OFDM for Terahertz Wireless Communication Systems
149
The objective of the DSP is to give the attenuations of the transmitted power as a
function of the frequency at which it is located. Thus, an “ideal” DSP would be flat
in an allocated frequency band (the frequency band available for a given application
for example) and would have an infinite attenuation outside this band. Thus, in this
case, there would be no loss of power on transmission. There would therefore be no
interference from this application on applications whose allocated frequencies are
close.
5.2.1 Analytical Calculation
To carry out this analytical calculation of the DSP, we use the expression of the signal
in continuous time, given in (5), of which we recall the expression:
x(t) =
i
N
k=1
{c i,k exp
j2π f k t
}
g(t − i T B ); 0 ≤ t ≤ T B
(11)
Thus, we obtain:
x(t)x
∗
(t − τ ) =
(i,q)∈Z 2
N
k, p=1
c i,k c
∗
p,q e ( j2π f k t) e ( j2π f p−k t)
g(t − i T B )g
∗
t − τ − qT B
(12)
Suppose further that the data c i,k are identically distributed independent (i.d.i), so
we have
c i,k c
∗
p,q
= σ
2
c δ i, p δ k,q .
E
x(t)x
∗
(t − τ )
= σ
2
c
i
N
k=1
exp
j2π f k t
g(t − i T B )g
∗
(t − τ − i T B )
(13)
This last equation shows us that the signal x is not stationary of order 2. In fact,
the autocorrelation function depends on both t and τ. Therefore, it is not possible
to use the definition of PSD given in Eq. (10). we note, however, that the function
Γ x (t, τ ) = E{x(t)x
∗
(t − τ )} has the following interesting property:
Γ x (t + T B , τ ) = Γ x (t, τ )
(14)
This relation is simply obtained by a change of variable on the sum index i. On
the other hand, we also have, like the c i,k its i.d.i.:
E{x(t)} = Cte
(15)
149
The objective of the DSP is to give the attenuations of the transmitted power as a
function of the frequency at which it is located. Thus, an “ideal” DSP would be flat
in an allocated frequency band (the frequency band available for a given application
for example) and would have an infinite attenuation outside this band. Thus, in this
case, there would be no loss of power on transmission. There would therefore be no
interference from this application on applications whose allocated frequencies are
close.
5.2.1 Analytical Calculation
To carry out this analytical calculation of the DSP, we use the expression of the signal
in continuous time, given in (5), of which we recall the expression:
x(t) =
i
N
k=1
{c i,k exp
j2π f k t
}
g(t − i T B ); 0 ≤ t ≤ T B
(11)
Thus, we obtain:
x(t)x
∗
(t − τ ) =
(i,q)∈Z 2
N
k, p=1
c i,k c
∗
p,q e ( j2π f k t) e ( j2π f p−k t)
g(t − i T B )g
∗
t − τ − qT B
(12)
Suppose further that the data c i,k are identically distributed independent (i.d.i), so
we have
c i,k c
∗
p,q
= σ
2
c δ i, p δ k,q .
E
x(t)x
∗
(t − τ )
= σ
2
c
i
N
k=1
exp
j2π f k t
g(t − i T B )g
∗
(t − τ − i T B )
(13)
This last equation shows us that the signal x is not stationary of order 2. In fact,
the autocorrelation function depends on both t and τ. Therefore, it is not possible
to use the definition of PSD given in Eq. (10). we note, however, that the function
Γ x (t, τ ) = E{x(t)x
∗
(t − τ )} has the following interesting property:
Γ x (t + T B , τ ) = Γ x (t, τ )
(14)
This relation is simply obtained by a change of variable on the sum index i. On
the other hand, we also have, like the c i,k its i.d.i.:
E{x(t)} = Cte
(15)