(New page: I am having difficulty getting an equivalent answer to the answer key on problem 5.31. Both methods seem reasonable but yield different results. My Solution First: <math>\quad x[n]=cos...)
 
 
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  <math>\quad x[n]=cos(\omega_0n) </math>
 
  <math>\quad x[n]=cos(\omega_0n) </math>
  
−
  <img alt="tex:\displaystyle\quad X(\omega)=\pi\delta(\omega-\omega_0)+\pi\delta(\omega+\omega_0)\phantom{mm}for\phantom{n}-\pi\leq\omega_0\leq\pi"/>
+
  <math>\quad X(\omega)=\pi\delta(\omega-\omega_0)+\pi\delta(\omega+\omega_0)-\pi\leq\omega_0\leq\pi</math>
  
−
  <img alt="tex:\displaystyle\quad y[n]=\omega_0cos(\omega_0n)"/>
+
  <math>\quad y[n]=\omega_0cos(\omega_0n)</math>
  
−
  <img alt="tex:\displaystyle\quad Y(\omega)=\omega_0\pi\delta(\omega-\omega_0)+\omega_0\pi\delta(\omega+\omega_0)\phantom{mm}for\phantom{n}-\pi\leq\omega_0\leq\pi"/>
+
  <math>\quad Y(\omega)=\omega_0\pi\delta(\omega-\omega_0)+\omega_0\pi\delta(\omega+\omega_0)-\pi\leq\omega_0\leq\pi</math>
  
−
<img alt="tex:\displaystyle\quad H(\omega)=\frac{Y(\omega)}{X(\omega)}=\frac{\omega_0\pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))}{\pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))}=\omega_0"/>
+
<math>\quad H(\omega)=\frac{Y(\omega)}{X(\omega)}=\frac{\omega_0\pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))}{\pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))}=\omega_0</math>
  
−
<img alt="tex:\displaystyle\quad h[n]=F^{-1}(H(\omega))=F^{-1}(\omega_0)=\omega_0\delta[n]\phantom{mmm}(since\phantom{n}\omega_0\phantom{n}is\phantom{n}a\phantom{n}constant.)"/>
 
  
 
Tom's reason this does not work:
 
Tom's reason this does not work:
  
−
  The reason that your solution does not work is because you are treating <img alt="tex:\omega_0" style="vertical-align: bottom;"> as a constant. However, <img alt="tex:\omega_0" style="vertical-align: middle;"> is actually <img alt="tex:\omega" style="vertical-align: middle;"> when you want to take the inverse transform and therefore it is a variable and not a constant. So when you write the integral it is of the form <img alt="tex: \int{x e^x}dx" style="vertical-align: middle;" /> and not <img alt="tex: \int{c e^x}dx" style="vertical-align: middle;" /> where c is a constant. I made the same mistake myself when I first tried it.
+
  The reason that your solution does not work is because you are treating <img alt="tex:\omega_0" style="vertical-align: bottom;"> as a constant. However, <math>\omega_0 </math>is actually <math>\omega</math> when you want to take the inverse transform and therefore it is a variable and not a constant. So when you write the integral it is of the form <math>\int{x e^x}dx</math> and not <math>\int{c e^x}dx </math> where c is a constant. I made the same mistake myself when I first tried it.
  
 
  Can anyone explain the solution key's answer though? I do not understand why it is the absolute value of w and why it is restricted from 0 to pi. I would have thought -pi/2 to pi/2 since if it's 0 to pi, then at pi/2 it would be division by zero. I also don't understand why the integral for the inverse transform is taken of -pi to pi when the solution key previously restricted it from 0 to pi.
 
  Can anyone explain the solution key's answer though? I do not understand why it is the absolute value of w and why it is restricted from 0 to pi. I would have thought -pi/2 to pi/2 since if it's 0 to pi, then at pi/2 it would be division by zero. I also don't understand why the integral for the inverse transform is taken of -pi to pi when the solution key previously restricted it from 0 to pi.
Line 25: Line 24:
 
  The real error is in the problem statement.  The system has no way of determining whether <img alt="tex:\omega_0" /> is positive or negative, because it sees its input as a sum of complex conjugate exponentials.  i.e. <img alt="tex:cos(\omega_0)=cos(-\omega_0)"/> and the system has no way of knowing which of the two was input, positive or negative.   
 
