What is sampling

Sampling is the process of converting a continuous-time signal $x(t)$ into a discrete-time signal by measuring its value at equally spaced instants $t = nT$:

$$x[n] = x(nT), \qquad n = 0, \pm1, \pm2, \dots$$

Here $T$ is the sampling period and $f_s = 1/T$ is the sampling rate. Mathematically it is modelled as multiplying $x(t)$ by an impulse train $p(t)=\sum_n \delta(t-nT)$. Sampling is the first step in analog-to-digital conversion and lets continuous signals be processed by digital systems. A band-limited signal can be rebuilt from its samples only if $f_s > 2f_M$ (sampling theorem); otherwise aliasing occurs.