Energy Signals and Power Signals

Introduction

Every signal falls into one of three categories based on its energy and power characteristics: energy signals (finite energy, zero average power), power signals (infinite energy, finite non-zero average power), or neither. This classification is not merely academic — it determines which analysis tools apply. Energy signals use the energy spectral density and their Fourier Transforms exist in the conventional sense. Power signals use the power spectral density and require statistical or time-averaged descriptions.

Understanding this distinction helps you choose the right framework for analysis: a single pulse is an energy signal (use Fourier Transform directly), but an ongoing periodic waveform or random noise is a power signal (use PSD and autocorrelation).

The physical intuition is straightforward. An energy signal exists for a limited time or decays sufficiently fast — it delivers a finite total amount of energy to a $1\,\Omega$ resistor. A power signal persists forever with sustained amplitude — it delivers infinite total energy but at a finite rate (finite watts). The classification determines whether you think in terms of total joules or average watts per hertz.

Definitions

Signal Energy

$$E = \int_{-\infty}^{\infty}|x(t)|^2 dt \quad \text{(continuous-time)}$$ $$E = \sum_{n=-\infty}^{\infty}|x[n]|^2 \quad \text{(discrete-time)}$$

The energy represents the total area under the squared magnitude of the signal. For electrical signals across a $1\,\Omega$ resistor, this gives energy in joules. The $|x(t)|^2$ accounts for complex-valued signals.

Signal Power (Time-Average)

$$P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2 dt$$ $$P = \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^2$$

Power is the time-averaged energy rate — how much energy the signal delivers per unit time, averaged over all time. For periodic signals, this simplifies to averaging over one period: $$P = \frac{1}{T_0}\int_0^{T_0}|x(t)|^2 dt$$

Classification

Important: A signal cannot be both an energy signal and a power signal. If energy is finite, power must be zero (the finite energy averages to zero over infinite time). If power is finite and non-zero, energy must be infinite. This mutual exclusivity is a direct mathematical consequence:

If $E < \infty$: $P = \lim_{T\to\infty}\frac{E}{2T} = 0$

If $0 < P < \infty$: $E = \lim_{T\to\infty} 2T \cdot P = \infty$

Examples with Detailed Computation

Energy Signals

Rectangular pulse: $x(t) = A$ for $|t| < T/2$, zero otherwise. $$E = \int_{-T/2}^{T/2}A^2\,dt = A^2 T \quad \text{(finite)}$$ $$P = \lim_{T_0\to\infty}\frac{A^2 T}{2T_0} = 0$$

Decaying exponential: $x(t) = e^{-at}u(t)$ with $a > 0$. $$E = \int_0^{\infty}e^{-2at}dt = \frac{1}{2a} \quad \text{(finite)} \qquad P = 0$$

Gaussian pulse: $x(t) = e^{-\pi t^2}$ $$E = \int_{-\infty}^{\infty}e^{-2\pi t^2}dt = \frac{1}{\sqrt{2}} \quad \text{(finite)} \qquad P = 0$$

Any absolutely integrable signal: If $\int|x(t)|dt < \infty$, then by Schwarz's inequality $E < \infty$ (energy signal).

Triangular pulse: $x(t) = (1 - |t|/T)$ for $|t| \leq T$, zero otherwise. $$E = \int_{-T}^{T}\left(1 - \frac{|t|}{T}\right)^2 dt = \frac{2T}{3}$$

Power Signals

Sinusoid: $x(t) = A\cos(\omega_0 t)$. $$E = \int_{-\infty}^{\infty}A^2\cos^2(\omega_0 t)dt = \infty$$ $$P = \frac{1}{T_0}\int_0^{T_0}A^2\cos^2(\omega_0 t)dt = \frac{A^2}{2} \quad \text{(finite, non-zero)}$$

Constant signal: $x(t) = c$. $$E = \infty \qquad P = c^2$$

Square wave with amplitude $\pm A$ and period $T_0$: $$P = \frac{1}{T_0}\int_0^{T_0}A^2\,dt = A^2$$

Periodic signal with period $T_0$: $P = \frac{1}{T_0}\int_0^{T_0}|x(t)|^2 dt$ (always finite if signal is bounded).

