Random Processes - Random Signals and Processes Primer with MATLAB - page 385

The obtained result shows that the autocorrelation function is also a periodic
function with the same period T .
(b) Next, we consider a process which has the periodic component A cos( o 0 t + y )
(the same as in (a)) and a zero-mean, nonperiodic component X 0 ( t ),

XðtÞ¼A cos ðo 0 t þ yÞþX 0 ðtÞ:

(6.63)

Both components are independent.
The autocorrelation function is:

R XX ðtÞ¼XðtÞXðt þ tÞ¼ A cos ðo 0 t þ yÞþX 0 ðtÞ ½ A cos ðo 0 ðt þ tÞþyÞþX 0 ðt þ tÞ ½

A 2
2

A 2
2

¼ cos ðo 0 t o 0 t o 0 tÞþ cos o 0 ð 2 t þ tÞþy ð Þ þ X 0 ðtÞX 0 ðt þ tÞ

þ X 0 ðtÞA cos o 0 ðt þ tÞþy ð Þ þ A cos ðo 0 t þ yÞX 0 ðt þ tÞ:

(6.64)

Knowing that the components of the process are independent, and using the
results from ( 6.62 ), we arrive at:

A 2
2

R XX ðtÞ¼ cos o 0 t þ R X 0 X 0 ðtÞ: (6.65)

The obtained result again shows that the autocorrelation function has a periodic
component with the same period as does a periodic component of the process.
(c) Finally, we consider a case in which the process has two independent periodic
components with periods T 1 and T 2 ,

XðtÞ¼A 1 cos ðo 1 t þ yÞþA 2 cos ðo 2 t þ yÞ; (6.66)

where A 1 , A 2 , o 1 , o 2 are constants, y is a uniform random variable in the range
[0, 2 p ], o 1 ¼ 2 p / T 1 , and o 2 ¼ 2 p / T 2 .
The autocorrelation function is:

R XX ðtÞ¼Ef A 1 cos ðo 1 t þ yÞþA 2 cos ðo 2 t þ yÞ ½

A 1 cos o 1 ðt þ tÞþy ½ ð Þ þ A 2 cos o 2 ðt þ tÞþy ð Þ g: (6.67)

Using the result described in the case (a) we can easily obtain:

A 1
2

A 2 2
2

R XX ðtÞ¼ cos o 1 t þ cos o 2 t: (6.68)

The obtained result shows that the autocorrelation function of the sum of
independent periodic components is equal to the sum of the autocorrelation
functions of the same periodic components.