wavefront segmentation - ulisboa

Chapter 4

Wavefront Segmentation

By transforming a severely multipath-distorted underwater channel into an equivalent

single-input single-output system with mild frequency selectivity, time reversal opens up

the possibility of using several modulation and coding methods that have been devel-

oped for flat-fading wireless channels. The physical process whereby a focal spot with

low multipath distortion emerges by coherent addition of path contributions seems some-

what redundant, in the sense that signals propagating through individual eigenrays are

themselves mildly-distorted replicas of the signal to be regenerated at the focus. Intu-

itively, one would expect an improvement in the spatial density of information if it were

possible to send different signals along the various eigenrays, thus creating a synchro-

nized multiuser communication scenario at the focus without introducing multipath. This

particular form of spatial modulation, termed wavefront segmentation, is the subject of

the present chapter. Methods are developed for detecting and extracting eigenray infor-

mation from distorted pulse shapes relying only on a few simple modeling hypotheses,

so that the crucial ability of time reversal to accurately beamform broadband signals in

poorly-characterized media is preserved.

According to the general results of Section 2.3.1 phase conjugation can automatically

cope with moving sources, which is of great practical interest for applications in commu-

nications involving mobile platforms. By resorting to delay-Doppler spread functions, the

system-theoretic concepts introduced in Chapter 3 are extended to incorporate motion-

induced Doppler. It is first verified that time reversal still produces multipath-free focusing

under this restricted analytical framework, as predicted by the general theory. Building on

this representation, a simple method is proposed to transparently compensate for Doppler

shifts at the focus, so that a uniformly-moving node experiences nearly Doppler-free re-

ception. The wavefront segmentation approach is adapted to handle moving sources, and

some simplifications are discussed to cope with the increase in computational complexity

relative to the static case. Interestingly, the availability of an additional Doppler dimen-

sion can actually simplify the task of segmenting eigenray data by eliminating wavefront

overlap when the source velocity vector induces distinguishable Doppler shifts in upward-

and downward-departing rays.

Before describing the wavefront segmentation approach for time-reversal arrays, generic

83

84 Wavefront Segmentation

spatial modulation and various channel decomposition strategies are briefly discussed.

4.1 Spatial Modulation in Underwater Channels

Propagation in underwater acoustic channels consistently occurs over multiple paths con-

necting the source and receiver. Unlike most terrestrial wireless channels, where path

gains are commonly viewed as random variables satisfying the WSSUS (Wide-Sense Sta-

tionary with Uncorrelated Scattering) assumption, underwater channels have a rich spatial

structure that can be exploited through spatial modulation, i.e., the controlled distribu-

tion of multiple communication signals through the available paths [65]. Given the severe

bandwidth constraints of underwater channels, taking advantage of the additional spa-

tial dimension when designing communication systems can potentially lead to significant

performance enhancements. In fact, the availability of several resolvable paths can be

interpreted as additional spatial bandwidth whose associated benefits are similar to those

of increased frequency bandwidth.

Spatial modulation as a tool for multiplexing communication signals and increasing

channel capacity is rooted in the notion of parallel channels in information theory. It

has recently been studied by Kilfoyle [65], who provides an overview of the main develop-

ments since the 1960’s. The material in this section highlights some of the most relevant

points discussed in [65] to place the proposed wavefront segmentation approach into proper

context.

4.1.1 Parallel Channels

In a parallel channel model, information can be simultaneously conveyed to the receiver

over a set of independent communication channels. These can be physically-separated

media, or generated by any kind of multiplexing scheme (e.g., in time, frequency or space)

where disturbances are fully decoupled. The data streams are partitioned, coded, and

their relative powers chosen so as to maximize the overall throughput. Some of the issues

related to the creation of such channels and the theoretical performance gains that can be

expected will now be discussed.

Information-Theoretic Results The ideal case of K parallel discrete memoryless

Gaussian channels with a common power constraint is analyzed in [19]. The objective

is to distribute the total transmitted power among the channels so as to maximize the

capacity. Each output Y is the sum of the input X and Gaussian noise Z. For channel k,

Yk = Xk + Zk , Zk ∼ N (0, Nk) , k = 1, . . . K , (4.1)

and the noise component is independent from channel to channel. Subject to the power

constraint E{∑K

k=1X2k

}

≤ P , the capacity optimization problem can be stated as

C = maxp(x1,... xK):E

∑Kk=1 X2

k≤P

I(X1, . . . XK ;Y1, . . . YK) , (4.2)

4.1 Spatial Modulation in Underwater Channels 85

where I(·; ·) denotes the mutual information between two sets of random variables and p

is the input joint probability density function. The optimal solution

C =1

2

K∑

k=1

log

(

1 +Pk

Nk

)

, Pk = E{X2k} ,

∑

k

Pk = P (4.3)

is achieved for Gaussian inputs (X1, . . . XK) ∼ N(

0, diag(P1, . . . PK))

. The power allot-

ment is found by water filling

Pk = (ν −Nk)+ , (x)+ =

{

x if x ≥ 0

0 if x < 0,(4.4)

where ν is chosen to satisfy the power constraint. For constant noise power Nk = N the

capacity is

C =K

2log

(

1 +P

NK

)

. (4.5)

The ratio of C to the single-channel capacity C ′ = 1/2 log(1 + P/N) can be plotted as

a function of P/N for different values of K, revealing that significant improvements are

possible at high SNR, whereas the ratio tends to 1 when P/N → 0. The interpretation

is that parallel channels are most effective in bandwidth-limited scenarios, as opposed to

power-limited scenarios [65]. Spatial modulation therefore makes sense under the practical

operating conditions of many short- and medium-range underwater acoustic links that are

of special interest for time-reversed communication.

