Digital Signal Processing Reference
In-Depth Information
where we have substituted for
P
(
e
jω
) using Eq. (10.44). The second equality
is obtained by using the identity (G.6) from Appendix G. Since the frequency
response function from
s
(
n
)to
s
(
n
)isgivenby
[
F
(
jω
)
H
(
jω
)
G
(
jω
)]
↓T
it therefore follows that this transfer function is unchanged when
G
(
jω
)
replaces
G
(
jω
)
.
Now consider the noise power spectrum which is given by
|
|
2
G
(
jω
)
at the output of
G
(
jω
)
.
At the output of the C/D converter this
becomes
|
2
|
G
(
jω
)
.
↓T
With
G
(
jω
) replaced by
G
(
jω
)wehave
|
2
↓T
| G
(
jω
)
=
|
2
P
(
e
jωT
)
F
∗
(
jω
)
H
∗
(
jω
)
|
↓T
=
|
2
P
(
e
jω
)
|
2
(from Eq. (G.6), Appendix G)
|
F
(
jω
)
H
(
jω
)
↓T
|
[
F
(
jω
)
H
(
jω
)
G
(
jω
)]
↓
T
2
=
|
2
↓T
.
|
F
(
jω
)
H
(
jω
)
Observe now that the last numerator can be expanded as follows:
F
(
j
(
ω
+2
πk
)
/T
)
H
(
j
(
ω
+2
πk
)
/T
)
G
(
j
(
ω
+2
πk
)
/T
)
T
k
1
2
F
(
j
(
ω
+2
πk
)
/T
)
H
(
j
(
ω
+2
πk
)
/T
)
G
(
j
(
ω
+2
πk
)
/T
)
T
k
T
k
1
2
1
2
≤
=
|
2
|
2
↓T
|
F
(
jω
)
H
(
jω
)
|
G
(
jω
)
,
↓T
where
T
=2
π/ω
s
.
Here the second line follows from the Cauchy-Schwartz
inequality. This proves that
| G
(
jω
)
|
2
|
2
↓T
≤
|
G
(
jω
)
.
↓T
That is, the noise power spectrum (at the output of the C/D converter)
cannot increase at any frequency
ω,
when we replace the receiver filter
G
(
jω
)
with
G
(
jω
).
10.4.1.A More general noise power spectrum
Lemma 10.2 assumed that
S
qq
(
jω
)=1
.
Now imagine that
S
qq
(
jω
) is arbitrary
but nonzero for all
ω.
Suppose we rearrange Fig. 10.6 as in Fig. 10.7(a), where
the noise
q
(
t
) has power spectrum equal to unity as in Lemma 10.2.
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