Signal power vs energy — formulas and Matlab/Python examples

Energy & Spectra (same on every page in this path)

1. Power & energy (you are here) → 2. Parseval → 3. RMS value

In one sentence: Energy signals have finite $latex \int|x|^2$; power signals have finite average power — pick the right definition before you normalize a BER or PSD sim.

Matlab/Python simulation: Compute energy/power on sampled vectors, then connect to RMS and Parseval checks in the Energy & Spectra path.

Search traffic for “power of a signal” and “energy of a signal” wants the formulas, the energy-vs-power classification, and a quick way to compute them. That is what follows.

Why not “area under the curve”?

In DSP we treat a signal as a function of time and ask for its size (strength). Signed area under $latex x(t)$ can cancel. Absolute area works but is awkward analytically. Squared magnitude $latex |x|^2$ matches an $latex L_2$ (Euclidean) notion of size used throughout detection and estimation — so that is the standard choice.

Energy of a signal

For a continuous-time complex signal,

\[E_x = \int_{-\infty}^{\infty} |x(t)|^2\,dt\]

If $latex x$ is real, $latex |x(t)|^2=x(t)^2$. This “energy” is a signal-processing measure of strength. When $latex x$ is a voltage across a $latex 1\,\Omega$ resistor it coincides with physical energy in joules; otherwise scale by the load (below).

Physical energy into a load $latex Z$

Voltage signal into impedance $latex Z$:

\[E = \frac{E_x}{Z} = \frac{1}{Z}\int_{-\infty}^{\infty} |x(t)|^2\,dt\]

Current signal into the same load:

\[E = Z\,E_x = Z\int_{-\infty}^{\infty} |x(t)|^2\,dt\]

Discrete-time energy

\[E_x = \sum_{n=-\infty}^{\infty} |x[n]|^2\]

The sum must converge: the sequence is square-summable. That is a finite-energy (energy) signal. An everlasting sinusoid is not square-summable — its energy is infinite — so we switch to power.

Power of a signal

Power is energy per unit time. For signals whose total energy diverges, take a long window, compute energy inside it, divide by the window length, then let the window grow:

\[P_x = \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^2\]

Equivalent textbook forms (different indexing of the same average):

\[\begin{aligned} P_x &= \lim_{N\to\infty}\frac{1}{2N}\sum_{n=-N}^{N-1}|x[n]|^2 \\ P_x &= \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}|x[n]|^2 \\ P_x &= \lim_{N\to\infty}\frac{1}{N_1-N_0+1}\sum_{n=N_0}^{N_1}|x[n]|^2 \end{aligned}\]

In continuous time the same idea is $latex P_x=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2\,dt$. For a real cosine $latex A\cos(\omega t+\phi)$, $latex P_x=A^2/2$.

Energy signal vs power signal

Energy signal pulse with finite energy versus everlasting sinusoid power signal — Figure 1: (a) Decaying pulse — finite $latex E_x$, zero average power. (b) Everlasting sinusoid — infinite energy, finite $latex P_x$.

Energy signal: $latex 0 < E_x < \infty$. Then $latex P_x=0$ (finite energy diluted over infinite time).
Power signal: $latex 0 < P_x < \infty$. Then $latex E_x=\infty$ (e.g. periodic sinusoids, everlasting DC).
Neither: both diverge — e.g. $latex x[n]=n$.
Never both: finite nonzero energy and finite nonzero power cannot hold together under these definitions.

Python: energy of a pulse, power of a sine

import numpy as np

# --- Energy of a finite exponential pulse ---
n = np.arange(0, 200)
x_pulse = np.exp(-0.05 * n)
E = np.sum(np.abs(x_pulse) ** 2)
print(f"Pulse energy E_x ≈ {E:.4f}  (finite)")

# --- Power of a cosine (use a long window) ---
A, N = 1.0, 100_000
n = np.arange(N)
x_cos = A * np.cos(2 * np.pi * 0.01 * n)
P = np.mean(np.abs(x_cos) ** 2)
print(f"Cosine power P_x ≈ {P:.6f}  (theory A²/2 = {A**2/2})")

Step-by-step Matlab verification of the power formulas: Computation of power of a signal in Matlab. For random noise, see simulation and analysis of white noise.

Start of path Next: Parseval →

FAQ

What is the difference between signal energy and average power? Energy is the integral (or sum) of $latex |x|^2$ over the whole support of a finite-energy signal. Average power is that energy divided by the observation length (or the limit of the same ratio for perpetual power signals). A finite pulse has finite energy and zero average power over infinite time; a never-ending sinusoid has finite average power and infinite energy.

How does Parseval relate to these definitions? Parseval’s theorem says you can compute the same energy (or average power, with consistent scaling) from the time waveform or from the spectrum. That is why window energy $latex \sum w^2$ appears in ENBW formulas — it is the frequency-domain noise bandwidth written in time-domain samples.

Which definition should I use in a Matlab/Python simulation? Match the definition to the question: use a sum of $|x[n]|^2$ for energy of a recorded block, divide by $latex N$ (or by $latex N/f_s$ if you want physical watts for a given impedance) for average power, and document whether your FFT is unitary so Parseval checks line up.

Reference

[1] Sanjay Lall, “Norm and Vector spaces,” Information Systems Laboratory, Stanford University.