The above result for memoryless channels and additive white Gaussian noise can be

extended to more realistic frequency-selective channels by splitting the passband into mul-

tiple narrowband components where all K transfer functions are approximately constant

[72].

Decomposition of Ocean Transfer Functions The problem of creating a set of in-

dependent communication channels over a physical medium immediately suggests the use

of some form of orthonormal decomposition as a mathematical tool. Indeed, in most

of the previous work surveyed in [65] optimal spatial modulation of narrowband signals

is approached through prolate spheroidal functions, as well as normal mode and eigen-

value/eigenvector decompositions.

The methods proposed in [65] are based on singular-value decomposition of the MIMO

transfer function between transmitter and receiver arrays withM and N elements, respec-

tively. The transfer matrix is denoted by[

G(ω, t)]

n,m= Fτ{gnm(t, τ)}, and the time scale

over which G(·, t) changes is assumed to be large compared to the support region of all

involved functions in the delay axis. Under these quasi-stationary conditions, which have

already been invoked in (3.3), the output of filtering blocks is obtained by straightforward

frequency-domain products. Given an input signal vector x(ω), the set of outputs at the

receiver array is then given by y(ω, t) = G(ω, t)x(ω). For each ω and t a singular value


... MIMOChannel

...

PSfrag replacements

v1

vS

M

a1(n)

aS(n)

uH1

uHS

N

σ1a1(n)

σSaS(n)G = UΣVH

Figure 4.1: SVD-based spatial modulation

decomposition (SVD) of the transfer matrix is performed

G(ω, t) = U(ω, t)Σ(ω, t)VH(ω, t) , UHU = VHV = IK , Σ = diag(σ1, . . . σK) ,(4.6)

where K ≤ min(M,N) is the number of (nonzero) singular values. The orthogonality

of singular vectors ensures that when the channel input equals the k-th right singular

vector, x(ω, t) = vk(ω, t), the corresponding scaled left singular vector is observed at the

output array, y(ω, t) = σkuk(ω, t). This suggests using sets of prefilters and postfilters

based on left and right singular vectors to create parallel channels, as shown in Figure 4.1.

The assignment of indices to singular vectors as a function of frequency is arbitrary, thus

providing some degrees of freedom in shaping the spectrum of individual filters. In practice

this mapping is defined at a discrete set of frequencies, and it may be desirable to form

smooth functions of frequency at all elements of both arrays to generate filter responses

with compact temporal support. Other possible mapping strategies are discussed in [65].

Even if G has full column or row rank for all ω and t, the number of usable parallel

channels may be somewhat lower than K due to the disparity of singular values. The num-

ber of effective spatial degrees of freedom (which multiplicatively increase those provided

by the bandlimited time-domain waveforms) can be estimated as S = (∑

k σk)2/

∑

k σ2k.

Several simplified time-invariant channels were simulated in [65] to better understand

the decomposition process. The broadband beampatterns associated with input and out-

put filters in Figure 4.1 were evaluated, providing enlightening interpretations in terms

of the underlying propagating paths. When the channel comprises a single eigenray, the

decomposition yields a single significant singular value, whose left and right singular vec-

tors beamform along the arrival and departure directions, respectively. In a channel with

two equal-strengh eigenrays the two sets of parallel channel filters also beamform along

the associated departure and arrival angles. Rather than attempting to select one of the

rays and null out the other, the decomposition places equal energy along both rays with

phases that add coherently for the desired channel, while destructively interfering for the

other. When there is a significant disparity in ray amplitudes the singular vectors tend to

beamform on individual rays, as one would intuitively expect. Physical interpretations in

terms of eigenrays become more involved in realistic ocean channels with multiple surface-

and bottom-reflected paths.

In addition to the deterministic SVD framework, [65] discusses several other decom-


position approaches that provide more robust performance over an ensemble of channels.

The rationale is that, while the specific propagation paths between two points usually per-

sist over long periods, the path coherence may change on time scales of miliseconds due to

many physical phenomena, and those variations should be gracefully handled by spatial

modulation schemes. Even for invariant media, robustness to source and receiver motion

could be built into the transmit and receive filters by designing them for the full range

of possible channel variations. The considered optimization criteria include (i) maximum

average power over a single parallel channel, (ii) strictly parallel channels in an average

sense, (iii) minimum weighted average crosstalk power over all realizations, (iv) maxi-

mum average signal to weighted average interference plus noise ratios, and (v) minimum

mean-square error.

