Cryptography Reference
In-Depth Information
the condition for the absence of ISI can be expressed from
X
t
0
(
f
)
by the follow-
ing relation:
+
∞
i
T
)=
T
X
t
0
(
f
−
(2.124)
i
=
−∞
This condition of absence of ISI is called the
Nyquist criterion
.
Let us recall that the transmission channel with spectrum
C
(
f
)
has a pass-
band
W
.
C
(
f
)=0
for
|
f
|
>W
(2.125)
Let us consider relation (2.124) for three situations.
1.
X
t
0
(
f
)
has a bandwidth
W<
1
/
2
T
. Relation (2.124) being a sum of
functions shifted by
1
/T
, there are no functions
X
t
0
(
f
)
that enable the
Nyquist criterion to be satisfied. The bandwidth
W
necessary for ISI-free
transmission is therefore higher than or equal to
1
/
2
T
.
2.
X
t
0
(
f
)
has a bandwidth
W
=1
/
2
T
. There is a single solution that satisfies
the Nyquist criterion:
X
t
0
(
f
)=
T
W
=0
elsewhere
|
f
|≤
(2.126)
or again:
X
(
f
)=
Tp
(
t
0
)exp(
−
j
2
πt
0
)
|
f
|≤
W
(2.127)
=
0
elsewhere
which, in the time domain, gives:
x
(
t
)=
p
(
t
0
)
sin [
π
(
t
−
t
0
)
/T
]
(2.128)
π
(
t
−
t
0
)
/T
This solution corresponds to a strictly speaking non-causal waveform
x
(
t
)
.
However, since the function
sin
y/y
decreases fairly rapidly as a function
of its argument
y
, it is possible, by choosing
t
0
large enough, to consider
x
(
t
)
as practically causal. With this solution, the eye diagram has a hor-
izontal aperture that tends towards zero and thus, any imprecision about
the sampling time can lead to errors even in the absence of noise. In con-
clusion, this solution is purely theoretical and therefore has no practical
applications.
3.
X
t
0
(
f
)
has a bandwidth
W>
1
/
2
T
. In this case, there are many solutions
that enable the Nyquist criterion to be satisfied. Among these solutions,
the most commonly used is the raised-cosine function defined by:
