![](https://csdnimg.cn/release/download_crawler_static/593198/bg5.jpg)
The asymptotic variances of the ML DOA and Doppler
estimates are given by
(23)
(24)
Proof: See Appendix A.
B. Unstructured Model
The ML solution for the unstructured model may be
similarly derived. The negative log-likelihood function for
is of the same form as (11) but with replaced by
. Minimization with respect to gives
(25)
where the subscript
is used to explicitly denote an
estimate obtained using the unstructured model. The cri-
terion resulting from substitution of (25) into (11) may be
simplified as follows:
(26)
(27)
(28)
from which it is clear that the solution for
is simply
(29)
The remaining term in (28) is a function of
only and can
be written as
(30)
(31)
The unstructured ML estimate of
is thus given by
(32)
Since it appears in the ML criteria for both the structured
and unstructured models, we make a note here regarding
the quantity
, where
(33)
Neither (16) nor (31) makes sense if
. To show
that this cannot occur, define
so that
Then
(34)
(35)
(36)
(37)
since
is a projection matrix. Equality in (37) can
hold only if
belongs to the range of . Due to the
presence of noise and interference in
, with
probability one and the ML criterion is well defined.
An interpretation for (32) can be obtained if the problem
is viewed as one of estimating the frequency of a single
sinewave observed in
distinct but dependent channels.
Equation (32) shows that the ML frequency estimate for
this problem is calculated by first spatially prewhitening
the data and then finding the largest peak in the magnitude
of the Fourier transform averaged over the
channels.
The derivation above also implies that in principle, array
calibration data is not needed in detecting the presence
or absence of a target in the radar data set. Given the
estimate
from (32), one could develop a statistical test
for detection based on the size of
where is a symmetric square root of . If the
array is imprecisely calibrated, such a test may outperform
(20)
SWINDLEHURST AND STOICA: ML METHODS IN SIGNAL PROCESSING 425