Cryptography Reference
In-Depth Information
With a high signal to noise ratio, a good approximation of the error probability
Peb
is given by:
2
L
L
E
b
LN
0
−
1
1
2
E
b
/LN
0
Peb
≈
for
>>
1
(2.180)
L
Note that diversity like the type presented here is a form of coding that uses
a repetition code and weighted decoding at reception. Figure 2.26 shows the
performance of 2-PSK and 2-FSK modulations in the presence of diversity.
Transmission on a slow-fading frequency-selective channel
Different transmission strategies are possible. We can use a waveform at emission
that is only slightly, or not at all, sensitive to the selectivity of the channel, or
we can correct the effects of the selectivity of the channel at reception.
Multicarrier transmission
Multicarrier transmission uses a multiplex of orthogonal carriers that are dig-
itally phase modulated (M-PSK), frequency modulated (M-FSK) or amplitude
and phase modulated (M-QAM). This waveform, called a "parallel" waveform
and known as
Orthogonal Frequency Division Multiplexing
(OFDM), enables us
to avoid the frequency selectivity of transmission channels.
We have seen that a channel is frequency selective for a modulated signal
with bandwidth
B
if its coherence bandwidth
B
c
≈
1
/T
M
is lower than
B
.Let
us recall, on the other hand, that the bandwidth
B
of a digitally modulated
signal is proportional to its symbol rate
1
/T
. To build a type of waveform that
is not sensitive to the frequency selectivity of the channel, we can proceed in the
following way.
Let us divide the bit rate
D
=1
/T
b
to be transmitted into
N
sub-rates
D
=1
/N T
b
, each elementary rate feeding a modulator with
M
states, with
carrier frequency
f
i
. The symbol rate of the modulated carriers is then equal
to
R
=1
/T
with
T
=
NT
b
log
2
(
M
)
.Choosing
N
large enough, the symbol rate
of the modulated carriers can become as low as required, and the bandwidth
B
of a modulated carrier then becomes far lower than the coherence bandwidth
B
c
of the channel. Proceeding thus, the channel is frequency non-selective for
the modulated carriers of the multiplex. For a multipath channel each mod-
ulated carrier is weighted by an attenuation that follows a Rayleigh law and
with a dephasing equidistributed on
[0
,
2
π
[
. Of course, not all the modulated
carriers are affected in the same way at the same time by the channel; some
are strongly attenuated whereas others are less so. The performance in terms
of error probability per modulated carrier is that of a non-frequency-selective
slow-fading Rayleigh channel.
To avoid large packets of errors at reception, we can make sure that informa-
tion data that follow each other are transmitted by carriers affected differently
by the channel, that is, spaced by a quantity at least equal to the coherence band
