Explanation of Sampling Theorem

The sampling theorem:

1. for x(nT) to be equally spaced samples of x(t), while n=0, +1, -1, +2, -2, ...

2. x(t) is band limited.
  X($ \omega $) = 0 for $ |\omega|>\omega_m $

3. $  2\pi/T = \omega_s > 2\omega_m $ (Nyquist Condition)

Then x(t) is uniquely recoverable.

Here is a block diagram of sampling and reconstruction using a LPF:

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