6.450 Principles of Digital Communication
Wednesday, November 7, 2002
MIT, Fall 2002
Handout #35
Due: Wednesday, November 14, 2001
Problem Set 10
Problem 10.1
The purpose of this problem is to show that if
{
N
j
;
j
∈
Z
}
is a sequence of Gaussian rv’s
N
(0
, σ
2
), then the sampling theorem expansion
N
(
t
) =
∑
j
N
j
sinc(
t

jT
) is a stationary
random process.
(a) Suppose that an
L
2
function
x
(
t
) is baseband limited to 1
/
(2
T
) for some
T >
0 and is
the inverse Fourier transform of ˆ
x
(
f
). Show that for any fixed
τ
,
x
(
t

τ
) can be expressed
as
x
(
t

τ
) =
X
j
∈
Z
x
(
jT

τ
) sinc
t

jT
T
(b) Show that
sinc
t

τ
T
=
X
j
∈
Z
sinc
τ

jT
T
sinc
t

jT
T
(c) Consider the stochastic process
N
(
t
) =
X
j
∈
Z
N
j
sinc
t

jT
T
where
{
N
j
;
j
∈
Z
}
is a set of iid
N
(0
, σ
2
) rv’s. Find
K
N
(
t, τ
) =
E
[
N
(
t
)
N
(
τ
)].
(d) Show that
{
N
(
t
)
}
is a stationary Gaussian random process and find the covariance
function
˜
K
N
(
t

τ
) =
K
N
(
t, τ
).
Note that the sample functions of
{
N
(
t
)
}
are not
L
2
,
although
N
(
t
1
)
, N
(
t
2
)
, . . . , N
(
t
k
) are jointly Gaussian.
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 Spring '08
 DanielLee

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