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[[Category:2013 Fall ECE 438 Boutin]][[Category:2013 Fall ECE 438 Boutin]][[Category:2013 Fall ECE 438 Boutin]][[Category:2013 Fall ECE 438 Boutin]][[Category:2013 Fall ECE 438 Boutin]]
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[[Category:HW10ECE438F13]][[Category:HW10ECE438F13]][[Category:HW10ECE438F13]][[Category:HW10ECE438F13]][[Category:HW10ECE438F13]]
  
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=Hw10_ECE438F13sln=
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== Question 1 ==
  
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(a)
  
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<table style="color:#000">
  
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Put your content here . . .
 
  
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<tr>
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<td>
  
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<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
  
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<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
  
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[[ 2013 Fall ECE 438 Boutin|Back to 2013 Fall ECE 438 Boutin]]
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<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
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<tr>
 +
 
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<td align="center" valign="center" width=30><math> \begin{align}  U_g \\\end{align}</math></td>
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</tr>
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<tr>
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<td align="center" valign="center" width=30><math> \begin{align}  V_g \\\end{align}</math> </td>
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</tr>
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</table></td>
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<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
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<td></td>
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<td> =  </td>
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<td></td>
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<td><math>\begin{align} \frac{1}{1+r_0} \\\end{align} </math></td>
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<td>
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<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
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<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
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<tr>
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<td align="center" valign="center" width=30><math> \begin{align}  1 \\\end{align}</math></td>
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<td align="center" valign="center" width=30><math> \begin{align}  -r_0 \\\end{align}</math></td>
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</tr>
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<tr>
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<td align="center" valign="center" width=30><math> \begin{align}  -r_0 \\\end{align}</math> </td>
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<td align="center" valign="center" width=30><math> \begin{align}  1 \\\end{align}</math></td>
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</tr>
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</table></td>
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<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
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<td> <math> \begin{align} Z^{1/2} \\\end{align}</math> </td>
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<td>
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<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
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<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
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 +
<tr>
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<td align="center" valign="center" width=30><math>\begin{align} 1 \\\end{align}  </math></td>
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<td align="center" valign="center" width=30><math> \begin{align} 0 \\\end{align} </math></td>
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</tr>
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 +
<tr>
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<td align="center" valign="center" width=30><math>\begin{align}  0 \\\end{align} </math> </td>
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<td align="center" valign="center" width=30><math> \begin{align}  Z^{-1} \\\end{align} </math></td>
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</tr>
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</table></td>
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<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
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 +
<td></td>
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<td><math>\begin{align} \frac{1}{1+r_1} \\\end{align} </math></td>
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<td>
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<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
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<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
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 +
<tr>
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<td align="center" valign="center" width=30><math> \begin{align}  1 \\\end{align}</math></td>
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<td align="center" valign="center" width=30><math> \begin{align}  -r_1 \\\end{align}</math></td>
 +
</tr>
 +
 
 +
<tr>
 +
 
 +
<td align="center" valign="center" width=30><math> \begin{align}  -r_1 \\\end{align}</math> </td>
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<td align="center" valign="center" width=30><math> \begin{align}  1 \\\end{align}</math></td>
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</tr>
 +
 
 +
</table></td>
 +
 
 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
 +
 
 +
<td> <math>\begin{align}Z^{1/2} \\\end{align} </math> </td>
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<td>
 +
 
 +
<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
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 +
<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
 +
 
 +
<tr>
 +
 
 +
<td align="center" valign="center" width=30><math>\begin{align} 1 \\\end{align}  </math></td>
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<td align="center" valign="center" width=30><math> \begin{align} 0 \\\end{align} </math></td>
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</tr>
 +
 
 +
<tr>
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 +
<td align="center" valign="center" width=30><math>\begin{align}  0 \\\end{align} </math> </td>
 +
<td align="center" valign="center" width=30><math> \begin{align}  Z^{-1} \\\end{align} </math></td>
 +
</tr>
 +
 
