Energy x(t)
The definition of energy E for a signal x(t) is as follows :
\(E=\int_{-\infty}^{+\infty}{\left|x(t)\right|^2dt}\)
Definition :
A signal is said to have non-zero finite energy if its energy remains finite as the interval considered varies from -∞ to +∞.
E\(=\int_{-\infty}^{+\infty}{\left|x(t)\right|^2dt}<\infty\)
The energy is always positive.
Example :
\(f\left(t\right)=Ae^{-\alpha\left|t\right|},\ \alpha>0\)
\(E=\int_{-\infty}^{+\infty}{\left|Ae^{-\alpha\left|t\right|}\right|^2dt=A^2\int_{-\infty}^{0}{e^{2\alpha t}dt}+A^2\int_{0}^{+\infty}{e^{-2\alpha t}dt}}\)
\(E=\frac{A^2}{\alpha}\) \(\implies\) A signal with finite energy
Observation :
All signals with finite energy, whether deterministic or random, are of the transient type.