Energy of a signal

Published May 19, 2022 - By Marco Garosi

You can find an introduction to signals here. This post is part of a series on Image and Signal Processing.

If you are looking for the power of a signal, you may read the dedicated note. Now, let’s dive into the energy of a signal to better understand it.

Energy of a signal

The energy of a signal is defined mathematically as:

E_f = \begin{cases} \int_{-\infty}^{+\infty} f^2(t) dt &\text{if } f \in \R \\ \int_{-\infty}^{+\infty} |f(t)|^2 dt &\text{if } f \in \mathbb{C} \end{cases}

Energy of a signal

Note that in the latter case $|f(t)|^2 = f(t) \bar{f(t)}$ , where $\bar{f(t)}$ is the complex conjugate of $f(t)$ .

The definition thus varies slightly depending on the type of signal: the ways you compute the energy of real signals and complex signals are a little different. You may want to read more about complex numbers.

A signal is said to be an energy signal (or with finite energy) if the integral $E_f$ converges and is not $0$ .

This condition is sufficient to ensure that $f$ has a Fourier Transform (though it is not necessary).
It tends to $0$ to infinity: $\lim_{t \to \infty} f(t) = 0$ .
Energy is measured in Joules [J].

If $f$ is a power signal then it cannot be an energy signal (and vice-versa). Some signals are not power nor energy signals — so beware!