(New page: Pick a note frequency <math>f_0=392Hz</math> {| |- | <math>x(t)=cos(2\pi f_0t)=cos(2\pi *392t)</math> |- | <math>when\ sample\ period\ T_1=\frac{1}{1000}</math> |- | <math>2f_0<\frac{1}{T...) |
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| <math>-\pi<-2\pi *\frac{392}{1000}<0</math> | | <math>-\pi<-2\pi *\frac{392}{1000}<0</math> | ||
| + | |} | ||
| + | <div align="left" style="padding-left: 0em;"> | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | \mathcal{X}_1(\omega) &=2\pi *\frac{1}{2} \left[\delta (\omega -2\pi *\frac{392}{1000}) + \delta (\omega + 2\pi *\frac{392}{1000})\right] \\ | ||
| + | &=\pi \left[\delta (\omega -2\pi *\frac{392}{1000}) + \delta (\omega + 2\pi *\frac{392}{1000})\right] \\ | ||
| + | \end{align} | ||
| + | </math> | ||
| + | </div> | ||
| + | |||
| + | graph x1w | ||
| + | |||
| + | {| | ||
| + | |- | ||
| + | | <math>for\ all\ \omega</math> | ||
| + | |- | ||
| + | | <math>\mathcal{X}_1(\omega)=\pi* rep_{2\pi} \left[\delta (\omega -2\pi *\frac{392}{1000}) + \delta (\omega + 2\pi *\frac{392}{1000})\right]</math> | ||
| + | |} | ||
| + | |||
| + | graph x1w_all | ||
| + | |||
| + | {| | ||
| + | |- | ||
| + | | <math>when\ sample\ period\ T_2=\frac{1}{500}</math> | ||
| + | |- | ||
| + | | <math>2f_0>\frac{1}{T_2}, \ Aliasing\ occurs.</math> | ||
| + | |- | ||
| + | |} | ||
| + | <div align="left" style="padding-left: 0em;"> | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | x_2(n) &=x(nT_2)=cos(2\pi *392nT_2)=cos(2\pi *\frac{392}{500}n) \\ | ||
| + | &=\frac{1}{2}\left( e^{-j2\pi *\frac{392}{500}n} + e^{j2\pi *\frac{392}{500}n} \right) \\ | ||
| + | \end{align} | ||
| + | </math> | ||
| + | </div> | ||
| + | {| | ||
| + | | <math>\pi<2\pi *\frac{392}{500}<2\pi</math> | ||
| + | |- | ||
| + | | <math>-2\pi<-2\pi *\frac{392}{500}<\pi</math> | ||
|} | |} | ||
Revision as of 20:04, 10 September 2010
Pick a note frequency $ f_0=392Hz $
| $ x(t)=cos(2\pi f_0t)=cos(2\pi *392t) $ |
| $ when\ sample\ period\ T_1=\frac{1}{1000} $ |
| $ 2f_0<\frac{1}{T_1}, \ No\ aliasing\ occurs. $ |
$ \begin{align} x_1(n) &=x(nT_1)=cos(2\pi *392nT_1)=cos(2\pi *\frac{392}{1000}n) \\ &=\frac{1}{2}\left( e^{-j2\pi *\frac{392}{1000}n} + e^{j2\pi *\frac{392}{1000}n} \right) \\ \end{align} $
| $ 0<2\pi *\frac{392}{1000}<\pi $ |
| $ -\pi<-2\pi *\frac{392}{1000}<0 $ |
$ \begin{align} \mathcal{X}_1(\omega) &=2\pi *\frac{1}{2} \left[\delta (\omega -2\pi *\frac{392}{1000}) + \delta (\omega + 2\pi *\frac{392}{1000})\right] \\ &=\pi \left[\delta (\omega -2\pi *\frac{392}{1000}) + \delta (\omega + 2\pi *\frac{392}{1000})\right] \\ \end{align} $
graph x1w
| $ for\ all\ \omega $ |
| $ \mathcal{X}_1(\omega)=\pi* rep_{2\pi} \left[\delta (\omega -2\pi *\frac{392}{1000}) + \delta (\omega + 2\pi *\frac{392}{1000})\right] $ |
graph x1w_all
| $ when\ sample\ period\ T_2=\frac{1}{500} $ |
| $ 2f_0>\frac{1}{T_2}, \ Aliasing\ occurs. $ |
$ \begin{align} x_2(n) &=x(nT_2)=cos(2\pi *392nT_2)=cos(2\pi *\frac{392}{500}n) \\ &=\frac{1}{2}\left( e^{-j2\pi *\frac{392}{500}n} + e^{j2\pi *\frac{392}{500}n} \right) \\ \end{align} $
| $ \pi<2\pi *\frac{392}{500}<2\pi $ |
| $ -2\pi<-2\pi *\frac{392}{500}<\pi $ |
