Side-by-side comparison
| Parameter | Power Spectral Density | Energy Spectral Density |
|---|---|---|
| Applicable Signal Type | Power signals — periodic or stationary random (finite average power, infinite energy) | Energy signals — finite-duration or finite-energy signals |
| Mathematical Definition | S_x(f) = lim(T→∞) E[|X_T(f)|²/T] W/Hz | E_x(f) = |X(f)|² J/Hz |
| Total Area Under Curve | Equals average power (watts) | Equals total energy (joules) |
| Fourier Transform Requirement | DTFT or PSD via autocorrelation (Wiener-Khinchin) | Standard Fourier transform exists (signal is square-integrable) |
| Example Signals | AWGN noise, sinusoid, random binary data stream | Single rectangular pulse, Gaussian pulse, finite-length ECG segment |
| Unit | W/Hz or V²/Hz | J/Hz or V²·s/Hz |
| Wiener-Khinchin Theorem | PSD = Fourier transform of autocorrelation R_x(τ) | ESD = |X(f)|² directly |
| Existence Condition | Signal must be wide-sense stationary (WSS) | ∫|x(t)|² dt < ∞ (square-integrable) |
| Practical Tool | Spectrum analyzer displaying dBm/Hz (e.g., Rigol DSA815) | FFT magnitude squared of a captured waveform segment |
| Cross-spectral Variant | Cross-PSD: S_xy(f) = FT of cross-correlation | Cross-ESD: X*(f)·Y(f) |
Key differences
The dividing line is energy: a sinusoid running forever has infinite total energy but finite average power (P = A²/2), so PSD applies. A 10 µs radar pulse has finite total energy, so ESD applies — compute |X(f)|² directly from the Fourier transform. The Wiener-Khinchin theorem links PSD to the autocorrelation function via R_x(τ) ↔ S_x(f), a tool that works only for WSS random processes — it breaks down for transient or deterministic finite-duration signals. The units give you the check: if your spectral measure integrates to watts, it's PSD; if it integrates to joules, it's ESD.
When to use Power Spectral Density
Use PSD for any signal that runs continuously or indefinitely — AWGN channel noise in a 5G NR receiver, thermal noise across a 50 Ω resistor at 290 K (N₀/2 = kT = 4×10⁻²¹ W/Hz), or a periodic clock signal.
When to use Energy Spectral Density
Use ESD for signals with a definite start and end — a single BPSK symbol burst, a captured 5 ms ultrasound echo, or a finite-length ECG heartbeat segment extracted for arrhythmia analysis.
Recommendation
Choose PSD for noise and continuous signals; choose ESD for pulses and transients. The Wiener-Khinchin theorem is your computational shortcut for PSD — never try to Fourier-transform an infinite-duration noise signal directly. Most GATE problems on this topic hinge on correctly identifying whether the signal has finite energy or finite power first.
Exam tip: GATE repeatedly tests Parseval's theorem in both forms: ∫|x(t)|²dt = ∫E_x(f)df for energy signals and (1/T)∫|x(t)|²dt = ∫S_x(f)df for power signals — write both forms and state which applies to which signal type.
Interview tip: Interviewers at communication system companies ask you to compute noise power in a bandwidth B given N₀ — the answer is N₀×B, which comes directly from integrating PSD over the band — show that you know PSD integrates to power, not energy.