Phase-Conjugate Arrays
Transcript of Phase-Conjugate Arrays
Chapter 2
Phase-Conjugate Arrays
In this chapter the operation of time-reversal mirrors — known as phase conjugate arrays
when dealing with quasi-monochromatic signals — is examined from the perspective of
array processing. The goal is to clarify how focusing occurs in terms of familiar spatial pro-
cessing concepts such as directivity functions, and to evaluate the sensitivity of this process
to various environmental and array design parameters. For mathematical tractability, a
simple homogeneous environment with perfectly-reflective surface, partially-reflective bot-
tom and constant sound speed is assumed, and acoustic propagation is modeled using ray
theory. The analysis highlights the nature of phase conjugation as a coherent interfer-
ence phenomenon between a set of aperture functions, but it also helps to understand the
remarkable focusing robustness that has been observed during sea trials.
Based on these results, strategies are discussed for reducing the number of array trans-
ducers that will be required for practical implementation of some variants of time reversal
proposed in this work. As in free space, it is shown that nonuniform arrays with randomly-
spaced sensors can offer substantial hardware savings with little impact on directivity,
hence preserving the focusing power of time-reversal mirrors.
2.1 Acoustic Propagation Models
Acoustic propagation models have been extensively used as a simulation tool to method-
ically evaluate the performance of sonar systems and other underwater acoustic systems
under controlled conditions. In spite of the practical difficulties in characterizing the prop-
erties of ocean environments, steady progress in modeling of transmission loss has greatly
enhanced the accuracy of such predictions [86]. The logical consequence of this would be to
try to expand the role of these models beyond simulation, incorporating them into actual
signal processing algorithms. Matched-field processing and ocean acoustic tomography
are two well-known examples where sophisticated propagation models have considerably
enhanced our ability to estimate parameters from real data [4].
In this section, a few popular acoustic propagation models are briefly reviewed, the
main goal being to point out their strengths and weaknesses in the context of underwater
communications, and hence justify the modeling approach that was followed in this work.
19
20 Phase-Conjugate Arrays
2.1.1 Deterministic Models
The ocean is an acoustic waveguide limited by the sea surface and seafloor, with the
speed of sound playing the role of the index of refraction in optics [59]. The medium
acts as an aberrating lens due to spatial and temporal variations in the sound speed
and boundaries. The usual starting point for computational methods in acoustics is the
Helmholtz equation, or reduced wave equation, which describes the complex amplitude of
the pressure field generated by a normalized harmonic point source with time dependence
ejωt [59] on a propagation medium with constant density. In Cartesian coordinates
[∇2 + k2(r)
]Gω(r, r
′) = −δ(r− r′) , (2.1)
where ω is the source angular frequency, r′ = (x′, y′, z′) is the source location, k(r) =
ω/c(r) is the wavenumber and c(r) is the sound speed. This equation is derived from
the basic relations of fluid mechanics using linear approximations, and is considered to
be quite accurate in most realistic scenarios involving medium and low-power acoustic
sources. The solution of (2.1) is commonly known as the medium Green’s function, and it
allows the effect of a source with arbitrary space-time dependence xt(r) to be determined
by superposition [59]. Denoting the Fourier transform of the source byXω(r), the resulting
pressure field in the time domain is
yt(r) =1
2π
∫
V
∫ ∞
−∞Xω(r
′)Gω(r, r′)ejωt dω dV ′ , (2.2)
where V denotes the volume occupied by the transmit aperture. From the perspective of
linear systems theory, it is clear that Gω(r, r′) represents the ocean transfer function from
point r′ to r [124].
2.1.2 Spectral Integration
Both spectral integral and normal mode models are particularly well-suited to stratified
media, i.e., boundary value problems where both the coefficients of the Helmholtz equa-
tion and the boundary conditions are independent of one or more space coordinates. In
underwater acoustics, the horizontally stratified or range-independent waveguide is the
most important geometry that satisfies these conditions [59].
Due to the axial symmetry of the problem, it is natural to choose a cylindrical co-
ordinate system where the downward-pointing z-axis passes through the source and the
r-axis is parallel to the interfaces, such that the pressure field is independent of the az-
imuth angle. For conciseness, the dependence of G on the frequency ω and source location
r′ = (0, zs) will be omitted. A Hankel transform is then applied in r to obtain
F (k, z) =
∫ ∞
0G(r, z)J0(kr)r dr , (2.3)
where J0 denotes the zeroth-order Bessel function of the first kind. The transformed field
F satisfies a second-order differential equation on the depth variable z. After solving this
2.1 Acoustic Propagation Models 21
new boundary value problem, the final pressure is computed with the inverse transform
G(r, z) =
∫ ∞
0F (k, z)J0(kr)k dk
≈√
2
πr
∫ ∞
0F (k, z) cos(kr − π/4)
√k dk ,
(2.4)
where a large-argument approximation to the Bessel function has been invoked [59]. An
FFT can be used to efficiently evaluate the inverse transform in a suitably defined range
grid.
2.1.3 Normal Modes
When the integrand F (k, z) in (2.4) is plotted as a function of the horizontal wavenumber
k, it is seen that it has a spiky behavior [86]. In fact, the integral is dominated by the values
of F (k, z) on a discrete set of points, which suggests approximating it by a summation
over the set of resonance frequencies. That is the essence of the method of normal modes.
The normal mode decomposition can also be derived from the Helmholtz equation
by the method of separation of variables, which postulates a solution to that differential
equation of the form G(r, z) = Φ(r)Ψ(z). Substituting into the wave equation, new dif-
ferential equations and boundary conditions are obtained for both Φ(r) and Ψ(z). The
latter turns out to be a classical Sturm-Liouville eigenvalue problem having an infinite
number of solutions characterized by a mode shape Ψn(z) (eigenfunction) and a horizon-
tal propagation constant krn (eigenvalue). The mode functions are orthonormal and form
a complete set, in the sense that they can be used to represent any function with arbitrary
accuracy [59]. For each mode n, the solution for the range function Φn(r) is a scaled
Hankel function whose exact form is determined by the radiation condition, which states
that energy should radiate outward as r →∞.
For a source time dependence of the form e−jωt the pressure field is given by
G(r, z) =j
4ρ(zs)
∞∑
n=1
Ψn(zs)Ψn(z)H(1)0 (krnr)
≈ je−jπ/4
ρ(zs)√8πr
∞∑
n=1
Ψn(zs)Ψn(z)ejkrnr√krn
,
(2.5)
where the Hankel function H(1)0 is related to the zeroth-order Bessel functions of the first
and second kind as H(1)0 (r) = J0(r) + jY0(r) [11]. The medium density ρ(z) appears
explicitly in (2.5) as it is allowed to vary with depth, e.g., in problems where sound
propagates in one or more sediment layers.
For consistency with the convention used for passband signals, where a complex enve-
lope modulates an exponential term ejωt, the Hankel function in (2.5) should have been
conjugated. This was not done due to the prevalence of the form (2.5) in the acoustics
literature, introducing a slight inaccuracy that has no impact on subsequent results.
22 Phase-Conjugate Arrays
2.1.4 Ray Methods
Ray models have been widely used to study acoustic propagation since the early 1960’s.
The derivation of ray models is based on a high-frequency approximation to the solution
of the wave equation, leading to somewhat coarse accuracy in problems involving complex
propagation in the water column and sediment layers. For that reason, full-wave models
are preferred by the research community for problems such as matched-field processing,
where actual propagation must be modeled as accurately as possible. However, field
computations by ray tracing can be hundreds of times faster than with full-wave models,
and ray diagrams provide much insight for validating and interpreting the results obtained
with more complex methods. Not surprisingly, ray methods are still used extensively in
operational environments, where computational time is a critical factor and environmental
uncertainty poses severe constraints on the attainable accuracy [59].
Mathematically, the derivation of ray models starts by assuming a solution to the
Helmholtz equation in the form of a series
G(r) = ejωτ(r)∞∑
m=0
Am(r)
(jw)m. (2.6)
Differentiating (2.6), substituting in (2.1) and equating terms of like order in ω, an infi-
nite sequence of equations for τ(r) and Am(r) is obtained. Invoking the high-frequency
approximation, only the first term in the ray series (2.6) is retained, thus eliminating all
but two of the equations, namely
|∇τ |2 = c−2(r) (2.7)
2∇τ · ∇A0 + (∇2τ)A0 = 0 . (2.8)
These are commonly known as the eikonal and transport equations, respectively. The
eikonal equation is solved by introducing a family of curves that are perpendicular to the
level curves of τ(r), i.e., the wavefronts [59]. It turns out that a ray trajectory is defined
by a system of four coupled differential equations
dr
ds= c ξ(s) ,
dξ
ds= − 1
c2dc
dr, (2.9)
dz
ds= c ζ(s) ,
dζ
ds= − 1
c2dc
dz, (2.10)
where r(s) =(r(s), z(s)
)is the position along the path and c ·
(ξ(s), ζ(s)
)its tangent.
The initial condition to integrate (2.9)–(2.10) is defined by the takeoff angle of the ray
at the source. The travel time along the ray satisfies dτ/ds = c−1(s), and can be easily
integrated along with the previous equations.
The final step is to associate an amplitude with each ray by solving the transport
equation. The expression is not very relevant in the present context and will be ommitted.
It can be shown that the pressure field is obtained by first dividing the energy of the point
source among a series of ray tubes formed by pairs of adjacent rays. The change in intensity
along a ray tube is then inversely proportional to the cross section of that tube [59].
2.1 Acoustic Propagation Models 23
The pressure field is calculated by coherently summing the contributions of all the
rays arriving at a single point (eigenrays). Amplitudes must be corrected by a 90◦ phase
shift whenever a ray crosses through caustics, i.e., regions where the section of a ray tube
vanishes and the predicted intensity is infinite. Gaussian beam tracing is a modification
to the standard technique described above that associates with each ray a beam with
a Gaussian intensity profile normal to the ray [87]. The beamwidth and curvature are
governed by two additional differential equations, which are integrated along with the usual
ray equations. The solutions so generated are free of singularities at caustics and abrupt
discontinuities at shadow zone boundaries. The approach also avoids explicit eigenray
computations, as the field at a given point can be calculated from nearby beams.
Ray models can be readily adapted to broadband problems with minor additional
computational cost and storage. Rather than computing field amplitudes, at each point
a list is kept of the delays and attenuations associated with the eigenrays. Note that, as
evidenced by (2.7) and (2.8), these parameters are independent of the frequency ω.
2.1.5 Parabolic Equations and Finite Difference Methods
Propagation models developed for range-independent waveguides can be modified to in-
corporate range dependence, but this entails a considerable increase in complexity. In an
attempt to obtain efficient solution methods in non-layered media, parabolic approxima-
tions to the original elliptic Helmholtz equation have been developed [59]. The crucial
advantage of parabolic equation (PE) methods is that they lead to an initial-value prob-
lem, so that one-pass solutions can be generated for the whole range under consideration,
given a source-field distribution over depth at the initial range.
