Digital Signal Processing Reference
In-Depth Information
reliably only for
a¼
2, that is, the Gaussian case. However, the robust
M
-estimator is
able to estimate the number of sources reliably for large range of
a
-values. Among the
robust
M
-estimators, MLT(1) has the best performance.
2.6.4 Subspace DOA Estimation for Noncircular Sources
We now describe the Root-MUSIC-like method presented in [13]. As usual, assume
that the signal
s
and noise
n
in the array model (2.1) are uncorrelated with zero-
mean. The method further requires the following additional assumptions.
(1) The array is ULA (in order to facilitate using polynomial rooting).
(2) Noise
n
is second-order circular and spatially white, that is
C
(
n
)
¼ s
2
I
and
P
(
n
)
¼
0.
(3) Sources signals
s
i
,
i ¼
1,
...
,
d
are uncorrelated in the sense that
C
(
s
)
¼
diag(
s
2
(
s
i
)) and
P
(
s
)
¼
diag(
t
(
s
i
)).
Under these assumptions,
C
(
z
)
¼ AC
(
s
)
A
H
þs
2
I
,
P
(
z
)
¼ AP
(
s
)
A
T
where as earlier
A ¼ A
(
u
) denotes the array response matrix. Further assume
that
(4)
P
(
s
)
¼C
(
s
)
F
, where
F¼
diag(
e
jf
i
).
Assumption (4) means that the circularity coefficient of the sources are equal to unity,
that is,
l
(
s
i
)
¼
1 for
i ¼
1,
...
,
d
, which by (2.3) implies that transmitted source signal
s
i
must be real-valued, such as AM or BPSK modulated signals, or the real part Re(
s
i
)
of the transmitted signal is a linear function of the imaginary part Im(
s
i
). If (1)-(4)
hold, then the covariance matrix for the augmented signal vector
z
is
C
(
s
)
H
A
A
F
A
A
F
þs
2
I:
C
(
z
)
¼
(2
:
34)
Now by performing eigenvalue decomposition
C
(
z
) we may find
d
dimensional signal
subspace and 2
k d
dimensional orthogonal noise subspace. Thus Root-MUSIC-like
direction finding algorithms can be designed; see [13] for details. By exploiting the
noncircularity property we obtain extra degrees of freedom since noncircularity
allows resolving more sources than sensors. Again, in the face heavy-tailed noise
or outlying observations, a robust estimate of the array covariance matrix
C
(
z
) and
pseudo-covariance matrix
P
(
z
) can be used instead of the conventional estimators,
C
and
P
. We wish to point out, however, that the four assumptions stated above are
not necessary for all subspace DOA estimation methods for noncircular sources; see
for example, [1].
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