Sampling Rate#

Functions of space, time, or any other dimension can be sampled, and similarly in two or more dimensions.

For functions that vary with time, let $s(t)$ be a continuous function (or “signal”) to be sampled, and let sampling be performed by measuring the value of the continuous function every $T$ seconds, which is called the sampling interval or sampling period. Then the sampled function is given by the sequence:

for integer values of $n$.

The sampling frequency or sampling rate, $f_s$, is the average number of samples obtained in one second, thus $f_s = \frac{1}{T}$, with the unit samples per second, sometimes referred to as “hertz”, for example 48 kHz is 48,000 samples per second

Reconstructing Continuous Function from Samples

The fidelity of a theoretical reconstruction is a common measure of the effectiveness of sampling. That fidelity is reduced when $s(t)$ contains frequency components whose cycle length (period) is less than 2 sample intervals

The original signal can be reconstructed from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a reconstruction filter.