Digital Signal Processing Reference
In-Depth Information
iSNR (
ω
)=
φ
X
(
ω
)
φ
V
1
(
ω
)
,
(2.30)
|X
(
ω
)
|
2
|V
1
(
ω
)
|
2
where
φ
X
(
ω
)=
E
and
φ
V
1
(
ω
)=
E
are the variances
of
X
(
ω
) and
V
1
(
ω
), respectively.
The output SNR is defined as
2
h
H
(
ω
)
d
(
ω,
cos0
◦
)
oSNR [
h
(
ω
)] =
φ
X
(
ω
)
h
H
(
ω
)
Φ
v
(
ω
)
h
(
ω
)
2
h
H
(
ω
)
d
(
ω,
cos0
◦
)
=
φ
X
(
ω
)
φ
V
1
(
ω
)
·
,
(2.31)
h
H
(
ω
)
Γ
v
(
ω
)
h
(
ω
)
where
v
(
ω
)
v
H
(
ω
)
Φ
v
(
ω
)=
E
(2.32)
and
Γ
v
(
ω
)=
Φ
v
(
ω
)
φ
V
1
(
ω
)
(2.33)
are the correlation and pseudo-coherence matrices of
v
(
ω
), respectively.
The definition of the gain in SNR is easily derived from the two previous
definitions, i.e.,
G
[
h
(
ω
)] =
oSNR [
h
(
ω
)]
iSNR (
ω
)
2
h
H
(
ω
)
d
(
ω,
cos0
◦
)
=
.
(2.34)
h
H
(
ω
)
Γ
v
(
ω
)
h
(
ω
)
Assume that the matrix
Γ
v
(
ω
) is nonsingular. In this case, for any two
vectors
h
(
ω
) and
d
(
ω,
cos0
◦
), we have
2
≤
h
H
(
ω
)
d
(
ω,
cos0
◦
h
H
(
ω
)
Γ
v
(
ω
)
h
(
ω
)
)
×
d
H
(
ω,
cos0
◦
−1
v
◦
)
Γ
(
ω
)
d
(
ω,
cos0
)
,
(2.35)
v
(
ω
)
d
(
ω,
cos0
◦
). Using the inequality
(2.35) in (2.34), we deduce an upper bound for the gain:
−1
with equality if and only if
h
(
ω
)
∝ Γ
G
[
h
(
ω
)]
≤ d
H
(
ω,
cos0
◦
−1
v
◦
)
Γ
(
ω
)
d
(
ω,
cos0
)
−1
v
◦
)
d
H
(
ω,
cos0
◦
≤
tr
Γ
(
ω
)
tr
d
(
ω,
cos0
)
−1
v
≤ M
tr
Γ
(
ω
)
,
(2.36)
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