Significance of RMS (Root Mean Square) value

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Loading... Root Mean Square (RMS) value is the most important parameter that signifies the \(size \; of \; a \;signal\).

Defining the term “size”:

In signal processing, a signal is viewed as a function of time. The term “size of a signal” is used to represent “strength of the signal”. It is crucial to know the “size” of a signal used in a certain application. For example, we may be interested to know the amount of electricity needed to power a LCD monitor as opposed to a CRT monitor. Both of these applications are different and have different tolerances. Thus the amount of electricity driving these devices will also be different.

A given signal’s size can be measured in many ways. Some of them are,

  • Total energy
  • Square root of total energy
  • Integral absolute value
  • Maximum or peak absolute value
  • Root Mean Square (RMS) value
  • Average Absolute (AA) value

RMS value

RMS value of a signal (\(x(t)\)) is calculated as the square root of average of squared value of the signal, mathematically represented as $$E_{RMS} = \sqrt{ \frac{1}{T} \int_{0}^{T} x(t)^2 dt} $$ For a signal represented as \(N\) discrete sampled values – \([x_0,x_1,\cdots,x_{N-1}]\), the RMS value is given as $$E_{RMS} = \sqrt{\frac{x_0^2+x_1^2+\cdots+x_{N-1}^2}{N}} $$ If the signal can be represented in Frequency domain as \(X(f)\), then as a result of Parseval’s theorem, the RMS value can be calculated as $$E_{RMS} = \sqrt{\sum \left| \frac{X(f)}{N} \right|^2}$$

Implementing in Matlab:

Significance of RMS value

  •  One of the most important parameter that is used to describe the strength of an Alternating Current (AC)
  • RMS value of an AC voltage/current is equivalent to the DC voltage/current that produces the same heating effect when applied across an identical resistor. Hence, it is also a measure of energy content in a given signal.
  • In statistics, for any zero-mean random stationary signal, the RMS value is same as the standard deviation of the signal. Example : \(Delay \; spread \) of a multipath channel is often calculated as the RMS value of the \(Power \; Delay \; Profile \) (PDP)
  • When two uncorrelated (or orthogonal ) signals are added together, such as noise from two independent sources, the RMS value of their sum is equal to the square-root of sum of the square of their individual RMS values.

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See also

Basics of – Power and Energy of a signal
Calculation of power of a signal and verifying it through Matlab.

Recommended Books on Signal Processing

  • Dilgush T I

    Yes, signals with no dc component or offset will have standard deviation same as the rms value.
    But many measuring instruments give standard deviation for the rms value, which is incorrect.

  • riaban

    Is RMS value of a signal same as its standard deviation?I have seen these two words being used interchangeably!

    • RMS value will be equal to standard deviation ONLY when the signal has zero-mean. See point number 3 under ‘significance of RMS values’ section in the above article. You could also relate the equations for calculating the RMS values and the equation for calculating the standard deviation to arrive at this conclusion.