 +
</table></td>
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 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
 +
 
 +
<td></td>
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<td><math>\begin{align} \frac{1}{1+r_2} \\\end{align} </math></td>
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<td>
 +
 
 +
<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
 +
 
 +
<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
 +
 
 +
<tr>
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 +
<td align="center" valign="center" width=30><math> \begin{align}  1 \\\end{align}</math></td>
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<td align="center" valign="center" width=30><math> \begin{align}  -r_2 \\\end{align}</math></td>
 +
</tr>
 +
 
 +
<tr>
 +
 
 +
<td align="center" valign="center" width=30><math> \begin{align}  -r_2 \\\end{align}</math> </td>
 +
<td align="center" valign="center" width=30><math> \begin{align}  1 \\\end{align}</math></td>
 +
</tr>
 +
 
 +
</table></td>
 +
 
 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
 +
 
 +
<td> <math>\begin{align}Z^{1/2} \\\end{align} </math> </td>
 +
<td>
 +
 
 +
<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
 +
 
 +
<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
 +
 
 +
<tr>
 +
 
 +
<td align="center" valign="center" width=30><math>\begin{align} 1 \\\end{align}  </math></td>
 +
<td align="center" valign="center" width=30><math> \begin{align} 0 \\\end{align} </math></td>
 +
</tr>
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 +
<tr>
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 +
<td align="center" valign="center" width=30><math>\begin{align}  0 \\\end{align} </math> </td>
 +
<td align="center" valign="center" width=30><math> \begin{align}  Z^{-1} \\\end{align} </math></td>
 +
</tr>
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 +
</table></td>
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 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
 +
 
 +
<td></td>
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<td><math>\begin{align} \frac{1}{1+r_3} \\\end{align} </math></td>
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 +
<td>
 +
 
 +
<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
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 +
<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
 +
 
 +
<tr>
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 +
<td align="center" valign="center" width=30><math> \begin{align}  1 \\\end{align}</math></td>
 +
<td align="center" valign="center" width=30><math> \begin{align}  -r_3 \\\end{align}</math></td>
 +
</tr>
 +
 
 +
<tr>
 +
 
 +
<td align="center" valign="center" width=30><math> \begin{align}  -r_3 \\\end{align}</math> </td>
 +
<td align="center" valign="center" width=30><math> \begin{align}  1 \\\end{align}</math></td>
 +
</tr>
 +
 
 +
</table></td>
 +
 
 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
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 +
<td>
 +
 
 +
<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
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 +
<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
 +
 
 +
<tr>
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 +
<td align="center" valign="center" width=30><math>\begin{align} U_L \\\end{align}  </math></td>
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</tr>
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<tr>
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 +
<td align="center" valign="center" width=30><math>\begin{align}  0 \\\end{align} </math> </td>
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</tr>
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 +
</table></td>
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<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
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 +
</tr>
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</table>
 +
 
 +
<table style="color:#000">
 +
<tr></tr><tr></tr>
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<tr>
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<td></td> <td></td> <td></td><td></td> <td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td>
 +
<td>=</td>
 +
<td></td>
 +
 
 +
<td><math>\begin{align} \frac{Z^{3/2}}{\pi_{k=0}^{3}(1+r_k)} \\\end{align} </math></td>
 +
 
 +
<td>
 +
 
 +
<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
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 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
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 +
<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
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 +
<tr>
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 +
<td align="center" valign="center" width=30><math>\begin{align} r_3(r_2(r_0r_1Z^{-1}+1)+(r_1+r_0Z^{-1})Z^{-1})Z^{-1}+r_0r_1Z^{-1}+(r_2r_1+r_2r_0Z^{-1})Z^-1+1 \\\end{align}  </math></td>
 +
</tr>
 +
 
