text: Given two signals  and , each of which possess power spectral densities  and , it is possible to define a cross power spectral density (CPSD) or cross spectral density (CSD).  To begin, let us consider the average power of such a combined signal.
  Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain
  where, again, the contributions of  and  are already understood.  Note that , so the full contribution to the cross power is, generally, from twice the real part of either individual CPSD. Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit  becomes the Fourier transform of a cross-correlation function.
  where  is the cross-correlation of  with  and  is the cross-correlation of  with .  In light of this, the PSD is seen to be a special case of the CSD for .  If  and  are real signals (e.g. voltage or current), their Fourier transforms  and  are usually restricted to positive frequencies by convention.  Therefore, in typical signal processing, the full CPSD is just one of the CPSDs scaled by a factor of two.
  For discrete signals  and , the relationship between the cross-spectral density and the cross-covariance is