As it was difficult to contribute something unique about sampling, I thought I would add a few examples using the theories.
Sampling Theorem: Let $ \omega_m $ be a non-negative number.
Let x(t) be a signal with $ \chi(\omega)=0 $ when $ |\omega|<\omega_m $.
Consider samples x(nT), for n=...,-2,-1,0,1,2,...
if $ T<\frac{1}{2}\frac{2\pi}{\omega_m} $, where T is the sampling period.
then x(t) can be uniquely recovered from its samples.
Example 1
P: A real-valued signal x(t) is known to be uniquely determined by its samples when the sampling frequency is $ \omega_s=10000\pi $. For what values of $ \omega $ is $ \chi(j\omega) $ guaranteed to be zero?
S: $ \omega_s=2\omega_m $
$ 10000\pi=2\omega_m $
$ \omega_m=5000\pi $
To guarantee $ \chi(j\omega)=0 $, $ |\omega|>\omega_m $,
so $ |\omega|>5000\pi $
Example 2
P: A continuous-time signal x(t) is obtained at the output of an ideal lowpass filter with cutoff frequency $ \omega_c=1000\pi $. If impulse-train sampling is performed on x(t), which of the following sampling periods would guarantee that x(t) can be recovered from its sampled version using an appropriate lowpass filter? a)$ T=0.5x10^{-3} $ b)$ T=2x10^{-3} $ c)$ T=1x10^{-4} $
S: a)$ \omega_c=1000\pi $ and to guarantee it can be recovered $ T<\frac{1}{2}\frac{2\pi}{\omega} $
$ 0.5x10{-3}<\frac{1}{2}\frac{2\pi}{1000\pi} $ $ 0.5x10{-3}<\frac{1}{1000} $ $ 0.5<1 $, true.
b)$ \omega_c=1000\pi $ and to guarantee it can be recovered $ T<\frac{1}{2}\frac{2\pi}{\omega} $ $ 2x10{-3}<\frac{1}{2}\frac{2\pi}{1000\pi} $ $ 2x10{-3}<\frac{1}{1000} $ $ 2<1 $, false.
c)$ \omega_c=1000\pi $ and to guarantee it can be recovered $ T<\frac{1}{2}\frac{2\pi}{\omega} $ $ 10{-4}<\frac{1}{2}\frac{2\pi}{1000\pi} $ $ 10{-4}<\frac{1}{1000} $ $ 0.1<1 $, true.