Generally speaking, the results of [65] using both simulated and real data prove the

concept that the spatial dimension may be exploited in practice through modulation to

improve the robustness and/or data rate of communication systems. The analytical de-

composition framework provided stable singular value estimates over periods of several

minutes during sea tests, and using the principal eigenvector resulted in a significant in-

crease in power transfer over the channel when compared with conventional beamforming.

This would be a significant advantage under power limited conditions even if only a sin-

gle spatial channel were used. In addition to a few variants of the SVD decomposition,

simpler methods were used to induce distinct spatial signatures in the various transmit-

ted data streams. Basically, these involved partitioning the set of transmitter elements

into sub-arrays and beamforming signals along pre-specified directions in each of them.

Although these alternative spatial modulation strategies were clearly inferior in terms of

power transfer efficiency, the resulting bit error rates were comparable to the ones obtained

with channel decomposition under ISI-limited conditions. The latter would presumably

have shown a significant advantage if less transmit power were available.

4.1.2 Spatial Diversity

Spatial modulation seeks to exploit the spatial coherence of propagating fields — acoustic,

electromagnetic, or other — to increase the achievable data rates. The channel parameters

are assumed as essentially deterministic quantities, giving rise to beamforming approaches

where signals are steered along particular directions to optimize the energy transfer be-

tween source and receiver. In traditional spatial diversity scenarios, on the other hand, a

stochastic model is inherently assumed for the channel parameters. Such models are typi-

cally derived assuming a large number of multipath reflections, so that signals do not have

a meaningful notion of direction of arrival. Capacity scales differently with the number of

array elements under both hypotheses [81].

As noted in [81], the distinction between diversity and directivity blurs in many ap-

plications. As bandwidth increases, multiple paths may become resolvable, rendering a

stochastic model for diversity invalid. On the other hand, channel identification for beam-

forming may become more difficult as the number of array elements increases, making the


assumption of perfect channel knowledge unrealistic.

In recent years there has been a clear emphasis on the notion of transmit/receive di-

versity among the communication engineering community, fueled by research on wireless

applications. Space-time coding, in particular, has produced many interesting results for

MIMO channels that arise from the use of multiple transmit and receive antennas. Almost

invariably an i.i.d. Rayleigh flat fading assumption is made, such that the elements of the

(memoryless) MIMO transfer function matrix are independent Gaussian random variables.

This model predicts large capacity improvements relative to the case of a single transmit-

ter/receiver [110, 77], and several signaling strategies are currently being developed to

try to capture this gain [53, 54]. The i.i.d. Rayleigh model leads to coding techniques

that do not take into account any coherence in the spatial structure of the underlying

field. However, intersensor correlation inevitably appears when the various propagation

paths become resolvable, and this leads to more modest capacity improvements than those

predicted by the ideal model above when dense arrays are used [48].

Other suboptimal procedures for exploiting diversity in Rayleigh fading channels are

presented in [119, 118, 121, 120]. These strategies emphasize simple linear processing

at the transmitter and/or receiver, and attempt to convert a flat-fading channel into

a marginally Gaussian white-noise channel. In particular, the precoding technique of

[121] for multiple transmit antennas requires no power or bandwidth expansion, and is

attractive in terms of robustness and delay considerations. This technique is suitable for

a single receiver element, and can be efficiently combined with trellis coding and other

popular error-correcting codes for bandwidth-constrained Gaussian channels. Moreover,

it can be used with the wavefront segmentation approach to be described in the sequel,

thus providing a way of creating equivalent memoryless channels even in highly dispersive

underwater environments.

Underwater communication systems resort to both beamforming and diversity combin-

ing as a way of obtaining redundant spatial information. When narrowband or broadband

beamformers are used, the goal is to explicitly obtain individual replicas of a transmitted

signal along various paths, which are then combined or selectively rejected [8]. This could

be characterized as bearing diversity, in contrast to plain spatial diversity that simply

benefits from distinct linear combinations of propagation paths at well-separated loca-

tions. Some of the multichannel receivers in the latter category are briefly described in

Sections C.2 and C.3 of Appendix C for single-user and multiuser communication, respec-

tively. One salient feature is the reduced sensitivity of equalization performance to the

placement of individual hydrophones when sparse receiver arrays are used. This desirable

behavior is commonly interpreted as a direct consequence of diversity gain but, as pointed

out in [66], that answer seems to be somewhat speculative given the scarceness of statis-

tical characterizations of spatial acoustic fields in the technical literature on underwater

communications.

Several other topics related to temporal, spectral and spatial diversity in ocean channels

are discussed in [66]. The design of underwater communication systems has traditionally


Channel #1

Channel #2

Channel #3

SourceMirror

Figure 4.2: Eigenray-based spatial modulation

been driven by advances in terrestrial wireless and wired data transmission, and the situ-

ation will undoubtedly persist in the future given the disproportion in research resources.

