# Uniform distributed modulo 1, exp factor

1. Mar 2, 2007

### svensl

Hello,

I would like to simplify the following equation:

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

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

I am aware of the theorem:
$$\lim_{N \to \infty} \frac{1}{N}\sum_{n=1}^{N}f \left( (x+na) \,\mathrm{ mod 1} \, \right) = \int_{0}^{1} f(y) dy$$
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.
$$\lim_{K \to \infty} \frac{1}{K}\sum_{n=0}^{K-1} e^{-j\omega n T_B}= \sum_{-\infty}^{\infty} \delta (f - \frac{n}{T_B})$$

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

Last edited: Mar 2, 2007