# Evaluate expressions without computing DTFT

Let $$X(\omega)$$ be the DTFT of the sequence $$x[n]$$ given by: $$x[n] = \{4, 2, -1, 5, -3, 1, -2, 4, 2\},\quad\text{for}\quad n \in [-6, 2]$$ I do want to compute

1. $$X(0)$$
2. $$X(\pi)$$
3. $$\displaystyle\int_{-\pi}^{\pi} X(\omega) d{\omega}$$
4. $$\displaystyle\int_{-\pi}^{\pi}|X(\omega)|^2 d{\omega}$$
5. $$\displaystyle\int_{-\pi}^{\pi} \bigg\lvert\frac{dX(\omega)}{d{\omega}}\bigg\rvert^2 d{\omega}$$

What I have tried

• For $$X(0)$$, I computed the sum and so it was 12.
• For $$X(\pi)$$ I did $$X(\pi) = x[9] = \sum_{n = 0}^{8} x[n]e^{-i\omega n} = (-1)^9 \times 12 = -12$$, because 0 corresponds to 0, 2 to $$\pi/8$$, -1 to $$\pi/4$$, ..., 2 to $$\pi$$.
• For $$\int_{-\pi}^{\pi} X(\omega) d{\omega}$$, I tried by Parseval Theorem, so it was $$2\pi x[n] x[n]*$$, but I don't know if it's correct. Last two I don't have any idea of how to do it.

I also have a similar exercise, but it is with DFT instead of DTFT, but it is pretty the same, isn't it?

• Can you include in your question (meaning edit your question instead of answering this comment with another comment) what exactly you know to be $X(\omega)$? You have told us what $x[n]$ is but I I am having a hard time figuring out what the DTFT of the sequence is. Dec 30, 2020 at 15:55
• I edited it. Sorry, it must have been because of my English. @DilipSarwate Dec 30, 2020 at 16:28

For the first two questions you just need to use the definition of the DTFT

$$X(\omega)=\sum_{n=-\infty}^{\infty}x[n]e^{-jn\omega}\tag{1}$$

and use $$\omega=0$$ and $$\omega=\pi$$, respectively.

For 3. you just need to use the definition of the inverse DTFT:

$$x[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}X(\omega)e^{jn\omega}d\omega\tag{2}$$

Finally, for 4. and 5. you need Parseval's theorem. In addition, for 5. you need to express the sequence corresponding to the derivative of $$X(\omega)$$ in terms of $$x[n]$$. You should be able to either derive this yourself or look it up in a table.

• I wrote it wrong, X(0) = 12 and X(\pi) = -12. Thanks. Dec 30, 2020 at 17:09
• @Danny: That's right. Dec 30, 2020 at 17:11
• @Danny: If the answer was helpful you can accept it by clicking on the green checkmark, thanks! Dec 30, 2020 at 21:42