Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain S x y ( f ) = lim T → → ∞ ∞ 1 T [ x ^ ^ T ∗ ∗ ( f ) y ^ ^ T ( f ) ] S y x ( f ) = lim T → → ∞ ∞ 1 T [ y ^ ^ T ∗ ∗ ( f ) x ^ ^ T ( f ) ] where, again, the contributions of S x x ( f ) and S y y ( f ) are already understood. Note that S x y ∗ ∗ ( f ) = S y x ( f ) , 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 T → → ∞ ∞ becomes the Fourier transform of a cross-correlation function. S x y ( f ) = ∫ ∫ − − ∞ ∞ ∞ ∞ [ lim T → → ∞ ∞ 1 T ∫ ∫ − − ∞ ∞ ∞ ∞ x T ∗ ∗ ( t − − τ τ ) y T ( t ) d t ] e − − i 2 π π f τ τ d τ τ = ∫ ∫ − − ∞ ∞ ∞ ∞ R x y ( τ τ ) e − − i 2 π π f τ τ d τ τ S y x ( f ) = ∫ ∫ − − ∞ ∞ ∞ ∞ [ lim T → → ∞ ∞ 1 T ∫ ∫ − − ∞ ∞ ∞ ∞ y T ∗ ∗ ( t − − τ τ ) x T ( t ) d t ] e − − i 2 π π f τ τ d τ τ = ∫ ∫ − − ∞ ∞ ∞ ∞ R y x ( τ τ ) e − − i 2 π π f τ τ d τ τ , where R x y ( τ τ ) is the cross-correlation of x ( t ) with y ( t ) and R y x ( τ τ ) is the cross-correlation of y ( t ) with x ( t ) . In light of this, the PSD is seen to be a special case of the CSD for x ( t ) = y ( t ) . If x ( t ) and y ( t ) are real signals (e.g. voltage or current), their Fourier transforms x ^ ^ ( f ) and y ^ ^ ( f ) are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full CPSD is just one of the CPSD s scaled by a factor of two. CPSD Full = 2 S x y ( f ) = 2 S y x ( f )