Most of the contributions from the underwater communications community have focused

on adaptating these techniques so that the variability and huge multipath dispersion of

many ocean channels can be successfully handled. However, there are unsettled issues

regarding the suitability of various diversity combining techniques to actual underwater

environments. In particular, it is not clear whether the common WSSUS assumption

borrowed from radio channels is realistic. Detailed spectral and spatial coherence mea-

surements are needed to answer questions concerning the available level of diversity and

the effectiveness of the approaches that are used to exploit it.

4.1.3 Spatial Modulation Properties of Time Reversal

The parallel channel paradigm of Section 4.1.1 emphasizes elimination of interchannel in-

terference and creation of truly independent data paths. A single-channel adaptive equal-

izer is still required after each receiver postfilter, as no provision is made for reducing the

(intrachannel) intersymbol interference. In spite of the improved power transfer efficiency

of SVD channel decomposition, experimental results showed that other ad hoc modulation

methods that posess no spatial orthogonality properties may lead to similar bit error rate

performance at the receiver. Moreover, the analysis of SVD-based transmitter and receiver

spatial filters on a few simple channels shows that the various waveforms propagate along

sets of eigenrays, and this suggests that imposing an association of this kind as a criterion

for spatial modulation could be a potentially fruitful approach.

The interpretation of the acoustic field generated by a time-reversal mirror during the

reciprocal phase as a set of beams projected along the original incoming path directions

shows that this device is able to simultaneously excite all the (eigen)rays connecting it

to the source location. If the spatial information stored at the mirror could somehow be

separated and associated with these rays, then it would be possible to selectively transmit

along a subset of them and implement the spatial modulation scheme described above

(Figure 4.2). That is the essence of the wavefront segmentation procedure to be developed

next.

Signals sent along the various rays by an acoustic mirror are intrinsically delayed by

time reversal to ensure simultaneous arrivals at the focus. That is the most crucial physical

process underlying the multipath compensation property, and is preserved even if those


... MISOChannel

PSfrag replacements

h1(t)

hS(t)

M

a1(n)

aS(n){gm(t)}Mm=1

g′1(t) δ(t−D)≡

Figure 4.3: Equivalent channel model in time-reversed spatial modulation

signals are not identical. Also essential for waveform regeneration — and unaffected by the

presence of distinct signals in a spatial modulation scheme — is the fact that waveforms

traveling along individual ray tubes are only mildly distorted. Actually, by transmitting

multiple delayed replicas of the same signal, a standard mirror achieves constructive in-

terference of rays at the focus that further increase the temporal peak-to-sidelobe ratio in

focused signals. That third factor in multipath compensation is lost when distinct signals

are used, but it is of relatively secondary importance.

The proposed spatial modulation technique based on time reversal therefore enables

several distinct signals to converge at the focus with low intersymbol interference and

accurately synchronized among themselves. The overall transfer function from the s-th

symbol stream to the focus in Figure 4.2 can be considered as the cascade of a mildly

frequency-selective term G′s(ω, t) and a common delay D that does not depend on s and

can be ignored from the perspective of channel modeling for communications (Figure 4.3).

With this comparatively benign and effectively instantaneous model it should be possible

to resort to some of the diversity techniques developed for Rayleigh flat-fading wireless

channels mentioned in Section 4.1.2. Spatial modulation using time-reversed eigenrays

acts as an inner processing block that exploits the coherence in underwater propagation,

leaving unstructured residual fluctuations to be handled by outer blocks.

As discussed in Section 3.2.3, phase conjugation is a relatively stable process that can

tolerate significant channel variations. A stochastic approach similar to the one used in

[65] for spatial modulation seems to be less crucial to ensure acceptable performance, and

accordingly the deterministic framework adopted in previous chapters will be retained.

The loss of coherence between eigenray contributions at the focus can be identified as

the main source of variability in a plain time-reversal mirror, particularly for rays whose

paths have very low spatial overlap, and are likely to undergo uncorrelated fluctuations.

Such interference between rays is excluded1 by the proposed spatial modulation approach,

where only fluctuations in the eigenrays themselves are relevant. The analysis of [23] for

1Actually, some of the rays may be used to carry the same message, in which case coherence loss

remains an issue. However, such groups will usually be formed by rays whose trajectories cannot easily be

separated as they nearly overlap in space, and under those circumstances their fluctuations are likely to

be correlated.


Focus #1

Focus #2

Focus #3

Figure 4.4: Simultaneous focusing at multiple depths

narrowband signals shows that time-reversed focusing is robust to intra-ray variations in

dynamic random media.

The motivation for multiple antenna precoding, as described in [121], is precisely to

avoid deep signal fades that occur when paths interfere destructively. Rather than send-

ing identical signals through all the transmit antennas (or, in this case, through all the

ray paths), the common symbol stream is convolved with different linear prefilters before

transmission, creating a form of spatial spreading. This reintroduces controlled intersym-

bol interference that can be exploited by a variety of traditional detection approaches at

the receiver. A simple linear equalizer filter is examined in [121], and it is shown that un-

der mild conditions its output equals the original symbol stream, scaled by a nonrandom

constant and corrupted by white noise that is uncorrelated with the message. The effects

of fading are thus transformed into a form of additive interference that is easier to handle

through coding. This approach involves no bandwidth expansion and is readily applicable

in time-reversed spatial modulation.

