Cryptography Reference
In-Depth Information
and
h
(
t
)
is a rectangular pulse of unit amplitude, but of width
2
T
.
h
(
t
)=1
if
t
T,T
[
=0
elsewhere
∈
[
−
(2.47)
MSK modulation can be seen as an amplitude modulation of the terms
πt
2
T
πt
2
T
cos
cos(2
πf
0
t
+
ϕ
0
)
and
−
sin
sin(2
πf
0
t
+
ϕ
0
)
by two bit streams
u
c
(
t
)=
i
2
iT
)
and
u
s
(
t
)=
i
c
2
i−
1
h
(
t
−
c
2
i
h
(
t
−
(2
t
+1)
T
)
whose transitions are shifted by
T
. Each bit stream enables a bit to be trans-
mitted every
2
T
seconds and thus, the binary rate of an MSK modulation is
D
=1
/T
b
with
T
=
T
b
.
MSK modulation is a particular case of CP M-FSK modulation since it is
linear. The psd is given by:
γ
S
(
f
)=
1
f
0
)+
1
4
γ
(
f
−
4
γ
(
−
f
−
f
0
)
(2.48)
with:
cos 2
πfT
1
2
γ
(
f
)=
16
A
2
T
π
2
(2.49)
16
f
2
T
2
−
Figure 2.11 shows the psd of the complex envelope of an MSK signal expressed
in dB as a function of the normalized frequency
fT
b
. We have also plotted the
psd of the complex envelope of a 4-PSK signal. In order for the comparison of
these two power spectral densities to make sense, we have assumed that the rate
transmitted was identical for these two modulations (that is,
T
=2
T
b
for the
4-PSK modulation).
The width of the main lobe of the power spectral density of an MSK mod-
ulation is
3
/
2
T
b
whereas it is only
1
/T
b
for 4-PSK modulation. Thus, for a
same rate transmitted, the main lobe of MSK modulation occupies 50% more
bandwidth than that of 4-PSK modulation. However, the envelope of the psd
of an MSK signal decreases in
f
−
4
whereas it only decreases in
f
−
2
for 4-PSK
modulation. One of the consequences of this is that the bandwidth
B
that con-
tains 99% of the power of the modulated signal for MSK is
1
.
2
/T
b
whereas it is
around
8
/T
b
for 4-PSK.
L-ary Raised Cosine modulation (L-RC)
For this modulation, the function
g
(
t
)
has the expression:
1
cos
2
πt
g
(
t
)=
2
LT
(1
−
LT
)
for
0
≤
t
≤
LT
(2.50)
=
0
elsewhere
The larger parameter
L
is, the faster the power spectral density of this modu-
lation decreases. For example, the psd is -40 dB for a
2
RC
(
h
=1
/
2
,
a
i
=
±
1)
