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Approximation in orthogonal functions with pulse shaping

  1. Dec 14, 2012 #1
    Hello!
    I have tried for a whole afternoon to solve this problem but I didn't succeed.
    Let [itex]\cos(2 \pi (f_0 + i/T_N) t + \phi_i)[/itex] and [itex]\cos(2 \pi (f_0 + j/T_N) t + \phi_j)[/itex] be two quasi-orthogonal functions:

    [itex]\int_{0}^{T_N} \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt \simeq \displaystyle \frac{T_N}{2} \delta (i - j)[/itex]

    where [itex]\delta()[/itex] is the Kroneker Delta, and the approximation is good if [itex]f_0 T_N \gg 1[/itex]. [itex]i,j = 0, 1, \ldots, N - 1[/itex].

    This is the theory of OFDM (Orthogonal Frequency Division Multiplexing) and it is partially explained in Andrea Goldsmith, Wireless Communications (Cambridge University Press) with these simbols and this notation.

    But what about

    [itex]\int_{0}^{T_N} g(t) \cos(2 \pi (f_0 + i/T_N) t + \phi_i) g(t) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt \simeq \displaystyle \frac{T_N}{2} \delta (i - j)[/itex]???

    [itex]g(t)[/itex] is called the pulse shaping function and it could have a raised-cosine form in [itex][0, T_N][/itex] (a sort of [itex]sin(x)/x[/itex]). I know only that the highest frequency component of [itex]g(t)[/itex] is about [itex]B_N = 1/T_N[/itex], with [itex]B_N \ll f_0[/itex].


    2. Relevant equations
    To demonstrate the first approximation, the book uses the Werner formulas and writes:

    [itex]\int_{0}^{T_N} \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt = [/itex]
    [itex]=\displaystyle \frac{1}{2} \int_{0}^{T_N} \cos(2 \pi (2 f_0 + (i + j)/T_N) t + \phi_i + \phi_j) + \displaystyle \frac{1}{2} \int_{0}^{T_N} \cos(2 \pi ((i - j)/T_N) t + \phi_i - \phi_j) dt \simeq [/itex]
    [itex]\simeq \displaystyle \frac{1}{2} \int_{0}^{T_N} \cos(2 \pi ((i - j)/T_N) t + \phi_i - \phi_j) dt = \frac{T_N}{2} \delta (i - j)[/itex]

    and I obviously agree fot [itex]f_0 T_N \gg 1[/itex]. But the book says nothing about the insertion of the [itex]g(t)[/itex] function.

    3. The attempt at a solution

    I tried to follow the same procedure:

    [itex]\int_{0}^{T_N} g^2 (t) \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt = [/itex]
    [itex]=\displaystyle \frac{1}{2} \int_{0}^{T_N} g^2 (t) \cos(2 \pi (2 f_0 + (i + j)/T_N) t + \phi_i + \phi_j) + \displaystyle \frac{1}{2} \int_{0}^{T_N} g^2 (t) \cos(2 \pi ((i - j)/T_N) t + \phi_i - \phi_j) dt[/itex]

    For [itex]i = j[/itex] I have no troubles. But for [itex]i \neq j[/itex]:
    the first integral is almost zero again because [itex]B_N \ll 2 f_0 + (i + j)/T_N[/itex] and so [itex]g^2(t)[/itex] is almost constant over each period of the integrating function; but what about the second integral? This simplifying hypothesis is not true anymore!
    In the second integral the cosine has a very small frequency and the [itex]g^2(t)[/itex] is variable over the period [itex]T_N/(i-j)[/itex].
    I hope this was the right section to post.
    Thank you anyway for your readings.
    Bye,

    Emily
     
  2. jcsd
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