Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 470

S YY ðoÞ¼S X 1 X 1 ðoÞþS X 2 X 2 ðoÞþS X 1 X 2 ðoÞþS X 2 X 1 ðoÞ: (7.96)

Consider that the processes X 1 ( t ) and X 2 ( t ) are uncorrelated. We have:

R X 1 X 2 ðtÞ¼R X 2 X 1 ðtÞ¼EfX 1 gEfX 2 g: (7.97)

Therefore, for uncorrelated processes, ( 7.95 ) becomes:

R YY ðtÞ¼R X 1 X 1 ðtÞþR X 2 X 2 ðtÞþ 2 EfX 1 gEfX 2 g: (7.98)

It is useful to have the expression of ( 7.98 ) for a case where

EfX 1 g¼EfX 2 g¼ 0 : (7.99)

From ( 7.98 ) and ( 7.99 ), we get:

R YY ðtÞ¼R X 1 X 1 ðtÞþR X 2 X 2 ðtÞ: (7.100)

An autocorrelation function of a sum of two at least WS stationary, uncorrelated,
zero-mean processes is equal to the sum of the autocorrelation functions of the
particular processes.

Using the Fourier transform of ( 7.100 ), we get a PSD of the sum:

S YY ðtÞ¼S X 1 X 1 ðtÞþS X 2 X 2 ðtÞ: (7.101)

A PSD of a sum of two at least WS stationary, uncorrelated, zero-mean processes
is equal to a sum of power spectral densities of particular processes.

The mean power of the sum is:

1

1
2 p

P YY ¼ Y 2 ¼ X 1 þ X 2 ¼ S YY ðoÞ d o;

1

1 1

1
2 p

1
2 p

¼ S X 1 X 1 ðoÞ d o þ S X 2 X 2 ðoÞ d o;

1 1

¼ P X 1 X 1 þ P X 2 X 2 : ð 7 : 102 Þ

The mean power of a sum of two at least WS uncorrelated zero-mean processes
is equal to a sum of mean powers of marginal processes.