Digital Signal Processing Reference
In-Depth Information
describe the interference between spatial frequency components of the array
signal. Their definition is similar to the definition of the cross-interference coef-
ficients (18.6) derived in Chapter 18 for the temporal spectrum analysis. In the
case of the spatial spectrum analysis they are given as
K
2
K
cos(2
π
Ω
m
d
k
) cos(2
π
Ω
n
d
k
)
(
A
m
C
n
)
=
,
k
=
1
K
2
K
cos(2
π
Ω
m
d
k
) sin(2
π
Ω
n
d
k
)
(
A
m
S
n
)
=
,
k
=
1
(20.5)
K
2
K
sin(2
π
m
d
k
) cos(2
π
(
B
m
C
n
)
=
Ω
Ω
n
d
k
)
,
k
=
1
K
2
K
sin(2
π
m
d
k
) sin(2
π
(
B
m
S
n
)
=
Ω
Ω
n
d
k
)
.
k
=
1
Another set of cross-interference coefficients, specifically coefficients
A
n
C
m
,
B
n
C
m
,
characterize interference acting in the inverse direction
from the spatial signal component
A
n
S
m
and
B
n
S
m
,
Ω
m
to the component
Ω
n
. It follows from
(20.5) that
A
n
C
m
=
A
m
C
n
,
B
n
C
m
=
A
m
S
n
,
(20.6)
A
n
S
m
=
B
m
C
n
,
B
n
S
m
=
B
m
S
n
.
Thus the cross-interference effect, impacting spatial spectrum analysis in the case
where sensors in the array are placed nonuniformly, is reflected by the following
matrix of the cross-interference coefficients:
⎡
⎣
⎤
⎦
(
A
1
C
1
)(
A
1
S
1
)(
A
1
C
2
)(
A
1
S
2
)
···
(
A
1
C
M
)(
A
1
S
M
)
(
B
1
C
1
)(
B
1
S
1
)(
B
1
C
2
)(
B
1
S
2
)
···
(
B
1
C
M
)(
B
1
S
M
)
(
A
2
C
1
)(
A
2
S
1
)(
A
2
C
2
)(
A
2
S
2
)
···
(
A
2
C
M
)(
A
2
S
M
)
Z
=
(
B
2
C
1
)(
B
2
S
1
)(
B
2
C
2
)(
B
2
S
2
)
···
(
B
2
C
M
)(
B
2
S
M
)
.
(20.7)
···
···
···
···
···
···
···
(
A
M
C
1
)(
A
M
S
1
)(
A
M
C
2
)(
A
M
S
2
)
···
(
A
M
C
M
)(
A
M
S
M
)
(
B
M
C
1
)(
B
M
S
1
)(
B
M
C
2
)(
B
M
S
2
)
···
(
B
M
C
M
)(
B
M
S
M
)
This matrix of the cross-interference coefficients is an essential characteristic
of the arrays of sensors with pseudo-random distances between them. Once the
pattern of sensors in the array is known, all coefficients of matrix
Z
, as well
as of the matrix inv(
Z
), can be calculated. This means that when a signal is
