Introduction to Acoustics

Fig. 5.37 Angle-versus-time representation of the simulated
data in Fig. 5.36 after time-delay beam-forming. All
sources appear clearly above the noise with their corresponding
arrival angle
on an array is correlated with replicas from a (waveguide)
propagation model for candidate locations ˆr, ˆz
(Fig. 5.40). Localization of a source is accomplished
with a resolution consistent with the modal structure
and SNR. The central difficulty with MFP is specifying
the coefficients and boundary conditions of the
acoustic wave equation for shallow water propagation,
i. e., knowing the ocean environment in order to generate
the replicas. An alternative to performing this
model-based processing is phase conjugation (PC), in
the frequency domain or, time reversal (TR) in the time
domain, in which the conjugate or time-reversed data
is used as source excitations on a transmit array colocated
with the receive array (Fig. 5.41a) [5.43]. The
PC/TR process is equivalent to correlating the data with
the actual transfer function from the array to the original
source location. In other words, both MFP and
PC are signal processing analogs to the mechanical
lens adjustment feedback technique used in adaptive
optics: MFP uses data together with a model (note
the feedback arrow in Fig. 5.40) whereas PC/TR is an
active form of adaptive optics simply retransmitting
phase-conjugate/time-reversed signal through the same
medium (e.g., see result of Fig. 5.41). Though time reversal
is thought of as as active process, it is presented
in this section because of its relation to passive MFP.
Ocean Time-Reversal Acoustics
Phase conjugation, first demonstrated in nonlinear optics
and its Fourier conjugate version, time reversal is
a process that has recently implemented in ultrasonic
laboratory acoustic experiments [5.44]. Implementation
of time reversal in the ocean for single elements [5.45]
and using a finite spatial aperture of sources, referred
to as a time-reversal mirror (TRM) [5.46], is now well
established.
The geometry of a time-reversal experiment is shown
in Fig. 5.41. Using the well-established theory of PC
and TRM in a waveguide, we just write down the result
of the phase-conjugation and time-reversal process,
respectively, propagating toward the focal position
Ppc(r, z,ω) =
J�
Gω(r, z, z j)G ∗ ω (R, z, zps)S ∗ (ω)
j=1
(5.86)
–60
–40
–20
0
20
40
60
Underwater Acoustics 5.6 SONAR Array Processing 183
0 0.01 0.02 0.03 0.04 0.05 0.06
Angle (deg)
Time (s)
and
Ptrm(r, z, t) = 1
(2π) 2
J�
��
G(r, z, t ′′ ; 0, z j, t ′ )
j=1
× G(R, z j, t ′ ; 0, zps, 0)
× S(t ′′ − t + T)dtdt ′′ , (5.87)
where S is the source function, G∗ ω (R, z, zps)
is the frequency-domain Green’s function and
G(R, z j, t ′ ; 0, zps, 0) is the time-domain Green’s function
(TDGF) from the probe source at depth zps to each
element of the SRA at range R and depth z j. Emphasizing
the time-domain process, G(r, z, t ′′ ; 0, z j, t ′ )isthe
0
–2
–4
–6
–8
–10
–12
–60 –40 –20 0 20 40 60
Array gain (dB)
Angle (deg)
Fig. 5.38 Comparison between coherent time-delay beam-forming
(in red) and incoherent frequency-domain beam-forming (in blue)
for the simulated data shown in Fig. 5.35. When data come from coherent
broadband sources, time-domain bema-forming show better
performance than frequency analysis
0
–2
–4
–6
–8
–10
Part A 5.6