Depending on the details of the approximation, it is possible to use a computationally-
efficient split-step Fourier algorithm to propagate the field. For other types of approxima-
tion that improve the numerical accuracy under wide-angle propagation, the solution must
be obtained by less efficient finite difference methods. In both cases it is essential that the
discretization step in range be small when compared with the acoustic wavelength.
Under specific conditions, the approaches mentioned previously are numerically effi-
cient for solving propagation problems in underwater acoustics. However, efficiency is
gained by sacrificing generality through various assumptions and approximations. For
modeling phenomena such as backscattering it is not possible to assume horizontal strati-
fication, as in wavenumber integration and normal mode methods, or one-way propagation,
as in parabolic equation methods. Therefore, there is still a need for models capable of
solving the full wave equation in inhomogeneous environments with complex geometries.
A number of numerical approaches are available for this purpose, based on discretization
of the differential equations over a computational mesh [59].
These discrete methods are computationally intensive due to the fact that the solution
must be able to represent the actual spatial and temporal evolution of the field either in
a volume or on a boundary. Their use in ocean acoustics is limited either to the solution
of special short-range problems, or as components in hybrid approaches, where they are
24 Phase-Conjugate Arrays
used to obtain a local representation of the acoustic field that is subsequently propagated
by one of the more efficient approaches described earlier.
2.1.6 Stochastic Models
Dynamical random inhomogeneities in the ocean can be attributed to phenomena such
as turbulence, internal waves and mesoscale eddies [14, 114]. They cause scattering of
sound and fluctuations of its intensity, reduce the coherence of sound waves and change
their frequency spectrum. The sea surface and bottom should also be regarded as random
rough surfaces, although the latter has no dynamic behavior and its randomness stems
from our inability to fully characterize its static properties.
Propagation of sound in a random inhomogeneous medium is described by a wave
equation in which the sound velocity is a random function of coordinates and sometimes
of time. This is a complicated statistical problem whose solution can only be obtained
by approximate methods [14, 57]. This type of statistical analysis is certainly relevant in
the context of phase conjugation, where some degradation in the focusing ability occurs
if the medium changes [23, 24]. However, such results are useful mainly in assessing
possible applications of phase-conjugate arrays, as too many parameters are unknown in
a real situation to incorporate physically-motivated stochastic propagation models into
signal processing algorithms. Moreover, in the digital transmission problems addressed
here it is assumed that phase conjugation occurs over periods of only a few seconds, in
which case the assumption of a frozen environment seems plausible. In fact, statistical
characterizations of fluctuations over such short time spans seem to be unavailable in the
technical literature.
For the reasons mentioned above, deterministic propagation models were favored in
this work. Receiver algorithms deal with dynamical changes in acoustic waveforms as
unstructured fluctuations that can be compensated with general-purpose signal processing
structures developed for other telecommunications applications [72, 91].
2.2 Selection of Modeling Tools
The structure of underwater modems that are currently used in the ocean, either for re-
search or in actual telemetry applications, is almost invariably based on black-box models
that have emerged from work on generic system identification [74]. Prior knowledge about
the environment or direct channel measurements are used only to define design parameters
such as the length of adaptive filters or their tracking bandwidth, the number of points
in signal constellations, the number and spacing of carriers in multi-tone systems or the
symbol rate [50]. It should be acknowledged that more sophisticated examples exist where
physical considerations play a greater role in defining the structure of the receiver. In [26],
for example, on-line measurements of Doppler shifts at the receiver are used for coefficient
allocation and initialization in a Doppler compensation structure. Radical changes to this
situation seem unlikely in the foreseeable future, but clear improvements in reliability
2.2 Selection of Modeling Tools 25
are needed before underwater modems gain widespread acceptance. The need for greater
robustness is even more obvious in coherent systems designed for high-speed transmis-
sion, whose performance in field experiments is highly dependent on the stability of the
multipath profile.
In this work, an attempt was made to incorporate a priori environmental information
in a more explicit way through propagation models. Great care must be exercised when
following this path for two main reasons.
Environmental Uncertainty Development of the propagation models described in
Section 2.1.1 has been motivated by problems in underwater acoustics involving low and
medium frequencies, seldom greater than a few hundred Hz. As the acoustic wavelengths
are large when compared with features of the environment such as fine-scale bottom
bathymetry, surface roughness and small-scale variations in sound velocity, these are ef-
fectively “lowpass filtered” out of the resulting pressure field. Pressure values are then
determined mostly by large-scale features that can be identified with reasonable accuracy
from carefully conducted environmental surveys. Under those circumstances, it makes
sense to use full-wave models when analyzing data, as good matching accuracy is poten-
tially possible.
The situation is quite different at telemetry frequencies in the tens of kHz, where wave-
lengths on the order of 10−1m imply that small-scale features, which cannot reallistically
be surveyed, have a significant impact on the acoustic field. Given the small wavelengths
involved, temporal fluctuations induced by source/receiver motion and medium variability
can cause large variations in the phase of bandpass acoustic signals, thus placing stringent
performance requirements on phase tracking subsystems used in coherent communication
links. Modeling those variations is important to understand adaptation issues in digital
receivers, but it seems unlikely that our understanding of such phenomena will ever reach
a point where useful predictions can be made in operational scenarios.
The problem of unmodeled phase jitter at telemetry frequencies places fundamental
limits on the accuracy of pressure fields that can be simulated with any propagation model.
The channel transfer functions computed with those models typically assume full spatial
and temporal coherence among all the rays or modes that are excited by the source, which
is inappropriate in the present context.
Computational Complexity Even if there existed strong reasons to favor one of the
models of Section 2.1.1 over the others in terms of accuracy, computational complexity
issues must be considered when choosing which one should be incorporated into signal
processing algorithms used at the transmitter or receiver.
Regarding normal mode models, a rough estimate of the required number of modes
needed to compute the field can be based on the assumption of constant sound speed and
perfectly-rigid bottom. Then the mode shapes are sinusoids with vertical wavenumbers
kzn = (n + 1/2)π/H, where H is the bottom depth [59]. Propagating modes have real
26 Phase-Conjugate Arrays
horizontal wavenumbers krn =√k2 − k2zn , and for c = 1500 ms−1, f = 10 kHz, and
H = 100 m, this leads to approximately 1.7 × 103 modes. Taking into account the fact
that field computations must be done at a relatively large set of frequencies within the
signal bandwidth to properly estimate the medium transfer function, the computational
power that would be needed for near real-time operation is found to be several orders of
magnitude larger than the one that could reasonably be expected in underwater equipment
using present-day technology.
Parabolic equation methods also lead to intensive computations as the acoustic field
must be propagated with range steps that are a fraction of the wavelength. For a plausible
situation where transmitter and receiver are separated by 1km in range, or about 6.7×103
wavelengths at 10 kHz, tens of thousands of steps are required at each frequency if the
initial field estimates are calculated near the source.
The complexity of ray methods, on the other hand, scales with frequency in a much
more favorable way. By itself, the computation of ray trajectories and attenuations, as
described by (2.7) and (2.8), does not depend on the source frequency. In field compu-
tations using the Gaussian beam approach of [87] the relevant ocean cross-section must
be densely covered with beams to ensure that pressure values are accurate. As the width
of those beams decreases with frequency, their number must increase accordingly within
the source angular range. Large numbers of beams at high frequency (say, 500 beams
with uniformly-spaced departure angles between ±15◦ at 10 kHz) are needed to model
micro-multipath due to small-scale changes in the refraction index and to account for
boundary reflection and scattering that can rapidly lead to intricate and highly irregular
ray patterns.
Based on these considerations, ray methods were deemed as best suited for modeling
the propagation of acoustic telemetry waveforms with reasonable complexity. The de-
lay/attenuation values that are generated by such codes can be linked in a very intuitive
and direct way to the practical experience of most people who have been exposed to
the subject of channel identification for telecommunications. For that reason ray data
seems to be better suited for incorporating time variability for simulation purposes, as a
post-processing step before actual signals are generated.
Acoustic rays have a strong physical significance at high frequencies. Small variations
in environmental parameters may drastically change the pressure values at any given
point in the ocean, but the structure of propagation paths that forms the “skeleton” of
the acoustic field remains relatively unaffected. In terms of the parameters defined in
Section 2.1.4, this means that ray trajectories and propagation delays are robust with
respect to modeling uncertainties, whereas path attenuations are not. In other words, this
work assumes that the shape of wavefronts — the loci of points with constant propagation
delay associated with a family of rays that undergo the same number of surface and bottom
reflections — can be reasonably well modeled. In contrast, attenuation values cannot be
predicted due to environmental uncertainty.
2.2 Selection of Modeling Tools 27
As a result of this restricted viewpoint, where propagation models are simply regarded
as tools that generate a database of wavefront shapes for a given description of the envi-
ronment, it is possible to relax the requirement of dense beam coverage that guarantees
small errors in pressure computations. This will hide fine effects such as micro-multipath,
but such phenomena cannot be reliably modeled and are best handled by general-purpose
adaptive compensation schemes at the transmitter or receiver. Reducing the number of
beams further simplifies real-time implementations, where the wavefront database can
be dynamically regenerated to reflect changes in estimated environmental parameters or
transmitter/receiver positions. This approach provides a possible research path for tighter
integration of communication and navigation systems when one of the channel endpoints
is mobile.
Conventional beamforming techniques that rely on calibrated arrays are excluded by
the assumption of unknown path gains [80], although this should not be interpreted as
asserting that coherent spatial processing becomes impossible. In fact, time reversal is an
inherently coherent technique where the deterministic parameters used for spatial filtering
are derived directly from received data. But trying to estimate a set of high-level parame-
ters θ, such as directions of arrival, and then using them to generate a beampattern based
on steering vectors with known functional dependence on θ is considered unrealistic.
No beamforming information is derived from the propagation model itself, which only
provides guidance for detection of wavefronts and retrieval of amplitudes from measured
data. Although plain time-reversal is a fully self-contained wave focusing technique that
does not require individual wavefronts to be manipulated, doing so enables interesting ex-
tensions described in later chapters. The practical feasibility of such detection/extraction
schemes critically depends on the presence of clear wavefronts in the received data. Sev-
eral reports in the technical literature describe field experiments where sparse impulse
responses have been observed in underwater communication channels, although almost
invariably too few sensors are used to inequivocally claim that distinguishable wavefronts
are present [31, 32]. While the validity of that crucial hypothesis lacks definitive con-
firmation in underwater environments, it should be remarked that other experiments on
time reversal for biomedical applications using ultrasonic arrays consistently reveal the
existence of multiple wavefronts [122].
Although common, sparsity is not the general rule in underwater channels. Measured
impulse responses sometimes have an almost continuous nature even at relatively short
ranges of a few km, usually attributed to diffuse reverberation in the ocean. In principle,
processing of wavefronts would be most problematic under those circumstances and lead to
poor focusing results. In fact, it doesn’t even seem clear whether such impulse responses
posess the necessary temporal stability for successful operation of a plain time-reversal
mirror.
