(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...)
 
 
(8 intermediate revisions by the same user not shown)
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Pick a note frequency <math>f_0=392Hz</math>
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*<span style="color:green"> You are welcome to add comments and questions here!</span> -[[User:zhao148|Zhao]]
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Pick a note frequency <span class="texhtml">''f''<sub>0</sub> = 392''H''''z''</span>  
  
 
{|
 
{|
 
|-
 
|-
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| <math>x(t)=cos(2\pi f_0t)=cos(2\pi *392t)</math>
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| <span class="texhtml">''x''(''t'') = ''c''''o''''s''(2π''f''<sub>0</sub>''t'') = ''c''''o''''s''(2π * 392''t'')</span>
 
|-
 
|-
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| <math>when\ sample\ period\ T_1=\frac{1}{1000}</math>
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| <math>a.\ Assign\ sampling\ period\ T_1=\frac{1}{1000}</math>
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|-
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| <math>2f_0<\frac{1}{T_1}, \ No\ aliasing\ occurs.</math>  
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|-
 
|-
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| <math>2f_0<\frac{1}{T_1}, \ No\ aliasing\ occurs.</math>
 
|}
 
|}
 
<div align="left" style="padding-left: 0em;">
 
<div align="left" style="padding-left: 0em;">
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x_1(n) &=x(nT_1)=cos(2\pi *392nT_1)=cos(2\pi *\frac{392}{1000}n) \\
 
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) \\
 
&=\frac{1}{2}\left( e^{-j2\pi *\frac{392}{1000}n} + e^{j2\pi *\frac{392}{1000}n} \right) \\
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\end{align}
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\end{align}</math>  
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</math>
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</div>  
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</div>
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{|
 
{|
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| <math>0<2\pi *\frac{392}{1000}<\pi</math>  
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|-
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| <math>0<2\pi *\frac{392}{1000}<\pi</math>
 
|-
 
|-
 
| <math>-\pi<-2\pi *\frac{392}{1000}<0</math>
 
| <math>-\pi<-2\pi *\frac{392}{1000}<0</math>
 
|}
 
|}
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<div align="left" style="padding-left: 0em;">
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<math>
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\begin{align}
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\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] \\
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&=\pi \left[\delta (\omega -2\pi *\frac{392}{1000}) + \delta (\omega + 2\pi *\frac{392}{1000})\right] \\
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\end{align}</math>
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</div>
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[[Image:Xw1 singleperiod.jpg]]
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{|
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|-
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| <math>for\ all\ \omega</math>
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|-
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| <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>
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|}
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[[Image:Xw1 multiperiod.jpg]]
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 +
{|
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|-
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| In this situation, no aliasing occurs. In the interval of <span class="texhtml">[ − π,π]</span>, which represents one period, the frequcy spectrum remains the same as Fig a-1.
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|}
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{|
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|-
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| <math>b.\ Assign\ sampling\ period\ T_2=\frac{1}{500}</math>
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|-
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| <math>2f_0>\frac{1}{T_2}, \ Aliasing\ occurs.</math>
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|}
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<div align="left" style="padding-left: 0em;">
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<math>
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\begin{align}
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x_2(n) &=x(nT_2)=cos(2\pi *392nT_2)=cos(2\pi *\frac{392}{500}n) \\
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&=\frac{1}{2}\left( e^{-j2\pi *\frac{392}{500}n} + e^{j2\pi *\frac{392}{500}n} \right) \\
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\end{align}</math>
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</div>
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{|
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|-
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| <math>\pi<2\pi *\frac{392}{500}<2\pi</math>
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|-
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| <math>-2\pi<-2\pi *\frac{392}{500}<\pi</math>
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|-
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| <math>\mathcal{X}_2(\omega)=\pi \left[\delta (\omega -2\pi *\frac{392}{500}) + \delta (\omega + 2\pi *\frac{392}{500})\right] </math>
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|-
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| <math>X_2(f)=\frac{1}{2}\left[\delta (f -\frac{392}{500}) + \delta (f + \frac{392}{500})\right]</math>
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|}
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[[Image:Xw2 singleperiod.jpg]]
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 +
{|
 +
|-
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| <math>for\ all\ \omega</math>
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|-
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| <math>\mathcal{X}_2(\omega)=\pi* rep_{2\pi} \left[\delta (\omega -2\pi *\frac{392}{500}) + \delta (\omega + 2\pi *\frac{392}{500})\right]</math>
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|-
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| <math>X_2(f)=\frac{1}{2}rep_2\left[\delta (f -\frac{392}{500}) + \delta (f + \frac{392}{500})\right]</math>
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|}
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[[Image:Xw2 multiperiod.jpg]]
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{|
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|-
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| In this situation, aliasing DO occurs. In the interval of <span class="texhtml">[ − π,π]</span>, which represents one period, the frequcy spectrum is different from Fig b-1.
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|}
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[[Image:Xf2 multiperiod.jpg]]
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----
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[[2010_Fall_ECE_438_Boutin|Back to 438 main page]]

Latest revision as of 11:27, 15 September 2010

  • You are welcome to add comments and questions here! -Zhao

Pick a note frequency f0 = 392H'z

x(t) = c'o's(2πf0t) = c'o's(2π * 392t)
$ a.\ Assign\ sampling\ 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} $

Xw1 singleperiod.jpg

$ 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] $

Xw1 multiperiod.jpg

In this situation, no aliasing occurs. In the interval of [ − π,π], which represents one period, the frequcy spectrum remains the same as Fig a-1.
$ b.\ Assign\ sampling\ 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 $
$ \mathcal{X}_2(\omega)=\pi \left[\delta (\omega -2\pi *\frac{392}{500}) + \delta (\omega + 2\pi *\frac{392}{500})\right] $
$ X_2(f)=\frac{1}{2}\left[\delta (f -\frac{392}{500}) + \delta (f + \frac{392}{500})\right] $

Xw2 singleperiod.jpg

$ for\ all\ \omega $
$ \mathcal{X}_2(\omega)=\pi* rep_{2\pi} \left[\delta (\omega -2\pi *\frac{392}{500}) + \delta (\omega + 2\pi *\frac{392}{500})\right] $
$ X_2(f)=\frac{1}{2}rep_2\left[\delta (f -\frac{392}{500}) + \delta (f + \frac{392}{500})\right] $

Xw2 multiperiod.jpg

In this situation, aliasing DO occurs. In the interval of [ − π,π], which represents one period, the frequcy spectrum is different from Fig b-1.

Xf2 multiperiod.jpg

Back to 438 main page

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