>a_1 = \frac{1}{2} a_{-1} = \frac{1}{2} and a_k = 0 for k not 1,-1 </math> | ||
| + | |- | ||
| + | |||
| + | |1 | ||
| + | |2\pi \delta[\omega] | ||
| + | | a_0 = 1 else a_k = 0 | ||
| + | |- | ||
| + | |||
| + | |<math>e^{\alpha t}u(t) </math> | ||
| + | |<math>\frac{1}{\alpha + j\omega} </math> | ||
| + | |<math> \frac{1}{\alpha T + j2\pi k} </math> | ||
| + | |- | ||
| + | } | ||
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<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/> | ||
| + | |- | ||
|Time Scaling | |Time Scaling | ||
| − | |<math>\mathfrak{F}(g(ct)) = | + | |<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. | ||
| + | |- | ||
| + | |||
| + | |Frequency Shifting | ||
| + | |<math>\mathfrak{F}^{-1}[X(j\omega + omega_0)] = x(t)e^{-j\omega_0t} </math> | ||
| + | |<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>=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}</math><br/> | ||
| + | |- | ||
| + | |||
| + | |Time Reversal | ||
| + | |<math>\mathfrak{F}[g(-t)] = G(-\omega) </math> | ||
| + | |<math>\mathfrak{F}[g(-t)] = \int_{-\infty}^{\infty}g(-t)e^{-j\omega t} dt</math><br/> | ||
| + | replace t with -t <br/> | ||
| + | <math>\mathfrak{F}[g(-t)] = -\int_{\infty}^{-\infty} g(t')e^{-j\omega t' | ||
| + | } dt' </math><br/> | ||
| + | <math> =\int_{-\infty}^{\infty}g(t') e^{-j\omega t'} dt' </math><br/> | ||
| + | <math> =G(-\omega) </math> | ||
| + | |- | ||
| + | |||
| + | |Complex Conjugate | ||
| + | |<math>\mathfrak{F}(g*(t) = G*(-j\omega) </math> | ||
| + | |<math> g*(t) = [\frac{1}{2\pi} \int_{-\infty}^{\infty} G(-\omega)e^{j\omega t} d\omega]^* </math><br/> | ||
| + | <math>=[\frac{1}{2\pi} \int_{-\infty}^{\infty} G*(-\omega)e^{j\omega' t} d\omega]^* </math><br/> | ||
| + | <math> x*(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} G*(-\omega')e^{j\omega' t} d\omega </math><br/> | ||
| + | <math> = \mathfrak{F}[G*(-\omega)]</math><br/> | ||
|- | |- | ||
} | } | ||
| + | |||
| + | ---- | ||
| + | |||
| + | [[2018_Spring_ECE_301_Boutin|Back to ECE301 Spring 2018 Prof. Boutin]] | ||
| + | |||
| + | --[[User:Plomada|Plomada]] 10:51 22 April 2018 (UTC) | ||
Latest revision as of 21:52, 22 April 2018
Table of CT Fourier Series Coefficients and Properties
Fourier series Coefficients
| Function | Fourier Series | Coefficients |
|---|---|---|
| $ sin(\omega_0t) $ | $ \frac{1}{2j}e^{j\omega_0 t} - \frac{1}{2j}e^{-j\omega_0 t} $ | $ a_1 = \frac{1}{2j} a_{-1} = -\frac{1}{2j} and a_k = 0 for k not 1,-1 $ |
| $ cos(\omega_0t) $ | $ \frac{1}{2}e^{j\omega_0 t} + \frac{1}{2}e^{-j\omega_0 t} $ | $ a_1 = \frac{1}{2} a_{-1} = \frac{1}{2} and a_k = 0 for k not 1,-1 $ |
| 1 | 2\pi \delta[\omega] | a_0 = 1 else a_k = 0 |
| $ e^{\alpha t}u(t) $ | $ \frac{1}{\alpha + j\omega} $ | $ \frac{1}{\alpha T + j2\pi k} $ |
| 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} $ |
| Time Reversal | $ \mathfrak{F}[g(-t)] = G(-\omega) $ | $ \mathfrak{F}[g(-t)] = \int_{-\infty}^{\infty}g(-t)e^{-j\omega t} dt $ replace t with -t |
| Complex Conjugate | $ \mathfrak{F}(g*(t) = G*(-j\omega) $ | $ g*(t) = [\frac{1}{2\pi} \int_{-\infty}^{\infty} G(-\omega)e^{j\omega t} d\omega]^* $ $ =[\frac{1}{2\pi} \int_{-\infty}^{\infty} G*(-\omega)e^{j\omega' t} d\omega]^* $ |
