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E
E

5
6
4
F
a
l
l
2
0
0
9
Homework #7
Due Wednesday, November 18, 2009
1.
“Digital Communications”, 5
th
Edition, Proakis & Salehi, Problem 10.2
2.
(Course Reader Problem 51) Consider an ISI channel, where the sequence of sampled
matched filter outputs is given by
=+ࢠ
Where
=±ݕ
ଵ
,ݕ
ଶ
,…,ݕ
ିଵ
²
,
is a Hermitian symmetric Toeplitz matrix with
±݅,݆²
element
[]
,
=݃
ି
, and where
ࢠ
denotes the noise contribution to the output of the matched filter.
Assume there exists a lower triangular matrix
such that
=
. This decomposition is called
Cholesky factorization
and
is referred to as the Cholesky factor of
G
.
L=
ۉ
ۈ
ۈ
ۈ
ۈ
ۇ
L
0
L
ଵ
L
00
L
ଶ
L
ଵ
0L
ଶ
L
0
L
ଵ
L
0
0
0
0
0
0
0
0
0
0
0
0
0
L
ଶ
L
ଵ
ଶ
L
0
L
ଵ
L
L
ଶ
L
ଵ
ଶ
L
0
L
ଵ
L
ی
ۋ
ۋ
ۋ
ۋ
ۊ
In which
L
=1
,
L
ଵ
=−1/2
, and
L
ଶ
=1/4
. Input signal set is {1, 1} and the output sequence is
y=±1.5,−1,2.5,−2,−2.5 ,0,1,0²
. Assume that the initial state is
x
ିଵ
=x
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This note was uploaded on 04/03/2010 for the course EE 564 at USC.
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