What is Parseval's theorem: Definition and 14 Discussions

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem.

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  1. M

    Engineering Parseval's Theorem - Average Power of the difference of functions

    Hi, So I have a quick conceptual question about Parseval's theorem in this application. In previous parts of this question, we have found the average powers of both f(t) and g(t) by integration and using the complex Fourier series respectively (not sure if this is relevant to my question)...
  2. Ineedhelp0

    I Parseval's theorem and Fourier Transform proof

    Given a function F(t) $$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$ I am looking for a proof of the following: $$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d...
  3. G

    MHB Solve b) Using Parseval's Theorem

    Hello good folks! I'm stuck trying to solve the problem b). In the theory book examples they are skipping steps and shortly states 'use algebra' and parsevals theorem to rewrite the Fourier series into the answer that is given. So I've tried to use parsevals theorem but I still can't rewrite...
  4. Charles Link

    Insights An Integral Result from Parseval's Theorem - Comments

    Charles Link submitted a new PF Insights post An Integral Result from Parseval's Theorem Continue reading the Original PF Insights Post.
  5. M

    Power signal calculation using Parseval's Theorem

    Homework Statement Hi guys, I have the following transmitted power signal: $$x(t)=\alpha_m \ cos[2\pi(f_c+f_m)t+\phi_m],$$ where: ##\alpha_m=constant, \ \ f_c,f_m: frequencies, \ \ \theta_m: initial \ phase.## The multipath channel is: $$h(t)=\sum_{l=1}^L \sqrt{g_l} \ \delta(t-\tau_l).$$...
  6. Gopal Mailpalli

    Fourier Series for Periodic Functions - Self Study Problem

    Self Study 1. Homework Statement Consider a periodic function f (x), with periodicity 2π, Homework Equations ##A_{0} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)dx## ##A_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)cos\frac{2\pi rx}{L}dx## ##B_{n} =...
  7. FeDeX_LaTeX

    Generalisation of Parseval's Theorem via Convolution Theorem

    Homework Statement [/B] Suppose we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{C} whose Fourier coefficients are known. Parseval's theorem tells us that: \sum_{n = -\infty}^{\infty}|\widehat{f(n)}|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^{2}dx, where...
  8. R

    Fourier Transform and Parseval's Theorem

    Homework Statement Using Parseval's theorem, $$\int^\infty_{-\infty} h(\tau) r(\tau) d\tau = \int^\infty_{-\infty} H(s)R(-s) ds$$ and the properties of the Fourier transform, show that the Fourier transform of ##f(t)g(t)## is $$\int^\infty_{-\infty} F(s)G(\nu-s)ds$$ Homework Equations...
  9. K

    Using Parseval's Theorem to evaluate an integral -- Help please

    Homework Statement By applying Parseval's (Plancherel's) theorem to the function are given by: f(x) = -1 for -2 ≤ x < 0 1 for 0 ≥ x < 2 0 otherwise determine the value of the following integral. ∫ dk sin^4(k)/(k^2) (Integral between ±infinity) Homework Equations...
  10. A

    Parseval's Theorem Homework: Fourier Sine Series

    Homework Statement I'm given the following function f(x) = \begin{cases} x &-2<x<2\\ f(x+4) &\mbox{otherwise} \end{cases} And I'm asked to find the Fourier sine series. Then I'm supposed to use Parseval's theorem to obtain a certain sum. Homework Equations Since I have a sine Fourier...
  11. T

    Evaluate infinite sum using Parseval's theorem (Fourier series)

    Homework Statement Show that: \sum_{n=1}^{\infty}\frac{1}{n^4} = \frac{π^4}{90} Hint: Use Parseval's theorem Homework Equations Parseval's theorem: \frac{1}{\pi}\int_{-\pi}^{\pi} |f(x)|^2dx = \frac{a_0^2}{2}+\sum_{n=1}^{\infty}(a_n^2+b_n^2) The Attempt at a Solution I've been trying to solve...
  12. M

    Fourier series and parseval's theorem

    A square wave has amplitude 3 and period 5. calculate its power? Using Fourier series for this square wave and Parseval’s theorem, calculate the power in a signal obtained by cutting out frequencies above 1 Hz in the square wave? i am able to obtain the Fourier series for the square wave...
  13. T

    Help regarding a question Parseval's Theorem

    Homework Statement The Fourier series for f(x) = x2 over the interval (−1/2, 1/2) is: f(x) = \frac{1}{12}-\frac{1}{\pi^2} (cos 2\pi x - \frac{1}{2^2}cos4\pi x + \frac{1}{3^2}cos6\pi x) ... Using Parseval's Theorem, show that \sum _{n = 1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}...
  14. N

    Parseval's Theorem and Fourier series

    Homework Statement Hi all. Please take a look at the lowest equation in this picture: http://img143.imageshack.us/img143/744/picture2ao8.png This is Parselvals Identity. Let us say that I am given a Fourier series of f(x), and I want to calculate the integral of f(x)^2 from -L to...
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