The absolute area of signal provides useful measures of its size. A signal \(x(t)\) is called absolutely integrable if it possesses finite absolute area:
\(\boxed{\int_{-\infty}^{\infty} \left|x(t)\right|\,dt < \infty}\quad\textrm{(for an absolutely integrable signal)}\)
All time-limited functions of finite amplitude have finite absolute area. The criterion of absolute integrability is often used to check for system stability or justify the existence of certain transforms.
Signal Energy
The instantaneous power \(p_{i}(t)\) (in watts) delivered to a \(1\,\Omega\) resistor may be expressed as \(p_{i}(t) = x^{2}(t)\) where the signal \(x(t)\) represents either the voltage across it or the current through it. The total energy \(E\) delivered to the \(1\,\Omega\) is called the signal energy (in joules) and is found by integrating the instantaneous power \(p_{i}(t)\) for all time, this is also known as the Parseval’s theorem.
\(\boxed{E = \int_{-\infty}^{\infty} p_{i}(t)\,dt = \int_{-\infty}^{\infty} \left|x(t)\right|^2\,dt}\)
The absolute value \(\left|x(t)\right|\) allows this relation to be used for complex-valued signals.
Signals with finite energy \((0 < E < \infty)\) are called an energy signal or square integrable. Energy signals have zero signal power \((P = 0)\) since finite energy is averaged over all (infinite) time. All time-limited signals of finite amplitude are energy signals.
Measures for Periodic Signals
Periodic signals are characterized by several measures:
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The duty cycle (ratio) \(D\) of periodic signal \(x(t)\) equals the ratio of its pulse width \(\mathrm{PW}\) and period \(T\).
\(\boxed{D = \frac{\mathrm{PW}}{T}}\)
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The average value \(x_{\textrm{av}}\) of periodic signal \(x(t)\) equals the average area per period.
\(\boxed{x_{\textrm{av}} = \frac{1}{T}\int_{T} x(t)\,dt}\)
where \(T\) is the period.
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The signal power \(P\) of periodic signal \(x(t)\) equals the average energy per period.
\(\boxed{P = \frac{1}{T}\int_{T} \left|x(t)\right|^2\,dt}\)
where \(T\) is the period.
Signals with finite power \((0 < P < \infty)\) and infinite energy \((E = \infty)\) are called power signals. The periodic signals and their combinations are the examples of power signals.
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The RMS value \(x_{\textrm{rms}}\) equals the square root of signal power \(P\) of periodic signal \(x(t)\).
\(\boxed{x_{\textrm{rms}} = \sqrt{P} = \sqrt{\frac{1}{T}\int_{T} \left|x(t)\right|^2\,dt}}\)
where \(T\) is the period.
The notation \(\int_{T}\) is used to mean integration over any convenient one-period duration \(T\).