170
CHAPTER 8
8.1
The Hermitian transpose of data matrix
A
is defined by
where
u
(
k,i
) is the output of sensor
k
in the linear array at time
i
, where
,
and
.
(a)
The matrix product
A
H
A
equals:
This represents the
M
-by-
M
spatial (deterministic) correlation matrix of the array with
temporal averaging applied to each element of the matrix. This form of averaging
assumes that the environment in which the array operates is temporally stationary.
(b)
The matrix product
AA
H
equals
A
H
u
1 1
,
(
)
u
1 2
,
(
)
…
u
1
n
,
(
)
u
2 1
,
(
)
u
2 2
,
(
)
…
u
2
n
,
(
)
u M
1
,
(
)
u M
2
,
(
)
…
u M n
,
(
)
=
. . .
. . .
. . .
k
1 2
…
M
,
,
,
=
i
1 2
…
n
,
,
,
=
A
H
A
u
1
i
,
(
)
u
* 1
i
,
(
)
i
=1
n
∑
u
1
i
,
(
)
u
* 2
i
,
(
)
i
=1
n
∑
…
u
1
i
,
(
)
u
*
M i
,
(
)
i
=1
n
∑
u
2
i
,
(
)
u
* 1
i
,
(
)
i
=1
n
∑
u
2
i
,
(
)
u
* 2
i
,
(
)
i
=1
n
∑
…
u
2
i
,
(
)
u
*
M i
,
(
)
i
=1
n
∑
…
u M i
,
(
)
u
* 1
i
,
(
)
i
=1
n
∑
u M i
,
(
)
u
* 2
i
,
(
)
i
=1
n
∑
…
u M i
,
(
)
u
*
M i
,
(
)
i
=1
n
∑
=
. . .
. . .
. . .
AA
H
=
u
*
k
1
,
(
)
u k
1
,
(
)
k
=1
M
∑
u
*
k
1
,
(
)
u k
2
,
(
)
k
=1
M
∑
…
u
*
k
1
,
(
)
u k n
,
(
)
k
=1
M
∑
u
*
k
2
,
(
)
u k
1
,
(
)
k
=1
M
∑
u
*
k
2
,
(
)
u k
2
,
(
)
k
=1
M
∑
…
u
*
k
2
,
(
)
u k n
,
(
)
k
=1
M
∑
u
*
k n
,
(
)
u k
1
,
(
)
k
=1
M
∑
u
*
k n
,
(
)
u k
2
,
(
)
k
=1
M
∑
…
u
*
k n
,
(
)
u k n
,
(
)
k
=1
M
∑
. . .
. . .
. . .