Random noise (stationary): Has finite average power determined by its variance. For zero-mean noise with variance $\sigma^2$: $P = \sigma^2$.

Unit step: $x(t) = u(t)$. $$P = \lim_{T\to\infty}\frac{1}{2T}\int_0^T 1\,dt = \lim_{T\to\infty}\frac{T}{2T} = \frac{1}{2}$$

Neither Energy nor Power

Ramp: $x(t) = t \cdot u(t)$. $$E = \int_0^{\infty}t^2\,dt = \infty$$ $$P = \lim_{T\to\infty}\frac{1}{2T}\int_0^T t^2\,dt = \lim_{T\to\infty}\frac{T^2}{6} = \infty$$

Growing exponential: $x(t) = e^{t}u(t)$. Both are infinite because the signal grows without bound.

Criterion: A signal is "neither" when its amplitude grows without limit (unbounded). Such signals are physically unrealizable — they require infinite power sources.

Spectral Representations

The distinction determines which spectral tool to use:

Energy signals → Energy Spectral Density (ESD): $\Psi_{xx}(\omega) = |X(j\omega)|^2$

• Parseval's theorem: $E = \frac{1}{2\pi}\int|X(j\omega)|^2 d\omega$
• $\Psi_{xx}(\omega)\,d\omega$ gives the energy in the infinitesimal band $[\omega, \omega+d\omega]$

Power signals → Power Spectral Density (PSD): $S_{xx}(\omega)$

• Power: $P = \frac{1}{2\pi}\int S_{xx}(\omega) d\omega$
• Obtained via Wiener-Khinchin theorem from autocorrelation
• $S_{xx}(\omega)\,d\omega$ gives the power in the infinitesimal band $[\omega, \omega+d\omega]$

Parseval's Theorem

For energy signals: $$E = \int_{-\infty}^{\infty}|x(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|X(j\omega)|^2 d\omega$$

This states that energy can be computed equivalently in time or frequency domains. The quantity $|X(j\omega)|^2$ is the energy spectral density — it shows how signal energy is distributed across frequencies.

Example: For $x(t) = e^{-3t}u(t)$, $X(j\omega) = \frac{1}{3+j\omega}$. $$E = \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{1}{9+\omega^2}\,d\omega = \frac{1}{2\pi}\cdot\frac{\pi}{3} = \frac{1}{6}$$

Verification: $E = \int_0^{\infty}e^{-6t}\,dt = \frac{1}{6}$ ✓

Bandwidth Considerations

Energy signals have a well-defined bandwidth related to where most of their spectral energy is concentrated. Common bandwidth definitions include:

• 3-dB bandwidth: frequency where $|X(j\omega)|^2$ drops to half its peak value
• Equivalent noise bandwidth: width of a rectangular spectrum with the same peak value and total area
• 99% energy bandwidth: frequency range containing 99% of total signal energy

Power signals use PSD-based bandwidth definitions, often specifying the frequency range where most average power resides.

Practical Implications

Finite-duration signals are always energy signals: Since $x(t) = 0$ outside a finite interval, the energy integral has finite limits and converges.

Periodic signals are always power signals: They repeat forever (infinite energy) but their power equals the average over one period.

Transient analysis vs steady-state analysis: Energy signals are analyzed with Fourier/Laplace Transforms directly. Power signals require PSD, autocorrelation, or Fourier Series (if periodic).

Mixed scenarios: In practice, signals often contain both transient (energy) and steady-state (power) components. For example, the startup of an oscillator has a transient that is an energy signal, while the steady-state oscillation is a power signal. Analysis must handle each component appropriately.

Key Takeaways

• Energy signal: $0 < E < \infty$, $P = 0$ — use Fourier Transform and energy spectral density
• Power signal: $E = \infty$, $0 < P < \infty$ — use PSD and autocorrelation
• A signal cannot be both energy and power simultaneously (mutual exclusivity)
• Finite-duration signals → energy signals; periodic signals → power signals
• Growing signals (unbounded amplitude) are neither energy nor power
• Parseval's theorem connects time-domain energy to frequency-domain energy spectral density
• The classification determines which spectral analysis framework applies
• For periodic signals: $P = \frac{1}{T_0}\int_0^{T_0}|x(t)|^2\,dt$ (average over one period)