28 Phase-Conjugate Arrays
2.2.1 Generation of Simulated Data
The primary tool used to calculate path delays and attenuations in subsequent simulations
is the Bellhop Gaussian beam ray tracer [87]. Given a source position and environment de-
scription, the propagation code determines for each specified range/depth a set of eigenray
delays τi and attenuations fi such that the passband impulse response is
g(t) =∑
i
fiδ(t− τi) . (2.11)
For analytical convenience, bandpass signals will be represented by complex envelopes
throughout this work. Given a real transmitted signal x(t) = Re{x(t)ejωct
}, where ωc =
2πfc is the carrier frequency, the complex envelope of the received signal at the channel
output is given by the convolution [91]
y(t) = x(t) ∗ g(t) (2.12)
g(t) =∑
i
fiδ(t− τi) , fi∆= fie
−jωcτi . (2.13)
Total coherence among rays is implicit in this expression. To simulate time variability and
coherence loss it is possible to post-process the ray tracer data and generate time-varying
attenuations [26]
y(t) =∑
i
fi(t)x(t− τi) (2.14)
fi(t) = fi[1 + si(t)
]ejνit . (2.15)
In (2.15) νi is a deterministic Doppler shift due to transmitter or receiver motion, while
si(t) is a sample of a Gaussian random process that models fluctuations associated with
the moving ocean surface and time-varying interference of rays in micro-multipaths. In
agreement with the previous discussion on the relative modeling accuracy of path delays
and attenuations, no variations are introduced in τi. More details on surface and bottom
reflections are given in Appendix A.
In addition to delays and attenuations, it is possible to compute departure, arrival and
reflection angles of rays from the output of the propagation code. This information is
very useful for simulating Doppler shifts during post-processing, and also for generating
benchmarking data with ideally-separated wavefronts.
2.3 Principles of Time-Reversed Acoustics
A long-standing problem in acoustical or optical wave propagation is related to wave-
front or phase distortion correction [16]. Phase conjugation originated in optics, and has
had a significant conceptual and practical impact precisely due to its ability to focus
monochromatic waves with very good accuracy even on poorly-characterized media [58].
The generalization of this technique to broadband signals, commonly known as time re-
versal, finds applications in ultrasonic materials testing (nondestructive flaw detection in
2.3 Principles of Time-Reversed Acoustics 29
solids) and biomedicine (tumor treatment by hyperthermia, destruction of kidney stones),
among others [29].
The devices that perform phase conjugation (or time-reversal) behave as retroreflective
mirrors, as they direct light or sound originating from a source back to that source. A
time-reversed replica of the original field is produced by the mirror, propagating in a recip-
rocal manner and focusing on the original source location even when the medium includes
complicated and unknown inhomogeneities. That property is highly relevant in under-
water acoustics, as refraction due to oceanic structures ranging in scale from centimeters
to kilometers has strong and undesirable effects on propagation [58]. Phase conjugation
offers a means for compensating these effects, as well as unknown array deformations, and
may therefore provide novel solutions to several problems associated with active trans-
mission and propagation in the ocean. In addition to underwater communications, the
high directivity of phase-conjugate mirrors may be instrumental in developing low-power
transponders and high-precision sonar systems [58].
Mathematically, the property of time-reversal stems from the fact that the wave equa-
tion only contains second-order time derivatives. If yt(r) is a solution of
[∇2 − c−2(r)
∂2
∂t2]yt(r) = xt(r) , (2.16)
for a volume source xt(r), then y−t(r) satisfies (2.16) for the time-reversed source x−t(r)
[16]. The time-reversed solution describes a pressure field that converges on the original
source location, and therefore generating y−t(r) across the whole volume is an optimal,
albeit highly impractical, way of focusing on the source without knowing its position. This
shows that phase conjugation (which is the frequency-domain equivalent of time reversal)
not only concentrates energy at the source location, but actually regenerates the original
waveforms. The implications for digital communications are obvious, as it provides a
transparent way of undoing the effects of multipath.
As imposing a time-reversed field on a volume is practically unfeasible, more realistic
approaches must consider 1D or 2D surfaces. In fact, Huygens’ principle provides justifica-
tion for reducing the time-reversal operation on a volume to a closed surface surrounding
it. While still highly idealized, much insight into the operation of time-reversal mirrors
can be gained by considering this kind of closed cavity. The major premise in Huygens’
principle states that, in order to determine the effect at time t1 of a phenomenon caused
by a given disturbance at t0 < t1, one may calculate the state at some intermediate instant
t′ and from that deduce the state at t1 [5]. A propagating wave is therefore treated as a
superposition of wavelets reradiated from a fictitious surface with amplitudes proportional
to the original amplitude. The mathematical statement of this principle is Helmholtz’s
integral, which relates the pressure field at a given point with an integral involving the
field values and their gradients on a surface [5, 111]. In terms of the Fourier transform of
yt(r) in (2.16), this is written as [58]
Yω(r) =
∫
S
[Gω(r, r
′)∇TYω(r′)− Yω(r
′)∇TGω(r, r′)]· dS′ , (2.17)
30 Phase-Conjugate Arrays
where the Green’s function Gω(r, r′) satisfies (2.1). Point r is outside the surface S, which
encloses all the sources, and the element dS′ points outward. The expression for r inside
the surface involves an additional integral over the volume sources [59]
Yω(r) =
∫
S
[Gω(r, r
′)∇TYω(r′)−Yω(r
′)∇TGω(r, r′)]·dS′−
∫
VXω(r
′)Gω(r, r′) dV ′ . (2.18)
The derivation of (2.17) and (2.18) requires the assumption of reciprocity of the acoustic
field, in which case Green’s function satisfies [59]
ρ(r2)Gω(r1, r2) = ρ(r1)Gω(r2, r1) . (2.19)
This makes it possible to exchange the two spatial arguments in a medium with constant
density ρ.
In an ideal time-reversal cavity the surface S does not disturb the original propagating
wave, it simply records the field values and their gradients at all points. During the “play-
back” phase the surface acts as a source, imposing the phase-conjugated values Y ∗ω (r′),
∇Y ∗ω (r′) in such a way that the field at an arbitrary point is given by
Zω(r) =
∫
S
[Gω(r, r
′)∇HYω(r′)− Y ∗ω (r
′)∇TGω(r, r′)]· dS′ . (2.20)
Strategies for approximately obtaining (2.20) using radiating monopoles and dipoles are
discussed in [58]. Noting that the conjugated field Y ∗ω (r) satisfies the inhomogeneous
Helmholtz equation[∇2 + k2(r)
]Y ∗ω (r) = X∗
ω(r) , (2.21)
one can repeat the steps in the derivation of (2.18) [59]. This involves multiplying (2.21)
by Gω(r, r′) and (2.1) by Y ∗ω (r), subtracting, integrating over the volume enclosed by the
time-reversal surface, and invoking Green’s theorem to obtain
Zω(r) = Y ∗ω (r) +
∫
VX∗
ω(r′)Gω(r, r
′) dV ′ . (2.22)
In particular, the response to a point source Xω(r) = −δ(r−rs) implies Yω(r) = Gω(r, rs),
and Zω(r) then equals the space-time transfer function of the conjugated field, which is
denoted by Gcω(r, rs) to emphasize its close relationship to the medium Green’s function
Gcω(r, rs) = G∗ω(r, rs)−Gω(r, rs) . (2.23)
Remarkably, (2.23) is independent of the surface S, which implies that the field generated
by a closed time-reversal cavity is independent of its size, shape and location, as long
as it completely surrounds the source. The response to an arbitrary source can still be
calculated by the integral (2.2), with Gω replaced by Gcω.
If only the first term were present in (2.23), then the field created by the mirror would
exactly equal the conjugate of the original one, and perfect focusing would be obtained at
2.3 Principles of Time-Reversed Acoustics 31
the source location rs. As shown in [58, 16] if Gω(r, r′) approximately equals the free-space
Green’s function in the vicinity of the source
Gω(r, r′) ≈ G0ω(r− r′) =
ejk|r−r′|
4π|r− r′| , (2.24)
then
Gcω(r, rs) ≈ −jsin k|r− rs|2π|r− rs|
. (2.25)
Whereas the source field is an outward-propagating spherical wave with smoothly-decreas-
ing amplitude from infinity to zero, (2.25) is finite everywhere inside the cavity — singu-
larities are physically impossible because no sources are present in the volume during the
reciprocal phase — and not monotonous. In the time domain, the kernel (2.25) describes
the difference of two spherical waves that respectively converge to and diverge from rs.
This can be interpreted as interference between traveling waves originating from oppo-
site sides of the surface, resulting in a standing wave. The width of the main lobe in
(2.25) is one wavelength, which limits the maximum resolution that can be attained with
time-reversed focusing.
At the focus Gcω(rs, rs) = −jω/(2πc(rs)) is recognized as a differentiator in the fre-
quency domain. Upon evaluation of (2.2) with Gω replaced by Gcω and Xω by X∗ω, this
implies that the field at rs is proportional to the time derivative of the original transmitted
waveform, reversed in time
zt(rs) = −1
2πc(rs)
dx−t(rs)
dt. (2.26)
Differentiation is relatively unimportant for the type of passband signals that are used
in underwater communications, where the bandwidth is much smaller than the carrier
frequency and complex envelopes vary smoothly over time. For x(t) = Re{x(t)ejωct
}
dx(t)
dt= Re
{dx(t)
dtejωct + ωcx(t)e
j(ωct+π/2)}
, (2.27)
and the first term can be neglected, yielding the original envelope x(t) up to a complex
scaling factor that is transparently compensated at the receiver.
2.3.1 Time Reversal of Broadband Moving Sources
Moving point sources can be handled as particular instances of the general framework for
the ideal time-reversal cavity. The case of a transmitter with time dependence x(t) moving
at constant velocity along the trajectory r0 + vt will be of special interest. The source
density in the spatial and spectral domain is
Xω(r) =
∫ ∞
−∞xt(r)e
−jωt dt =
∫ ∞
−∞x(t)δ(r− r0 − vt)e−jωt dt . (2.28)
32 Phase-Conjugate Arrays
(a) (b)
Figure 2.1: Time reversal of a moving source (a) Forward transmission (b) Focusing alongthe time-reversed trajectory
Similarly to the static case (2.26), the broadband time-reversed field is obtained by super-
position using (2.23). In [58] this is shown to be
zt(r) =1
2π
∫
V
∫ ∞
−∞X∗
ω(r′)Gcω(r, r
′)ejωt dω dV ′ =x(−t+ τ−)− x(−t+ τ+)
R(−t)d(t) , (2.29)
where
τ± =γ2R(−t)
c
(−β cos θ−t ±
√
1− β2 sin2 θ−t
), β = |v|/c , (2.30)
d(t) = 4π
√
1− β2 sin2 θ−t , γ = 1/√
1− β2 . (2.31)
In (2.29) R(t) = |r− r0 − vt| is the distance between the source and field point at time t,
and θt is the angle between v and r − r0 − vt. Both the numerator and denominator of
(2.29) vanish when R(−t) = 0, i.e., along the time-reversed source trajectory, leading to a
sharply-peaked field. The limit of (2.29) along that trajectory is similar to (2.26), namely
zt(r0 − vt) = − γ2
2πc
dx(−t)dt
. (2.32)
Apart from the unimportant differentiation operation for passband communication signals,
(2.32) shows that the original waveform is still regenerated along the time-reversed focal
trajectory (Figure 2.1). As noted in [58], no Doppler shifts are apparent in (2.32) because
the observation point r0 − vt is in motion. At a fixed location, however, up and down
shifts become evident. The former can be identified with wavefronts originating from the
portion of the time-reversal cavity that the source is approaching, and the latter with the
opposite half.