  The real error is in the problem statement.  The system has no way of determining whether <img alt="tex:\omega_0" /> is positive or negative, because it sees its input as a sum of complex conjugate exponentials.  i.e. <img alt="tex:cos(\omega_0)=cos(-\omega_0)"/> and the system has no way of knowing which of the two was input, positive or negative.   
  
−
  If we take to problem statement literally, then <img alt="tex:\omega_0" /> must be restricted to zero, because:
+
  If we take to problem statement literally, then <math> omega_0 </math> must be restricted to zero, because:
  
−
  <img alt="tex:cos(-\omega_0n)\rightarrow-\omega_0cos(-\omega_0n)" />
+
  <math>cos(-\omega_0n)\rightarrow-\omega_0cos(-\omega_0n)</math>
  
−
  which is equal to:  <img alt="tex:cos(\omega_0n)\rightarrow-\omega_0cos(\omega_0n)" />
+
  which is equal to:  <math>cos(\omega_0n)\rightarrow-\omega_0cos(\omega_0n)</math>
  
−
  but:  <img alt="tex:cos(\omega_0n)\rightarrow\omega_0cos(\omega_0n)" />
+
  but:  <math>cos(\omega_0n)\rightarrow\omega_0cos(\omega_0n)</math>
  
−
  thus: <img alt="tex:\omega_0=-\omega_0" /> and the only way this is true is when <img alt="tex:\omega_0=0" />
+
  thus: <math>\omega_0=-\omega_0 </math> and the only way this is true is when <math>\omega_0=0</math>
  
 
  To deal with this issue, the solution key solved the problem using the following instead:
 
  To deal with this issue, the solution key solved the problem using the following instead:
  
−
  <img alt="tex:cos(\omega_0n)\rightarrow\left|\omega_0\right|cos(\omega_0n)" />
+
  <math>cos(\omega_0n)\rightarrow\left|\omega_0\right|cos(\omega_0n)</math>
  
  
Line 45: Line 44:
 
  <div style="color: #22a; border: 2px solid #22a; margin: 40px; padding: 10px;">
 
  <div style="color: #22a; border: 2px solid #22a; margin: 40px; padding: 10px;">
  
−
  From the given information, it is clear that when the input to the system is a complex exponential of frequency <img alt="tex:\displaystyle\omega_0"/> the output is a complex exponential of the same frequency but scaled by the <img alt="tex:\displaystyle\left|\omega_0\right|"/>.  Therefore, the frequency response of the system is
+
  From the given information, it is clear that when the input to the system is a complex exponential of frequency <math>\omega_0</math> the output is a complex exponential of the same frequency but scaled by the <img alt="tex:\displaystyle\left|\omega_0\right|"/>.  Therefore, the frequency response of the system is
  
−
  <img alt="tex:\displaystyle\quad H(\omega)=\left|\omega\right|,\phantom{mmm}for\phantom{n}0\leq\left|\omega\right|\leq\pi ."/>
+
  <math>\quad H(\omega)=\left|\omega\right|,for \leq\left|\omega\right|\leq\pi </math>
  
 
  Taking the inverse Fourier transform of the frequency response, we obtain
 
  Taking the inverse Fourier transform of the frequency response, we obtain
  
−
  <img alt="tex:\displaystyle\quad h[n]=\frac{1}{2\pi}\int_{-\pi}^\pi H(\omega)e^{j\omega n}d\omega"/>
+
  <math>\quad h[n]=\frac{1}{2\pi}\int_{-\pi}^\pi H(\omega)e^{j\omega n}d\omega </math>
  
−
  <img alt="tex:\displaystyle\quad\phantom{h[n]}=\frac{1}{2\pi}\int_{-\pi}^0 -\omega e^{j\omega n}d\omega+\frac{1}{2\pi}\int_0^\pi \omega e^{j\omega n}d\omega"/>
+
  <math>\quad {h[n]}=\frac{1}{2\pi}\int_{-\pi}^0 -\omega e^{j\omega n}d\omega+\frac{1}{2\pi}\int_0^\pi \omega e^{j\omega n}d\omega</math>
  
−
  <img alt="tex:\displaystyle\quad\phantom{h[n]}=\frac{1}{\pi}\int_0^\pi \omega cos(\omega n)d\omega"/>
+
  <math>\quad {h[n]}=\frac{1}{\pi}\int_0^\pi \omega cos(\omega n)d\omega </math>
  
−
  <img alt="tex:\displaystyle\quad\phantom{h[n]}=\frac{1}{\pi}\left(\frac{cos(n\pi)-1}{n^2}\right)"/>
+
  <math>\quad {h[n]}=\frac{1}{\pi}\left(\frac{cos(n\pi)-1}{n^2}\right)</math>
  
 
  </div>
 
  </div>

Latest revision as of 20:46, 6 April 2008

I am having difficulty getting an equivalent answer to the answer key on problem 5.31. Both methods seem reasonable but yield different results.

