Properties of Signals

Energy and Power

The total energy consumed by $x(t)$ during $t_1\le t\le t_2$ is

$$E={\int }_{t_1}^{t_2}{\left|x(t)\right|}^2\mathrm{d}t⟹P=\frac{E}{t_2-t_1}$$

Similarly, for discrete signal $x[n]$ during $n_1\le n\le n_2$, the energy consumption is

$$E=\sum _{n=n_1}^{n_2}{\left|x[n]\right|}^2⟹P=\frac{E}{n_2-n_1+1}$$

As for infinite case

$$E_{\infty }={\int }_{-\infty }^{\infty }{\left|x(t)\right|}^2\mathrm{d}t⟹P_{\infty }=\underset{T\to \infty }{\lim }\frac{1}{2T}{\int }_{-T}^T{\left|x(t)\right|}^2\mathrm{d}t$$

and

$$E_{\infty }=\sum _{n=-\infty }^{+\infty }{\left|x[n]\right|}^2⟹P_{\infty }=\underset{N\to \infty }{\lim }\frac{1}{2N+1}\sum _{n=-N}^N{\left|x[n]\right|}^2$$