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:

Ef={+f2(t)dtif fR+f(t)2dtif fC 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)f(t)ˉ|f(t)|^2 = f(t) \bar{f(t)}, where f(t)ˉ\bar{f(t)} is the complex conjugate of f(t)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 EfE_f converges and is not 00.

  • This condition is sufficient to ensure that ff has a Fourier Transform (though it is not necessary).
  • It tends to 00 to infinity: limtf(t)=0\lim_{t \to \infty} f(t) = 0.
  • Energy is measured in Joules [J].

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

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