2.3.2 Open Mirrors
Although the monopole/dipole approach of (2.20) leads to an elegant expression for the
time-reversed field that closely reflects the original transmission, simpler implementations
2.3 Principles of Time-Reversed Acoustics 33
may be required in practice. One possibility is to record the original field values and apply
them to monopole radiators
Gcω(r, rs) =
∫
SG∗ω(r
′, rs)Gω(r, r′) dS . (2.33)
Contrary to (2.23), this time-reversed field is not totally independent of the surface shape,
although the focusing performance may still be perfectly acceptable in many cases. Going
one step further, practical feasibility requires that the closed surface in (2.33) be replaced
by a 2D or 1D open array. As reasoned in [58], the performance of a continuous open
array is determined mostly by the shape and position of its boundary, rather than the
shape of the surface itself. Such tolerance to deformation is very appealing in the ocean,
as it enables the deployment of non-rigid arrays.
Evaluating (2.20) or (2.33) at r = rs, one concludes that the field amplitude at the
focus is proportional to the acoustic power intercepted by the array during the original
transmission [58]. This general result, also noted in [68], suggests that focusing is influ-
enced mainly by local properties of the medium in the vicinity of the mirror and the focal
point. As a useful guideline derived from this result, a practical array should be oriented
so as to maximize the amount of energy flowing through it. Increasing the array size is
only helpful up to a point where the structure captures most of the available power in the
sound field.
The focusing properties of an open mirror differ significantly from those of a closed
time-reversal cavity [58]. In the latter the field arrives from all directions, producing a
standing wave and a focal spot whose dimensions are on the order of a wavelength. In
open mirrors the field is a traveling wave, and the physical size of the focus is related to
the ratio of mirror dimensions to the water column depth and source-mirror distance. This
is confirmed in Section 2.4 and Appendix B for uniform discrete arrays in homogeneous
environments.
The principle of multipath compensation in the ocean using an open mirror is perhaps
most easily illustrated in terms of ray propagation. As a result of refraction and reflec-
tion, the acoustic signals received at the mirror arrive through several paths with different
attenuations and delays. When the waveforms are time reversed and retransmitted, the
reciprocal field retains the same spatial configuration, which means that a perfect mirror
transmits energy along the same directions of impinging rays. Time-reversal also implies
that the latest arrivals will be retransmitted first, and these delays are such that they
automatically yield simultaneous arrivals at the focus. As all contributions add up coher-
ently, the focused signal at this position is a nearly perfect time-reversed replica of the
original transmission. Multipath propagation is actually beneficial from this perspective,
as it allows more energy to reach the mirror, and a stronger focus results [92, 25]. The
analysis of Section 2.4 will confirm the validity of this argument for arrays of discrete
monopole transducers.
34 Phase-Conjugate Arrays
2.3.3 Discrete Arrays
Practical mirrors for ocean applications are implemented as arrays of discrete source/re-
ceiver transducers [68, 69, 25]. As the pressure field is only sampled at a finite set of
points, some degradation in field magnitude and sharpness near the focal point inevitably
occurs when compared with the continuous mirrors discussed previously. Although there
is no theoretical reason for preferring 1D, 2D or 3D sensor arrangements, practical consid-
erations seem to dictate the choice of 1D linear arrays in virtually all reported sea tests.
By contrast, 2D planar or non-planar arrays are common in ultrasonics experiments [29].
Using the monopole approach of (2.33), the field is given by
Gcω(r, rs) =M∑
m=1
G∗ω(rm, rs)Gω(r, rm) , (2.34)
where rm is the position vector of the m-th mirror sensor. General results regarding the
loss of focusing power and invariance to array deformations relative to continuous mirrors
are not available. However, some theoretical analyses of discrete arrays and experimental
evidence indicate that approximate invariance holds for many sensor configurations.
In [68] results from a time-reversal experiment in the ocean using a vertical linear array
are reported, and an asymptotic analysis of that mirror based on normal mode theory is
presented. Significant energy concentration and pulse compression were demonstrated
with a 455Hz source over a distance of more than 6 km, in a mildly range-dependent area
with a depth of about 100 m. This experiment proved that acoustic time reversal at low
frequencies could be accomplished in the sea, under radically different conditions from
ultrasonic laboratory tests. Measurements seem to indicate that the temporal stability of
phase conjugation is much longer than antecipated. In one of the experiments a single
50ms pulse was sent once from the source, and the time-reversed received waveforms were
then retransmitted every 10 s over a period of more than an hour. In spite of variations
in wave height and sound speed profile, a clear focus was observed at the source location
throughout the experiment. These observations agree with the theoretical perturbation
analysis developed in [68], which predicts that the focal spot is dominated by the mean
field, whereas fluctuations are diffuse and their effect becomes more apparent in other
areas of lower acoustic pressure. As long as the coherent part of the medium Green’s
function remains stable, focusing is preserved. An even more impressive experimental
demonstration of the long-term stability of time reversal is reported in [69], where it was
found that recorded probe pulses up to one week old still produced a significant focus at
the original source location. These experiments also confirmed theoretical results which
predict that the properties of the focal spot depend on the source depth, with best results
being obtained in areas where both the sound speed profile and the acoustic field are more
stable.
The robustness of time reversal to environmental fluctuations described in [68] can
also be intuitively justified in terms of ray propagation. The focal spot is created by
constructive interference of narrow acoustic beams that are configured by the mirror along
2.3 Principles of Time-Reversed Acoustics 35
the direction of incoming rays. Even though the mirror operates in nearfield beamforming
mode, these beams should still be present at the range of the source for moderate variations
of the propagation parameters. In a broadband context, the precise timing of signals sent
along these beams should not be significantly affected either, thus preserving the automatic
multipath compensation ability afforded by synchronous arrivals. Fluctuations are most
likely to result in phase variations among the beams, but even if coherence is completely
lost simple incoherent combination may be enough to create a strong focus.
It is interesting to note the analogy pointed out in [68] between matched-field processing
and phase conjugation. In the former several replica fields are synthetically created based
on a model of the environment for a set of unknown parameters, and matched to the
measured pressure by some form of correlation akin to (2.34) at r = rs. The optimal
parameter vector is chosen as the one that leads to the highest correlation. Formally,
phase conjugation is similar in the sense that the ocean itself generates a replica field
that is (hopefully) very accurately matched to the original one. Analyzing the effect of
environment variations and source/focus positioning errors in phase conjugation is similar
to the study of mismatch in matched-field processing.
Narrowband Performance in Dynamic Random Media Some authors have stud-
ied the effect of dynamic variations in the refractive index during the time that elapses
between the source-to-mirror transmission and the arrival of the time-reversed response
back at the source location [23, 64, 24]. These findings are based mainly on statistical
analysis of propagation in the presence of internal wave random refraction, and are di-
rectly applicable when a single propagation path links the source and the time-reversal
mirror. These formulations do not account for a sound channel that would cause determin-
istic multipath propagation, which somewhat limits their usefulness in typical underwater
communication scenarios. However, it is argued in [23] that this limitation is not as severe
as one might expect, as the statistical results can be extended to provide conservative
estimates in the presence of multipath by appropriately summing the contributions from
each deterministic path. This is essentially the same heuristic reasoning expressed above
regarding the robustness of time reversal.
Somewhat surprisingly, it is shown in [23] that a phase-conjugate array operating in
a static random medium should often be able to focus better than an identical array im-
mersed in a homogeneous medium. Random volume scattering creates virtual source/array
images that increase the effective array aperture, and hence the fraction of acoustic en-
ergy that it intercepts. Regarding dynamic behavior, internal wave-induced fluctuations
may actually aid focusing for short time delays on the order of one minute at ranges of
a few km by suppressing sidelobes [24]. Naturally, retrofocusing is degraded for longer
delays, as deeper changes in the refractive structure become apparent. Sharper focusing
is obtained at higher frequencies as more propagating modes become available (provided
that the array is dense enough to adequately sample them), but the tolerance to dynamic
variations decreases due to faster coherence loss in random scattering.
36 Phase-Conjugate Arrays
No results are available for telemetry frequencies of tens of kHz, but extrapolating
from the low-frequency predictions mentioned above it seems plausible that a coherence
period on the order of one minute could be attained at a range of 1 or 2 km. As data
packets typically last for less than 10 s, it should be possible to implement a physical
communication protocol that includes transmission of probe pulses to the mirror before
sending data in the reverse direction.
2.4 Focusing Performance of Discrete Arrays
While theoretical results such as those derived in [68, 23] are very useful for gaining
physical insight into the operation of time-reversal mirrors, they do not address issues
that are highly relevant for the design of practical arrays for communications, such as
the required number of sensors, array length, and sensor placement strategies. In this
section discrete arrays are analyzed from that perspective, so that design guidelines can
be extracted. The approach is based on ray propagation modeling, while similar results
are derived in Appendix B for normal modes.
2.4.1 Image Method
Suppose that a homogeneous layer is bounded by the free surface z = 0 above and by the
bottom z = H below. Although the Helmholtz equation (2.1) is satisfied by the free-space
Green’s function (2.24), that solution does not in general satisfy the boundary conditions
at the interfaces. To account for reflections, additional free-space terms must be added,
corresponding to virtual images of the original source reflected on the surface and bottom.
As a result, the total field can be written as an infinite series [14]
Gω(r, r′) =
∞∑
l=0
(−αB)l[
G0ω(r− r′l0)−G0ω(r− r′l1)− αBG0ω(r− r′l2) + αBG0ω(r− r′l3)]
=
∞∑
l=−∞
(−αB)|l|[
G0ω(r− r′l0)−G0ω(r− r′l1)]
,
(2.35)
where the surface reflection coefficient has a value of −1, the bottom reflection coefficient
is denoted by αB, and
r′l0 = (r′, z′ + 2Hl) , r′l1 = (r′,−z′ − 2Hl) ,
r′l2 = (r′, z′ − 2H(l + 1)) , r′l3 = (r′,−z′ + 2H(l + 1)) .
If αB = 1, (2.35) corresponds to a pressure-release surface G|z=0 = 0 and perfectly-
rigid bottom ∂G/∂z|z=H = 0. The straightforward generalization 0 ≤ αB ≤ 1 models
a partially-reflective bottom whose reflection coefficient is independent of the angle of
incidence. Figure 2.2 depicts the contributions of several images to (2.35).
