Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 471

7.4.2 Multiplication of Random Process with a Sinusoidal Signal

There are many practical applications where one may need to multiply a random
process with a sinusoidal function, as shown in Fig. 7.16 .
We can describe a time and spectral characteristics of the output process Y ( t )

YðtÞ¼XðtÞU 0 cos o 0 t: (7.103)

The autocorrelation function is:

R YY ðt; t þ tÞ¼EYðtÞYðt þ tf g
¼EXðtÞU 0 cos o 0 t ½ Xðt þ tÞU 0 cos o 0 ðt þ tÞ ½

g ;

f

¼ U 0 cos o 0 t cos o 0 ðt þ tÞEXðtÞXðt þ tÞ f g: (7.104)

Using the definition for the autocorrelation function ( 6.32 ) and the trigonometric
relation from Appendix D , we arrive at:

U 0

R YY ðt; t þ tÞ¼

2 R XX ðt; t þ tÞ cos o 0 t þ cos ð 2 o 0 t þ o 0 tÞ

½ : (7.105)

If the process X ( t ) is at least WS stationary, then

R XX ðt; t þ tÞ¼R XX ðtÞ; (7.106)

and ( 7.105 ) reduces to:

U 0
2 R XX ðtÞ cos o 0 t þ cos ð 2 o 0 t þ o 0 tÞ

R YY ðt; t þ tÞ¼ ½ : (7.107)

Note that the autocorrelation function ( 7.107 ) is a function of time and it is
necessary to find the time average for it:

T=

ð

2

1
T

AR YY ðt; t þ tÞ f g ¼ lim

T!1

R YY ðt; t þ tÞ d t;

T= 2

T= 2

ð

U 0 U 0

1
T

¼

2 R XX ðtÞ cos o 0 t þ 2 R XX ðtÞ lim

cos ð 2 o 0 t þ o 0 tÞ d t;

T!1

T= 2

U 0
2 R XX ðtÞ cos o 0 t;
¼ R YY ðtÞ:

¼

ð 7 : 108 Þ