Short note: Nyquist theorem

The Nyquist (sampling) theorem states that a continuous-time signal band-limited to a highest frequency $f_M$ can be exactly recovered from its samples if it is sampled at a rate

$$f_s > 2 f_M$$

• $2f_M$ is the Nyquist rate; $T_N = 1/(2f_M)$ is the Nyquist interval.
• In the frequency domain, sampling repeats the spectrum $X(j\omega)$ at every multiple of $\omega_s$. With $\omega_s > 2\omega_M$ the copies do not overlap.
• Recovery is done with an ideal low-pass filter (gain $T$, cutoff between $\omega_M$ and $\omega_s - \omega_M$).
• If $f_s < 2f_M$, copies overlap and aliasing occurs; high frequencies appear as low ones and the signal is lost.
• In practice an anti-aliasing low-pass filter is placed before the sampler and $f_s$ is kept above $2f_M$ (e.g. audio with $f_M = 20$ kHz is sampled at 44.1 kHz).