Determining the Discrete Fourier Transformation of signal.

Question

I'm studying signal analysis and now I'm at the Fourier series/transformations.

I'm following an example but can't understand the last step.

This problem is divided in two steps.

FIRST

From this Fourier Transformation (FT)

I have to show the expression of the Discrete Time Fourier Transformation (DTFT) and from what frequency I can reconstruct the signal without aliasing. This is done this way:

I can understand all of these steps so no problem here.

SECOND I have to determine the Discrete Fourier Transformation (DFT) assuming $f_s=50Hz$ and that the DFT assumes a window with $500$ samples. This is done this way:

Can someone please point me in the right direction? I can't understand the red dotted square.

Answer 1

All that happens in the red dotted square is plugging $f_s=50$ and $\omega=k \pi/5$ into the formula for $X_{DTFT}$ written above. It should be noted that $X_{DTFT}$ is periodic with period $\omega_s=2\pi f_s=100\pi$, and the formula we have for it is written in terms of $\omega- k\omega_s$. This means we sometimes have to subtract an appropriate multiple of the period from $\omega$ to place the difference $\omega- k\omega_s$ into a range where the formula applies.

• If $0\le k\le 200$, then $0\le \omega\le 40\pi$, hence $X_{DTFT}(\omega)=f_s(40\pi -\omega)= 50(40\pi - k\pi/5)$.
• If $300\le k< 500$, then $60\pi \le \omega<100\pi$, which can also be written as $-40\pi \le \omega-100\pi <0$. Hence $X_{DTFT}(\omega)=f_s(\omega-100\pi+40\pi)= 50( k\pi/5-60\pi)$.
• If $200< k< 300$, then $40\pi< \omega< 60\pi$ (the second inequality is also $\omega-100\pi<-40\pi$), hence $X_{DTFT}(\omega)=0$.