The availability of several nearly flat-fading equivalent channels in the ocean provides

an appealing framework for leveraging some recent developments in space-time coding for

wireless channels. While suboptimal techniques such as transmit precoding can provide

performance benefits in practical systems with low complexity, it is not self-evident that

the availability of multiple independent transmit paths leads to increased capacity relative

to simpler channels where only a single path or a single transducer are present. In fact, for

MIMO Gaussian channels with S inputs and R outputs capacity scales with min(S,R), so

no fundamental improvements can be obtained as long as a single receiver is used [110].

A similar result holds under Rayleigh flat fading for perfect channel estimation at the

receiver [110] and asymptotically for unknown channel but slow time variations [77].

It is possible to increase the number of receivers in the equivalent MIMO channel model

using a variant of time reversal described in [2] and depicted in Figure 4.4. There exist

now multiple focal spots, each exchanging waveforms with the time-reversal array using

the protocol of Section 3.1.2. During the forward phase probe pulses are sequentially sent

from the focal points to the mirror, which stores the waveforms that are needed to focus

at the various depths. For properly oriented transducers the time-reversed field intended


for a given receiver will nearly vanish at all the remaining ones, provided that their depth

separation is sufficiently large. This property stems from the results presented in Sections

2.4 and B.2, which show that the time-reversed field converges on a focal point with the

same kind of angular dependence of the original source. Somewhat greater interference

will occur if the focal points in Figure 4.4 are not vertically aligned. The mirror can

simultaneously transmit independent messages to each receiver due to the linearity of the

medium, hopefully creating a MIMO channel with increased capacity when eigenray-based

spatial modulation is used. It should be remarked that the sets of eigenrays associated

with the various focal depths are likely to be very similar, and if their fluctuations are

correlated this will have a negative impact on capacity [48].

Dynamic Refocusing Sequentially repeating the forward phase of the time-reversal

communication protocol for every possible focal location can quickly become cumbersome.

An interesting alternative would be to derive all required waveforms at the mirror from

the signals sent by a single source. Some dynamic refocusing techniques such as the the

so-called wavefront tilting approach of [21, 109] seem to be compatible with the wavefront

segmentation procedure that will be described next, but this possibility has not been

explored.

An alternative well-known technique for changing the focal range in underwater phase

conjugation is based on the theory of invariants, which predicts that the interference

structure in oceanic waveguides is characterized by the existence of lines of maximum

intensity having constant slope in the frequency-range plane [97]. The expression that

relates range and frequency shifts ∆r, ∆ω relative to their nominal values r, ω is

∆ω

ω= β

∆r

r, (4.7)

where β is an invariant whose precise value depends on the nature of the waveguide and

must be determined experimentally. Some variants of the basic concept incorporating

focusing and nulling constraints in both range and depth are described in [99, 67]. In

spite of the successful demonstration of variable range focusing in the ocean [55], this

technique does not seem to be well suited for underwater telemetry, where a constant

carrier frequency is desired. Moreover, as the value of waveguide invariants is small (|β| ≈ 1

in a few scenarios considered in [97]), unreasonably large frequency shifts would be needed

for the ranges and frequencies that are relevant to underwater communication. In the

present context (4.7) is perhaps more appropriately interpreted as confirming that the

Doppler compensation technique developed in Section 4.5 has negligible impact on the

location of the focus.

4.2 Wavefront Segmentation

From (3.7) and (3.17), the PAM pulse shape received at the m-th mirror transducer in

the time-invariant case is denoted by hm(t) = q(t) ∗ gm(t) = q∗(−t) ∗ gm(t). Depicting

4.2 Wavefront Segmentation 93

Direct path

Surface bounce

Bottom bounce

DelaySensor index (depth)

Figure 4.5: Wavefront signatures in |hm(t)|

|hm(t)|, m = 1, . . . M as a function of t and m reveals the presence of multiple wavefronts

associated with the rays that impinge upon the array (Figure 4.5). Due to linearity and

the retroreflective property, each wavefront contains the set of amplitudes and delays

across the array that are required to steer a broadband beam along a given direction.

If these wavefronts could be individually separated, then beamforming information would

become available at the mirror to transmit along any incoming ray direction, while ensuring

synchronous arrivals at the focus.

According to the discussion of Section 2.2, detecting the presence of incoming wave-

fronts and then synthesizing the segmented waveforms using a model-based approach seems

to be unfeasible in practice regardless of which specific mathematical models are adopted.

Rather than trying to re-create the waveforms needed for modulation in the various spatial

channels, the approach taken here identifies which regions of the delay-index plane of Fig-

ure 4.5 are associated with each wavefront, and then extracts the associated signals using

masks. The price paid for this simple but robust approach is reduced spatial resolution,

which may limit the number of equivalent flat-fading channels that can be created.

