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!

Homework Help: Fourier transform: signal with filter

  1. Nov 21, 2016 #1
    Hi Guys,

    I'm having trouble with the following:

    A finite-time signal is the result of a filter G(t) applied to a signal. The filter is simply “on” (1) for t ∈ [0,T] and off (“0”) otherwise. If x(t) is the signal, and x(ω),its Fourier transform, compute the Fourier transform of the filtered signal. Next, take a simple sine for x(t), x(t) = sin(ω0t), and compute the Fourier transform for the finite-time signal. Write the result, it must involve the filter, and integrations should stretch [−∞,∞]

    I don't really know what to do exactly, with the first problem.

    I can try calculating the Fourier transform of the filter:

    G(ω)= ∫0T e-iωtdt = -1/(iω)⋅(e-iωT-1)

    The Fourier transform of the signal is: x(ω)

    The convolution theorum says that the convolution of two functions is the product of the Fourier-transformed functions. Which makes: G(ω)x(ω).

    But I have the idea that this isn't right. Could one of you guys assist me?

  2. jcsd
  3. Nov 21, 2016 #2


    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    The convolution theorem also works the other way around: the Fourier transform of the product of a step function and some other function is the convolution of their Fourier transforms.

    By the way: do use the template, don't erase it. It helps you order to your thinking and us to help you better
  4. Nov 21, 2016 #3
    Ok. If I understand you correctly, you mean:

    Fourier{x(t)g(t)} = 1/2π ⋅ X(ω)⊗G(ω) ==>

    Writing -1/(iω)⋅(e-iωT-1) to -1/iω⋅e-iωT/2(e-iωT/2-eiωT/2) = T⋅e-iωT/2⋅sinc(ωT/2)

    Fourier{x(t)g(t)}=1/2π⋅∫-∞ X(w-w')⋅T⋅sinc(ω'T/2) dω' ??

    Sorry, I will do that next time, thanks!
  5. Nov 21, 2016 #4


    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    Looks reasonable (all the contributions are there -- didn't check the gory details. Most of the time I use a table like this)
    I take it you mean ##\ x(\omega-\omega_0) \ ## ?

    Now for the second part ...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted