Cryptography Reference
In-Depth Information
For 2-PSK modulation, there is an exact expression of the bit error proba-
bility
Peb
. Assuming the binary data
iid
, this error probability is equal to:
1
2
Pr
+
1
Peb
=
{
r
1
>
0
|
φ
j
=
π
}
2
Pr
{
r
1
<
0
|
φ
j
=0
}
Output
r
1
of the demodulator is:
E
b
+
b
r
1
=
±
where
E
b
=
A
2
T/
2
is the energy received per information bit transmitted and
b
is an AWGN, with zero mean and variance equal to
N
0
/
2
.
0
∞
(
r
1
+
√
E
b
)
2
)
dr
1
1
2
1
√
πN
0
1
N
0
Peb
=
exp(
−
exp
(
r
1
−
√
E
b
)
2
dr
1
0
+
2
1
√
πN
0
1
N
0
−
−∞
Introducing the complementary error function, the bit error probability
Peb
is
equal to:
2
erfc
E
b
1
Peb
=
(2.74)
N
0
Case of 4-PSK modulation
For this modulation, phase
φ
j
takes four values
π/
4
,
3
π/
4
,
5
π/
4
,
7
π/
4
.
With each state of the phase are associated two binary data. For equiprob-
able phase states, the MAP criterion leads to the following decision rules:
φ
j
=
4
if
r
1
>
0;
r
2
>
0
φ
j
=
3
4
if
r
1
<
0;
r
2
>
0
φ
j
=
5
4
r
1
<
0;
r
2
<
0
if
φ
j
=
7
4
if
r
1
>
0;
r
2
<
0
Considering the following Gray coding:
π
4
→
3
π
4
→
5
π
4
→
7
π
4
→
11
01
00
10
The estimation of the binary data can be performed by separately comparing
outputs
r
1
and
r
2
of the demodulator to a threshold fixed to zero. The coherent
receiver for a 4-PSK modulation is represented in Figure 2.16.
