# What kind of signal is the signum function? A power signal or an energy signal?

I'm studying "signals and systems". We can indentify two kinds of signals: power signal and energy signal.

The definitions are:

Energy of a time-continous signal $$E_x = \int_{-\infty}^{+\infty}{|x(t)|}^2dt$$

Power of a time-continous signal $$P_x = \lim_{Z\rightarrow+\infty}{\frac{1}{2Z}\int_{-Z}^{+Z}{|x(t)|}^2dt}$$

An energy signal satisfies $0<E_x<+\infty$ and $P_x=0$.

A power signal satisfies $0<P_x<+\infty$ and $E_x = +\infty$.

Now, if we have a signum function

$$sgn(t)=\left\{\begin{matrix} 1 & if\;t \ge 0 \\ -1 & if\;t < 0 \end{matrix}\right.$$

and we calculate the formulas, then we obtain:

$$P_x = 0 \;\; E_x=0$$

My question is, as the title said: the signum function which kind of signal is?

• signum is hardly time continuous. (massive jump at t = 0) – ratchet freak Sep 3 '15 at 10:01
• Also the vertical bars indicate absolute amplitude which is a constant 1 which makes $E_{sgn} = +\infty$ and $P_{sgn} = 1$ – ratchet freak Sep 3 '15 at 10:50
• Correct! I forgot the absolute value! Thank you for answering anyway to my stupid question! I will pay more attention next time. – GenKs Sep 3 '15 at 11:02
• @ratchetfreak That function is time continuous. That means there are no discontinuities in time. – Greg Schmit Feb 6 '17 at 0:50

First point is that signum is not continuous with the jump at $t=0$.
Second because of the absolute value in the integral it has the same $E$ and $P$ as a constant signal $1$.
This means that $E_{sgn}=+\infty$ and $P_{sgn}=1$