Random Processes - Random Signals and Processes Primer with MATLAB - page 384

If the mean value of the process is zero, XðtÞ ¼ 0, then

ðtÞ ¼ s 2 R XX ð 0 Þ ¼ X 2

X ;

(6.56)

where s X is the variance of the process.
For t ¼ 0, from ( 6.33 ), we can write,

ðtÞ ¼ s X : R XX ð 0 Þ ¼ XðtÞXðt þ 0 Þ ¼ X 2 (6.57)

P.4 If the mean value of the process is not zero, then the autocorrelation function
has a constant term which is equal to the squared mean value:

2

R XX ðtÞ ¼ R X 0 X 0 ðtÞþXðtÞ (6.58)

where X 0 ( t ) is a zero-mean process obtained from the process X ( t ) by subtracting the
mean value c .
Consider the WS process X ( t ), where a mean value is a constant c , E { X ( t )} ¼ c ,
We can write:

XðtÞ ¼ c þ X 0 ðtÞ;

(6.59)

The autocorrelation function of the process X ( t ) is

g ¼ E c þ X 0 ðtÞ c þ X 0 ðt þ tÞ R XX ðtÞ ¼ E XðtÞXðt þ tÞ f f ½ ½ g

¼ E c 2 þ X 0 ðtÞX 0 ðt þ tÞþcX 0 ðtÞþcX 0 ðt þ tÞ

¼ c 2 þ E X 0 ðtÞX 0 ðt þ tÞ g ¼ c 2

f þ R X 0 X 0 ðtÞ : (6.60)

P.5 If a random process is a periodic process, then its autocorrelation function also
has a periodic component of the same period as the process itself.

(a) Consider a periodic process X ( t ) ¼ A cos( o 0 t + y ) where A , and o 0 ¼ 2 p / T ,
where T is a period, are constants, and y is a uniform random variable in the
range [0, 2 p ].
The autocorrelation function is:

R XX ðtÞ ¼ X ð t Þ X ð t þ t Þ ¼ A cos ð o 0 t þ y Þ A cos o 0 ð t þ t Þþ y ð Þ

Þ :

(6.61)

A 2
2

A 2
2

¼

cos ðo 0 to 0 to 0 tÞþ cos o 0 ð 2 t þ tÞþy

ð

We can easily find that the second term is equal to zero, resulting in:

A 2
2

R XX ðtÞ ¼ cos o 0 t: (6.62)