Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 486

where T is a time constant

T ¼ RC: (7.166)

Using ( 7.129 ), ( 7.162 ), and ( 7.165 ), we arrive at:

2

S YY ðoÞ¼ ð oT Þ

p
1 þðoTÞ

2 dðo þ o 0 Þþdðo o 0 Þ

½ : (7.167)

There is no delta function for o ¼ 0 and consequently the output process Y ( t )
has no DC component (the same result comes from the fact that the capacitor does
not pass a DC component).

EYðtfg¼ 0 : (7.168)

The variance is obtained from

1

ðtÞ ¼R YY ð 0 Þ¼

1
2 p

s YY ¼EY 2

S YY ðoÞ d o

1

2
4

3
5 :

1 1

2 2

p
2 p

ðoTÞ ðoTÞ

¼

2 dðoþo 0 Þ d oþ 2 dðoo 0 Þ d o

ð 7 : 169 Þ

1 þðoTÞ 1 þðoTÞ

1 1

Using the characteristics of delta function from ( 2.72 ), we arrive at:

" #

2 2 2

1
2

ðo 0 TÞ ðo 0 TÞ ðo 0 TÞ

s 2

YY ¼

2 þ

¼

2 :

(7.170)

2

1 þðo 0 TÞ 1 þðo 0 TÞ 1 þðo 0 TÞ

Exercise E.7.10 Find the autocorrelation function of the output process Y ( t ) from
Exercise E.7.9 and confirm that the mean and variance of the process Y ( t ) are given
as in ( 7.168 ) and ( 7.170 ), respectively.

Answer The autocorrelation function is the inverse Fourier transform of the PSD

( 7.167 )

1

2

1
2 p

p
1 þðoTÞ

ðoTÞ

R YY ðtÞ¼F 1 e jot d o; f S YY ðoÞ g ¼

2 dðoþo 0 Þþdðoo 0 Þ

½

1

2 3

1 1

2 e jot
1 þðoTÞ

2 e jot
1 þðoTÞ

1
2

ðoTÞ ðoTÞ

4 5 :

¼

2 dðoþo 0 Þ d oþ 2 dðoo 0 Þ d o

ð 7 : 171 Þ

1 1