Digital Signal Processing Reference
In-Depth Information
q
(
n
)
0
x
(
n
)
0
y
(
n
)
0
y
(
n
)
1
s
(
n
)
0
s
(
n
)
0
s
(
n
)
1
x
(
n
)
1
s
(
n
)
1
F
s
(
n
)
A
G
s
(
n
)
R
x
(
n
)
M
− 1
q
(
n
)
P
− 1
s
(
n
)
M
s
(
n
)
M
− 1
y
(
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)
P
− 1
y
(
n
)
−
1
precoder
channel
equalizer
Figure 18.2
. The zero-padded transceiver system redrawn in matrix-vector notation.
The expressions for MMSE and symbol error probability derived in earlier
chapters can also be applied here directly. We will consider optimal zero-padded
precoders of four types. These are
1. Pure MMSE systems,
2. MMSE systems with zero-forcing (ZF-MMSE systems),
3. Pure MMSE systems with orthonormal precoder, and
4. ZF-MMSE systems with orthonormal precoder.
In the orthonormal case, a scalar multiplier
α
is used at the transmitter to satisfy
the power constraint as explained in Chap. 15. It is assumed that extra unitary
matrices
U
and
U
†
are inserted to minimize error probability as in Chap. 16.
So the expression for the symbol error probability has the form
⎧
⎨
⎩
A
√
E
ave
c
Q
(ZF-MMSE case)
P
e,min
=
(18
.
3)
1
E
ave
−
A
1
σ
s
c
Q
(pure-MMSE case),
where
E
mmse
/M.
The second expression holds under the assumption that
bias has been removed before symbol detection (Sec. 16.3). Here the constants
c
and
A
depend on the constellation used. For example, in a
b
-bit PAM system
E
ave
=
A
=
3
σ
s
2
2
b
2
−b
)
,
c
=2(1
−
and
(18
.
4)
−
1
The expression for the error
E
mmse
depends on whether we have the zero-forcing
constraint or not, and whether the precoder is constrained to be orthonormal or
Toeplitz structure.
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