Spectral Analysis - Signal Processing of Random Physiological Signals - page 170

FIGURE 15.2 : The relationship between the phase angle to cross-spectral terms

The magnitude of the cross-spectrum is calculated from (15.22), and the phase is calcu-
lated from (15.23).

C XY ( f )

G XY ( f ) =

Q XY + (15.22)

Q XY ( f )
C XY ( f )

tan −1

θ XY ( f ) = (15.23)

Special note should be made of the fact that the autospectrum does not retain
any phase information whereas the cross-spectrum contains phase information. Keep in
mind that engineers look for relationships in both magnitudes or in phase between an
input stimulus and the output response of a system. Phase relations are often expressed
as leading or lagging a reference signal. The relationship between the phase angle to
cross-spectral terms is shown in Fig. 15.2. Since phase is calculated with a tangential
function, the relationship is expressed in quadrants.

15.4 PROPERTIES OF SPECTRAL DENSITY FUNCTIONS
For both autospectra and cross-spectra, the Power Spectral Density functions are “ Even ”
Functions and the Power Spectra are always “Positive” valued functions, G xx > 0.

15.5 FACTORS AFFECTING THE SPECTRAL DENSITY
FUNCTION ESTIMATION
There are several factors that adversely affect the spectral density function estimation,
regardless of the method of estimation that is employed, if these factors are not taken
into account.