My Solution First: 

$ \quad x[n]=cos(\omega_0n)  $

$ \quad X(\omega)=\pi\delta(\omega-\omega_0)+\pi\delta(\omega+\omega_0)-\pi\leq\omega_0\leq\pi $

$ \quad y[n]=\omega_0cos(\omega_0n) $

$ \quad Y(\omega)=\omega_0\pi\delta(\omega-\omega_0)+\omega_0\pi\delta(\omega+\omega_0)-\pi\leq\omega_0\leq\pi $

$ \quad H(\omega)=\frac{Y(\omega)}{X(\omega)}=\frac{\omega_0\pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))}{\pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))}=\omega_0 $


Tom's reason this does not work: 

The reason that your solution does not work is because you are treating <img alt="tex:\omega_0" style="vertical-align: bottom;"> as a constant. However, $ \omega_0  $is actually $ \omega $ when you want to take the inverse transform and therefore it is a variable and not a constant. So when you write the integral it is of the form $ \int{x e^x}dx $ and not $ \int{c e^x}dx  $ where c is a constant. I made the same mistake myself when I first tried it.

Can anyone explain the solution key's answer though? I do not understand why it is the absolute value of w and why it is restricted from 0 to pi. I would have thought -pi/2 to pi/2 since if it's 0 to pi, then at pi/2 it would be division by zero. I also don't understand why the integral for the inverse transform is taken of -pi to pi when the solution key previously restricted it from 0 to pi.

Ross's Reason This Does Not Work: 

The real error is in the problem statement.  The system has no way of determining whether <img alt="tex:\omega_0" /> is positive or negative, because it sees its input as a sum of complex conjugate exponentials.  i.e. <img alt="tex:cos(\omega_0)=cos(-\omega_0)"/> and the system has no way of knowing which of the two was input, positive or negative.  

If we take to problem statement literally, then $  omega_0  $ must be restricted to zero, because:

$ cos(-\omega_0n)\rightarrow-\omega_0cos(-\omega_0n) $

which is equal to:  $ cos(\omega_0n)\rightarrow-\omega_0cos(\omega_0n) $

but:  $ cos(\omega_0n)\rightarrow\omega_0cos(\omega_0n) $

thus: $ \omega_0=-\omega_0  $ and the only way this is true is when $ \omega_0=0 $

To deal with this issue, the solution key solved the problem using the following instead:

$ cos(\omega_0n)\rightarrow\left|\omega_0\right|cos(\omega_0n) $


Answer Key's Solution: 

Exactly as it says...

From the given information, it is clear that when the input to the system is a complex exponential of frequency $ \omega_0 $ the output is a complex exponential of the same frequency but scaled by the <img alt="tex:\displaystyle\left|\omega_0\right|"/>.  Therefore, the frequency response of the system is

$ \quad H(\omega)=\left|\omega\right|,for \leq\left|\omega\right|\leq\pi  $

Taking the inverse Fourier transform of the frequency response, we obtain

$ \quad h[n]=\frac{1}{2\pi}\int_{-\pi}^\pi H(\omega)e^{j\omega n}d\omega  $

$ \quad {h[n]}=\frac{1}{2\pi}\int_{-\pi}^0 -\omega e^{j\omega n}d\omega+\frac{1}{2\pi}\int_0^\pi \omega e^{j\omega n}d\omega $

$ \quad {h[n]}=\frac{1}{\pi}\int_0^\pi \omega cos(\omega n)d\omega  $

$ \quad {h[n]}=\frac{1}{\pi}\left(\frac{cos(n\pi)-1}{n^2}\right) $

Theirs seems logically correct to me (except for the absolute value part), but mine seems mathematically correct.  Where is the problem.

From mireille.boutin.1 Fri Oct 19 14:53:56 -0400 2007 From: mireille.boutin.1 Date: Fri, 19 Oct 2007 14:53:56 -0400 Subject: One problem with your answer Message-ID: <20071019145356-0400@https://engineering.purdue.edu>

Dividing by zero, or by infinity, is not recommended.

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