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
Parseval’s theorem
The Parseval’s theorem expresses the energy of a signal in time-domain in terms of the average energy in its frequency components.
Suppose if the x[n] is a sequence of complex numbers of length N : xn={x0,x1,…,xN-1}, its N-point discrete Fourier transform (DFT): Xk={X0,X1,…,XN-1} is given by
![X[k] = \displaystyle{\sum_{n=0}^{N-1} x[n] e^{-j\frac{2 \pi}{N} k n}}](https://s0.wp.com/latex.php?latex=X%5Bk%5D+%3D+%5Cdisplaystyle%7B%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+x%5Bn%5D+e%5E%7B-j%5Cfrac%7B2+%5Cpi%7D%7BN%7D+k+n%7D%7D+&bg=ffffff&fg=000&s=1&c=20201002)
The inverse discrete Fourier transform is given by
![\tilde{x}[n] = \displaystyle{ \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j \frac{2 \pi}{N} kn}}](https://s0.wp.com/latex.php?latex=%5Ctilde%7Bx%7D%5Bn%5D+%3D+%5Cdisplaystyle%7B+%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+X%5Bk%5D+e%5E%7Bj+%5Cfrac%7B2+%5Cpi%7D%7BN%7D+kn%7D%7D+&bg=ffffff&fg=000&s=1&c=20201002)
Suppose if x[n] and y[n] are two such sequences that follows the above definitions, the Parseval’s theorem is written as
![\boxed{ \displaystyle{\sum_{n=0}^{N-1} x[n] y^{\ast}[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] Y^{\ast}[k]}}](https://s0.wp.com/latex.php?latex=%5Cboxed%7B+%5Cdisplaystyle%7B%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+x%5Bn%5D+y%5E%7B%5Cast%7D%5Bn%5D+%3D+%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+X%5Bk%5D+Y%5E%7B%5Cast%7D%5Bk%5D%7D%7D+&bg=ffffff&fg=000&s=1&c=20201002)
where,
Deriving Parseval’s theorem
![\begin{aligned} \sum_{n=0}^{N-1} x[n] y^{\ast}[n] &= \sum_{n=0}^{N-1} x[n] \left(\frac{1}{N} \sum_{k=0}^{N-1} Y[k] e^{j\frac{2 \pi}{N} k n} \right )^\ast \\ &= \frac{1}{N}\sum_{n=0}^{N-1} x[n] \sum_{k=0}^{N-1} Y^\ast[k] e^{-j\frac{2 \pi}{N} k n} \\ &= \frac{1}{N} \sum_{k=0}^{N-1} Y^\ast[k] \cdot \sum_{n=0}^{N-1} x[n] e^{-j\frac{2 \pi}{N} k n} \\ &= \frac{1}{N} \sum_{k=0}^{N-1} X[k] Y^\ast[k] \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D+%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+x%5Bn%5D+y%5E%7B%5Cast%7D%5Bn%5D+%26%3D+%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+x%5Bn%5D+%5Cleft%28%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+Y%5Bk%5D+e%5E%7Bj%5Cfrac%7B2+%5Cpi%7D%7BN%7D+k+n%7D+%5Cright+%29%5E%5Cast+%5C%5C+%26%3D+%5Cfrac%7B1%7D%7BN%7D%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+x%5Bn%5D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+Y%5E%5Cast%5Bk%5D+e%5E%7B-j%5Cfrac%7B2+%5Cpi%7D%7BN%7D+k+n%7D+%5C%5C+%26%3D+%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+Y%5E%5Cast%5Bk%5D+%5Ccdot+%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+x%5Bn%5D+e%5E%7B-j%5Cfrac%7B2+%5Cpi%7D%7BN%7D+k+n%7D+%5C%5C+%26%3D+%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+X%5Bk%5D+Y%5E%5Cast%5Bk%5D+%5Cend%7Baligned%7D+&bg=ffffff&fg=000&s=1&c=20201002)
Energy content
Given a discrete-time sequence length N : xn={x0,x1,…,xN-1}, according to Parseval’s theorem, the energy content of the signal in the time-domain is equivalent to the average of the energy contained in its frequency components.
