Parseval's theorem Definition and 14 Threads

I'm an undergraduate trying to understand quantum physics. So, I'm trying to understant the plane wave basis for a general 3D box. I understood that the plane wave basis is used to define a period boundary to model real-world situations where the space in which the experiment is performed is...

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...

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...

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

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: $$h(t)=\sum_{l=1}^L \sqrt{g_l} \ \delta(t-\tau_l).$$...

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} =...

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...

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...

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...

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...

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...

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...

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}...

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...