Side-by-side comparison
| Parameter | Energy | Power Signal |
|---|---|---|
| Total energy E | 0 < E < ∞ | E = ∞ |
| Average power P | P = 0 (E is finite over all time) | 0 < P < ∞ |
| Duration | Finite duration or rapidly decaying (e.g., exponential) | Infinite duration — exists for all time |
| Typical waveforms | Rectangular pulse, Gaussian pulse, decaying exponential e^{−at}u(t) | Sinusoids, square waves, periodic signals, white noise |
| Energy formula (CT) | E = ∫_{-∞}^{∞} |x(t)|² dt | E → ∞; use P = lim_{T→∞} (1/2T) ∫_{-T}^{T} |x(t)|² dt |
| Energy formula (DT) | E = Σ_{n=−∞}^{∞} |x[n]|² | P = lim_{N→∞} (1/2N+1) Σ_{n=−N}^{N} |x[n]|² |
| Fourier Transform exists? | Yes, always (finite energy → square-integrable) | Exists in the generalised sense using impulses (δ functions) |
| Real example | Single EEG spike, one radar pulse from a pulsed radar | |
| Continuous carrier in AM radio, 230 V AC mains supply | ||
| Neither energy nor power? | Does not apply | Signals like x(t) = t (ramp) are neither energy nor power |
| GATE test | Compute E; if finite, it is an energy signal | If E = ∞, compute P; if finite, it is a power signal |
Key differences
If ∫|x(t)|² dt converges to a finite number, the signal is an energy signal and its average power is zero. If that integral diverges but the time-averaged version (1/2T)∫|x|² dt converges as T → ∞, it is a power signal. A decaying exponential Ae^{−at}u(t) with a > 0 has energy A²/2a — finite, so it is an energy signal. A unit step u(t) has infinite energy but average power of 0.5 W (normalised), so it is a power signal. The ramp x(t) = t is neither: both E and P diverge.
When to use Energy
Classify a signal as an energy signal when it is a transient — for example, a single radar pulse of amplitude 100 V lasting 1 µs has energy E = (100)² × 1×10⁻⁶ = 0.01 J, clearly finite.
When to use Power Signal
Classify a signal as a power signal when it is periodic or stationary random — for example, a 230 V rms, 50 Hz sinusoidal mains supply delivers 230²/R watts continuously and has infinite total energy.
Recommendation
For every S&S exam, choose the energy signal classification when the signal has finite duration or decays to zero. Choose power signal when the waveform is periodic or has no decay. Calculate E first; if it diverges, calculate P. Never guess from the shape alone.
Exam tip: Examiners give x(t) = e^{−3t}u(t) and ask energy — compute ∫₀^∞ e^{−6t} dt = 1/6 J; also state that P = 0 since E is finite — both answers are expected.
Interview tip: Interviewers at signal processing companies ask why Fourier Transform of a power signal needs impulse functions — explain that infinite energy means the signal is not square-integrable, so we use the generalised FT with δ(f − f0) for sinusoids.