Root-Mean-Square

📐 Definition

For samples $\{x_i\}_{i=1}^{N}$ ,

\operatorname{RMS}(x) = \sqrt{\frac{1}{N}\sum_{i=1}^{N} x_i^{2}}

For complex data, RMS uses $|x_i|^{2}$ inside the sum.

Requires finite second moment. Output is nonnegative real.

Always satisfies $\operatorname{RMS}(x) \ge |\mu|$ (Cauchy–Schwarz). Invariance to sign flips. Scales linearly with scalar multiplication.

If all samples equal $c$ , RMS equals $|c|$ . For zero-mean signals, $\operatorname{RMS}(x) = \sqrt{\sigma^{2}}$ .

Constant samples $x_i=c$ give $\operatorname{RMS}=|c|$ .
Zero-mean signals satisfy $\operatorname{RMS}^2=\sigma^2$ (under matching conventions).

Mean and variance provide complementary first and second moments; energy norms extend RMS to continuous fields.

Oakfield reports RMS magnitude directly in its field statistics:

SimFieldStats.rms is computed as $\sqrt{\mathrm{mean}(|u|^2)}$ for real or complex fields.
Used as a scale indicator for visualization, threshold selection, and stability monitoring.

RMS is a standard quadratic average used in physics and engineering to measure effective magnitude of oscillatory signals.

It is widely used as an energy-proportional magnitude measure and as a normalization target in numerical pipelines.