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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?

Thanks

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# Uniform distributed modulo 1, exp factor

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