Digital Signal Processing Reference
In-Depth Information
j
ω
G
( )
d
e
(a)
ω
−π
π
j
ω
T/L
G
( )
d
e
(b)
ω
−
L
π
/T
L
π
/
T
gap
(c)
G
(
j
ω)
c
ω
−σ
−π
/
T
π
/
T
σ
“don't care” band
j
ω
(d)
G
( )
d
e
ω
−π
π
Figure 4.36
. (a) Example of a digital postfilter response
G
d
(
e
jω
); (b) the scaled
response
G
d
(
e
jωT/L
)
,
with oversampling ratio
L
; (c) example of a nonideal lowpass
postfilter
G
c
(
jω
) with excess bandwidth; (d) example of a digital postfilter
G
d
(
e
jω
)
with a “don't care” region.
4.10 MMSE equalization
The equalizers introduced in Secs. 4.6 and 4.8 work under the zero-forcing
constraint. This constraint guarantees that the transfer function between the
transmitted signal
s
(
n
) and the received signal
s
(
n
)isidentity. Itispossibleto
relax this constraint, and design the equalizer such that the mean square error
between
s
(
n
)and
s
(
n
),
|
2
]
,
E
=
E
[
|
s
(
n
)
−
s
(
n
)
(4
.
66)
is minimized. Such an equalizer is called a minimum mean square error equal-
izer, or
MMSE
, equalizer. The theory of such equalizers comes from the theory
of
Wiener
filtering (Appendix F). MMSE equalizers offer smaller mean square
reconstruction error than zero-forcing equalizers. With the detector appropri-
ately designed, this eventually results in smaller error probability, as we shall
repeatedly observe in later chapters.
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