ECEN 661  Modulation Theory
Homework #6 Solutions
1.
(a)
Given the sequence of matched filter outputs,
, the ML detector will find the
sequence which optimizes
.
The trellis diagram and the appropriate branch metrics for the MLSE is shown below.
(b) The energy transmitted per bit is
.
The minimum distance will occur when two sequences differ in two bits. For example,
and
. Then the resulting precoded sequences
(assuming
) would be
and
. This will result
in transmitted waveforms that differ (by an amplitude of 2A) during two different symbol inter
vals. The total energy in the difference waveforms will be
. The resulting
asymptotic efficiency is
.
(c) For this format, each path has
.
nearest neighbors.
X
( )
t
d
n
1
–
(
)
T
s
nT
s
∫
r t
( )
r
n
2
ω
c
t
(
)
cos
MLSE
(Viterbi
Algorithm)
a
ˆ
n
r
r
1
r
2
r
3
…
,
,
,
(
)
=
λ
a
( )
r
n
1
–
(
)
b
n
n
∑
r
n
1
–
(
)
a
n
a
n
1
–
⊕
n
∑
=
=
a
n
1
–
0
=
a
n
1
–
1
=
…
…
r
1
/
a
1
0
=
r
1
/
a
1
1
=
r
–
1
/
a
1
0
=
r
–
1
/
a
1
1
=
r
2
/
a
2
0
=
r
2
/
a
2
1
=
r
–
2
/
a
2
0
=
r
–
2
/
a
2
1
=
Each branch is labelled with “branch metric / input bit.”
E
b
s
2
t
( )
t
d
0
T
s
∫
A
2
T
s
2

=
=
a
0 0 0 0 0
…
,
,
,
,
,
(
)
=
a
'
0 1 0 0 0
…
,
,
,
,
,
(
)
=
a
0
0
=
b
0 0 0 0 0
…
,
,
,
,
,
(
)
=
b
'
0 1 1 0 0
…
,
,
,
,
,
(
)
=
d
min
2
4
A
2
T
s
8
E
b
=
=
η
d
min
2
4
E
b

2
=
=
N
d
min
N
1
–
(
)
N
2
–
(
)
…
1
+
+
+
N N
1
–
(
)
2

N
2
=
=
=
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(d) For each path there are
nearest neighbors which cause one bit error,
neighbors
which cause 2 errors,
neighbors which cause 3 bit errors, ..., and 1 neighbor which causes
bit errors. Hence the average number of bit errors is
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 Spring '11
 Miller
 Trigraph, Pallavolo Modena, Sisley Volley Treviso, Associazione Sportiva Volley Lube, Piemonte Volley

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