Using the arguments of Reference 5 which show that the spectral density and the 

 autocorrelation are Fourier transforms, it can be shown that the cross correlation and the 

 cross spectral density are also Fourier transforms: 



Since Pi2(f) i^ complex, we can write 



where it follows from the definition of Pi2if) ^^^^ ^^^ '"^^1 functions U^2if) and V^2if) ^"^^ 

 even and odd, respectively. Accordingly, we can write 



/?J2('^)= I f/j 2 (/) cos cjr (// - / Fj2(/) sin w Ttf/ 

 



This last equation provides the basis for an experimental method of measuring the real 

 and imaginary parts of the cross spectral density. If the cross correlation is measured with 

 zero time delay, then 



^12(0) = j ^i2(f)df 



If the cross correlation is measured in a narrow band of frequencies A/ as shown in Figure 3, 

 then 



where rR^^i^) denotes the cross correlation measured in the narrow frequency band. The 

 real part of the normalized cross spectral density u^^^if) is then 



U,2if) 



iP^{f)P2{f)f' 



