Signal measures for discrete signals are often based on summations. Summation is the discrete-time equivalent of integration.
For discrete signal \(x[n]\) and integer index \(n\),
The discrete sum \(S_D\),
\(\boxed{S_D = \sum_{n = -\infty}^{\infty} x[n]}\)
The absolute sum \(S_A\),
\(\boxed{S_A = \sum_{n = -\infty}^{\infty} \left|x[n]\right|}\)
Signals for which the absolute sum \(S_A\) is finite are called absolutely summable.
The cumulative sum \(s_C\),
\(\boxed{s_{C}[n] = \sum_{k = -\infty}^{n} x[k]}\)
For non-periodic signals, the signal energy \(E\) is a useful measure. It is defined as the sum of the squares of the signal values.
For integer index \(n\),
\(\boxed{E = \sum_{n = -\infty}^{\infty} \left|x[n]\right|^2}\)
The absolute value \(\left|x[n]\right|\) allows this relation to be used for complex-valued signals.
Measures for periodic signals are based on averages since their signal energy \(E\) is infinite. The average value \(x_{\textrm{av}}\) and signal power \(P\) of a discrete periodic signal \(x[n]\) with period \(N\) are defined as the average sum per period and average energy per period, respectively.
\(\boxed{x_{\textrm{av}} = \frac{1}{N}\sum_{n = 0}^{N - 1} x[n]}\quad\textrm{(for discrete periodic signals)}\)
\(\boxed{P = \frac{1}{N}\sum_{n = 0}^{N - 1} \left|x[n]\right|^2}\quad\textrm{(for discrete periodic signals)}\)
where period \(N\) and index \(n\) are always integers.
Signals with finite energy \((0 < E < \infty)\) are called energy signals (or square summable). Signals with finite power \((0 < P < \infty)\) are called power signals.