 +
<tr>
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 +
<td align="center" valign="center" width=30><math>\begin{align}-r_0-r_1Z^{-1}-(r_3(r_2r_0+r_2r_1Z^{-1}+r_0r_1Z^{-1}+Z^{-2}))Z^{-1}-r_2r_0r_1Z^{-1}+r_2Z^{-2}\\\end{align} </math> </td>
 +
</tr>
 +
 
 +
</table></td>
 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
 +
<td><math>\begin{align}  U_L \\\end{align} </math></td>
 +
 
 +
</tr>
 +
</table>
 +
 
 +
<table style="color:#000">
 +
<tr></tr><tr></tr>
 +
<tr>
 +
<td></td> <td></td> <td></td><td></td> <td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td>
 +
<td>=</td>
 +
<td></td>
 +
 
 +
<td><math>\begin{align} \frac{Z^{3/2}}{\pi_{k=0}^{3}(1+r_k)} \\\end{align} </math></td>
 +
 
 +
<td>
 +
 
 +
<table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
 +
 
 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td>
 +
 
 +
<td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
 +
 
 +
<tr>
 +
 
 +
<td align="center" valign="center" width=30><math>\begin{align}1+(r_0r_1+r_1r_2+r_2r_3)Z^{-1}+(r_0r_1r_2r_3+r_0r_2+r_1r_3)Z^{-2}+r_0r_3Z^{-3}\\\end{align}  </math></td>
 +
</tr>
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 +
<tr>
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 +
<td align="center" valign="center" width=30><math>\begin{align}-r_0-(r_0r_1r_2+r_0r_2r_3+r_1)Z^{-1}-(r_0r_1r_3+r_1r_2r_3+r_2)Z^{-2}-r_3Z^{-3}\\\end{align} </math> </td>
 +
</tr>
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 +
</table></td>
 +
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;</td></tr></table></td>
 +
 
 +
<td><math>\begin{align} U_L \\\end{align} </math></td>
 +
</tr>
 +
</table>
 +
 
 +
<table style="color:#000">
 +
<tr></tr><tr></tr>
 +
<tr>
 +
 
 +
<td><math>\begin{align} V(Z) \\\end{align} </math></td> <td></td>
 +
<td>=</td>
 +
<td></td>
 +
 
 +
<td><math>\begin{align} \frac{U_L}{U_g} \\\end{align} </math></td>
 +
<td></td>
 +
<td>=</td>
 +
<td></td>
 +
<td><math>\begin{align} \frac{\pi_{k=0}^{3}(1+r_k)}{1+(r_0r_1+r_1r_2+r_2r_3)Z^{-1}+(r_0r_1r_2r_3+r_0r_2+r_1r_3)Z^{-2}+r_0r_3Z^{-3}}\\\end{align} </math></td>
 +
</tr>
 +
</table>
 +
 
 +
 
 +
 
 +
 
 +
b)  From the transfer function, we can see that there are three poles. If the poles are all real, there are three formants.
 +
If one pole is real and the other are complex pole pair, there are two formants.
 +
 
 +
 
 +
c) We can change the area of each segments. In this way,<math>r_k</math> will change , thus the root of the denominator of V(Z) is changed, which determines the location of formants.
 +
 
 +
 
 +
[[2013 Fall ECE 438 Boutin|Back to 2013 Fall ECE 438 Boutin]]
 +
 
 +
[[Category:2013_Fall_ECE_438_Boutin]]

Revision as of 14:48, 13 November 2013


Question 1

(a)

 
$ \begin{align} U_g \\\end{align} $
$ \begin{align} V_g \\\end{align} $
 
 = $ \begin{align} \frac{1}{1+r_0} \\\end{align} $
 
$ \begin{align} 1 \\\end{align} $ $ \begin{align} -r_0 \\\end{align} $
$ \begin{align} -r_0 \\\end{align} $ $ \begin{align} 1 \\\end{align} $
 
 $ \begin{align} Z^{1/2} \\\end{align} $
 
$ \begin{align} 1 \\\end{align} $ $ \begin{align} 0 \\\end{align} $
$ \begin{align} 0 \\\end{align} $ $ \begin{align} Z^{-1} \\\end{align} $
 
