Digital Signal Processing Reference
In-Depth Information
10.4.2.B Choice of the alias-free(
T
) prefilter
F
a
(
jω
)
The preceding results hold for any choice of the alias-free(
T
)region
A.
In par-
ticular, suppose
A
is chosen such that if
ω ∈A
then
|
|
2
|
2
S
qq
(
j
(
ω
+
kω
s
))
H
(
jω
)
S
qq
(
jω
)
≥
|
H
(
j
(
ω
+
kω
s
))
(10
.
53)
for any integer
k
. That is, we define the region
to be the best alias-free(
T
)
band of the channel (i.e., the portion where the channel gain divided by the
noise spectrum is maximized). Then replacement of
F
(
jω
)with
F
a
(
jω
)
does
not increase
the transmitted power!
A
Proof.
The transmitted power is given by Eq. (E.42) in Appendix E. With
F
(
jω
) replaced by
F
a
(
jω
) this power becomes
∞
F
a
(
jω
)
1
T
2
S
ss
(
e
jωT
)
dω
2
π
−∞
F
(
j
(
ω
+
kω
s
))
H
(
j
(
ω
+
kω
s
))
2
S
qq
(
jω
)
|H
(
jω
)
|
2
S
ss
(
e
jωT
)
dω
2
π
=
S
qq
(
j
(
ω
+
kω
s
))
A
k
F
(
j
(
ω
+
kω
s
))
H
(
j
(
ω
+
kω
s
))
2
S
qq
(
jω
)
S
ss
(
e
jωT
)
dω
2
π
=
×
S
qq
(
j
(
ω
+
kω
s
))
|
H
(
jω
)
|
2
A
k
F
(
j
(
ω
+
kω
s
))
2
S
ss
(
e
jωT
)
dω
2
π
≤
(from Eq. (10.53))
A
k
F
(
j
(
ω
+
kω
s
))
2
S
ss
(
e
j
(
ω
+
kω
s
)
T
)
dω
2
π
=
(since
ω
s
T
=2
π
)
A
k
∞
−∞
|
|
2
S
ss
(
e
jωT
)
dω
=
F
(
j
(
ω
)
2
π
.
The last equality follows from the alias-free(2
π/ω
s
) property of
A
(which
ensures that the regions
are nonoverlapping for different values of
k,
and cover the entire frequency axis as
k
varies over all integers).
{A
+
kω
s
}
Thus, the optimal system can be assumed to be such that the prefilter has the
form
F
a
(
jω
) and the equalizer has the form
B
a
(
jω
) (followed by the digital
equalizer
P
(
z
)
,
which is located to the right of the C/D converter).
10.4.3 The all-digital equivalent
The transceiver system with anti-alias filters
F
a
(
jω
)and
B
a
(
jω
) is shown in Fig.
10.10. Since
F
a
(
jω
)and
B
a
(
jω
) are restricted to the alias-free(
T
) band
A
we
can write
F
a
(
jω
)=
H
id
(
jω
)
F
d
(
e
jωT
)
,
B
a
(
jω
)=
H
id
(
jω
)
B
d
(
e
jωT
)
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