The Parseval’s theorem (a.k.a Plancherel 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 discrete-time sequence of complex numbers of length N : xn={x0,x1,…,xN-1}, its N-point discrete Fourier transform (DFT)[1] : Xk={X0,X1,…,XN-1} is given by
\[X[k] = \sum_{n=0}^{N-1} x[n] e^{-j\frac{2 \pi}{N} k n} \]
The inverse discrete Fourier transform is given by
Time-domain and frequency domain representations are equivalent manifestations of the same signal. Therefore, the energy of the signal computed from time domain samples must be equal to the total energy computed from frequency domain representation.
Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.
Root Mean Square (RMS) value is the most important parameter that signifies thesize 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
The inverse discrete Fourier transform is given by
Suppose if x[n] and y[n] are two such sequences that follows the above definitions, the Parseval’s theorem is written as
where, indicates conjugate operation.
Deriving Parseval’s theorem
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.
If the samples x[n] and X[k] are real-valued, then
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 discrete sampled values – , the mean square xMS value is given as
Applying Parseval’s theorem, the mean square value can also be computed using frequency domain components X[k]
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 discrete sampled values – , the mean square xRMS value is given as
Applying Parseval’s theorem, the root mean square value can also be computed using frequency domain components X[k]
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.
Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Necessary cookies are absolutely essential for the website to function properly. These cookies ensure basic functionalities and security features of the website, anonymously.
Cookie
Duration
Description
cookielawinfo-checbox-analytics
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checbox-analytics
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checbox-functional
11 months
The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checbox-functional
11 months
The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checbox-others
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checbox-others
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checkbox-necessary
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookies is used to store the user consent for the cookies in the category "Necessary".
cookielawinfo-checkbox-performance
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Performance".
viewed_cookie_policy
11 months
The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. It does not store any personal data.
Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features.
Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors.
Analytical cookies are used to understand how visitors interact with the website. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc.