Which signal type is each density used for?

ESD for energy signals (finite E), PSD for power signals (finite P)

ESD for power signals, PSD for energy signals

Which signal has an ESD and which has a PSD?

e^(-t)u(t) has ESD 1/(1+ω^2); cos(ω0 t) has impulsive PSD

cos(ω0 t) has ESD; e^(-t)u(t) has PSD

For real signals both densities are:

Real, non-negative and even, with phase lost

2080 Chaitra · 4 marks

Energy vs power spectral density

Short answer

Energy spectral density Ψ(ω)=|X(jω)|^2 applies to energy signals; total energy E=(1/2π)∫Ψ dω, unit J/Hz. Power spectral density S(ω)=lim_{T→∞}|X_T(jω)|^2/T applies to power signals; total power P=(1/2π)∫S dω, unit W/Hz. Both are real, non‑negative and even for real signals.

Differentiate between energy and power spectral density.

Energy spectral density (ESD) shows how the energy of an energy signal is distributed over frequency, while power spectral density (PSD) shows how the average power of a power signal is distributed over frequency.

Point	Energy spectral density	Power spectral density
Used for	Energy signals ( $0 < E < \infty$ , $P = 0$ )	Power signals ( $0 < P < \infty$ , $E = \infty$ )
Definition	$\Psi(\omega) = \lvert X(j\omega)\rvert^2$	$S(\omega) = \lim_{T\to\infty}\frac{\lvert X_T(j\omega)\rvert^2}{T}$
Total quantity	$E = \frac{1}{2\pi}\int \Psi(\omega)d\omega$	$P = \frac{1}{2\pi}\int S(\omega)d\omega$
Unit	J/Hz	W/Hz
FT pair with	Energy autocorrelation $R(\tau)=\int x(t)x(t+\tau)dt$	Power autocorrelation $R(\tau)=\lim\frac{1}{T}\int_{-T/2}^{T/2} x(t)x(t+\tau)dt$
Typical signals	Pulses, decaying exponentials	Periodic signals, DC, random noise
Periodic signal case	Not defined	Impulses: $2\pi\sum \lvert a_k\rvert^2\delta(\omega-k\omega_0)$

Both are real, non-negative and even in $\omega$ for real signals, and both lose phase information.

Example: $x(t) = e^{-t}u(t)$ is an energy signal with ESD $\frac{1}{1+\omega^2}$ ; $x(t) = \cos\omega_0 t$ is a power signal with PSD $\frac{\pi}{2}[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$ .

✓ Answer checked by Claude Opus 5.5 (high effort)

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