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Uniform distributed modulo 1, exp factor

  1. Mar 2, 2007 #1

    I would like to simplify the following equation:

    [tex]T_s sinc\left( \frac{\omega}{2} T_s\right) \sum_{n=0}^{K-1}
    \mathrm{<}\sigma+n\Delta\mathrm{>} e^{-j\omega n T_B}[/tex]

    The sequence [tex]\mathrm{<}\sigma+n\Delta\mathrm{>}[/tex] is a uniformly distributed modulo 1 sequence.
    Delta and sigma are two constants.

    I am aware of the theorem:
    [tex]\lim_{N \to \infty} \frac{1}{N}\sum_{n=1}^{N}f \left( (x+na)
    \,\mathrm{ mod 1} \, \right) = \int_{0}^{1} f(y) dy[/tex]
    but don't know how to factor in the exponential term.

    Without the modulo 1 sequence in the sum the exponential term will simplify to a comb function which is periodic with TB.
    [tex]\lim_{K \to \infty} \frac{1}{K}\sum_{n=0}^{K-1}
    e^{-j\omega n T_B}= \sum_{-\infty}^{\infty} \delta (f - \frac{n}{T_B})[/tex]

    However, I want to know how the multiplication of the modulo one sequence will effect the summation over all the exponential terms.
    When plotting the equation in matlab I get the expected dirac pulses at multiples of TB, but also artifacts that are related to the constant Delta at different frequencies. I would like to re-write the above equation such that it tells me where spectral lines will occur and what their magnitude will be.

    Can anyone help?

    Last edited: Mar 2, 2007
  2. jcsd
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