^*
^*
^*
^*
^
^
^
^
X0 ( f )X0 ( f )
X0 ( f )X1 ( f )
X0 ( f )X2 ( f )
...
X0 ( f )XM 1 ( f )
^*
^*
^*
^*
^
^
^
^
X1( f )X0 ( f )
X1 ( f )X1 ( f )
X1 ( f )X2 ( f )
...
X1 ( f )XM 1 ( f )
^*
^*
^*
^*
^
^
^
^
X2 ( f )X0 ( f )
X2 ( f )X1 ( f )
X2 ( f )X2 ( f )
...
X2 ( f )XM 1 ( f )
^
R( f ) =
(23)
...
...
...
XM 1( f )XM 1( f )
*
^*
^*
^*
^
^
^
^
^
XM 1( f )X0 ( f )
XM 1( f )X1 ( f )
XM 1( f )X2 ( f )
...
Figure 4. Spatial correlation matrix structure.
tively insensitive to λ, but it should be kept as
2.3.3. Spatial correlation matrix conditioning
The structure of the spatial correlation matrix is
small as possible since it lowers the signal-to-noise
given by eq 23 (Fig. 4). In the programmed imple-
ratio (SNR) of the beam response.
mentation, each element of this matrix is normal-
ized according to
2.3.4. Variance of the cross-power matrix
The OBAFFT method allows a degree of control
*
over the bias/variance trade-off in the estimate of
X i(f)X j (f)
.
(24)
R i j(f) =
the cross-power matrix. The bias is mitigated by
X i(f) X j(f)
the choice of window taper and the number of
points in each block, while the variance is a func-
If the signal vector is perfectly coherent, then the
tion of the number and degree of overlap between
diagonal elements of R will contain unity values,
blocks. The independence of each block is reduced
resulting in a maximum beam power response of
by increasing the overlap. The characteristics of
1 when applying the BT method. This allows quali-
the window taper applied to each block also affect
tative judgments to be made on the reliability of
the block`s independence for a given degree of
the wavenumber estimate, which is particularly
overlap. This partial dependence/independence
important when dealing with nonideal signal types
may be exploited to decrease the variance of a
or inhomogeneous propagation environments.
given spectral estimate.
When using narrowband BT processing with this
As an illustration of the variance reduction that
normalization, a peak power response < 0.5 is
can be obtained using this strategy, a signal (dt =
often unreliable.
1/1023) was generated by superimposing two sine
The existence of the R 1 matrix is of critical
waves with line spectra at 20 and 23 Hz and
importance when applying the ML method (see eq
unitary peak-to-peak amplitudes. The second sine
18). Capon (1969) notes that if the number of blocks
wave series is time-shifted (relative to the first
(N) in the estimate is less than the number of array
signal) by 0.0455 s (at 20 Hz in the frequency
domain this is an approximately 32 phase shift).
sensors (M), then R is of order M and rank N and
is thus singular. This is frequently the case when
A unique pseudo-random Gaussian "noise" series
processing signals of short duration. To guarantee
with zero mean and a maximum peak-to-peak
the nonsingularity of the estimated spatial corre-
amplitude of 1 is added to each of the two sine
lation matrix we add a small amount of incoherent
wave signals. This yields an SNR of 1 for each
noise (λ) to the elements of the spatial correlation
spectral line in the series. The OBAFFT spectral
matrix. Capon (1969) suggests
estimation process was applied at 20 Hz for each
sine wave plus noise signal, and the cross-power
R′ = 1 λ R + λI .
(25)
operation was carried out. Note that the presence
of the 23-Hz spectral line in the resulting series
The choice of λ is determined by trial and error (the
allows simulation of phase and amplitude bias
results reported here often used values < 104). In
effects. The 20-Hz OBAFFT estimate of the cross
practice the accuracy of the beam response is rela-
power between the two noisy signals was ob-
7
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