Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 467

P.2 The following relation between S XY (o) and S YX (o) holds:

S XY ðoÞ¼S YX ðoÞ or S XY ð f Þ¼S YX ðf Þ: (7.79)

We rewrite here the relation ( 6.113 ) for a cross-correlation function

R XY ðtÞ¼R YX ðtÞ: (7.80)

Placing ( 7.80 ) into ( 7.75 ), we get:

1 1

R YX ðtÞ e jot d t ¼ R YX ðtÞ e joðtÞ d t

S XY ðoÞ¼

1 1

1

R YX ðtÞ e joðtÞ d t ¼ S YX ðoÞ: ¼ (7.81)

1

P.3 The following relations hold for S XY ( o ) and S YX ( o ):

S XY ðoÞ¼S XY ðoÞ;
S XY ðoÞ¼S XY ðoÞ: (7.82)

Similarly, we have:

S YX ðoÞ¼S YX ðoÞ;
S YX ðoÞ¼S YX ðoÞ:

(7.83)

From ( 7.75 ), we write:

1 1

R XY ðtÞ e jðoÞt d t ¼ R XY ðtÞ e ðjÞot d t ¼ S XY ðoÞ; S XY ðoÞ¼

1 1

1 1

R YX ðtÞ e jðoÞt d t ¼ R YX ðtÞ e ðjÞot d t ¼ S YX ðoÞ:

S YX ðoÞ¼

(7.84)

1 1

P.4 If two processes (that are at least WS stationary, both with zero means) are
uncorrelated, then their cross-spectral densities are zero

S XY ðoÞ¼S YX ðoÞ¼ 0 : (7.85)

For uncorrelated processes from ( 6.135 ) , we have:

R XY ðtÞ¼XY ¼ 0

(7.86)

and ( 7.85 ) follows from the definition ( 7.75 ).