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

Power of a signal

The power of a signal is defined mathematically as:

Pf={limT+1TT2T2f2(t)dtif fRlimT+1TT2T2f(t)2dtif fC P_f = \begin{cases} \lim_{T \to +\infty} \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} f^2(t) dt &\text{if } f \in \R \\ \lim_{T \to +\infty} \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} |f(t)|^2 dt &\text{if } f \in \mathbb{C} \end{cases}

Power 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 power 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 a power signal (or with finite power) if the integral PfP_f converges and is not 00.

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

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