Digital Signal Processing Reference
In-Depth Information
error ratios (SER) before and after bias removal are given by
β
k
=
σ
s
σ
s
E
k,br
E
k
,
k,br
=
(16
.
35)
If the receiver matrix
G
is chosen as the MMSE matrix (for a given precoder
F
,
channel
H
,
and noise statistics), then
s
k
is an MMSE estimate of
s
k
.A
fundamental result in this case is that the above two SERs are related as follows:
β
k,br
=
β
k
−
1
.
(16
.
36)
This is proved in Appendix 16.A (Lemma 16.6) at the end of the chapter. This
relation can be rewritten as
1
E
k,br
1
E
k
−
1
σ
s
=
(16
.
37)
The
k
th symbol error probability after bias removal is therefore given by
3
A
1
E
k,br
=
c
A
1
,
1
σ
s
P
k,br
=
c
Q
Q
E
k
−
(16
.
38)
where
c
and
A
are the constants defined earlier (e.g., as in Eq. (16.6) for QAM).
Thus the average error probability after bias removal is
1
E
k
−
A
.
M−
1
c
M
1
σ
s
P
br
=
Q
(16
.
39)
k
=0
16.3.3 Optimality of bias-removed MMSE estimate
For a fixed precoder
F
and channel
H
,
consider again the optimization of the
equalizer
G
.
Since there is no constraint on
G
(e.g., no unitarity constraint,
etc.), we can optimize one row at a time. For example, assume the
k
th row
g
k
is chosen such that its output is an MMSE estimate:
4
s
⊥
=
αs
+
τ.
(16
.
40)
For simplicity, the argument (
n
) is omitted, and the subscript
k
is deleted tem-
porarily. In Eq. (16.40),
τ
is a combination of noise and interference terms and
will be assumed to be zero-mean, and statistically independent of
s
. The above
estimate is used by the detector to identify the transmitted symbol
s.
In this
process, the detector first removes the bias term automatically:
s
⊥,br
=
s
+
τ
α
.
(16
.
41)
3
This assumes that the error terms in Eq. (16.30), which come from both noise and inter-
ference, are Gaussian. The assumption is acceptable if
M
is large (in view of the central limit
theorem).
4
The subscript
⊥
is a reminder that MMSE estimates satisfy the orthogonality condition
(Sec. F.2.1 in Appendix F).
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