Consider a ternary communication system where the source
produces three possible symbols: 0, 1, 2.
a) Assign three modulation signals
s
1
t
s
1
t
,
s
2
t
s
2
t
, and
s
3
t
s
3
t
defined on
t∈
0
T
t
0
T
to these symbols, 0, 1, and 2, respectively. Make sure that
these signals are not orthogonal and assume that the symbols
have an equal probability of being generated.
b) Consider an orthonormal basis
ψ
1
t
ψ
1
t
,
ψ
2
t
ψ
2
t
, ...,
ψ
N
t
ψ
N
t
to represent these three signals. Obviously
NN could be either 1, 2, or 3.
Now consider two different receivers to decide which one of
the symbols were transmitted when
r
t
=
s
m
t+
N
t
r
t
s
m
t
N
t
is received where
m=123
m
1
2
3
and
N
t
N
t
is a zero mean white Gaussian process with
S
N
f=
N
0
2
S
N
f
N
0
2
for all ff. What is
f
r

s
m
(
t
)
f
r

s
m
(
t
)
and what is
f
Y

s
m
(
t
)
f
Y

s
m
(
t
)
?
Find the probability that
m
^
≠m
m
^
m
for both receivers.
P
e
=Pr
m
^
≠m
P
e
m
^
m
.