How Nyquist criterion prevents aliasing

The Nyquist criterion says a signal band-limited to $f_m$ must be sampled at $f_s \ge 2f_m$. It prevents aliasing because it keeps the spectral copies created by sampling from overlapping.

How it works: Sampling at rate $f_s$ makes the spectrum periodic:

$$X_s(f) = f_s\sum_{k=-\infty}^{\infty}X(f - kf_s)$$

The original band occupies $-f_m$ to $f_m$. The first copy starts at $f_s - f_m$. There is no overlap when

$$f_s - f_m \ge f_m \;\Rightarrow\; f_s \ge 2f_m$$

 (a) fs > 2fm : copies separate, guard band
     ___         ___         ___
    /   \       /   \       /   \
 --+--+--+-----+--+--+-----+--+--+--> f
  -fm 0 fm  fs-fm fs      2fs
            |<->| guard band
     [ LPF cutoff fs/2 recovers centre copy ]

 (b) fs = 2fm : copies just touch (ideal LPF needed)
     ___ ___ ___
    /   V   V   \
 --+---+---+---+--> f
       0   fs  2fs

 (c) fs < 2fm : copies overlap -> aliasing
     ____ ____
    /    X    \
 --+---+---+----> f
       0   fs

Explanation:

• In (a), the original spectrum stays separate, so an ideal low-pass filter with cutoff $f_s/2$ removes the copies and gives back $x(t)$ exactly.
• In (b), recovery is possible only with an ideal brick-wall filter.
• In (c), the tail of the copy at $f_s$ folds into the band $0$ to $f_s/2$. Those components appear at false frequencies $|f - f_s|$ and cannot be separated by any filter. This is aliasing.

Example: A 3 kHz tone needs $f_s \ge 6$ kHz. At $f_s = 8$ kHz it is recovered correctly; at $f_s = 4$ kHz it appears as $|3 - 4| = 1$ kHz.

In practice, an anti-aliasing low-pass filter (cutoff $f_s/2$) is placed before the sampler to make sure the signal really is band-limited.