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# sm8_39

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```Problem 8.39 Assume the narrow-band process X(t) described in Problem 8.38 is Gaussian with zero
mean and variance .
2
X\u3c3
(a) Calculate .
2
X\u3c3
(b) Determine the joint probability density function of the random variables Y and Z obtained by
observing the in-phase and quadrature components of X(t) at some fixed time.

Solution
(a) The variance is given by

( ) ( )
watts
hbhb
dffSRX
3
11
2
112
2
12
2
1
2
12
0
2211
2
=
\u239f\u23a0
\u239e\u239c\u239d
\u239b \u22c5\u22c5+\u22c5\u22c5=
\u239f\u23a0
\u239e\u239c\u239d
\u239b +=
== \u222b\u221e\u221e\u2212\u3c3

(b) The random variables Y and Z have zero mean, are Gaussian and have variance . If
Y and Z are independent, the joint density is given by
2
X\u3c3

( )
\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b +\u2212=
\u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212\u22c5\u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212=
2
22
2
2
2
2
2
,
2
exp
2
1
2exp2
1
2exp2
1,
XX
XXXX
ZY
zy
zyZYf
\u3c3\u3c0\u3c3
\u3c3\u3c3\u3c0\u3c3\u3c3\u3c0

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page\u20268-49```