Consider the following system
Assume that
N
τ
N
τ
is a white Gaussian process with zero mean and spectral height
N
0
2
N
0
2
.
If b b is "0" then
X
τ
=A
p
T
τ
X
τ
A
p
T
τ
and if b b is "1" then
X
τ
=(−A)
p
T
τ
X
τ
A
p
T
τ
where
p
T
τ={1 if 0≤τ≤T0 otherwise
p
T
τ
1
0
τ
T
0
. Suppose
Prb=1=Prb=0=1/2
b
1
b
0
12
.
- Find the probability density function
Z
T
Z
T
when bit "0" is transmitted and also when bit "1" is transmitted.
Refer to these two densities as
f
Z
T
H
0
z
f
Z
T
H
0
z
and
f
Z
T
H
1
z
f
Z
T
H
1
z
,
where
H
0
H
0
denotes the hypothesis that bit "0" is transmitted and
H
1
H
1
denotes the hypothesis that bit "1" is transmitted.
- Consider the ratio of the above two densities;
i.e.,
Λz=f
Z
T
H
0
zf
Z
T
H
1
z
Λ
z
f
Z
T
H
0
z
f
Z
T
H
1
z
(1) - Find a γγ that minimizes
Pr
b
^
≠b
b
^
b
.
