Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 487

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

" #

2 e jo 0 t
1 þðo 0 TÞ

2 e jo 0 t
1 þðo 0 TÞ

2

e jo 0 t þ e jo 0 t

2

1
2

ðo 0 TÞ

2 þ ð o 0 T Þ

ðo 0 TÞ

R YY ðtÞ¼ ¼ ;

2 2

1 þðo 0 TÞ

2

ðo 0 TÞ

¼

2 cos ðo 0 tÞ:

ð 7 : 172 Þ

1 þðo 0 TÞ

The autocorrelation function ( 7.171 ) does not have a constant term and

consequently

EYðtfg¼ 0 : (7.173)

The variance is:

2

ðtÞ EYðtfg¼ EY 2 ðtÞ ¼ R YY ð 0 Þ¼ ð o 0 T Þ s YY ¼ EY 2

2 :

(7.174)

1 þðo 0 TÞ

The obtained results ( 7.173 ) and ( 7.174 ) are equal to the results ( 7.168 ) and
( 7.170 ), respectively.

Exercise E.7.11 If the input process X ( t ) from Exercise E.7.9 has a Gaussian PDF
what will be the PDF of the output process Y ( t )?

Answer Since the input process is subjected to a LTI filtering, it follows that the
input Gaussian process is subjected to a linear transformation. As a consequence,
the output process is also a Gaussian process (see Sect. 4.3 ).

The process Y ( t ) is a WS stationary (with a constant mean and autocorrelation
function that depends only on t ) and, consequently, its PDF is independent of time.
This means that the PDF of the output process is:

; s YY Þ: f Y ðyÞ¼Nð 0 (7.175)

where the variance is given in ( 7.174 ).

Exercise E.7.12 Consider that the input to the RC filter shown in Fig. 7.27 is a
white noise with the PSD S XX ( o ) ¼ N 0 /2. Find the PSD and the autocorrelation
function of the process at the output of the filter.

Fig. 7.27 RC filter in
Exercise E.7.12