When ray theory is used to model acoustic propagation, the impulse response between
r and r′ is approximated by a series of eigenray contributions as in (2.11). The complex
2.4 Focusing Performance of Discrete Arrays 37
PSfrag replacements
Surface image
Bottom image
Surface
Bottom
z′ − 2H
−z′ + 2H
−z′
z′H
r
z′00
z′01
z′02 (z′−10)
z′03 (z′−11)
z′10
z′11
Figure 2.2: Field computation by the image method
conjugate of the medium Green’s function is then given by the transfer function,
G∗ω(r, r′) =
∑
i
fie−jωτi . (2.36)
Direct comparison with (2.24) and (2.35) yields the ray parameters
flp = (−1)p (−αB)|l|
4π|r− r′lp|, p ∈ {0, 1} , τlp =
|r− r′lp|c
. (2.37)
As acoustic rays are straight in the homogeneous medium considered here, these delays
and attenuations can be determined graphically from Figure 2.2 based on the path lengths
and the number of surface and bottom interactions. In fact, the arrows in the figure can
be thought of as representing “straightened rays” whose incoming angles are inverted at
every reflection, as the path is traced back from r to r′. The same geometric procedure
could be used to expand r′ into a series of virtual images even when rays are bent due to
inhomogeneities in the medium. This viewpoint provides intuitive interpretations of the
beamforming behavior of time-reversal arrays, and will be used in subsequent sections.
2.4.2 Phase-Conjugate Field
Propagation from a source to each mirror transducer and back generates an intricate
pattern of eigenrays. Discarding contributions from rays that undergo more than NB
bottom reflections, the series (2.35) may then be written as
Gω(r, r′) =
[α
−α
]H[
G(0)0ω (r, r
′)
G(1)0ω (r, r
′)
]
, (2.38)
38 Phase-Conjugate Arrays
PSfrag replacements
Array
za
r − ra
Source
Surface
Bottom
H
(a)
PSfrag replacements Real (0, 0)
Virtual (0, 1)
Virtual (1, 0)
Virtual (−1, 1)
Virtual (−1, 0)
za
−za + 2H
−za
za − 2H Bottom image
Surface image
(b)
Figure 2.3: Array expansion (a) Waveguide propagation (b) Real/virtual images
where
α =
(−αB)|−NB |
...
(−αB)|NB |
, G
(p)0ω (r, r
′) =
G0ω(r− r′−NB ,p)...
G0ω(r− r′NB ,p)
, p ∈ {0, 1} . (2.39)
According to (2.34), the phase-conjugate field is now obtained as
Gcω(r, rs) =M∑
m=1
G∗ω(rm, rs)Gω(r, rm)
= αH[
B(0,0)ω (r, rs) + B(1,1)ω (r, rs)− 2Re{
B(0,1)ω (r, rs)}]
α .
(2.40)
By reciprocity, the spatial arguments in G(p)0ω (r, r
′) may be interchanged, yielding the
following expression for the beamforming matrices Bω in (2.40)
B(p,q)ω (r, r′) =M∑
m=1
G(p)0ω (r, rm)G
(q)H0ω (r′, rm) . (2.41)
At this point, clarity is gained if the array is expanded into a series of images, as shown
in Figure 2.3. Let ra = (ra, za) be a convenient reference point for the array. Coordinate
systems will be placed at all l, p images of ra, as defined in Section 2.4.1, and their z
axis oriented so that the coordinates of the associated image sensors are independent
2.4 Focusing Performance of Discrete Arrays 39
of l, p. Displacements in the l-th element[G(p)0ω (r, rm)
]
l= G0ω(r − rmlp
) will now be
expressed in frame l, p as G0ω(rlp− rm), where rlp will henceforth denote the new position
vector of the original field point r. Writing sensor coordinates homogeneously throughout
all reference frames will simplify the interpretation of the conjugated field in terms of
directivity functions.
To proceed, the source is assumed to be in the far field of each array image, so that a
plane wave approximation to the free-space Green’s function can be used [125]
|r− r′| ≈ |r| − 〈r′, r/|r|〉 , |r′| ¿ |r| (2.42)
G0ω(r− r′) ≈ ejk|r|
4π|r| e−jk〈r′,r〉 , r = r/|r| , (2.43)
where 〈·, ·〉 denotes the inner product of two vectors. Strictly speaking, acoustic mirrors
operate more effectively in the near field, but nonetheless the approximation (2.43) is
useful in understanding some of the spatial directivity issues involved. Each element of
the beamforming matrices (2.41) now has the form1
[B(p,q)ω (r, r′)
]
m,n= Cω(|rmp|, |r′nq|)Dω(rmp − r′nq) , −NB ≤ m,n ≤ NB , (2.44)
where
Cω(r, r′) =
ejk(r−r′)
(4π)2rr′, Dω(r) =
M∑
m=1
e−jk〈rm,r〉 . (2.45)
In (2.45), Dω is recognized as an array directivity function [125], although its argument
in (2.44) does not necessarily have unit norm. When rmp = r′nq each term in the sum is
equal to unity, and the contributions from all elements add in phase in this direction. In
other directions the terms are not in phase, and the field is smaller. This implies that,
when viewed as a function of rmp, the array is automatically phased to steer a beam in
the direction of r′nq regardless of the sensor positions rm. Beamforming is automatic even
if these elements have random, unknown positions.
The beamforming matrix element (2.44) accounts for the influence at rmp of a beam-
pattern steered towards rsnq , plus losses due to the geometrical spreading term Cω. Field
calculations then require evaluating the influence of each virtual (and real) array on all
images of the target ocean section. Figure 2.4 depicts the relevant directions for p = 1,
q = 0, m = −1 and n = −1. According to (2.40) the total field Gcω is obtained as a sum
over array images and target images, weighted by the reflection coefficients. If the array
aperture and sensor density are large enough so that Dω is narrow and has a single main
lobe, then near the range of the source one may ignore the beamforming matrix elements
where the endpoints of rmp and rsnq in Figure 2.4b are not on the same image of the ocean
cross-section. This includes B(0,1)ω and all off-diagonal terms in B
(0,0)ω and B
(1,1)ω . With
1With a minor abuse of notation, it is convenient to let row and column indices in beamforming matricesrun from −NB (top/left) to NB (bottom/right).
40 Phase-Conjugate Arrays
PSfrag replacements
r−1,1
rs−1,0
Virtual (−1, 1)
Virtual (−1, 0)
PSfrag replacements
r−1,1
rs−1,0
(a) (b)
Figure 2.4: Interpretation of beamforming matrix entries (a) Source and target directions(b) Spatial directivity
PSfrag replacements
Field point
Source
Figure 2.5: Phase conjugation as coherent sum of source-aligned beampatterns
that simplification the mirror is seen to operate in purely retrodirective mode, steering
beams in the directions of incoming rays during the first transmision
Gcω(r, rs) =∑
−NB≤l≤NB
p∈{0,1}
α2|l|B Cω(|rlp|, |rslp |)Dω(rlp − rslp) . (2.46)
Figure 2.5 illustrates how the field is obtained as a coherent sum of beampatterns steered
towards the source position. The large-scale envelope of the focal region is mostly de-
termined by the beampattern Dω, while the fine-scale structure results from interference
of beams, and is also strongly influenced by Cω. This figure shows in a compelling way
how phase conjugation takes advantage of surface and bottom reflections to implicitly
obtain an equivalent aperture that may be considerably larger than the physical mirror
size, yielding a stronger and sharper focus than in free space.
At r = rs all beams interfere constructively, generating large pressure values. At other
ranges the field will be weaker because (i) the beampatterns are no longer in phase and
(ii) the spherical term Cω in (2.44) becomes complex for |r| 6= |rs|, further degrading the
2.4 Focusing Performance of Discrete Arrays 41
constructive addition of energy that would be needed to focus the field.
As the source range increases the angular spread of those NB rays that effectively
contribute to the acoustic field decreases. As a result, differential phase variations in
the terms of (2.46) become smaller, and the transition from in-focus to blurred regions
becomes smoother. This effect will be characterized in more detail for a uniform linear
array.
2.4.3 Uniform Linear Vertical Arrays
For a uniform linear vertical array of total length L(M − 1)/M whose sensors are placed
at rm = (0, z0 +mL/M), m = 1, . . . M , the directivity function may be evaluated as
Dω(rlp − rslp) = ejkz′0(sinβlp−sin θlp) sincd
( kL
2M(sinβlp − sin θlp),M
)
, (2.47)
where z′0 = z0 + L/2(1 + 1/M) is the depth of the array middle point,
sincd(ω,N)∆=
sinωN
sinω= N
sinc(ωN/π)
sinc(ω/π)(2.48)
is the periodic discrete sinc function and βlp, θlp are the arrival angles at the l, p array
image associated with r and rs, respectively
rlp = (cosβlp, sinβlp) , rslp = (cos θlp, sin θlp) . (2.49)
To simplify the statistical analysis of Section 2.5 with randomly-spaced sensors, it will be
convenient to normalize depths as x = 2z/L and write (2.47) as
Dω(rlp − rslp) = ejx′0u sincd
( u
M,M
)
, u = kL
2(sinβlp − sin θlp) . (2.50)
Due to the choice of axis and ordering of layers βl0 ≈ βl1, θl0 ≈ θl1 for large l. As the level
of secondary sidelobes in (2.47) does not depend on M , a similar behavior is expected
in the vicinity of the focal point. Amplitude weighting (shading) could be used at the
array in transmit mode to obtain more desirable focusing properties [23], in which case
the time-reversed field (2.34) would be written as
Gcω(r, rs) =M∑
m=1
Wω(rm)G∗ω(rm, rs)Gω(r, rm) , (2.51)
where Wω represents a possibly frequency-dependent weighting function. This strategy
was not pursued in this work.
Focus Vertical Size When represented in cartesian coordinates the width of the main
lobe in directivity patterns changes as the steering angle is varied, becoming wider for
steeper angles approaching endfire [125]. It is also clear that, due to amplitude scaling by
the reflection factors in (2.46), the field near the focus will be dominated by the l = 0,
p = 0, 1 terms. It will then be assumed that, at range rs, the field variation in the
42 Phase-Conjugate Arrays
immediate vicinity of the focus depth zs follows2 the behavior of Dω, and is therefore sinc-
like. The steering angles for the real and surface-reflected arrays, θ00 and θ01, are typically
close to 0, hence the shape of the beampatterns will be very similar when represented as a
function of depth. These beampatterns are exactly in phase at (rs, zs), but as the depth z
moves away from the focus the two will start to interfere destructively, eventually creating
a deep null in the pressure field.
This effect is conceptually similar to the one described in [117], where horizontal passive
localization accuracy based on normal mode amplitude matching is interpreted in terms of
a so-called mode interference distance. This is defined as the displacement relative to the
nominal source range where the complex exponential factor in (2.5) leads to destructive
interference between the lower and higher-order (filtered) modes. Here, an equivalent role
is played by ejkz′0(sinβlp−sin θlp) in (2.47), with the l = 0, p = 0, 1 images acting as modes.
