Databases Reference
In-Depth Information
As discussed in Section
2.4
, the unconstrained maximum-likelihood (ML) estimate
of
R
xx
is the augmented sample covariance matrix
S
xx
. The ML estimate of
R
xx
under
the constraint
R
xx
=
0
is
S
xx
0
0
xx
S
0
=
.
(3.92)
Hence, the GLR (
3.91
) can be expressed as
S
xx
exp
I
2
/
M
det
S
−
1
0
tr
S
−
1
0
M
2
2
/
M
=
λ
=
−
S
xx
−
(3.93)
det
I
xx
S
xx
S
−
1
=
(3.94)
xx
S
xx
S
−∗
I
det
−
2
S
xx
det
S
xx
=
(3.95)
n
k
i
)
=
(1
−
=
1
−
ρ
1
.
(3.96)
i
=
1
k
i
}
{
In the last line,
denotes the estimated circularity coefficients, which are com-
puted from the augmented sample covariance matrix
S
xx
. The estimated circularity
coefficients are then used to estimate the degree of impropriety
ρ
1
. Our main finding
follows.
Result 3.8.
The estimated degree of impropriety
ρ
1
is a test statistic for a GLRT for
impropriety.
Equations (
3.95
) and (
3.96
) are equivalent formulations of this GLR. A full-rank
implementation of this test relies on (
3.95
) since it does not require computation of
S
−
1
xx
. However, a reduced-rank implementation considers only the
r
largest estimated
circularity coefficients in the product (
3.96
).
This test was first proposed by
Andersson and Perlman (1984
), then in complex
notation by
Ollila and Koivunen (2004
), and the connection with canonical correlations
was established by
Schreier
et al
. (2006
).
Andersson and Perlman (1984
) also show that
the estimated degree of impropriety
n
1
n
k
i
ρ
3
=
(3.97)
i
=
1
ρ
1
is the
locally most powerful
(LMP) test for impropriety. An LMP test has
the highest possible power (i.e., probability of detection) for
H
1
close to
H
0
, where all
circularity coefficients are small.
Intuitively, it is clear that
ρ
1
and
ρ
3
behave quite differently. While
ρ
1
is close
to 1 if at least one circularity coefficient is close to 1,
rather than
ρ
3
is close to 1 only if all
circularity coefficients are close to 1.
Walden and Rubin-Delanchy (2009
) reexamined
testing for impropriety, studying the null distributions of
ρ
1
and
ρ
3
, and deriving a
distributional approximation for
ρ
1
. They also point out that no uniformly powerful
(UMP) test exists for this problem, simply because the GLR and the LMP tests are
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