Cryptography Reference
In-Depth Information
where
g
e
(
t
)=
g
c
(
t
)+
jg
s
(
t
)
is the baseband-equivalent waveform of the emission
filter. The output
e
(
t
)
of the emission filter is equal to:
e
(
t
)=
A
i
c
i
z
(
t
−
iT
)
(2.106)
where
z
(
t
)=
h
(
t
)
⊗
g
e
(
t
)
is, in the general case, a complex waveform while
h
(
t
)
is real.
2.3.2 Intersymbol interference
After passing through the emission filter, the modulated signal has a bandwidth
W
and, thus, the signal at the output of the transmission channel has the
expression:
r
(
t
)=
e
(
t
)+
b
(
t
)
(2.107)
where
b
(
t
)
is a complex AWGN, with zero mean and power spectral density
equal to
2
N
0
.
The coherent receiver uses a reception filter followed by a sampler at time
t
0
+
nT
,where
t
0
can be chosen arbitrarily. The output of the reception filter
with impulse response
g
r
(
t
)
has the expression:
y
(
t
)=
A
i
iT
)+
b
(
t
)
c
i
x
(
t
−
(2.108)
where:
x
(
t
)=
z
(
t
)
⊗
g
r
(
t
)
b
(
t
)=
b
(
t
)
g
r
(
t
)
Sampling signal
y
(
t
)
at time
t
0
+
nT
, we obtain:
y
(
t
0
+
nT
)=
A
i
⊗
i
)
T
)+
b
(
t
0
+
nT
)
c
i
x
(
t
0
+(
n
−
(2.109)
Considering that in the general case
x
(
t
)=
p
(
t
)+
jq
(
t
)
is a complex waveform,
the sample
y
(
t
0
+
nT
)
can again be written in the form:
y
(
t
0
+
nT
)=
Ac
n
p
(
t
0
)+
A
i
=0
c
n−i
p
(
t
0
+
iT
)
+
jA
i
(2.110)
c
n−i
q
(
t
0
+
iT
)+
b
(
t
0
+
nT
)
The first term
Ac
n
p
(
t
0
)
represents the desired information for the decoding of
the symbol
c
n
, the following two terms being Intersymbol Interference (ISI)
terms. Let us examine the outputs of the two components of the receiver, called
the in-phase and quadrature components, corresponding to the real part and
to the imaginary part of
y
(
t
0
+
nT
)
respectively. We can notice that the in-
phase component (respectively the quadrature component) depends on symbols
