I recently took up studying elementary stochastic control theory and I have trouble comprehending why exactly is the (cross) spectral density defined in many texts as
$\phi_{xy}(\omega)={1\over{2\pi}} \sum\limits^\infty_{k=-\infty}r_{xy}(k)e^{-ik\omega}$
and
$r_{xy}(k)=\int^\pi_{-\pi}e^{ik\omega}\phi_{xy}(\omega)d\omega$
I've been accustomed to the DTFT definition where the $1\over2\pi$ is a factor in the inverse transformation. Is there particular reason why in most texts I've read about defining the spectral density through covariance, the $1\over2\pi$ is always associated with the summation term?
Especially with white noise this gives $\phi(\omega)={\sigma^2\over{2\pi}}$, and for me it would seem more reasonable to have the variance come straight out from the Fourier transform, without the division by $2\pi$.