Sampling

As discussed in the previous module about sampling, the conversion of an analog or continuous-time signal $x\left(t\right)$ to the digital domain is carried out by a C-to-D convertor as indicated in the figure below.

The convertor runs at a sampling rate of $f_s=1/T_s$ [samples/second], in which $T_s$ is the inter-sample distance. In the discrete domain the signals are represented as sequences of numbers called samples. A sample value of a typical discrete-time signal or sequence is denoted as $x[n\cdot T_s]$. Square brackets are used to denote the difference with continuous-time signals. Furthermore, in most cases the notation $T_s$ is skipped and the samples are denoted by $x\left[n\right]$. Although the independent variable $n$ need not necessarily represent ‘time’, $n$ may for example correspond to a spatial coordinate or distance, $x\left[n\right]$ is generally referred to as a function of time with the argument $n$ being an integer in the range $-\infty$ and $+\infty$. It should be noted that $x\left[n\right]$ is defined only for integer values of index $n$ and is undefined for non-integer values of index $n$. Usually a real-valued signal $x\left[n\right]$ is represented in a graph, from which the correspondence with the underlying continuous-time signal becomes clear in case the sampling rate $f_s$ meets the Nyquist criterion.