Spectral density#Cross-spectral density

The cross-power spectrum ( more precisely, cross-power density spectrum ) is based on the calculated as a basis in the signal analysis Fourier spectra. It is calculated analogously to the auto power spectrum using the equation

Since X * (f) and Y (f ) is not a complex conjugate to each other, the result is the complex.

, Instead of the Fourier transform X (f) and Y (f), the short-time spectra formed over the period T XT (f) and YT ( ​​f) is used, the cross power spectrum obtained from the expected value of the spectrum of products:

For further processing, the one-sided spectrum is usually used:

For f = 0 and

For f > 0

For interpretations of the phase response of the cross-power spectrum is at best interesting. However, since it is identical to the phase transition of the frequency response, the cross-power spectrum is generally used only as an important basis for the calculation of further signal analysis functions. Again, in analogy to the auto power spectrum and the cross-power spectrum as the Fourier transform of a correlation function, here the cross-correlation function can be calculated:

In digital signal analysis, however, this calculation process is not common.

• Signal processing