Energy signal vs power signal

Signals fall into two mutually exclusive useful 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.

For the purposes of this course, every signal we care about is either an energy signal or a power signal, and which one 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.