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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 Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. |  Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. |  Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. |
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