PBT f,k = A *TRA′ ,
(12)
source wavefield the bias is substantially smaller
for ML compared with the BT method (De Graaf
where R is the estimated spatial correlation ma-
and Johnson 1985).
trix.
In the ML beamformer, am(f) are defined by the
2.3. Estimation and conditioning of the
properties of the spatial correlation matrix so that
correlation matrix
spatial energy leakage is minimized. This is done
The spatial correlation matrix (R) is constructed
by minimizing the weighted array power function
from the frequency domain representation of the
under the constraint that a fixed gain be main-
signal vector x. The properties of the spatial corre-
tained at the observation wavenumber. Johnson
lation matrix determine the quality of the
(1982) sets up the problem as
beampower function. To estimate R accurately,
we use a tapered overlapped block-averaging FFT
(OBAFFT). This is a widely utilized procedure for
PML f,k
W*T RW = 0 ,
estimating signals with intricate spectra (Welch
=
W
W
1967, Capon et al. 1967). The OBAFFT method
allows one to make trade-offs between the bias
subject to W*T A ′ = 1 .
(13)
and variance in the estimate of R.
It is important to guarantee the nonsingularity
A′ is given by eq 11, and W is defined as the
of the spatial correlation matrix when applying
weighted steering vector whose elements are
the ML processing procedure. This can be done by
the block averaging process or it can be forced by
Wm = amA′m .
(14)
applying a small amount of additive noise to the
correlation matrix (Capon 1969). A normalization
The solution to eq 13 is (Johnson 1982, Capon 1969)
procedure that produces ones along the diagonal
of the spatial correlation matrix is also useful in
R 1A ′
making qualitative assessments of the accuracy of
.
(15)
W=
the wavenumber spectrum.
A′
*T R 1A ′
2.3.1. Tapered overlapped block-averaged FFT
Note that eq 15 is a function of the spatial correla-
The mathematical procedure used to form the
tion matrix. The effect of designing the weighted
OBAFFT is as follows. The time-domain signal
steering vector using the observed signal proper-
vector obtained from a passive array with M sen-
ties is that we produce a wavenumber window
sors and components such as that of eq 1 has the
function that is adaptive to the signal-in-noise
form
field and the specific array geometry. The result-
x0(t)
ant beampower function is
x1(t)
x2(t)
PML f,k = W*T RW .
(16)
.
(18)
x=
Substituting eq 15 into 16 and using the estimation
for the cross-power matrix, we obtain
xM1(t)
1
PML f,k =
,
(17)
Let x have N samples at intervals of ∆t. Now
1
A′*T R A′
define a block of this sequence with a length L,
such that L < N; further, let each block of L samples
where ^ indicates an estimate based on available
be overlapped by a percentage p. The number of
data.
blocks will be
This is the working form of the ML beamformer
used in this paper. It should also be noted that the
NL
,
(19)
nblks = 1 +
ML beamformer is also termed a minimum vari-
nshft
ance method, and in the case of only one coherent
where nshft = INT L 1 p and INT is the near-
signal it is also spatially unbiased (Capon 1969), in
the sense that the bias tends toward zero as the
est integer operator. Thus, nblks gives the total
number of sensors becomes large. In a multiple-
number of overlapped segments (or blocks) that
5
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