Digital Signal Processing Reference
In-Depth Information
and using (7.12), we obtain the gain derived in Chapter 3 [eq. (3.55)], i.e.,
1
−
cos[
ωτ
0
(1
− α
1,1
)]
1
−
cos[
ωτ
0
(cos
θ
N
′
G
NS,1
[
h
(
ω
)] =
− α
1,1
)]
.
(7.22)
The gain of
h
(
ω
) for a point noise source is
2
h
H
(
ω
)
d
(
ω,
1)
G
NS,1
[
h
(
ω
)] =
|h
H
(
ω
)
d
(
ω,
cos
θ
N
)
|
2
1
(
a
1,0
+
a
1,1
cos
θ
N
)
2
,
=
(7.23)
which corresponds to the theoretical gain of the first-order DMA.
Clearly, the first-order DMAs derived in this section are strictly equivalent
to the ones derived in Chapter 3.
7.3 Second-Order Differential Arrays
For second-order DMAs, we need to inverse the matrix
Ψ
3
, which is given by
1
−
3
1
2
02
−
1
0
−
1
2
2
−1
3
Ψ
=
.
(7.24)
1
2
We deduce that the corresponding filter is
−
a
2,2
(
ωτ
0
)
2
+
3
a
2,1
a
2,0
2
ωτ
0
2
a
2,2
(
ωτ
0
)
2
−
2
a
2,1
ωτ
0
h
(
ω
)=
−
a
2,2
(
ωτ
0
)
2
+
a
2,1
2
ωτ
0
.
1
−
a
2,0
(
ωτ
0
)
2
a
2,2
−
3
ωτ
0
a
2,1
2
a
2,2
−
2+
2
ωτ
0
a
2,1
a
2,2
1
−
ωτ
0
a
2,1
2
a
2,2
=
−
a
2,2
(
ωτ
0
)
2
(7.25)
Replacing 1 +
x
+
x
2
/
2 by
e
x
in (7.25), we get
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