Cryptography Reference
In-Depth Information
2.3.3 Condition of absence of ISI: Nyquist criterion
The absence of ISI is translated by the following conditions:
p
(
t
0
+
iT
)=0
∀
i
=0
(2.116)
i
(2.117)
which can again be written by using the complex signal
x
(
t
)=
p
(
t
)+
jq
(
t
)
q
(
t
0
+
iT
)=0
∀
x
(
t
0
+
iT
)=
p
(
t
0
)
δ
0
,i
∀
i
(2.118)
where
δ
0
,i
is the Kronecker symbol.
Let us introduce the sampled signal
x
E
(
t
)
, defined by:
x
E
(
t
)=
x
(
t
)
i
δ
(
t
−
t
0
−
iT
)
(2.119)
We can notice that the impulse train
u
(
t
)=
i
iT
)
is periodic, of
period
T
. It can therefore be decomposed into the Fourier series:
δ
(
t
−
t
0
−
exp
T
t
0
exp
T
t
T
i
u
(
t
)=
1
j
2
π
i
j
2
π
i
−
−
(2.120)
Since we are seeking to determine the minimal bandwidth
W
necessary to trans-
mit the ISI-free modulated signal, it is wise to work in the frequency domain.
Taking the Fourier transform denoted
X
E
(
f
)
of relation (2.119) and taking into
account the previous expression of
u
(
t
)
, we obtain:
exp
T
t
0
X
(
f
T
i
X
E
(
f
)=
1
j
2
π
i
i
T
)
−
−
(2.121)
The sampled signal, according to relation (2.119), can also be written:
x
E
(
t
)=
i
x
(
t
0
+
iT
)
δ
(
t
−
t
0
−
iT
)
(2.122)
which, after Fourier transform and taking into account the condition of absence
of ISI, becomes:
X
E
(
f
)=
p
(
t
0
)exp(
−
j
2
πft
0
)
(2.123)
The equality of the relations (2.121) and (2.123) gives:
exp
j
2
π
(
f
T
)
t
0
X
(
f
i
i
T
)=
Tp
(
t
0
)
−
−
i
Putting:
X
t
0
(
f
)=
X
(
f
)
p
(
t
0
)
exp(
j
2
πft
0
)