Digital Signal Processing Reference
In-Depth Information
=
span
i
i
...
i
span
e
e
...
e
,
(14 . 30)
1
2
M
1
2
M
where
e
i
represents the
i
t h
column of
e
. If we adopt the approach in [43] for interference
DOA estimation, then we define
1
2
WaW
c
=
()
−
,
(14 . 31)
D
a
σ
where σ
2
represents the power of the AWGN. Substituting equations (14.26) and (14.29)
into (14.31), the cancellation weight vector for the
k
t h
satellite can be expressed as
1
1
−
1
H
−
1
H
w
=
a
()
θ
−
eea
Λ
()
θ
=
Ia
()
θ
−
e
Λ
ea
(()
θ
ck
k
k
k
k
2
2
σ
σ
(14 . 32)
M
2N
1
∑
1
∑
1
= −
I
e e
H
−
ee
H
a
k
(),
θ
i
i
j
j
σ
2
σ
2
σ
2
i
n
i
=
1
jM
= +
1
where
I
is the identity matrix. Typically,
1
1
2 2
σσ
σσ
i
,
=
,
n
2
2
and equation (14.32) can be simplified to
2N
M
M
1
∑
1
∑
1
=
( )
∑
1
H
H
H
w
= −
I
ee
a
()
θ
=
ee
a
()
θ
e
a
()
θ
i
.
e
(14 . 33)
ck
j
j
k
ii
k
i
k
2
2
2
2
σ
σ
σ
σ
jM
=+
1
i
=
1
i
=
1
The above vector lies in the interference subspace. If
D
>
M
, then the columns of the
cancellation weight matrix span the
M
interference subspace, i.e.,
=
.
span
i
i
…
i
span
w
w
…
w
(14 . 34)
1
2
M
c
1
c
2
cD
It is noted that if the interferers are LHCP, the eigenvectors of the covariance matrix
in the presence of weak GPS signals are presented as LHCP. Accordingly, the coefficients
e
i
H
a
k
(θ) in (14.33) become zero, and the corresponding weight vector
w
ck
becomes zero as
well. As a result, the DOAs of the interferers cannot be estimated from
W
a
or
W
c
. This
can be viewed as a drawback of using the adaptive weights for DOA estimation. If the
interferers are RHCP, horizontally polarized, or vertically polarized, the above coeffi-
cients are nonzero, and equation (14.34) is valid.
If the exact order statistics are available, then one would generate the nulling weights,
proceed with eigendecomposition of
W
c
, and obtain a corresponding MUSIC spectrum,
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