Sampling and Aliasing

Sampling converts a continuous signal into discrete values taken every $T_s = 1/f_s$ seconds.

Nyquist theorem: $f_s \ge 2 f_{max}$. If $f_s < 2f_{max}$, higher frequencies fold back as false lower frequencies. This is aliasing, and it cannot be removed after sampling. Alias frequency: $f_a = |f - n f_s|$.

Example: a 700 Hz signal sampled at 1000 Hz appears as 300 Hz.

Anti-aliasing prefilter

A low-pass filter placed before the ADC removes frequency components above $f_s/2$, so aliasing cannot occur. Cut-off is set at or slightly below $f_s/2$.

First-order vs second-order filter

Point	First order	Second order
Roll-off	20 dB per decade	40 dB per decade
Components	One R and one C	Two R, two C (or op-amp stage)
Cut-off sharpness	Gradual	Sharper
Aliasing rejection	Poor, needs high $f_s$	Better
Complexity	Simple	Moderate
Use	Basic smoothing	Practical anti-aliasing

Exam tip

State Nyquist, give the alias formula and place the filter before the ADC in the diagram.