Digital Signal Processing Reference
In-Depth Information
where
11 1
···
1
01 2
··· N
012
2
··· N
2
. . .
.
.
.
.
012
N
···N
N
Ψ
N+1
=
(7.4)
is a Vandermonde matrix of size (
N
+ 1)
×
(
N
+ 1) and
T
a
N,0
a
N,1
ωτ
0
(
ωτ
0
)
2
...
N
!
a
N,N
2!
a
N,2
a
N+1
(
ω
)=
(7.5)
(
ωτ
0
)
N
is a vector of length
N
+ 1.
In the next sections, we will study and design different order differential
arrays based on this approach.
7.2 First-Order Differential Arrays
The system of two linear equations corresponding to first-order DMAs is
a
1,0
a
1,1
ωτ
0
11
01
h
(
ω
)=
,
(7.6)
for which the solution is
a
1,0
a
1,1
1
−
ωτ
0
h
(
ω
)=
a
1,1
ωτ
0
.
(7.7)
−
1
Replacing 1 +
x
by
e
x
in (7.7), we get
−
ωτ
0
a
1,0
/a
1,1
−
1
h
(
ω
)=
a
1,1
ωτ
0
e
.
(7.8)
We then deduce that the equivalent filter is
−
ωτ
0
a
1,0
/a
1,1
1
(
ω
)=
1
ω
−e
′
h
,
(7.9)
since
C
1
=
−
a
1,1
τ
0
is frequency independent. In the rest of this section, we
will consider the filter
h
′
(
ω
) as the practical beamformer that should be
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