Signals are classified as either finite-energy or finite-power based on whether their total energy or average power is bounded. Energy signals have zero average power and decay to zero, while power signals have non-zero average power and may be periodic or persist indefinitely. This classification determines appropriate analysis methods and norms.
Start with simple periodic signals (sine waves) to compute average power, then examine exponential pulses to understand energy signals. Compare total energy calculations using integration.
When you measure how much "stuff" a signal carries, you have two fundamentally different ways to count. The first is to add up the total contribution over all time — this is signal energy, defined as E = ∫|x(t)|² dt from −∞ to ∞ (or the discrete-time sum Σ|x[n]|²). The second is to ask about the long-run rate of contribution — this is average power, P = lim(T→∞) (1/2T) ∫|x(t)|² dt. A signal is an energy signal if E is finite (and therefore P = 0), and a power signal if P is finite and nonzero (meaning E = ∞). These two classes are mutually exclusive and together cover most signals of practical interest.
The intuition is straightforward when you think about decaying versus persistent signals. A single pulse that goes to zero as t→∞ — like a decaying exponential e^(−at)u(t) — concentrates all its energy in a finite region of time. You can integrate it out and get a number. Its average power over all of infinite time averages to zero, because the energy gets spread over an ever-growing interval. Contrast this with a sine wave sin(2πft): it never dies out. If you tried to compute its total energy, the integral diverges. But if you compute its average power over one period and then ask "what does this look like over a long time?", you get exactly 1/2 — a stable, nonzero number. The sine wave is a power signal.
A useful check is the behavior at infinity: if x(t) → 0 as |t| → ∞ fast enough that the area under |x(t)|² converges, you are looking at an energy signal. If x(t) persists — constant amplitude, periodic, or stationary random — you are looking at a power signal. This distinction directly controls which analysis tools apply. Energy signals fit naturally into the Fourier transform framework (the transform exists in a classical sense if energy is finite); power signals require the power spectral density framework using autocorrelation and limiting arguments. Using the wrong tool — trying to take the Fourier transform of a sine wave directly — produces distributional objects (Dirac deltas) that require careful handling.
It is worth noting that not every signal fits into either class. A signal like x(t) = t grows without bound, making both its total energy and its average power infinite — neither category applies. Such signals arise in unstable systems and require separate treatment. The classification also extends naturally to discrete-time signals: a finite-length sequence always has finite energy (and is therefore an energy signal), while an infinite periodic sequence like cos[ωn] is a power signal. This framework gives you the vocabulary and the boundary conditions for every signal-processing technique that follows — Fourier analysis, filtering, correlation, and spectral estimation all depend on knowing which class of signal you are working with.