In time-frequency analysis, there are fundamental formulas expressing the mean and variance of the Fourier frequency of signals,s, originally defined in the Fourier frequency domain, in terms of integrals against the density |s(t)|²  in  the time domain. In the literature, the existing formulas are only for smooth signals, for it is the classical derivatives of the phase  and  amplitude  of  the  signals  that  are  involved.  The  two  representations  of  the  covariance  also  rely  on  the  classical  derivatives  and  thus  are  restrictive.  In  this  fundamental  study,  by  introducing  a  new  type  of  derivatives,  called Hardy–Sobolev  derivatives,  we  extend  the  formulas  to  signals  in  the  Sobolev  space  that  do  not  usually  have  classical derivatives.  We  also  investigate  the  corresponding  formulas  for  periodic  (infinite  discrete)  and  finite  discrete  signals.