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 )
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