Energy signal vs power signal

Signals split into two mutually exclusive categories, based on whether energy or average power gives a finite nonzero measure of their “size.”

• Energy signal: finite nonzero energy $E_x={\int }_{-\infty }^{\infty }|x(t){|}^{2}dt$. Examples: any rectangular pulse, triangular pulse, decaying exponential, anything that goes to zero fast enough at $\pm \infty$ for the integral to converge.
• Power signal: finite nonzero average power $P_x={\lim }_{T\to \infty }(1/T){\int }_{-T/2}^{T/2}|x(t){|}^{2}dt$. Examples: sinusoids, any periodic signal, the unit step.

Why these categories don’t overlap

If a signal has finite energy, then dividing by an unbounded interval makes its average power go to zero — so it’s not a power signal (which requires nonzero power).

If a signal has finite nonzero power, then integrating over an unbounded interval gives infinity for the energy — so it’s not an energy signal.

The two categories are exclusive: each signal is at most one of them.

Signals that are neither

The dichotomy isn’t exhaustive. Some signals have neither finite energy nor finite power: an exponentially growing signal $e^t$ has infinite energy ($e^{2t}$ integrated to $\infty$) and infinite power (the average grows without bound). We can’t measure its “size” with either definition.

Almost every signal we deal with is either an energy signal or a power signal, and which one it is tells you which measure to use.

How to tell at a glance

• Decays to zero (faster than $1/\sqrt{t}$ tails) at both $\pm \infty$ → probably an energy signal.
• Persists or repeats forever without dying out → probably a power signal.
• Grows without bound → probably neither.

If in doubt, compute the energy integral. If it diverges, compute the power limit instead.