Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 258

Notice that R(s)isCase even, with an Case absolute maximum at s = 0.
MATLAB: The above question can be solved on MATLAB r as follows:

x ¼ stepfun ð t ; 0 Þ stepfun ð t ; 1 Þ ;y ¼ xcorr ð x Þ Ts;

Tutorial 24

Q: A white noise signal n(t) is passed through a RC circuit as shown. Find the
autocorrelation function of the output noise.

R

n o ( t )

+

+

n i ( t )

C

_

_

Solution:

1
jxC
R þ 1
jxC

H ð f Þ¼ X C
R þ X C

1
1 þ j2pfCR ;

¼ ¼

R n ð s Þ¼ g G n ð f Þ¼ g = 2 F

WKT

2 d ð s Þ;

1
R 2 C 2

1 þ 4p 2 f 2 R 2 C 2 ¼ g

g = 2

G n 0 ð f Þ¼ H ð f j j 2 G n ð f Þ¼

2

R 2 C 2 þð 2pf Þ 2

1

1
RC

¼ g
4RC

2

RC 2 þ 2p ð 2 :

1

4RC e jj

g¼ g

) R n 0 ð s Þ¼ F 1

f

G n 0 ð f Þ

RC [Tables; Wiener-Kinchin Theorem].
Note that R n ð s Þ¼E n ð t Þ n ð t þ s f g , where E is the statistical expectation functional
[Compare with the deterministic signal in Tutorial 22].

η
4 RC

R n o ( τ )

τ

0