Digital Signal Processing Reference
In-Depth Information
show that the autocorrelation of minimized error in the memoryless case
is indeed given by (4.83).
4.10.
Consider Fig. 4.37 and assume we have a memoryless MIMO channel with
transfer function matrix
C
=
10
2
,
where
>
0
.
Assume the signal and noise are zero-mean processes with
constant power spectra
S
ss
(
z
)=
I
and
S
qq
(
z
)=
σ
q
I
.
1. Find an expression for the MMSE equalizer filter
G
,
the error auto-
correlation
R
ee
, and the total mean square error
E
(trace of
R
ee
).
2. Simplify the expression for
when
σ
q
becomes arbitrarily small for
E
fixed
.
3. Simplify the expression for
E
when
becomes arbitrarily small for
fixed
σ
q
.
4.11.
In Problem 4.10 suppose we use a zero-forcing equalizer
G
instead of an
MMSE equalizer.
1. Find an expression for
G
,
the error autocorrelation
R
ee
, and the total
mean square error
(trace of
R
ee
).
2. For
σ
q
=0
.
01 and
=0
.
1, compute the total mean square error
E
.
How does this compare with the MMSE value obtained from Problem
4.10?
3. For
σ
q
=0
.
01 and
=0
.
001, compute the total mean square error
E
.
How does this compare with the MMSE value obtained from Problem
4.10?
E
4.12.
In Fig.
4.37 assume the MIMO channel is FIR with transfer function
matrix
1+
z
−
1
.
1
− z
−
1
C
(
z
)=
1
2
1
− z
−
1
1+
z
−
1
Assume the signal and noise are zero-mean processes with constant power
spectra
S
ss
(
z
)=
I
and
S
qq
(
z
)=
σ
q
I
,
with
σ
q
=0
.
1. Find expressions for the MMSE equalizer filter
G
(
z
) and the error
power spectrum
S
ee
(
z
).
2. Assume we use a zero-forcing equalizer
G
(
z
) instead. Find expres-
sions for
G
(
z
)and
S
ee
(
z
) (i.e., find all the elements
G
km
(
z
)and
S
km
(
z
) explicitly).
3. By comparing the expressions for the error power spectra
S
ee
(
z
)in
the above two cases, can you argue explicitly that the MSE for the
MMSE solution is smaller than the MSE for the ZF solution (which
of course should be the case by the definition of MMSE)?
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