RMS

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In the case of a set of <math>n</math> values <math>\{x_1,x_2,\dots,x_n\}</math>, the RMS value is given by this formula:

<math>

x_{\mathrm{rms}} = \sqrt{ \frac{1}{n} \left( x_1^2 + x_2^2 + \cdots + x_n^2 \right) } </math>

The corresponding formula for a continuous function (or waveform) <math>f(t)</math> defined over the interval <math>T_1 \le t \le T_2</math> is

<math>

f_{\mathrm{rms}} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {[f(t)]}^2\, dt}}, </math>

and the RMS for a function over all time is

<math>

f_\mathrm{rms} = \lim_{T\rightarrow \infty} \sqrt {{1 \over {T}} {\int_{0}^{T} {[f(t)]}^2\, dt}}. </math>

The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a series of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus.

In the case of the RMS statistics of a random process, the expected value is used instead of the mean.

See also