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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)\ $

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 $ \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.


comments:

One problem with your answer --mireille.boutin.1, Fri, 19 Oct 2007 14:53:56 Dividing by zero, or by infinity, is not recommended.

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