The segmentation method to be developed in this section splits the distorted pulse

shapes {hm(t)}Mm=1 into groups of waveforms {hm,s(t)}Mm=1, s = 1, . . . S. If the received

replicas are perfectly separated, then all components associated with a given wavefront

will be assigned to the same group s. In that case one may write

hm,s(t) = q(t) ∗ gm,s(t) = q∗(−t) ∗ gm,s(t) (4.8)

hm(t)∆=

S∑

s=1

hm,s(t) = q∗(−t) ∗S∑

s=1

gm,s(t) , (4.9)

where gm,s(t) denotes the contribution of the s-th group of wavefronts to the medium im-

pulse response. This condition will be approximately verified with practical segmentation

provided that the overlap between wavefronts in different groups is small, so that any

truncation effects induced by segmentation have small impact on the underlying convolu-

tive structure. With no additional filtering, the “multiuser” PAM signal generated by the


mirror at the m-th sensor is

xm(t) =∑

k

S∑

s=1

as(k)h∗m,s(kTb − t) , (4.10)

where as(k) denotes the complex symbol sequence transmitted in the s-th set of wavefronts,

i.e., the s-th spatial channel. Using (3.20) the received signal at the focus is given by the

sum of convolutions

z(t) =M∑

m=1

xm(t) ∗ gm(t) =S∑

s′=1

M∑

m=1

xm(t) ∗ gm,s′(t)

=∑

k

S∑

s,s′=1

as(k)q(t− kTb) ∗(

M∑

m=1

g∗m,s(−t) ∗ gm,s′(t))

.

(4.11)

The term inside brackets in (4.11) can be interpreted as the response of a broadband

beamformer whose spatio-temporal response is matched to the wavefronts in the set s. If

beamwidths are narrow and the grouping of segmented wavefronts is performed such that

the directions in s and s′ are well separated in space, then

q(t) ∗(

M∑

m=1

g∗m,s(−t) ∗ gm,s′(t))

≈ δ(s− s′) q(t) ∗(

M∑

m=1

g∗m,s(−t) ∗ gm,s(t))

≈ Csδ(s− s′) q(t) .

(4.12)

The last equality in (4.12) follows from the multipath compensation property of time re-

versal. Indeed, signals arrive at the focus simultaneously even if some of the eigenrays

are suppressed by segmentation, and therefore a modified version of (3.12) is still approx-

imately valid for a truncated set of paths. Naturally, the scaling factor Cs will be smaller

than in a nonsegmented mirror, as fewer terms contribute to the pressure at the focus.

Finally,

z(t) =∑

k

(

S∑

s=1

Csas(k))

q(t− kTb) . (4.13)

According to (4.13), the received signal at the focus is a PAM sequence with no inter-

symbol interference, where the pulse amplitudes are obtained by a weighted sum of S

simultaneously transmitted data symbols. This is the same model discussed in Section

4.1.3, which opens up the possibility of using a wide array of techniques developed for

Rayleigh fading channels in underwater environments.

Unlike plain time-reversal mirrors, constructive interference in segmented focusing is

mainly important among beams that comprise a given set s. Segmentation is done so that

the impinging directions are similar within each set, which suggests that phase perturba-

tions should be correlated because the propagation paths traversed by these wavefronts

are similar. One would therefore expect constructive interference to be approximately

preserved over reasonably large time intervals. Under these assumptions, the observation

model (4.13) seems plausible even when weak channel fluctuations occur, if the gains Cs

are regarded as random variables.


4.2.1 Wavefront Detection

In line with the signal model of Section 3.1, each impulse response is assumed to be a

sum of P path contributions gm(t) =∑P

p=1 gm,p(t). In turn, each term is decomposed

into a quasi-deterministic macro-multipath component fm,p(t) = fm,pδ(t − τm,p), where

the delay τm,p can be reasonably well predicted, and a stochastic component ψm,p(t) such

that gm,p(t) = fm,p(t) ∗ ψm,p(t).

Much work has been published on the subject of direction of arrival estimation, but

almost invariably either perfect wavefront coherence is assumed, or else only rather struc-

tured perturbations are allowed [35, 108]. In the present case stochastic components have a

short time span, but are otherwise almost completely unstructured, leading to an ill-posed

estimation problem. Given these limitations, it seems more reasonable to detect wave-

fronts based on their shape and energy alone, instead of relying on coherent techniques

that require poorly-justified perturbation modeling.

Consider a one-dimensional slice through the delay-index plane {hm(t)} parametrized

by vector θ

u(θ) =[

h1(τ1(θ)) . . . hM (τM (θ))]T. (4.14)

The delays τm define the shape and location of a wavefront, and they are uniquely deter-

mined from the minimal parameter vector θ. The independence assumptions on ψm,p(t)

allow the elements of u(θ) to be considered as independent random variables, regardless

of whether an actual wavefront described by θ exists. However, the variance of an element

um(θ) will be affected by the presence or absence of a wavefront at delay τm(θ), which

can be used as the basis for a statistical detection test. These random variables will be

assumed Gaussian to simplify the derivations.

