Cryptography Reference
In-Depth Information
Neglecting the coecients that weight the complementary error functions, M-
PSK modulation requires the relation
E
b
/N
0
to be increased by:
10 log
1)
π
2
dB
3
M
2
2(
M
−
to obtain the same error probability as M-QAM modulation. If, for example, we
compare the performance of 16-PSK modulation with that of 16-QAM modula-
tion, we note that the former requires about 4 dB more for the relation
E
b
/N
0
to obtain the same error probabilities.
Frequency shift keying - M-FSK
For an M-FSK modulation, the modulator provides signals of the form:
s
j
(
t
)=
Ah
(
t
)cos(2
πf
j
t
+
ϕ
j
)
(2.83)
where the frequencies
f
j
are chosen in such a way that the
Ms
j
(
t
)
signals are
orthogonal. The space defined by these signals therefore has dimension
N
=
M
and the vectors
ν
j
(
t
)
are of the form:
ν
j
(
t
)=
2
T
cos(2
πf
j
t
+
ϕ
j
)
j
=1
,
2
,
···
M
(2.84)
Assuming the information data
d
i
iid
,the
s
j
(
t
)
signals have the same probability.
In addition, they have the same energy
E
and thus, the MAP criterion leads to
the following decision rule:
M
M
s
j
(
t
)
if
r
p
s
jp
>
r
p
s
np
∀
n
=
j
(2.85)
p
=1
p
=1
where
s
jp
is equal to:
s
j
(
t
)
ν
p
(
t
)
dt
=
A
T
T
s
jp
=
2
δ
jp
(2.86)
0
Taking into account the expression of
s
jp
, decision rule (2.85) is simplified and
becomes:
=
j
(2.87)
The optimal coherent receiver for an M-FSK modulation is shown in Figure 2.19.
s
j
(
t
)
if
r
j
>r
n
∀
n
Conditionally to the emission of the signal
s
j
(
t
)
,the
M
outputs of the de-
modulator are of the form:
r
j
=
E
s
+
b
j
r
p
=
b
p
∀
p
=
j
