Cryptography Reference
In-Depth Information
In this form,
s
l
is not an inverse discrete Fourier transform. Introducing virtual
symbols in the following way:
c
2
N−n, i
=
c
n, i
n
=1
,
···
,
(
N
−
1)
c
N,i
=
{
c
0
,i
}
(2.187)
and replacing
c
0
,i
by
, we can show that the sampled signal
s
l
can effec-
tively be written in the form of an inverse discrete Fourier transform:
{
c
0
,i
}
c
n, i
exp
j
2
π
nl
2
N
2
N−
1
s
l
=
A
(2.188)
n
=0
On each interval of duration
T
we perform an inverse discrete Fourier transform
on
2
N
points that enable the
N
M-QAM baseband signals to be obtained.
A digital-analogue converter followed by a frequency transposition enables the
modulated signals to be obtained on a carrier frequency.
At reception, after amplification and transposition of the OFDM signal into
baseband, the decoding of the modulation symbols
c
n,i
=
a
n,i
+
jb
n,i
can also
be performed by a discrete Fourier transform.
We have seen that the duration
T
of the modulation symbols increases with
N
and, for large values of
N
, can become comparable to the coherence time
t
c
of
the transmission channel. In this case, the hypothesis of a slow-fading channel is
no longer satisfied. There are therefore limits to the choice of parameter
N
.For
a multipath channel, if the choice of
N
does not enable
B<B
c
to be obtained,
the channel remains frequency-selective in relation to the modulated carriers,
and intersymbol interference appears.
To avoid residual intersymbol interference without increasing
N
,wecanuse
the guard interval principle. For a multipath channel, the propagation paths
are received at the receiver with delays
τ
n
. Calling
τ
Max
the largest of these
delays, guard interval
Δ
must satisfy the following inequality:
Δ
≥
τ
Max
(2.189)
We put:
T
=
t
s
+Δ
In the presence of a guard interval, the modulation symbols are still of duration
T
but the discrete Fourier transform at reception is realized on the time intervals
[
iT
+Δ
,
(
i
+1)
T
[
. Proceeding thus, we can check that on this time interval
only the modulation symbol transmitted between
iT
and
(
i
+1)
T
is taken into
account for the decoding: there is therefore no intersymbol interference.
The introduction of a guard interval has two consequences. The first is that
only a part of the energy transmitted on emission is exploited on reception.
Indeed, we transmit each modulation symbol over a duration
T
and we recover
this same symbol from an observation of duration
t
s
=
T
−
Δ
. The loss, expressed