If the variances of um(θ) under hypothesis H0 (noise-only) and H1 (signal present)

were known and independent of m, the optimal likelihood ratio test would simply compare

|u(θ)|2 =∑M

m=1|um(θ)|2 with a threshold [61]. The detection problem considered here is

more complex than that, as slices may — and often do — intersect wavefronts. In that case

only an unknown subset of the elements of u(θ) belong to one or more different wavefronts

for choices of θ other than the true values Θ = {θ1, . . . θP }. To avoid having to jointly

search for all parameter vectors, whose number P is not even known a priori, it is assumed

that wavefronts are temporally narrow and sparsely distributed across the array, so that

the number of um(θ) that belong to any wavefront for θ /∈ Θ is negligible. Therefore

u(θ) is either taken along the main crest of an actual wavefront, where variances should

be similar for any index m, or it can essentially be considered as a noise-only vector.

Incoherent Detection Algorithm Motivated by the simple form of the optimal test

statistic above, and considering the simplifying assumptions, the following ad hoc tech-

nique is proposed to detect wavefronts:

1. The range of valid parameters Dθ is discretized.


2. The cost function

J(θ) = |u(θ)|2 (4.15)

is computed for (a subset of) those grid points. This is the step where a model of the

environment is used to map parameter vectors into slices on the delay-depth plane.

3. The maximum of J is found and the corresponding abcissa is stored.

4. The detected wavefront is then approximately removed from the data by applying a

mask to generate new impulse responses h′m(t).

5. The whole wavefront extraction process ir repeated sequentially until the residual

energy in the masked impulse responses is sufficiently low.

The cost function contains peaks in the vicinity of the nominal set Θ, although their

widely-varying amplitudes preclude a simple thresholding operation to determine the wave-

front parameters. The dynamic range of these maxima could be decreased if cylindrical

propagation losses were compensated when extracting each u(θ). The width of both the

wavefronts and the peaks of J directly depends on the temporal support of the stochastic

components ψm,p(t) and the bandwidth of the signaling pulses. Due to the nonzero width

of those peaks, multiple parameter vectors can be associated with a single physical wave-

front. This poses no problem as long as these vectors are grouped together during the

segmentation step. It should also be noted that the energy accumulation criterion leads

to smooth variations in J(θ), which simplifies the search for maxima.

Masked impulse responses can be conveniently generated with Gaussian functions as

h′m(t) = hm(t)(

1− e−(t−τm(θ))2

2b2)

, (4.16)

where the mask width b can be chosen a priori based on the assumed temporal support for

the stochastic components and PAM pulses. It may also be computed for every detected

wavefront to reflect the width of the peak in J .

Plane Waves Depending on the propagation model that is used to extract u(θ), itera-

tively computing the cost function J may become too computationally intensive. In the

special case of a uniform linear time-reversal array in shallow water with focal range of

several hundred meters, wavefronts may be considered as approximately planar. In the

simplest parameterization θ = [τ φ]T contains the angle of arrival relative to the array

axis, φ, and the delay measured at a reference sensor, τ , such that

τm(θ) = τ + (m− 1)d

csinφ , (4.17)

where d is the intersensor separation. For a given angle φ, the cost function∑M

m=1|hm(t−

τ − (m − 1)dc sinφ)|2 can be efficiently evaluated for all values of the discretized delay

variable τ using the FFT.


Source range rs = 2 km Constellation {−1,+1}Source depth zs = 70 m Signaling rate 1/Tb = 2 kbaudBottom depth H = 130 m SNR ∞Carrier frequency f = 10 kHz PAM pulse q(t) Root-raised cosineBottom reflectivity αB = 0.6 Pulse rolloff 20%RMS surface roughness σ = 0.1 m

Table 4.1: Wavefront segmentation simulation parameters

Note that this plane wave approximation is only possible because a purely energetic

criterion is used for detection. Even in the absence of random variations, complex ampli-

tudes across spherical wavefronts do not conform to a plane wave model at the operating

ranges of interest.

4.2.2 Segmentation

After the detection step, the extracted parameter vectors are grouped into S sets and

used to generate the segmented impulse responses hm,s(t). This grouping operation is

currently performed heuristically, with θi, θj being assigned to the same set if (i) the

vectors are “close”, meaning that the associated wavefronts have significant overlap and

cannot easily be separated, or (ii) the wavefront directions are similar, to preserve the

spatial orthogonality property of sets required by (4.12).

Similarly to the detection phase, a Gaussian mask is created for each θ as

ϕm(t;θ) = e−(t−τm(θ))2

2b2 . (4.18)

Denoting by I(s) the set of parameter vector indices that belong to set s, the segmented

impulse response is obtained as

hm,s(t) = hm(t)

∑

i∈I(s) ϕm(t;θi)∑S

s=1

∑

i∈I(s) ϕm(t;θi). (4.19)

4.2.3 Simulation Results

The environmental and communication parameters used in the simulations are given in

Table 4.1. Ray tracing in this range-independent environment was based on the same

sound-speed profile of Figure 3.4b, and surface reflection was modeled as a deterministic

angle-dependent coefficient equal to the average specular component (A.1), with RMS

surface roughness σ = 0.1 m.

