# Cross-service spectrum

The cross power spectrum (more precisely: cross power density spectrum ) is based on the Fourier spectra calculated as a basis in the signal analysis . It is calculated analogously to the car power spectrum according to the equation

${\ displaystyle S_ {XY} (f) = X ^ {*} (f) \ cdot Y (f) \,}$

Since X * (f) and Y (f) are generally not complex conjugate to one another, the result remains complex .

If instead of the Fourier transforms X (f) and Y (f) the short-term spectra X T (f) and Y T (f) formed over the period T are used, the cross-power spectrum results from the expected value of the spectrum products:

${\ displaystyle S_ {XY} (f) = E \ {X_ {T} ^ {*} (f) \ cdot Y_ {T} (f) \} \,}$

The one-sided spectrum is usually used for further processing:

${\ displaystyle G_ {XY} = S_ {XY} \,}$

for f = 0 and

${\ displaystyle G_ {XY} (f) = 2S_ {XY} (f) \,}$

for f> 0.

The phase curve of the cross-power spectrum is at most interesting for interpretations. However, since it is identical to the phase response of the frequency response , the cross-power spectrum is generally only used as an important basis for calculating further signal analysis functions. Again in analogy to the car power spectrum, the cross power spectrum can also be calculated as a Fourier transform of a correlation function, here the cross correlation function :

${\ displaystyle S_ {XY} (f) = \ int _ {- \ infty} ^ {\ infty} R_ {xy} (\ tau) e ^ {- \ mathrm {i} 2 \ pi f \ tau} \, d \ tau \,}$

However, this calculation process is not common in digital signal analysis.

## literature

• Bahaa EA Saleh, Malvin Carl Teich: Fundamentals of Photonics . 2nd edition, Wiley-VCH, Weinheim 2008, ISBN 978-3-527-40677-7 .