What is aliasing; how to prevent it

Aliasing is the distortion that occurs when a signal is sampled at a rate lower than twice its highest frequency ($f_s < 2f_m$). The shifted copies of the spectrum overlap, so high-frequency components fold back and appear as lower frequencies. After this, the original signal cannot be recovered by any filter.

Why it happens

Sampling with an impulse train of rate $f_s$ makes the spectrum periodic:

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

Each copy occupies $kf_s - f_m$ to $kf_s + f_m$. The copies are separate only if $f_s - f_m \ge f_m$, i.e. $f_s \ge 2f_m$.

 fs > 2fm (no aliasing):
   ___       ___       ___
  /   \     /   \     /   \
-+--+--+---+--+--+---+--+--+--> f
  -fm 0 fm    fs        2fs

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

Example: A 700 Hz tone sampled at 1000 Hz gives samples identical to a 300 Hz tone ($|700 - 1000| = 300$ Hz). The 700 Hz tone has been "aliased" to 300 Hz.

Effects

• False frequencies appear in the reconstructed signal.
• In images, aliasing causes jagged edges and moiré patterns; in video, wheels seem to turn backwards.

Prevention

1. Sample above the Nyquist rate: choose $f_s > 2f_m$, usually with a margin (e.g. CD audio uses 44.1 kHz for a 20 kHz band).
2. Anti-aliasing (pre-)filter: pass the analog signal through a low-pass filter with cutoff $f_s/2$ before the sampler. Real signals are never perfectly band-limited, and noise has high frequencies; the filter removes these before they can fold back.
3. Guard band: since practical filters are not ideal, keep $f_s$ comfortably above $2f_m$ so that the filter's transition band fits between $f_m$ and $f_s - f_m$.
4. Oversampling: sample at a much higher rate, then filter and decimate digitally; this relaxes the analog filter requirement.

x(t) -> [Anti-aliasing LPF, fc = fs/2] -> [Sampler fs] -> [ADC]