![\displaystyle{\sum_{n=0}^{N-1} x[n] x^\ast[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] X^{\ast}[k]}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+x%5Bn%5D+x%5E%5Cast%5Bn%5D+%3D+%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+X%5Bk%5D+X%5E%7B%5Cast%7D%5Bk%5D%7D+&bg=ffffff&fg=000&s=1&c=20201002)
If the samples x[n] and X[k] are real-valued, then
![x[n] x^\ast[n] = |x[n]|^2](https://s0.wp.com/latex.php?latex=x%5Bn%5D+x%5E%5Cast%5Bn%5D+%3D+%7Cx%5Bn%5D%7C%5E2&bg=ffffff&fg=000&s=0&c=20201002)
![\boxed{ \displaystyle{\sum_{n=0}^{N-1} \left| x[n] \right|^2 = \frac{1}{N} \sum_{k=0}^{N-1} \left| X[k] \right|^2 }}](https://s0.wp.com/latex.php?latex=%5Cboxed%7B+%5Cdisplaystyle%7B%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+%5Cleft%7C+x%5Bn%5D+%5Cright%7C%5E2+%3D+%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+%5Cleft%7C+X%5Bk%5D+%5Cright%7C%5E2+%7D%7D++&bg=ffffff&fg=000&s=1&c=20201002)
Mean Square value
Mean square value is the arithmetic mean of squares of a given set of numbers. For a complex-valued signal set represented as
![[x_0,x_1,\cdots,x_{N-1}]](https://s0.wp.com/latex.php?latex=%5Bx_0%2Cx_1%2C%5Ccdots%2Cx_%7BN-1%7D%5D&bg=ffffff&fg=000&s=0&c=20201002)
![\displaystyle{ x_{MS} = \frac{ |x_0|^2 + |x_1|^2 + \cdots + |x_{N-1}|^2}{N} = \frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2 }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+x_%7BMS%7D+%3D+%5Cfrac%7B+%7Cx_0%7C%5E2+%2B+%7Cx_1%7C%5E2+%2B+%5Ccdots+%2B+%7Cx_%7BN-1%7D%7C%5E2%7D%7BN%7D+%3D+%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+%7Cx%5Bn%5D%7C%5E2+%7D+&bg=ffffff&fg=000&s=1&c=20201002)
Applying Parseval’s theorem, the mean square value can also be computed using frequency domain components X[k]
![\displaystyle{ x_{MS} : \quad \frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N^2} \sum_{k=0}^{N-1} \left| X[k] \right|^2}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+x_%7BMS%7D+%3A+%5Cquad+%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+%7Cx%5Bn%5D%7C%5E2+%3D+%5Cfrac%7B1%7D%7BN%5E2%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+%5Cleft%7C+X%5Bk%5D+%5Cright%7C%5E2%7D+&bg=ffffff&fg=000&s=1&c=20201002)
RMS value
RMS value of a signal is calculated as the square root of average of squared value of the signal. For a complex-valued signal set represented as
![\displaystyle{ x_{RMS} = \sqrt{\frac{ |x_0|^2 + |x_1|^2 + \cdots + |x_{N-1}|^2}{N}} = \sqrt{\frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2 }}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+x_%7BRMS%7D+%3D+%5Csqrt%7B%5Cfrac%7B+%7Cx_0%7C%5E2+%2B+%7Cx_1%7C%5E2+%2B+%5Ccdots+%2B+%7Cx_%7BN-1%7D%7C%5E2%7D%7BN%7D%7D+%3D+%5Csqrt%7B%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+%7Cx%5Bn%5D%7C%5E2+%7D%7D+&bg=ffffff&fg=000&s=1&c=20201002)
Applying Parseval’s theorem, the root mean square value can also be computed using frequency domain components X[k]
![\displaystyle{ x_{RMS}: \quad \sqrt{\frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2} = \sqrt{\frac{1}{N^2} \sum_{k=0}^{N-1} \left| X[k] \right|^2}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+x_%7BRMS%7D%3A+%5Cquad+%5Csqrt%7B%5Cfrac%7B1%7D%7BN%7D+%5Csum_%7Bn%3D0%7D%5E%7BN-1%7D+%7Cx%5Bn%5D%7C%5E2%7D+%3D+%5Csqrt%7B%5Cfrac%7B1%7D%7BN%5E2%7D+%5Csum_%7Bk%3D0%7D%5E%7BN-1%7D+%5Cleft%7C+X%5Bk%5D+%5Cright%7C%5E2%7D%7D+&bg=ffffff&fg=000&s=1&c=20201002)
Implementing in Matlab:
Following Matlab code demonstrates the calculation of RMS value for a random sequence using time-domain and frequency domain approach. Figure 1, depicts the simulation results for RMS values for some well-known waveforms.
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.*conj(x))) %RMS value from time domain samples
RMS2 = sqrt(sum(X.*conj(X))/length(x)^2) %RMS value from frequency domain representation
%Result: RMS1 = 0.9814, RMS2 = 0.9814
%Matlab has inbuilt 'rms' function, it can also be used.
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.
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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.
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.