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:

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N=100; %length of the signal x=randn(1,N); %a random signal to test X=fft(x); %Frequency domain representation of the signal RMS1 = sqrt(mean(x.^2)) %RMS value from time domain samples RMS2 = sqrt(sum(abs(X/N).^2)) %RMS value from frequency domain representation %Result: RMS1 - RMS2 = 1.1102e-16 |

### 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.