Root Mean Square - Definition

Definition

The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean (average) of the squares of the original values (or the square of the function that defines the continuous waveform).

In the case of a set of values, the RMS value is given by this formula:

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

The corresponding formula for a continuous function (or waveform) defined over the interval is

f_{\mathrm{rms}} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {}^2\, dt}},

and the RMS for a function over all time is

f_\mathrm{rms} = \lim_{T\rightarrow \infty} \sqrt {{1 \over {T}} {\int_{0}^{T} {}^2\, dt}}.

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, as shown by Cartwright.

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

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