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Problem 5.31

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:

$x[n]=cos(\omega_0n)\$
$X(\omega)=\pi\delta(\omega-\omega_0)+\pi\delta(\omega+\omega_0) \;\; for \; -\pi\leq\omega_0\leq\pi$
$y[n]=\omega_0cos(\omega_0n)\$
$Y(\omega)=\omega_0\pi\delta(\omega-\omega_0)+\omega_0\pi\delta(\omega+\omega_0) \;\; for \; -\pi\leq\omega_0\leq\pi$
$H(\omega)=\frac{Y(\omega)}{X(\omega)}=\frac{\omega_0\pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))}{\pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))}=\omega_0$
$h[n]=F^{-1}(H(\omega))=F^{-1}(\omega_0)=\omega_0\delta[n] \;\;\; (since \; \omega_0 \;is \; a \; constant.)$

Tom's reason this does not work:

The reason that your solution does not work is because you are treating $\omega_0\$ 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 $\omega\$ and why it is restricted from 0 to $\pi\$. I would have thought $\frac {-\pi}{2}$ to $\frac {\pi}{2}$ since if it's 0 to $\pi\$, then at $\frac {pi}{2}$ 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 $\omega_0\$ is positive or negative, because it sees its input as a sum of complex conjugate exponentials. i.e. $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 0, 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)\$

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 $\left|\omega_0\right|\$. Therefore, the frequency response of the system is

$H(\omega)=\left|\omega\right|,\;\;\; for \; 0\leq\left|\omega\right|\leq\pi$.

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

\begin{align} h[n]&=\frac{1}{2\pi}\int_{-\pi}^\pi H(\omega)e^{j\omega n}d\omega \\ &=\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 \\ &=\frac{1}{\pi}\int_0^\pi \omega cos(\omega n)d\omega \\ &=\frac{1}{\pi}\left(\frac{cos(n\pi)-1}{n^2}\right) \end{align}

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