Density Function of Weighted Sum of Chi-Square Variables with Trigonometric Weights

We have investigated a weighted chi-square distribution of the variable $\xi$ which is a weighted sum of squared normally distributed independent variables whose weights are cosines of angles ${\phi }_k=2\pi k\mathrm{/}N$, where $k\in \{0,1,\mathrm{.}\mathrm{.}\mathrm{.},N-1\}$ and $N$ is the number of the freedom degrees. We have found the exact expression for the density function of this distribution and its approximation for large $N$. The distribution is compared with the Gaussian distribution.