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 $T <\frac{2\pi}{\omega_0}$.
Answer: True
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?
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?