EE564
Fall 2009
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
)
,
is a Hermitian symmetric Toeplitz matrix with
(݅, ݆)
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
0
0
0
0
L
ଶ
L
ଵ
0
L
ଶ
L
0
L
ଵ
L
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
L
ଶ
L
ଵ
0
L
ଶ
0
0
0
0
0
0
0
0
L
0
L
ଵ
L
0
0
0
0
L
ଶ
L
ଵ
0
L
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, 0)
. Assume that the initial state is
x
ିଵ
= x
ିଶ
= 0
. Find the
optimum MAP decoded sequence.
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 '08
 staff
 Triangular matrix, Viterbi algorithm, sequence detection

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