Multiple choice on sampling and aliasing

I found some multiple choices in a well known book . The problem is that I don't get the answers in some and I want to do so.

Question 1:

The signal x(t) with Fourier transform $$X(j\omega) = u(\omega)- u(\omega -\omega_0 )$$ can undergo impulse-train sampling without aliasing, provided that the sampling period $$T <\frac{2\pi}{\omega_0}$$.

Okay here's what I think: $$x(t)$$ should be sampled with at least:

\begin{align} ωs &> 2B \\ &> 2(B_1-B_2)\\ &>2(\omega_0 - 0) \\ &> 2\omega_0 \\ \frac{2\pi}{T_s} &> 2\omega_0 \\ T_s&<\frac{\pi}{\omega_0} \end{align}

What do I miss?

Question 2:

Answer: $$50\pi$$

For $$x(t)$$, $$\omega_{\text{max}} = 50π + 50π = 100π$$ and so $$ω_s > 2\omega_{\text{max}} = 200π$$. So , if we actually sample with $$\omega_s<\omega_{\text{Nyquist}} = 200π$$ , then there is no chance to avoid aliasing for all $$X(j\omega)$$ spectrum but we can gain a proper copy of it as long as $$\omega_0 < \frac{\omega_s}{2} = 75π$$. Again what do I miss?

• Have you considered accepting any answers to your previous questions? Dec 19 '20 at 12:45

Question 1

Perhaps the easiest way to understand this is to draw $$X(j\omega)$$:

With the corresponding aliases after sampling in blue:

So long as the blue rectangle do no overlap you would not have aliasing. At this point it should be more clear that you need $$\omega_s \gt \omega_0 \\ T_s \lt \frac{2\pi}{\omega_0} \\$$

Question 2

Again, plotting $$G(j\omega)$$ (red), together with $$75X(j\omega)$$ and an alias image after sampling by $$\omega_s$$ (dashed blue) can help visualize the situation:

Indeed, because $$\omega_{\max} > \frac{1}{2}\omega_s$$ you have some aliasing. However the question is not whether you can avoid aliasing, but rather to find the value $$\omega_0$$ below which the aliasing does not affect the spectrum. In the above graph you may notice that the red and dashed blue curves overlap up to $$50\pi$$. So $$\omega_0 = 50\pi$$.