Cryptography Reference
In-Depth Information
Let us introduce the power transmitted at the output of the emission filter:
+
∞
+
∞
+
∞
1
4
f
0
)
df
+
1
4
γ
e
(
f
+
f
0
)
df
=
1
2
P
=
γ
e
(
f
−
γ
e
(
f
)
df
(2.148)
−∞
−∞
−∞
Replacing
γ
e
(
f
)
by its expression, we obtain:
+
∞
A
2
T
A
2
T
P
=
p
(
t
0
)
CS
α
(
f
)
df
=
p
(
t
0
)
(2.149)
−∞
Using expressions (2.147) and (2.149), the error probability is equal to:
2
erfc
PT
Pe
a
n
=
1
(2.150)
2
N
0
The energy
E
b
used to transmit an information bit
d
n
is:
E
b
=
PT
b
(2.151)
where
T
b
is the inverse of the bit rate of the transmission.
For a 4-PSK modulation,
T
=2
T
b
and the bit error probability
d
n
is finally:
2
erfc
E
b
Pe
d
n
=
1
(2.152)
N
0
The error probability in the presence of Nyquist filtering for a 4-PSK modulation
is identical to that obtained for a transmission on an infinite-bandwidth channel.
This result is also true for the other linear modulations of the M-ASK, M-PSK
and M-QAM type.
To conclude this section, we can say that filtering according to the Nyquist
criterion of a linear modulation makes it possible to reduce the bandwidth nec-
essary for its transmission to
(1 +
α
)
R
m
,where
R
m
is the symbol rate. This
filtering does not degrade the performance of the modulation, that is, it leads to
the same bit error probability as that of a transmission on an infinite bandwidth
channel.
2.4
Transmission on fading channels
2.4.1 Characterization of a fading channel
Let us consider a transmission over a multipath channel where the transmitter,
which is mobile compared to the receiver, provides a non-modulated signal
s
(
t
)=
A
exp(
j
2
πft
)
with amplitude
A
and frequency
f
.Signal
s
(
t
)
propagates by being