 $ \begin{align} \frac{1}{1+r_1} \\\end{align} $
 
$ \begin{align} 1 \\\end{align} $ $ \begin{align} -r_1 \\\end{align} $
$ \begin{align} -r_1 \\\end{align} $ $ \begin{align} 1 \\\end{align} $
 
 $ \begin{align}Z^{1/2} \\\end{align} $
 
$ \begin{align} 1 \\\end{align} $ $ \begin{align} 0 \\\end{align} $
$ \begin{align} 0 \\\end{align} $ $ \begin{align} Z^{-1} \\\end{align} $
 
 $ \begin{align} \frac{1}{1+r_2} \\\end{align} $
 
$ \begin{align} 1 \\\end{align} $ $ \begin{align} -r_2 \\\end{align} $
$ \begin{align} -r_2 \\\end{align} $ $ \begin{align} 1 \\\end{align} $
 
 $ \begin{align}Z^{1/2} \\\end{align} $
 
$ \begin{align} 1 \\\end{align} $ $ \begin{align} 0 \\\end{align} $
$ \begin{align} 0 \\\end{align} $ $ \begin{align} Z^{-1} \\\end{align} $
 
 $ \begin{align} \frac{1}{1+r_3} \\\end{align} $
 
$ \begin{align} 1 \\\end{align} $ $ \begin{align} -r_3 \\\end{align} $
$ \begin{align} -r_3 \\\end{align} $ $ \begin{align} 1 \\\end{align} $
 
 
$ \begin{align} U_L \\\end{align} $
$ \begin{align} 0 \\\end{align} $
 
= $ \begin{align} \frac{Z^{3/2}}{\pi_{k=0}^{3}(1+r_k)} \\\end{align} $
 
$ \begin{align} r_3(r_2(r_0r_1Z^{-1}+1)+(r_1+r_0Z^{-1})Z^{-1})Z^{-1}+r_0r_1Z^{-1}+(r_2r_1+r_2r_0Z^{-1})Z^-1+1 \\\end{align} $
$ \begin{align}-r_0-r_1Z^{-1}-(r_3(r_2r_0+r_2r_1Z^{-1}+r_0r_1Z^{-1}+Z^{-2}))Z^{-1}-r_2r_0r_1Z^{-1}+r_2Z^{-2}\\\end{align} $
 
 $ \begin{align} U_L \\\end{align} $
= $ \begin{align} \frac{Z^{3/2}}{\pi_{k=0}^{3}(1+r_k)} \\\end{align} $
 
$ \begin{align}1+(r_0r_1+r_1r_2+r_2r_3)Z^{-1}+(r_0r_1r_2r_3+r_0r_2+r_1r_3)Z^{-2}+r_0r_3Z^{-3}\\\end{align} $
$ \begin{align}-r_0-(r_0r_1r_2+r_0r_2r_3+r_1)Z^{-1}-(r_0r_1r_3+r_1r_2r_3+r_2)Z^{-2}-r_3Z^{-3}\\\end{align} $
 
 $ \begin{align} U_L \\\end{align} $
$ \begin{align} V(Z) \\\end{align} $ = $ \begin{align} \frac{U_L}{U_g} \\\end{align} $ = $ \begin{align} \frac{\pi_{k=0}^{3}(1+r_k)}{1+(r_0r_1+r_1r_2+r_2r_3)Z^{-1}+(r_0r_1r_2r_3+r_0r_2+r_1r_3)Z^{-2}+r_0r_3Z^{-3}}\\\end{align} $



b) From the transfer function, we can see that there are three poles. If the poles are all real, there are three formants. If one pole is real and the other are complex pole pair, there are two formants.


c) We can change the area of each segments. In this way,$ r_k $ will change , thus the root of the denominator of V(Z) is changed, which determines the location of formants.


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