Digital Signal Processing Reference
In-Depth Information
1. Compute the precoder and equalizer matrices
F
and
G
in the optimal
transceiver structure with optimal bit allocation.
2. With
P
e
(
k
)=10
−
7
for all
k, σ
q
=0
.
01, and
b
= 8 bits, what is the
optimal bit allocation? What is the coding gain due to bit allocation
(Sec. 14.7).
Hint:
Circulants are diagonalized by the DFT matrix.
14.6.
Repeat Problem 14.5 for the circulant channel
⎡
⎤
411
141
114
⎣
⎦
,
H
=
which also has positive eigenvalues. Why is the coding gain so different
from that in Problem 14.5?
14.7.
For this problem assume
P
e
(
k
)=10
−
7
for all
k
, and channel noise variance
σ
q
=0
.
01
.
Assume block size
M
=4
,
and the average number of bits
b
=8
.
The distribution of the bits
b
k
in Eq. (14.41) depends on the distribution
of the channel singular values
{
σ
h,k
}
.
1. Find a set of singular values
{
σ
h,k
}
such that all
b
k
calculated from
Eq. (14.41) are positive.
2. Find a set of singular values
{
σ
h,k
}
such that only one
b
k
calculated
from Eq. (14.41) is positive.
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