Power of x(t)
The normalised (average) power of a signal is defined as the time average of the signal's energy.
\(P=\lim\limits_{x \to \infty} \frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}{\left|x(t)\right|^2dt}\)
If x(t) is periodic with a period T, the average signal power, is the average energy per period, is given by :
\(P=\frac{1}{T}{\int_{-\frac{T}{2}}^{\frac{T}{2}}{\left|x(t)\right|^2dt}}\)
Definition :
A signal x(t) is said to have finite non-zero power if the previous equation remains finite from 0 to +∞.
\(0<P=\lim\limits_{x \to \infty}\frac{1}{T_0}{\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}{\left|x(t)\right|^2dt}<}\infty\)
Observation :
Signals with finite average power are not physically realisable.
Example :
Consider a signal x(t) = 5V for -∞ < t < +∞. Is it a power or energy signal ?
\(E=\int_{-\infty}^{+\infty}{5^2dt=\infty}\)
\(P=\lim\limits_{x \to \infty}{\frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}{5^2dt}}=\lim\limits_{x \to \infty}{\frac{1}{T_0}25\left[T_0\right]}=25\ W \implies\) it is a power signal.