Discrete signals may arise naturally or as a consequence of sampling continuous signals (typically at a uniform sampling interval \(t_s\)). A sampled or discrete signal \(x[n]\) is just an ordered sequence of values corresponding to the integer index \(n\) that embodies the time history of the signal. It contains no direct information about the sampling interval \(t_s\), except through the index \(n\) of the sample locations.
A discrete signal \(x[n]\) is plotted as lines against the integer index \(n\).
For example, \(\displaystyle x[n] = \{1,\,2,\,\overset{\Downarrow}{4},\,8,\,\ldots\}\)
where marker \((\Downarrow)\) indicates the origin \(n = 0\) and ellipses \((\ldots)\) denote infinite extent on either side.
A discrete signal \(x[n]\) is called right-sided if it is zero for \(n < N\), causal if it is zero for \(n < 0\), left-sided if it is zero for \(n > N\), and anti-causal if it is zero for \(n \geq 0\).
A discrete periodic signal repeats every \(N\) samples.
\(\boxed{x[n] = x[n \pm kN]}\)
The period \(N\) is the smallest number of samples that repeats. Unlike its analog counterpart, the period \(N\) of discrete signals is always an integer.
The common period \(N\) of a linear combination of periodic discrete signals is given by the least common multiple (LCM) of the individual periods.