1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How Fourier Expansion indicates the Amplitude w.r.t a certain frequency?

  1. Sep 7, 2011 #1
    1. The problem statement, all variables and given/known data

    I wanna know how could I extract the amplitude(of the sinusoid component) of a random continuous wave w.r.t a certain frequency response? The teacher said the Fourier Expansion can do that but I'm really confused by the limits and integrals.

    2. Relevant equations

    [itex]F(\omega)=\int f(t) e^{-j \omega t} dt[/itex]

    3. The attempt at a solution

    I tried the very easy example and wanna extract the amplitude where the frequency matches(say [itex]\omega = \omega_0 [/itex]). [itex]f(t)=A \cdot cos\omega_0t,\hat{f}(t)=A \cdot e^{j \omega_0 t},\hat{F}(\omega)=\int \hat{f}(t) \cdot e^{-j \omega t} dt [/itex], range ([itex]-\infty ,\infty [/itex]), but it turned out to be [itex]Re\{ \hat{F}( \omega ) \}=A \cdot \frac{sin(\omega_0 - \omega ) (t_2-t_1)}{\omega_0 - \omega}[/itex], where [itex]t_2=\infty,t_1=-\infty[/itex] , it's weird if I follow the basic operation of sin function, I got [itex]Re\{ \hat{F}( \omega ) \}=2 \cdot A \cdot \frac{sin(\omega_0 - \omega ) \infty}{\omega_0 - \omega}[/itex], and then although applying that [itex]lim \frac{sinx}{x} -> 1[/itex] while x->0, it's 2A, besides I don't even know if this's right.

    I have no idea what happened...

    Any help will be appreciated !!!
    Last edited: Sep 7, 2011
  2. jcsd
  3. Sep 7, 2011 #2
    Terribly sorry that my teacher has corrected my mistakes, the Amplitude should be [itex]|F(\omega)|[/itex] instead of [itex]Re \{ \hat{F}(\omega) \}[/itex],but the calculation becomes even harder, I'm still trying on this question.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Fourier Expansion indicates Date
DSB-SC signals Wednesday at 6:49 PM
Thermal expansion of a ring Feb 27, 2018
Fourier Conduction Law Nov 16, 2017