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.


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


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


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.


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)


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


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)


(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


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.


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


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.


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)


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


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).


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


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


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|.


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−


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


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.


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)


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


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


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


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.


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


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


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.

Phase-Conjugate Arrays

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