Not surprisingly, the general evolution of the pressure field in the vicinity of the focus
agrees with that of the indicator function of [117] around the true source range, both for
the results in this section and in Appendix B. Undesirable sidelobes appear by similar
constructive interference phenomena.
Consider a point at the range of the focus r = (rs, zs+∆z), with |∆z|, zs ¿ rs. Defining
εlp = βlp − θlp, the corresponding arrival angle at the real array is now approximated in
terms of θlp in the argument of Dω
sin ε00 = sin(
tan−1zs +∆z
rs− tan−1
zsrs
)
≈ rsr2s + z2s
∆z ≈ ∆z
rs(2.52)
sin(θ00 + ε00
)= sin θ00 cos ε00
︸ ︷︷ ︸
≈1
+cos θ00︸ ︷︷ ︸
≈1
sin ε00 ≈ sin θ00 +∆z
rs. (2.53)
The same argument can be repeated for the surface-reflected image, where the inverted
orientation of the depth axis leads to
sin(θ01 + ε01
)≈ sin θ01 −
∆z
rs. (2.54)
Taking argCω ≈ 0 and using (2.53)—(2.54) in (2.47), the contribution of these two terms
to the time-reversed field (2.46) is
2 cos(
kz′0∆z
rs
)
sincd( kL
2M
∆z
rs,M
)
. (2.55)
A pressure null will be created when the left factor vanishes
∆z = ± πrs2kz′0
→ z = zs ±λrs4z′0
. (2.56)
Based on this vertical displacement a farfield approximation to the position vector mag-
nitudes can be obtained and used in (2.45) to conclude that the argument of the complex
exponential in Cω is indeed close to 0.
2This simplification ignores the loss of coherence due to Cω, which is reasonable for |rlp| ≈ |rslp|.
2.4 Focusing Performance of Discrete Arrays 43
If the array is close to the surface, so that z0 = 0, a pressure null will be created when
∆z = ±rsλ/(2L), which is half the width of the main lobe for the individual directivity
patterns Dω. This is natural, as the 0, 0 and 0, 1 images jointly act as an array with
double the original length. The familiar dependence of lobe width with the array-size to
wavelength ratio is observed in (2.56) [125]. The vertical extent of the focus also widens
with increasing source-mirror range, rs, and its amplitude decreases due to Cω, which
is consistent with the smoother transition from focused to blurred zones, as discussed
previously. A similar dependence of focal spot dimensions on environment and array
parameters is observed in expressions (B.17) and (B.24), derived in Appendix B using
modal analysis.
The beampatterns may exhibit grating lobes if the intersensor separation is larger than
half a wavelength. In terms of the graphical representation of Figure 2.4, this means that
a beampattern in layer mp may generate large pressure values for image nq of the field
point. The physical interpretation is that, in addition to the directions of incoming rays,
the mirror transmits the desired signal along a set of rays that result from spatial aliasing.
Waveforms propagating along these spurious paths lack the precise timing that ensures
simultaneous arrivals at the focus, thus increasing the residual multipath. That is not a
major concern as long as the sensor spacing does not exceed a few wavelengths, because
then there is a large angular separation between the main (desired) lobe and the grating
lobes such that∣∣|m| − |n|
∣∣ À 1. Transmitted rays that are spatially aliased to low-order
layers are associated with incoming rays at steep angles, whose amplitude is very small.
Reciprocally, strong incoming paths may only generate aliased rays at steep angles, which
are greatly attenuated at the focus due to multiple reflections. Even if significant secondary
lobes do exist in the water column, the kind of constructive interference among rays that
creates the main focal point will not be observed. Experimental studies have shown that
the time-reversal mirror is surprisingly effective even when the sensor separation is on the
order of ten wavelengths [69].
As a numerical example, Figure 2.6 shows the contribution of the real array and its
surface-reflected image as a function of depth, as well as the total acoustic pressure calcu-
lated with (2.40) for rs = 5km, zs = 40m, f = 4kHz, c = 1500ms−1, H = 100m, αB = 0.3
and NB = 10. The array has 50 sensors spaced 2m appart, spanning depths 1, 3, . . . 99m.
The main lobe width is consistent with the value ∆z = ±10 m obtained from (2.56).
Intersymbol Interference Intersymbol interference in a discrete-time PAM signal used
for coherent signaling is a function of the discrete pulse shape h(n) at the receiver. The
following definition is commonly used [93]
ISI(h) =
∑+∞n=−∞|h(n)|2 − |h(n)|2max
|h(n)|2max. (2.57)
Although (2.57) is not easily expressed in the frequency domain, it is related to the flatness
of the channel impulse response that distorts the transmitted ISI-free pulse shape. Assum-
ing a continuous PAM received signal with symbol interval Tb [90], yc(t) =∑
k a(k)hc(t−
44 Phase-Conjugate Arrays
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4x 10
−6
Depth (m)
|P|
Direct ReflectedSum
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5x 10
−6
Depth (m)
|P|
(a) (b)
Figure 2.6: Numerical simulation of pressure fields (a) Contribution of real array andsurface-reflected image (b) Total
0 10 20 30 40 50 60 70 80 90 100−40
−30
−20
−10
0
10
Depth (m)
ISI (
dB)
Figure 2.7: Intersymbol interference as a function of depth
kTb), the discrete signal is obtained by sampling at instants t0 + nTb, where the fixed
sampling offset t0 is arbitrary. This results in an equivalent discrete PAM pulse shape
h(n) = hc(t0 + nTb).
Figure 2.7 shows a numerical example where the signal hc(t) is the convolution of the
ocean impulse response, obtained from (2.40) by inverse Fourier synthesis, and a raised-
cosine pulse with rate 1/Tb = 500 baud, rolloff factor β = 100%, and carrier frequency
fc = 4 kHz. The time offset t0 is selected at each point (r, z) from a set of 8 possible
values so as to minimize the ISI measure (2.57). The evolution of ISI with depth confirms
that energy focusing significantly reduces the effects of multipath in a broadband context.
The frequency fluctuations in the ocean transfer function are similar in magnitude for all
depths, but they are superimposed on a much larger average amplitude near the focus,
leading to larger |h(n)|2max and dramatically reducing the ISI measure. The issue of ISI
compensation will be much more thoroughly addressed in subsequent chapters.
2.5 Sparse Time-Reversal Arrays
For conventional array designs where all elements are spaced uniformly, there exists an
upper limit to the intersensor separation if grating lobes are not permitted to appear in the
visible region [75]. As a result, the required number of transducers becomes unreasonably
high if very narrow beamwidths — hence large apertures — are desired. As discussed
in Section 2.4, that issue is very relevant in time-reversal arrays, where grating lobes
can destroy the retroreflective property and lead to an increase in residual intersymbol
interference at the focus.
2.5 Sparse Time-Reversal Arrays 45
Fortunately, the visible region is quite narrow in the scenarios of interest in underwater
communications, i.e., if grating lobes are separated from the main beampattern lobe by
more than a few degrees, then their impact at the focus will be negligible because acous-
tic energy propagating along these spurious directions will be greatly attenuated due to
multiple reflections. It is then possible to use intersensor separations on the order of ten
wavelengths — or about 0.5–1.5m at telemetry frequencies of 10–30kHz — and still obtain
acceptable focusing. Even with such sparse separation the number of elements required in
an array spanning a significant fraction of the water column in coastal areas, where depths
greater than 100 m are common, may be very large. Building these arrays is technically
challenging, particularly because transducers designed to operate in both transmit and
receive mode are larger than simple hydrophones and require high-power electronics to
drive them.
Several authors have shown that nonuniformly-spaced arrays can achieve the same
resolution of uniform arrays of comparable size, but with a significant reduction in the
number of sensors [75, 18]. As in much of the array processing literature, recent research
on nonuniform arrays has been mainly devoted to the problem of direction of arrival
estimation. Specifically, most of the published work is concerned either with the derivation
of performance bounds in (usually linear) arrays with arbitrary sensor distribution (see
[18] and references therein), or the development of optimal/effective sensor placement
strategies from the point of view of estimation accuracy [84, 1, 34, 71]. Almost invariably,
the proposed methods rely on narrowband assumptions and calibrated arrays, making
them unsuitable for underwater applications in the present context.
Contrary to the deterministic sensor placement framework that is adopted in the refer-
ences mentioned above, the approach followed in [75] emphasizes random element spacing
and seems to be better suited for flexible underwater arrays. This statistical approach was
developed primarily for very large arrays used in astronomy — results in [75] are illustrated
for 102 to 105 sensors —, where nonuniform spacing can dramatically reduce the number
of elements by several orders of magnitude. From that perspective time-reversal mirrors
have comparably few sensors, and are therefore expected to benefit only moderately from
random spacing.
Specifically, [75] studies the probabilistic properties of an antenna array when its el-
ements are placed at random over an aperture according to a given distribution. The
results turn out to be very general, and applicable to any particular member of a large
class of useful distribution functions. Once the conditons for obtaining the desired antenna
performance with large probability are found, actual element spacings can be determined
by the Monte Carlo method. Designing the array then amounts to playing a game in
which the odds in favor of success are judged to be sufficiently large before any actual
evaluation of sensor positions is attempted [75]. Interestingly, [75] emphasizes that the
problem of beampattern synthesis is a deterministic one, and the statistical approach is
merely a useful tool that circumvents the lack of effective optimization methods. While
recent advances in convex optimization theory have made it possible to effectively compute
46 Phase-Conjugate Arrays
optimal or near-optimal solutions to the original problem [71], the probabilistic framework
still seems appealing in a random propagation medium.
2.5.1 Properties of Random Beampatterns
For a vertical linear array of length L with M sensors positioned at rm = (0, zm) the
free-space beampattern (2.45), phased towards a particular angle θ, is written as
Dω(u) =M∑
m=1
ejuxm , xm =2
Lzm , u = k
L
2(sinβ − sin θ) . (2.58)
The reference frame for the array is chosen so that the normalized sensor depths satisfy
|xm| ≤ 1. In the probabilistic framework {xm} are assumed to be independent random
variables with a common pdf p(x) and characteristic function φ(u) = E{ejux} such that
p(x) = 0 , |x| > 1 ,
∫ 1
−1p(x) dx = 1 . (2.59)
It follows from the definition of characteristic functions that the mean beampattern in
(2.58) is identical to that which would be obtained by taking p(x) as a continuous aperture
excitation
E{Dω(u)} =M∑
m=1
E{ejux} = Mφ(u) . (2.60)
Given the Fourier transform relation between p(x) and φ(u), choosing a suitable pdf that
(approximately) induces a desired beampattern φ(u) is relatively simple. In most cases of
practical interest φ(u) is real, hence the function p(x) has even symmetry. The following
main points should be highlighted in the statistical characterization of the beampattern
[75]:
1. Finite support of p(x) implies that the sidelobe maxima of φ(u) decrease monotoni-
cally, in contrast with uniform arrays, where the beampattern is a periodic function
of u.
