Decide which
s
m
t
s
m
t
from the set of
s
1
t…
s
m
t
s
1
t
…
s
m
t
signals was transmitted based on observing
r=r1r2⋮rN
r
r
1
r
2
⋮
r
N
,
the vector composed of demodulated
received signal, that is, the vector of projection of the
received signal onto the NN bases.
m
^
=argmax
1≤m≤M
Pr
s
m
t was transmitted |
r was observed
m
^
1
m
M
r
was observed
s
m
t
was transmitted
(1)
Note that
Pr
s
m
|
r
≜
Pr
s
m
twas transmitted|
r was observed
=
f
r
|
s
m
Prsm
f
r
r
s
m
≜
r
was observed
s
m
t
was transmitted
f
r
|
s
m
s
m
f
r
(2)
If
Prsm was transmitted=1M
s
m
was transmitted
1
M
, that is information symbols are equally likely to
be transmitted, then
argmax
1≤m≤M
Prsm|
r
=argmax
1≤m≤M
f
r
|
s
m
1
m
M
r
s
m
1
m
M
f
r
|
s
m
(3)
Since
rt=
s
m
t+
N
t
r
t
s
m
t
N
t
for
0≤t≤T
0
t
T
and for some
m=12…M
m
1
2
…
M
then
r=sm+η
r
s
m
η
where
η=η1η2⋮ηN
η
η
1
η
2
⋮
η
N
and
ηn
η
n
's are Gaussian and independent.
∀
r
n
,
r
n
∈R:
f
r
|
s
m
=12π
N
0
2N2e−∑
n
=1Nrn−sm,n22
N
0
2
r
n
r
n
f
r
|
s
m
1
2
N
0
2
N
2
n
1
N
r
n
s
m
n
2
2
N
0
2
(4)
m
^
=argmax
1≤m≤M
f
r
|
s
m
=argmax
1≤m≤M
ln
f
r
|
s
m
=argmax
1≤m≤M
(−(N2lnπ
N
0
))−1
N
0
∑
n
=1Nrn−sm,n2=argmin
1≤m≤M
∑
n
=1Nrn−sm,n2
m
^
1
m
M
f
r
|
s
m
1
m
M
f
r
|
s
m
1
m
M
N
2
N
0
1
N
0
n
1
N
r
n
s
m
n
2
1
m
M
n
1
N
r
n
s
m
n
2
(5)
where
Drsm
D
r
s
m
is the
l
2
l
2
distance between vectors
rr
and
sm
s
m
defined as
Drsm
≜
∑n=1Nrn−sm,n2
D
r
s
m
≜
n
1
N
r
n
s
m
n
2
m
^
=argmin
1≤m≤M
Drsm=argmin
1≤m≤M
∥r∥2−2〈(r,sm)〉+∥sm∥2
m
^
1
m
M
D
r
s
m
1
m
M
r
2
2
r
s
m
s
m
2
(6)
where
∥r∥
r
is the
l
2
l
2
norm of vector
rr
defined as
∥r∥
≜
∑n=1Nrn2
r
≜
n
1
N
r
n
2
m
^
=argmax
1≤m≤M
2〈(r,sm)〉−∥sm∥2
m
^
1
m
M
2
r
s
m
s
m
2
(7)
This type of receiver system is known as a
correlation (or correlator-type) receiver.
Examples of the use of such a system are found
here. Another type of
receiver involves linear, time-invariant filters and is known
as a
matched
filter receiver. An analysis of the performance of a
correlator-type receiver using antipodal and orthogonal binary
signals can be found in
Performance Analysis.
