Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

8.7 CROSS-POWER SPECTRAL DENSITY

The cross-PSD for random processes X(t) and Y(t) can be derived using the same approach used for SXX(ω). From Parseval's theorem, the power can be written as follows:

(8.106)

where

(8.107)

Substituting Fourier transforms gives

(8.108)

It is clear from the derivation for SXX(ω) that the steps are essentially the same here, leading to the following result which we state as a definition.

Definition: Cross-Power Spectral Density The cross-PSD of wide-sense stationary processes X(t) and Y(t) with cross-correlation function is

(8.109)

where ω is radian frequency.

(Note that some books define the cross-correlation function to be .) The other forms of the PSD with arguments f and s, as well as those for random sequences with arguments jω and z, can also be defined for the cross-PSD.

The cross-PSD of random processes {X(t), Y(t)} has the following properties. ...