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Signal Energy and Power

Basic Definition

Given the continuous time signal \(x(t)\) and discrete time signal \(x[n]\).

The instantaneous signal power is defined as

\[ \begin{align*} P(t) &= |x(t)|^2 \\ P[n] &= |x[n]|^2 \end{align*} \]

The signal energy is defined as

\[ \begin{align*} E(t_0, t_1) &= \int_{t_0}^{t_1} |x(t)|^2 \,\mathrm{d}t \\ E(n_0, n_1) &= \sum_{n=n_0}^{n1} |x[n]|^2 \end{align*} \]

The average signal power is defined as

\[ \begin{align*} P(t_0, t_1) &= \frac{1}{t_1-t_0} \int_{t_0}^{t_1} |x(t)|^2 \,\mathrm{d}t \\ P(n_0, n_1) &= \frac{1}{n_1-n_0+1} \sum_{n=n_0}^{n1} |x[n]|^2 \end{align*} \]

Usually, the limits are taken over an infinite time interval:

\[ \begin{align*} E_\infty &= \int_{-\infty}^{\infty} |x(t)|^2 \,\mathrm{d}t \\ E_\infty &= \sum_{n=-\infty}^{\infty} |x[n]|^2 \\ P_\infty &= \lim_{T \rightarrow \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 \,\mathrm{d}t\\ P_\infty &= \lim_{N \rightarrow \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2 \end{align*} \]

Reference