Cryptography Reference
In-Depth Information
reflected on different obstacles and, thus, the receiver receives
M
copies of the
signal transmitted, each copy being affected by an attenuation
ρ
n
(
t
)
,withdelay
τ
n
(
t
)
and a Doppler frequency shift
f
n
(
t
)
. The attenuations, delays and Doppler
frequencies are functions of time in order to take into account the time-varying
channel. To simplify the notations, in the following we will omit variable
t
for
the attenuations, the delays and the Doppler frequencies.
Let
r
(
t
)
be the response of the transmission channel to the signal
s
(
t
)
,which
is generally written in the form:
M
ρ
n
exp
j
2
π
(
f
n
+
f
)(
t
τ
n
)
r
(
t
)=
−
(2.153)
n
=1
Making
s
(
t
)
appear, the received signal can again be written:
M
ρ
n
exp
j
2
π
(
f
n
t
(
f
n
+
f
)
τ
n
))
s
(
t
)
r
(
t
)=
−
(2.154)
n
=1
and thus the frequency response of the transmission channel is defined by:
M
ρ
n
exp
−
f
n
t
+
f
n
τ
n
)
c
(
f, t
)=
j
2
π
(
fτ
n
−
(2.155)
n
=1
The multipath channel is generally frequency selective, that is, it does not trans-
mit all the frequency components of the signal placed at its input in the same
way, certain components being more attenuated than others. The channel will
therefore create distortions of the transmitted signal. In addition, their evolution
over time can be more or less rapid.
To illustrate the frequency selectivity of a multipath channel, we have plotted
in Figure 2.25 the power spectrum of the frequency response of this channel for
M
=2
, in the absence of a Doppler frequency shift (
f
n
=0)
and fixing
τ
1
to
zero.
2
=
ρ
1
(1 +
α
cos 2
πfτ
2
)
2
+
α
2
sin
2
2
πfτ
2
|
c
(
f
)
|
(2.156)
with
α
=
ρ
2
/ρ
1
.
Two parameters are now introduced: coherence bandwidth
B
c
and coherence
time
t
c
that allow the transmission channel to be characterized in relation to
the frequency selectivity and its evolution speed.
Coherence bandwidth
There are several definitions of the coherence bandwidth but the most common
definition is:
1
T
m
B
c
≈
(2.157)
