Digital Signal Processing Reference
In-Depth Information
Lemma 10.3.
Form of G
(
jω
). In Fig. 10.6 assume that the noise power
spectrum is
S
qq
(
jω
)
>
0 for all
ω.
Given some combination of filters
F
(
jω
)and
G
(
jω
)
,
suppose we replace
G
(
jω
)with
♠
G
1
(
jω
)=
P
(
e
jωT
)
F
∗
(
jω
)
H
∗
(
jω
)
S
qq
(
jω
)
,
(10
.
47)
where
F
(
jω
)
H
(
jω
)
G
(
jω
)
↓T
P
(
e
jω
)=
|
|
2
/S
qq
(
jω
)
↓T
.
(10
.
48)
F
(
jω
)
|
2
|
H
(
jω
)
Then the signal component of
s
(
n
) is unchanged, and the noise power spectrum
at the output of the C/D converter does not increase for any
ω.
♦
Remarks
1.
Parts of the optimal filter.
Thus the signal component at the output of Fig.
10.8(c) is identical to that in Fig. 10.8(a), and the noise spectrum does
not increase at any frequency. The filter (10.47) has the form
G
1
(
jω
)=
B
(
jω
)
P
(
e
jωT
)
.
This is product of a transversal part
P
(
e
jωT
) (periodic in
ω
) and a non-
transversal part
B
(
jω
)
.
2.
Digital filter part.
In view of the noble identity shown in Fig. G.1 of
Appendix G, the transversal part
P
(
e
jωT
) can be realized by implementing
a digital filter
P
(
z
) with frequency response
P
(
e
jω
) after the C/D unit (i.e.,
after sampling at the receiver). See Fig. 10.8(c).
3.
Matched filter interpretation.
With
P
(
e
jωT
) so moved, the optimum filter
G
1
(
jω
) has the remaining factor
B
(
jω
)=
F
∗
(
jω
)
H
∗
(
jω
)
S
qq
(
jω
)
.
(10
.
49)
This can be regarded as a matched filter corresponding to the cascaded
filter
F
(
jω
)
H
(
jω
) with additive noise spectrum
S
qq
(
jω
)
.
Search WWH ::
Custom Search