2. The real and imaginary parts of Dω(u) are asymptotically independent and jointly
normal, while |Dω(u)| has a Rice distribution. For large values of M it is practically
certain that Dω(u) approximately equals φ(u) for any u. However, pointwise con-
vergence to φ(u) does not guarantee that the beampattern response as a whole is
satisfactory. To study the global properties of |Dω(u)| the directivity function must
be modeled as a random process.
3. For a given pdf the required number of sensors is directly related to the sidelobe
level and, to a much lesser degree, to the aperture dimension or the particular form
of p(x). One can improve upon the resolution of a uniform antenna by several orders
of magnitude if the sensors are allowed to be nonuniformly spread over an aperture
10 or 100 times larger than the original one. As long as M is sufficiently high, the
risk of obtaining a much higher sidelobe level is very slight.
2.5 Sparse Time-Reversal Arrays 47
4. The half-power beamwidth of a beampattern is defined as the smallest positive root
of
|Dω(u0)| = M/√2 .
The distribution of u0 is highly concentrated around the half-power beamwidth of
φ(u), therefore the resolution of a randomly-spaced array will almost surely be very
close to that of an array with continuous excitation p(x).
5. The squared norm of the deviation ofDω(u) from the desired pattern across a bearing
interval U is defined as
|Dω − φ|2 =∫
U|Dω(u)− φ(u)|2 du . (2.61)
For arrays with very large apertures and almost any pdf of practical interest the
probability Pr{|Dω−φ| < k|φ|} has a threshold behavior when viewed as a function
of k. For average sensor spacing d = L/M and |p|2 =∫ 1−1|p(x)|2 dx, that probability
is nearly one for k2 > d/|p|2 and zero otherwise.
6. With high probability the loss in directivity index of Dω(u) relative to the desired
pattern with continuous excitation φ(u) does not exceed 20 log(1 +√
d/|p|2) dB.The directivity loss is much lower when referred to a uniform discrete array with
amplitude tapering. This quantity is approximately proportional to the number of
sensors.
2.5.2 Random Phase-Conjugate Arrays
As the elements of the beamforming matrices (2.44) only depend on the sensor positions
through the directivity function Dω, (2.58) can be readily applied when the array is linear
and vertical
E{[
B(p,q)ω (r, r′)]
m,n
}= Cω(|rmp|, |r′nq|)Mφ(u) , u = k
L
2(sinβmp − sin θnq) . (2.62)
It is then clear that the mean conjugated field Gcω = E{Gcω} is still given by (2.40), with
(2.44) replaced by (2.62). As in (2.60), the mean field is identical to the one that would
be created by a continuous aperture with excitation p(x), which excludes the existence
of grating lobes. Naturally, the latter can be present when the pdf is used to generate a
specific realization of M discrete sensor depths. The free-space beampattern φ(u) induced
by p(x) should ensure that the conditions for retroreflective operation depicted in Figure
2.5 are met, so that the average field can be written similarly to (2.46)
Gcω(r, rs) =∑
−NB≤l≤NB
p∈{0,1}
α2|l|B Cω(|rlp|, |rslp |)Mφ(u) , u = k
L
2(sinβlp − sin θlp) . (2.63)
48 Phase-Conjugate Arrays
The variance of the phase-conjugate field is given by
σ2(r, rs) = E{|Gcω(r, rs)− Gcω(r, rs)|2
}
= E{∣∣αH
[∆B(0,0)ω (r, rs) + ∆B(1,1)ω (r, rs)− 2Re
{∆B(0,1)ω (r, rs)
}]α∣∣2}
=∑
p,q,u,v∈{0,1}
(−1)(p−q)−(u−v)αHE{∆B(p,q)ω (r, rs)αα
H∆B(u,v) ∗ω (r, rs)}α ,
(2.64)
where
[∆B(p,q)ω (r, r′)
]
m,n=
[B(p,q)ω (r, r′)
]
m,n− E
{[B(p,q)ω (r, r′)
]
m,n
}
= Cω(|rmp|, |r′nq|)(Dω(u)−Mφ(u)
),
(2.65)
and u is the same argument of (2.62). The expected product of terms E{∆B(p,q)m,n ∆B(u,v) ∗i,l }
in (2.64) involves the free-space covariance function E{(D(u)−Mφ(u)
)(D(v)−Mφ(v)
)∗}.
Due to the i.i.d. assumption on sensor positions
E{Dω(u)D∗ω(v)} =
M∑
m=1
E{ejxm(u−v)}+∑
1≤m,n≤Mm6=n
E{ejxmu}E{ejxnv}∗
= Mφ(u− v) +M(M − 1)φ(u)φ∗(v) ,
(2.66)
hence
E{(D(u)−Mφ(u)
)(D(v)−Mφ(v)
)∗}= E{Dω(u)D
∗ω(v)} −M2φ(u)φ∗(v)
= M(φ(u− v)− φ(u)φ∗(v)
).
(2.67)
The most relevant point to note here is that (2.67) depends linearly on the number of
sensors M , and the same will be true for E{∆B(p,q)m,n ∆B(u,v) ∗i,l } and ultimately the field
covariance (2.64). This shows that, for a given physical configuration and placement pdf,
the variance of the normalized field Gcω/M decreases to zero with 1/M , as in free-space.
For large M , the response of individual mirror realizations will therefore be close to the
average pressure Gcω with high probability.
For sufficiently large u, v the second term in (2.67) can be neglected, and the crossco-
variance of beamforming matrix entries then depends mostly on argument differences
E{∆B(p,q)m,n ∆B(u,v) ∗i,l } ≈ e−jk
((|rmp|−|r′nq |)−(|riu|−|r
′lv |)
)
(4π)4|rmp||r′nq||riu||r′lv|
×Mφ(
kL
2
[(sinβmp − sin θnq)− (sinβiu − sin θlv)
])
.
(2.68)
Significant values in (2.68) will be obtained when sinβmp − sin θnq ≈ sinβiu − sin θlv,
or equivalently rmp − r′nq ≈ riu − r′lv. For given m,n, p, q this will happen when the
indices i, l, u, v define two pairs of vectors that occupy similar relative positions, i.e., the
number of virtual layers between rmp, r′nq and riu, r
′lv is the same (see also Figure 2.4). This
implies that E{∆B(p,q)m,n ∆B(p,q)ω } will only have non-negligible values along the subdiagonal
2.5 Sparse Time-Reversal Arrays 49
containing element m,n. In general, large values in E{∆B(p,q)m,n ∆B(u,v)ω } for p, q 6= u, v will
also occur along single subdiagonals.
Even when the approximation (2.68) is invoked to reduce the number of elements in the
sum (2.64) as discussed above, the resulting covariance expression is not simple and pro-
vides little insight into the factors affecting the variability of the phase-conjugate field. In
addition to the pointwise convergence property to Gcω that stems from (2.67), the follow-
ing heuristic argument helps to justify the plausibility of using randomly-spaced sensors
in a phase-conjugate array. As shown by (2.40), (2.44) and Figure 2.5, the conjugated
field is generated by a bank of parallel beampatterns placed at 4NB + 2 images of the
ocean cross-section and steered towards the physical source location rs. These beampat-
terns are identical when expressed in terms of the direction cosine u [125], although their
shapes differ in (r, z) coordinates due to differences in steering angles. Each column of the
beamforming matrices B(p,q)ω quantifies the effect of one of these beampatterns in (a subset
of) the images of the field point r under consideration. Calculating the contribution of
a specific beampattern at all depths for constant range is perhaps most easily visualized
as a folding operation; The beampattern is evaluated over the (subset of) layers as if it
were in free space, then weighted by the spreading term Cω and the appropriate reflection
coefficient at each depth, and finally folded onto the physical ocean cross-section.
The beamwidth and sidelobe level of Gcω are therefore intimately related to those
of the common directivity function Dω, in agreement with the analysis of Section 2.4.3
for the case of a uniform array. The results of [75] enumerated in Section 2.5.1 ensure
that, for large enough M , any realization of sensor depths will lead to Dω(u) ≈ Mφ(u),
the half-power beamwidth being determined primarily by the array aperture and the
sidelobe level by the number of sensors. Due to the absence of grating lobes in the
“visible range”, the folding operation generates a beampattern across the water column
with characteristics similar to those of Dω(u). Based on the number and attenuation of
layers, it is possible to establish conservative bounds on the secondary lobes of Dω(u) that
will ensure a pre-specified maximum sidelobe level in folded beampatterns. Similarly, the
total field (2.40) is formed by a weighted sum of (depth-synchronized) folded beampatterns,
whose beamwidths and sidelobe levels can be bounded to guarantee that Gcω meets a given
set of spatial directivity specifications.
Building on the free-space case of [75], the above argument can be used to establish
the uniform convergence of Gcω to Gcω with the same tradeoffs between resolution and
aperture size/number of sensors. Although very loose bounds on both Dω(u) and the
folded beampatterns suffice to obtain the desired convergence result, that approach does
not yield useful expressions for predicting the focusing performance with moderate values
of M . While the complexity of the expressions for the mean field and its covariance
prevented the derivation of useful analytical results, one could reasonably expect the phase-
conjugate field to be dominated by contributions from the real array and its surface-
reflected image. As shown in Figure 2.6 for a uniform array, these two beampatterns will
be almost identical in the cases of interest and differ little from the one generated by an
50 Phase-Conjugate Arrays
0 10 20 30 40 50 60 70 80 90 100
−140
−120
−100
−80
−60
−40
−20
z (m)
|P| (
dB)
Figure 2.8: Phase-conjugate field obtained with a uniform linear vertical array of constantlength with variable number of sensors
0 10 20 30 40 50 60 70 80 90 100−150
−100
−50
0
z (m)
|P| (
dB)
0 10 20 30 40 50 60 70 80 90 100−150
−100
−50
0
z (m)|P
| (dB
)
(a) (b)
Figure 2.9: Expected field with random spacing (a) Uniform pdf (b) Square-cosine pdf
array in free space, as long as the source is not located very close to the surface or bottom.
The results of [75] should then be applicable to characterize the envelope of Gcω as a
function of array parameters with reasonable accuracy. The fine structure of the pressure
field is determined by coherent interference between these two beampatterns, and in the
vicinity of the focal region depends mainly on the vertical distance between the two array
images, rather than the specific shape of Dω.
