Cryptography Reference
In-Depth Information
where erf
(
x
)
represents the error function defined by:
x
2
√
π
u
2
)
du
erf
(
x
)=
exp(
−
0
In Figure 2.12, we have plotted
g
(
t
)
as a function of the normalized variable
t/T
, for different values of the normalized passband
B
N
=
B
g
T
.Wenotethat
the graph of the function
g
(
t
)
has been shifted by
2
T
for
B
N
=0
.
2
and by
1
.
5
T
for
B
N
=0
.
3
.
Figure 2.12 - Variation of the function
g
(
t
)
for two values of
B
N
.
The term
B
N
allows the time spreading of function
g
(
t
)
to be fixed. Thus for
B
N
=0
.
2
, this function is approximately of width
4
T
whereas its width is only
3
T
for
B
N
=0
.
3
.When
B
N
tends towards infinity, it becomes a recatngu-
lar pulse with width
T
(the case of MSK modulation). GMSK modulation is
therefore a partial response continuous phase modulation (
L>
1
).
On the interval
[
iT,
(
i
+1)
T
[
, the phase
φ
(
t
)
of the GMSK signal is equal to:
i
i−L
nT
)+
π
2
φ
(
t
)=
π
a
n
q
(
t
−
a
n
(2.55)
n
=
i
−
L
+1
n
=
−∞
where
L
=3
if
B
N
=0
.
3
and
L
=2
if
B
N
=0
.
2
.
Thus on an interval
[
iT,
(
i
+1)
T
[
, the phase
φ
(
t
)
of the GMSK signal depends on
symbol
a
i
but also on the symbols prior to symbol
a
i
(
a
i−
1
,a
i−
2
,
,a
i−L
+1
)
.
This non-linear modulation presents an memory effect that gives it good spectral
properties.
···
GMSK modulation, with a normalized passband
B
N
=0
.
3
was
