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) $
<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"/>
<img alt="tex:\displaystyle\quad y[n]=\omega_0cos(\omega_0n)"/>
<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"/>
<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"/>
<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:
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.
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 <img alt="tex:\omega_0" /> must be restricted to zero, because:
<img alt="tex:cos(-\omega_0n)\rightarrow-\omega_0cos(-\omega_0n)" />
which is equal to: <img alt="tex:cos(\omega_0n)\rightarrow-\omega_0cos(\omega_0n)" />
but: <img alt="tex:cos(\omega_0n)\rightarrow\omega_0cos(\omega_0n)" />
thus: <img alt="tex:\omega_0=-\omega_0" /> and the only way this is true is when <img alt="tex:\omega_0=0" />
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)" />
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 <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
<img alt="tex:\displaystyle\quad H(\omega)=\left|\omega\right|,\phantom{mmm}for\phantom{n}0\leq\left|\omega\right|\leq\pi ."/>
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"/>
<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"/>
<img alt="tex:\displaystyle\quad\phantom{h[n]}=\frac{1}{\pi}\int_0^\pi \omega cos(\omega n)d\omega"/>
<img alt="tex:\displaystyle\quad\phantom{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.
