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Integral Approximation

  1. Mar 21, 2013 #1
    Hello,

    If tau<<T which of the following relations are true:

    [tex]\int_{\tau/(1+a)}^{(T+\tau)/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt=\int_{0}^{T/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt[/tex]

    or

    [tex]\int_{\tau/(1+a)}^{(T+\tau)/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt\simeq\int_{0}^{T/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt[/tex]

    Thanks
     
  2. jcsd
  3. Mar 23, 2013 #2
    Your assumtion is not enough to show one of the relations is true. It seems you need to show that the integrand is periodic with [itex] T/(1+\alpha)[/itex].
     
  4. Mar 23, 2013 #3
    OK, the integral after substituting for v(t) will look like:

    [tex]\int_{\tau/(1+a)}^{(T+\tau)/(1+a)}e^{j2\pi\frac{m-k}{T}t[1+a]}\,dt[/tex]

    Is that enough to tell now?

    Thanks
     
  5. Mar 23, 2013 #4
    If that is the final version of the integral then it is enough to say that the equality holds.
     
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