The implementation and theory of correlator-type receivers can
be found in
Detection.
Example 1
m
^
=2
m
^
2
since
Drs1>Drs2
D
r
s
1
D
r
s
2
or
∥s1∥2=∥s2∥2
s
1
2
s
2
2
and
<r,s2>><r,s1>
r
s
2
r
s
1
.
Example 2
Data symbols "0" or "1" with equal probability. Modulator
s
1
t=st
s
1
t
s
t
for
0≤t≤T
0
t
T
and
s
2
t=-st
s
2
t
s
t
for
0≤t≤T
0
t
T
.
ψ
1
t=stA2T
ψ
1
t
s
t
A
2
T
,
s11=AT
s
11
A
T
, and
s21=-AT
s
21
A
T
∀m,m=12:rt=
s
m
t+
N
t
m
m
1
2
r
t
s
m
t
N
t
(1)
r1=AT+
η
1
r
1
A
T
η
1
(2)
or
r1=-AT+
η
1
r
1
A
T
η
1
(3)
η
1
η
1
is Gaussian with zero mean and variance
N
0
2
N
0
2
.
m
^
=argmaxATr1-ATr1
m
^
A
T
r
1
A
T
r
1
, since
AT>0
A
T
0
and
Pr
s
1
=Pr
s
1
s
1
s
1
then the MAP decision rule decides.
s
1
t
s
1
t
was transmitted if
r1≥0
r
1
0
s
2
t
s
2
t
was transmitted if
r1<0
r
1
0
An alternate demodulator:
rt=
s
m
t+
N
t
⇒r=sm+η
r
t
s
m
t
N
t
r
s
m
η
(4)