sm8_53

Prévia do material em texto

Problem 8.53 A baseband signal is disturbed by a noise process N(t) as shown by 
 
 ( ) )(3.0sin)( tNtAtX += π 
 
where N(t) is a stationary Gaussian process of zero mean and variance σ2. 
 
(a) What are the density functions of the random variables X1 and X2 where 
 
11
)( == ttXX 
 
22
)( == ttXX 
 
(b) The noise process N(t) has an autocorrelation function given by 
 ( )τστ −= exp)( 2NR 
 
What is the joint density function of X1 and X2, that is, ? ),( 21, 21 xxf XX
 
Solution 
(a) The random variable X1 has a mean 
 [ ] ( )[ ]
[ ]
( )π
π
π
3.0sin
)()3.0sin(
)(3.0sin)(
1
11
A
tNA
tNAtX
=
+=
+=
E
EE
 
 
Since X1 is equal to N(t1) plus a constant, the variance of X1 is the same as that of N(t1). In 
addition, since N(t1) is a Gaussian random variable, X1 is also Gaussian with a density 
given by 
{ }21 2/)(exp21)(1 σµσπ −−= xxf X 
 
where [ )( 11 tXE= ]µ . By a similar argument, the density function of X2 is 
 
{ }22 2/)(exp21)(2 σµσπ −−= xxf X 
 
where ( )πµ 6.0sin2 A= . 
 
 
Continued on next slide 
 
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only 
to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that 
permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. 
page…8-66 
Problem 8-53 continued 
 
(b) First note that since the mean of X(t) is not constant, X(t) is not a stationary random 
process. However, X(t) is still a Gaussian random process, so the joint distribution of 
N Gaussian random variables may be written as Eq. (8.90). For the case of N = 2, this 
equation reduces to 
 
{ }2/)()(exp
2
1)( 12/1
Tf µxµxxX −Λ−−Λ=
−
π 
 
where Λ is the 2x2 covariance matrix. Recall that cov(X1,X2) =E[(X1-µ1)(X2-µ2)], so that 
 
( ) (
( ) ⎥⎦
⎤⎢⎣
⎡
−
−=
⎥⎦
⎤⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡=Λ
22
22
2212
2111
1exp
)1exp(
)0()1(
)1()0(
,cov,cov
),cov(),cov(
σσ
σσ
NN
NN
RR
RR
XXXX
XXXX
)
 
 
If we let ρ = exp(-1) then 
 
)1( 24 ρσ −=Λ 
 
and 
 
⎥⎦
⎤⎢⎣
⎡
1−
−
−=Λ
−
ρ
ρ
ρσ
1
)1(
1
22
1 
 
Making these substitutions into the above expression, we obtain upon simplification 
 
⎭⎬
⎫
⎩⎨
⎧
−
−−−−+−−−= )1(2
))((2)()(exp
12
1),( 22
2211
2
22
2
11
2221, 21 ρσ
µµρµµ
ρπσ
xxxxxxf XX 
 
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only 
to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that 
permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. 
page…8-67