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: Signal filtering in digital transmission

  1. Jul 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Let's suppose a set of numerical values {an} sent with a rectangular function, period T and with amplitude an.
    The signal can be expressed as follows:
    [tex]x(t)= \sum a_n \Pi_T (t-nT)[/tex]
    To optimize detection at reception, the signal x(t) is processed through a filter whose impulse response is:
    [tex]h(t)= sin(\frac{\pi t}{T} \Pi_T (t-\frac{T}{2})[/tex]
    We write y(t) the output signal

    1/ If we only have a0=A, calculate y(t) and illustrate the computed elements graphically.

    2/ Express y(t) in the general case. Plot x(t) and y(t) if the values to be transmitted are ...,0,0,0,3,5,-2,1,-3,0,0,0,... What are the optimal moments to detect {an} from y(t) ?

    3/ Express h(t) as a convolution:
    Determine h0(t), calculate H0(f) and give a graphical representation. What are the modulus and phase of the transfer function H(f) ?

    4/ Let’s suppose the {an} are obtained after an audio signal sampling at Fe=1/T, we have an = s(nT). Give the spectrum of x(t), X(f) with a graphical representation. Express the mean value of x(t) with respect to the one of s(t), Ps. Is the filtering of x(t) by h(t) a good way to get back s(t) from x(t)? Why?

    2. Relevant equations

    [tex]x(t)= \sum a_n \Pi_T (t-nT)[/tex]

    [tex]h(t)= sin(\frac{\pi t}{T} \Pi_T (t-\frac{T}{2})[/tex]


    3. The attempt at a solution

    Well, this is not exactly for me but for one of my students who’s having a hard time. Except he’s my student in eletronics laboratories not in signal processing and since he begged me to help him, here i am getting back in signal processing. :(

    My ideas:

    h(t) is the multiplication of a sine with a rectangular function translated by T/2 so I can have their common area, plot it and do my convolution graphically. But I can’t carefully write the mathematical operation.
    By writing the convolution :
    [tex]y(t)=\int sin(\frac{\pi \tau}{T}) \Pi_T (\tau-\frac{T}{2}) A \Pi_T (t - \tau) d\tau[/tex]
    But this doesn’t seem easy to compute. I thought of coming through it with a Fourier transform to get two sinc with the rectangular functions and a dirac with the sine. But I’m not sure about the carefullness of my operation.

    Actually once I understand this question and the math behind it, I may be able to go on by myself. The second question is a generalization of the first and the optimal detection instants are trivial.

    For the third question the dirac makes me think of a T/2 delay so h(0) would be for me the product of a sine and something doing the rectangular function but I’m not sure. Again, once started I should be able to go on by myself.

    I hope this thread doesn’t seem too improper with respect to the policy here, but I’m starting to see my student’s face decomposed.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted