Evaluate expressions without computing DTFT

Question

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?

Thanks in advance.

Answer 1

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.