Digital Signal Processing Reference
In-Depth Information
Problems
Note.
Unless mentioned otherwise, assume
σ
s
=1
.
18.1.
Consider again the channel in Problem 17.1, but assume that we do zero
padding instead of cyclic prefixing. With
L
= 4 and
M
= 5, and with all
parameters (including the numerical value of power
p
0
) as in Problem 17.1,
compute the MMSE values (Sec. 18.2) with and without zero forcing. How
do these values compare with what you obtain for the cyclic-prefix system
of Problem 17.1? Assume the precoder is not restricted to be orthonormal.
18.2.
Repeat Problem 18.1 for the channel in Problem 17.2.
18.3.
Repeat Problem 18.1 for the case where the zero-padded system has its
precoder restricted to be orthonormal (but scaled appropriately to satisfy
the power constraint).
18.4.
Figure 18.9 showed the error probability as a function of block size for
three examples of channels which use zero padding. With all parameters
as described in Sec. 18.3 for Fig. 18.9, repeat these plots for the modified
channels
C
(
z
)=
β
(1 +
ρz
−
1
)
,C
(
z
)=
β
(1 +
ρz
−
1
)
2
,
and
C
(
z
)=
β
(1 +
ρz
−
1
)
3
,
for
ρ
=0
.
99. Repeat for
ρ
=0
.
9
.
In each case assume
β
is such that
the channel energy
|
|
2
= 6 as in Sec. 18.3. Explain qualitatively
how and why the behavior is different from Fig. 18.9.
c
(
n
)
18.5.
Repeat Problem 18.4 for
C
(
z
)=
β
(1 +
ρz
−
1
)
,
where
ρ
=1
/
0
.
9and
β
such
that
|
|
2
= 6. How does this compare with the case
ρ
=0
.
9(with
β
c
(
n
)
adjusted again such that
|
|
2
=6)?
c
(
n
)
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