Cryptography Reference
In-Depth Information
For a frequency non-selective channel, we have
B<B
c
. Noting that band
B
is proportional to
1
/T
, this leads to
T>T
m
, or again
T>>τ
n
n
.Inthe
expression of
r
(
t
)
, we can therefore neglect
τ
n
in front of
T
which gives:
∀
n
=1
ρ
n
i
M
r
(
t
)=
A
a
i
h
(
t
−
T
)cos(2
πf
0
t
+
ϕ
n
(
t
))
−
i
T
)sin(2
πf
0
t
+
ϕ
n
(
t
))
b
i
h
(
t
−
with:
ϕ
n
(
t
)=
f
n
t
(
f
0
+
f
n
)
τ
n
−
Putting:
M
M
a
c
(
t
)=
ρ
n
cos
ϕ
n
(
t
)
and
a
s
(
t
)=
ρ
n
sin
ϕ
n
(
t
)
n
=1
n
=1
and:
a
c
(
t
)
a
c
(
t
)+
a
s
(
t
)
a
s
(
t
)
a
c
(
t
)+
a
s
(
t
)
cos
φ
(
t
)=
and
sin
φ
(
t
)=
signal
r
(
t
)
can again be written:
r
(
t
)=
Aα
(
t
)
i
a
i
h
(
t
−
iT
)cos(2
πf
0
t
+
φ
(
t
))
−
i
iT
)sin(2
πf
0
t
+
φ
(
t
))
(2.158)
b
i
h
(
t
−
with
α
(
t
)=
a
c
(
t
)+
a
s
(
t
)
.
For a frequency non-selective multipath channel, the modulated M-QAM
signal only undergoes an attenuation
α
(
t
)
and dephasing
φ
(
t
)
.
Modelling the attenuations
ρ
n
, the delays
τ
n
, and the Doppler frequencies
f
n
by mutually independent random variables then, for large enough
M
and for a
given
t
,
a
c
(
t
)
and
a
s
(
t
)
tend towards non-correlated random Gaussian variables
(central limit theorem). The attenuation
α
(
t
)
,foragiven
t
, follows a Rayleigh
law and the phase
φ
(
t
)
is equidistributed on
[0
,
2
π
[
.
exp
α
2
σ
α
p
(
α
)=
2
α
σ
α
−
α
≥
0
(2.159)
with
σ
α
=
E
α
2
.
The attenuation
α
(
t
)
can take values much lower than unity and, in this
case, the information signal received by the receiver is very attenuated. Its level
is then comparable to, if not lower than, that of the noise. We say that the
transmission channel shows deep Rayleigh fading.
If band
B
occupied by the modulated signal is higher than the coherence
band, the channel is frequency selective. Its frequency response, on band
B
,is
