Introduction to Acoustics

182 Part A Propagation of Sound
Part A 5.6
a)
b)
exp (i kr)
Array data
l = 1 2 3
T
N channels
L segments
T secs long
di–1 di di+1
Windowed
FFT
Windowed
FFT
è
Array
Wavefront
R 1 1
R 1 N
R 1
R
R L 1
R L N
R L
FFT vectors
Data matrix
Fig. 5.35a,b Array processing. (a) Geometry for plane-wave beamforming.
(b) The data is transformed to the frequency domain in
which the plane-wave processing takes place. The cross-spectraldensity
matrix (CSDM) is an outer product of the data vector for
a given frequency
0
1
2
3
4
5
6
side-lobes can be in the physical angle region,
thereby interfering with acoustic propagating
signals.
• Array gain: defined as the decibel ratio of the signalto-noise
ratios of the array output to a single phone
7
0 0.01 0.02 0.03 0.04 0.05 0.06
Array (m)
Time (s)
Fig. 5.36 Depth-versus-time representation of simulated broadband
coherent data received on a vertical array in the presence of ambient
noise. The wavefronts corresponding to different sources are nearly
undistinguishable
2
1.5
1
0.5
0
–0.5
output. If the noise field is isotropic, the array gain
is also termed the directionality index.
5.6.3 Adaptive Processing
There are high-resolution methods to suppress sidelobes,
usually referred to as adaptive methods since the
signal processing procedure constructs weight vectors
that depend on the received data itself. We briefly describe
one of these procedures: the minimum-variance
distortionless processor (MV), sometimes also called
the maximum-likelihood method (MLM) directional
spectrum-estimation procedure.
We seek a weight vector wMV applied to the matrix
K such that its effect will be to minimize the output of the
beam-former (5.77) except in the look direction, where
we want the signal to pass undistorted. The weight vector
is therefore chosen to minimize the functional [5.38].
From (5.72)
F = wMVKwMV + α(wMVw − 1) . (5.83)
The first term is the mean-square output of the array
and the second term incorporates the constraint of unity
gain by means of the Lagrangian multiplier α. Following
the method of Lagrange multipliers, we obtain the MV
weight vector,
wMV = K −1w . (5.84)
wK −1 w
This new weight vector depends on the received data as
represented by the cross-spectral density matrix; hence,
the method is adaptive. Substituting back into (5.77)
gives the output of our MV processor,
BMV (θs) =[w(θs)K −1 (θtrue)w(θs)] −1 . (5.85)
The MV beam-former should have the same peak value
at θtrue as the Bartlett beam-former, (5.77) but with sidelobes
suppressed and a narrower main beam, indicating
that it is a high-resolution beam-former. Examples are
showninFig.5.39. Of particular practical interest for
this type of processor is the estimation procedure associated
with Fig. 5.35band(5.80). One must take sufficient
snapshots to allow the stable inversion of the CSDM.
This requirement may conflict with source motion when
the source moves through multiple beams along the time
interval needed to construct the CSDM.
5.6.4 Matched Field Processing, Phase
Conjugation and Time Reversal
Matched field processing (MFP) [5.42] is a generalization
of plane-wave beam-forming in which data