Short note: Discrete data function

A discrete data function (discrete-time signal) is a function whose independent variable takes only discrete (integer) values. It is a sequence of numbers $x[n]$, where $n = \ldots, -2, -1, 0, 1, 2, \ldots$; it is not defined between integer values of $n$.

How it arises

• By sampling a continuous-time signal every $T_s$ seconds: $x[n] = x(nT_s)$, e.g. a digital audio signal with $f_s = 44.1$ kHz.
• Naturally discrete data: daily rainfall, monthly sales, number of students per year.

Ways to represent it

1. Functional: $x[n] = (0.5)^n$ for $n \ge 0$, 0 otherwise.
2. Sequence: $x[n] = \{1, 2, \underline{3}, 1\}$ (underline/arrow marks $n=0$).
3. Tabular: a table of $n$ and $x[n]$.
4. Graphical: stem (lollipop) plot.

 x[n]       o
        o   |
        |   |   o
    o   |   |   |
 ---+---+---+---+---> n
   -1   0   1   2

Features: processed by digital computers and DSP chips; operations are sums and differences instead of integrals and derivatives; DT sinusoids are periodic only if $\omega/2\pi$ is rational; frequency is unique only over a $2\pi$ range. If the amplitude is also quantised, it becomes a digital signal.