# Relation between power spectral density and mean absolute value

The root mean square

$$\sigma_{x} = \sqrt{\frac{1}{T}\int_0^T x^2(t) \, \mathrm{d}t}$$

of a finite zero-mean random signal $$x(t)$$ in the range $$0 < t < T$$

is related to the signal's power spectral density $$S_{xx}(\omega)$$ via the relationship

$$\int_{-\infty}^{\infty} S_{xx}(\omega) \, \mathrm{d}\omega = \sigma_x^2$$

However, I was wondering if a similar relationship exists between the power spectral density and the mean of the absolute value of the signal, i.e.

$$\mathrm{mean}\left({|x|}\right) = \frac{1}{T}\int_0^T |x(t)| \, \mathrm{d}t$$