Numerical Results The performance of a time-reversal array with randomly-spaced
sensors is illustrated for a range-independent scenario with rs = 1.5 km, zs = 80 m,
f = 4kHz, c = 1500ms−1, H = 100m, αB = 0.3 and NB = 10. Both uniform and square-
cosine densities were considered as sensor placement strategies [75], with nonzero support
in the depth interval [1m, 99m]. Analytical expressions for the densities and characteristic
functions are given in Table 2.1. Figure 2.8 shows the deterministic phase-conjugate field
evaluated using (2.40) for uniform linear vertical arrays with M = 5, 10, 15, 20, 30, 50, 70,
and 100 sensors, evenly-spaced between 1 m and 99 m. The focusing effect is still clearly
visible when 50 sensors are used (5.3λ spacing), but becomes severely degraded for lower
values of M . Figure 2.9 shows the average time-reversed acoustic field for the densities
of Table 2.1, evaluated using (2.40) with the expected beamforming matrices (2.62). The
Table 2.1: Densities and characteristic functions
Uniform Square-Cosine
p(x), |x| < 1 12 cos2 πx
2φ(u) sinu
usinu
u[
1−(uπ )2]
2.6 Summary and Discussion 51
0 10 20 30 40 50 60 70 80 90 100−120
−100
−80
−60
−40
−20
z (m)
|P| (
dB)
0 10 20 30 40 50 60 70 80 90 100−120
−100
−80
−60
−40
−20
z (m)
|P| (
dB)
(a) (b)
Figure 2.10: Average field in Monte Carlo simulations (a) Uniform pdf (b) Square-cosinepdf
corresponding free-space responses φ(u) are also superimposed on these plots. Several
curves are shown for different values of M to simplify the comparison with Monte Carlo
simulations, although it is clear from the previous discussion that M only introduces a
gain in the mean field. These results confirm that the large-scale evolution of the field is
determined by φ(u), although the detailed behavior depends on the interference pattern
between array images. In particular, the acoustic field between the pressure nulls at 77 m
and 83 m is almost identical in Figures 2.8 and 2.9.
The time-reversed field of Figure 2.9b seems to be more suitable for coherent commu-
nication applications, as it creates a broader region of high acoustic energy around the
focus. As shown in Appendix B, the extent of the low ISI zone may be estimated by
considering the joint evolution of the acoustic field for the higher and lower frequencies
in PAM signaling pulses. From that perspective, concentrating energy in a broad main
lobe maximizes the region where field components within the signal bandwidth behave
coherently, leading to low spectral pulse distortion and mild ISI.
Figure 2.10 shows the average acoustic fields that were obtained for the previously
considered values ofM in 500 Monte Carlo simulations. The results are in good agreement
with the theoretical mean values of Figure 2.9, even for the lowest values of M . The
difference in residual sidelobe level may be partly attributed to model discrepancies, as
the curves of Figure 2.10 were obtained with the exact beamforming matrices (2.41),
whereas the ideal responses of Figure 2.9 were based on a plane wave approximation.
Naturally, individual time-reversed fields vary considerably, especially when few sensors
are used, but beampatterns with globally desirable features are obtained with reasonably
high probability for M > 30.
The same results of Figure 2.10 are represented in Figure 2.11 using a linear scale,
showing that the mean field does indeed increase linearly with M .
2.6 Summary and Discussion
This chapter focused on generic design issues for time-reversal mirrors, with particular
emphasis on the focusing power of discrete arrays. Before actually introducing the prin-
ciples of time-reversed acoustics, propagation models were briefly discussed to motivate
52 Phase-Conjugate Arrays
050
100
050
100
0
0.02
0.04
0.06
z (m)M
|P|
050
100
050
100
0
0.02
0.04
0.06
z (m)M
|P|
(a) (b)
Figure 2.11: Average field evolution with the number of sensors (a) Uniform pdf (b)Square-cosine pdf
the set of analytical tools adopted throughout this work. Although much progress has
been achieved in modeling acoustic propagation at low frequencies, it is reasoned that so-
phisticated models are of limited usefulness at frequencies of tens of kHz, where the main
factors affecting the accuracy of computed acoustic fields are due to intrinsically high a
priori relative uncertainties, rather than shortcomings in the mathematical description of
wave propagation. Ray models were deemed suitable in the context of this work, as they
are computationally effective and sufficiently precise at telemetry frequencies.
The theory of phase conjugation, or time reversal, was first described for an ideal
cavity, where very general and simple expressions can be obtained for the acoustic field.
The pressure at the focal spot is given by the time derivative of the original transmitted
signal, which essentially amounts to amplitude scaling for the kind of passband waveforms
used in digital communication. Even in the case of a moving source signals are focused and
regenerated along the time-reversed trajectory regardless of any (lossless) inhomogeneities
in the medium. This is a desirable property when operating in the ocean, where it is hard
to ensure that mobile nodes remain stationary throughout the communication process.
Linear arrays of discrete monopole transducers were subsequently considered as more
practical options to be used in ocean environments. A ray-based directivity analysis
was carried out for the case of a monochromatic source immersed in a simple medium
with constant sound speed, perfectly-reflective surface and partially-reflective bottom.
The time-reversed field was shown to exhibit a sinc-like dependence near the focus. A
similar result was obtained in Appendix B using modal analysis and a slightly different
environment model where mode shapes can be easily computed in closed form. The
vertical extent of the focal spot, defined as the distance between the two pressure nulls
surrounding the main lobe, depends directly on the wavelength and source-mirror range,
and inversely on the array length. This result is very similar to known expressions for array
beampatterns in free space, providing formal justification for applying well-established
engineering criteria when designing time-reversal arrays.
2.6 Summary and Discussion 53
Due to the image method used to account for ray reflections, the acoustic field in
the ocean cross-section is expressed as the result of coherent interference between the
beampatterns of several vertically-stacked virtual arrays, located at the reflected images
of the original mirror and steered towards the physical source. These beampatterns will
be narrow and nearly horizontal when the mirror operates over medium ranges in the
ocean, and under those conditions they will interfere to produce a focal spot that is
much more elongated horizontally than vertically. This is confirmed by the expression
for the horizontal focus size obtained with modal analysis, which additionally shows that
this quantity is proportional to the square of the vertical size for a given wavelength. The
parallel beampattern representation lends support to the notion that multipath is actually
beneficial for time-reversed focusing under static conditions, as it creates an equivalent
(amplitude-shaded) super-array that is longer than the real mirror.
Naturally, empirical beampatterns in the ocean may differ significantly from the ones
derived for simplified environments. The tests reported in [68, 69], for example, show that
strong secondary sidelobes are sometimes present in the water column at the source range.
Nevertheless, the general depth dependence of the conjugated pressure field is commensu-
rate with what would be expected from the analysis carried out in this chapter. Taking
into account the fact that virtually all phase conjugation experiments use vertical uniform
arrays, the lack of results on the horizontal extent of the focal spot are understandable,
as it would be nearly impossible to sample a potentially large vertical ocean cross-section
over a sufficiently short period to ensure that temporal variations in the pressure field are
negligible. However, one should mention that the disparity in horizontal versus vertical
size of the focus is suggested by other simulation studies [2, 24].
Having established a framework where the time-reversed field is expressed in terms
of free-space beampatterns, it seems natural to investigate whether the latter can be
synthesized in alternative ways, while preserving the overall interference pattern. Formally,
the set of mirror images can be understood as a large array where individual elements are
(virtual) linear apertures, such that the ensemble satisfies a kind of product theorem [125]
where the overall directivity function factors into terms that separately account for the
position of virtual images and their individual responses. Due to this decoupling property
the two effects can be studied almost independently, and any effective design strategy for
free-space arrays is, in principle, suitable for designing acoustic mirrors that operate in
ocean waveguides.
The technique that was proposed for reducing the number of sensors needed to gener-
ate a desired mirror response is based on the theory of linear arrays with randomly-spaced
elements. The approach itself is classic, and only provides dramatic savings for extremely
narrow apertures, where thousands of sensors would be required with uniform array ge-
ometries. This is certainly one of the topics that should be significantly improved in future
work, and is especially relevant as an enabling technique for applying the spatial modula-
tion concepts of Chapter 4 in practice, as very narrow directivity functions must be used.
In the case of plain (i.e., nonsegmented) mirrors there is no need for more than, say, 100
54 Phase-Conjugate Arrays
sensors, and while that number is in itself very ambitious for present-day technology, the
savings afforded by random spacing are somewhat scarce. In principle, it would seem more
promising to approach the design of nonuniform mirrors from a deterministic perspective,
formulating it as an optimization problem. However, given the nature of uncertainties in
steering vectors, robustness constraints would have to be incorporated, and it is currently
not clear how this could be achieved.
By contrast, the philosophy of random sensor placement is intuitively more appealing
in the presence of uncertainties. For a sufficiently large number of sensors in free space,
any specific realization will exhibit desirable features with high probability, such as main
lobe beamwidth, directive gain and sidelobe level. Considering the broadness of the class
of suitable sensor placement realizations, the directivity properties of any particular one
are not expected to change drastically due to perturbations in the propagation medium.
This conjecture was not proved, but served as a motivation for pursuing the probabilistic
design of nonuniform mirrors in inhomogeneous underwater environments.
Simulation results show that the sensor count can indeed be reduced in homogeneous
waveguides by as much as 30 to 40% relative to uniformly-spaced arrays with about 100
elements. The average beampatterns that were obtained are in good agreement with
expected mean values, showing the familiar tradeoff between energy concentration around
the main lobe and secondary sidelobe level at other depths as a function of the position pdf.
In fact, this pdf can be thought of as serving the same purpose of windows in conventional
filter designs. Superimposed on the smooth large-scale behavior of the pressure field,
more rapid variations are observed due to the interference pattern of virtual mirror images.
These depend only on the mirror location and waveguide structure, and are nearly identical
for any sensor distribution function. Due to the complex coupling between beampattern
images, it was not possible to obtain simple closed-form expressions for the statistical
properties of the total pressure field besides its mean value. Although it was empirically
verified that realizations with more than about 30 sensors lead to beampatterns that are
close to the average with acceptably high probability, only a limiting argument for dense
arrays was presented to justify this convergence.
From the point of view of communications, it makes more sense to examine focusing
in terms of residual intersymbol interference, rather than simple energy concentration at
a single frequency. Intersymbol interference is intrinsically a time-domain phenomenon,
and expressing the adopted ISI metric in the frequency domain proved to be unfeasible.
Consequently, it was not possible to analytically calculate the dimensions of the focal zone
in terms of ISI, as done for single-frequency beampatterns, although numerical values can
certainly be evaluated and plotted as a function of depth and range. Not surprisingly, the
evolution of ISI is very similar to that of the acoustic pressure for frequencies within the
signal bandwidth. In fact, the discrepancy between the beampatterns at the upper and
lower edges of the frequency band of interest provides a useful heuristic for estimating the
spatial distribution of ISI, as described in Appendix B.
In the case of nonuniform arrays, this suggests that it is preferable to choose a pdf that
2.6 Summary and Discussion 55
leads to a broad main lobe and lower pressure values at other depths, as this will tend to
maximize the effective (large-scale) size of the focal spot both in terms of ISI and SNR.
Notice that pressure nulls and ISI local maxima will still exist inside this broad region due
to the fine-scale interference pattern of mirror images. Actually, the interval delimited
by the first pressure null above and below the source depth is almost independent of the
chosen pdf, as can be seen from Figure 2.9. The issue of main lobe versus sidelobe level
is less relevant in the horizontal direction, where the spatial scale of pressure variations is
much larger.
56 Phase-Conjugate Arrays