Plain Mirror Figure 4.6 shows the distorted pulse shapes at the array after the trans-

mitted signal has propagated over a range of 2 km, with ISI spanning about 70 symbol

intervals at 2 kbaud. Wavefronts are labeled according to the sequence of surface (S)

and bottom (B) bounces, with D denoting the direct path. The pulse shape at the focus

and associated constellations are shown in Figure 4.7 for a plain mirror that performs


Figure 4.6: Received pulse shapes at mirror

−4000

−2000

0

2000

4000 020

4060

80100

120140

0

0.2

0.4

0.6

0.8

1

depth (m)frequency (Hz)

(a)

−2000 −1500 −1000 −500 0 500 1000 1500 20000

50

100

150

200

250

300

frequency (Hz)

mag

nitu

de

M = 130M = 520

(b)

(c)

Figure 4.7: Plain mirror (a) Stored spectra (b) Pulse spectra at focus (c) Constellationsat focus

no wavefront segmentation. Results are presented for arrays that span the whole wa-

ter column with M = 130 and 520 uniformly-spaced transducers. Such a large number

of receive-transmit sensors is admittedly unreasonable using present-day technology, and

the concentration of constellation points indicates that both values of M are unneces-

sarily high in this case. Nonetheless, they have been retained to properly evaluate the

performance of wavefront segmentation in the absence of beampattern artifacts induced

by spatial aliasing. In any case the intersensor separation is about 6.7 wavelengths for

M = 130 and 1.7 for M = 520, well above the half-wavelength value that is commonly

used in spatially-coherent array processing applications.

As the analytical and simulation results of previous chapters have shown, array length

is the most relevant design parameter in plain mirrors, and good focusing can be obtained

even with intersensor separations of tens of wavelengths, as long as the array intercepts

most of the energy in the water column. According to the classical spatial analysis of Sec-

tion 2.4, focusing performance is largely determined by the strongest transmitted beam-


pattern, whose main lobe path over the water column does not undergo surface or bottom

reflections. Any acoustic energy sent along grating lobe directions is strongly attenuated

by multiple reflections, and has only a moderate impact at the focus.

By contrast, non-redundant information is ideally sent over reflected eigenrays when

wavefront segmentation is used, and practical beampatterns at the transmitter must ap-

proximate this desired behavior. In a vertical discrete mirror these beampatterns are

steered away from array broadside, and for large intersensor separation it becomes likely

that a grating lobe will send energy through a stronger, albeit nonsynchronized, path. To

avoid such a situation, which effectively destroys the multipath compensation property of

spatial channels, discrete arrays must be denser than the ones used for plain time reversal.

Practical application of the proposed technique would then require a reduction in array

length and/or nonuniform placement strategies as described in Section 2.5 to reduce the

number of sensors while retaining most of the desirable beampattern features.

Wavefront Detection and Classification Even at a relatively short range of 2 km

the wavefronts of Figure 4.6 are approximately planar, and can be parametrized by angle-

of-arrival and delay according to (4.17) in the incoherent detection algorithm of Section

4.2.1. Figure 4.8a shows the segmentation cost function (4.15), evaluated using a subset

of 130 sensors spaced uniformly along the array for both values ofM considered here. The

delay separation between wavefronts that can be individually resolved using the incoherent

detection algorithm is typically much larger than the baseband sampling period used for

pulse estimation, Tb/L, across most of the array. It is therefore also possible to decimate

the estimated pulse shapes in the delay axis to reduce the complexity of iteratively com-

puting J(θ) as segmentation masks are generated. Once the wavefront parameters are

known, masks can be created for the received pulses with full resolution in both depth

and delay axes.

For ease of interpretation, the correspondences between the various peaks of Figure

4.8a and the arrival patterns of Figure 4.6 are explicitly indicated. Note that high values of

J for large τ are due to FFT wrap-around, and these delays should actually be interpreted

as negative. Figures 4.8b–f show the identified parameter vectors after 1, 2, 3, 6 and 9

iterations, superimposed on the modified cost functions where the effect of previously-de-

tected wavefronts has been removed using Gaussian masks. The width parameter in (4.16)

was chosen a priori as b2 = 50(Tb/L)2, yielding an effective time window of about ±15

samples for discrete-time processing of baseband signals. After 6 iterations more than 90%

of the total energy is accounted for, and using this subset of parameters would have caused

virtually no degradation in performance due to truncation of multipath components.

Figures 4.9b–d depict the normalized segmentation masks given by (4.19) for S =

3 spatially-modulated channels. These were obtained by ad hoc grouping of estimated

parameter vectors as shown in Figure 4.9a, such that wavefronts in the same class have

similar directions of arrival and the spatial orthogonality relation (4.12) is satisfied between

classes. The same width parameter b2 = 50(Tb/L)2 was used in (4.18), but its impact on


(a) (b)

(c) (d)

(e) (f)

Figure 4.8: Segmentation with planar wavefront model (a) Initial cost function (b)–(f)Detected wavefronts and modified cost functions after iterations 1, 2, 3, 6 and 9


(a) (b)

(c) (d)

Figure 4.9: Generation of three spatially modulated channels (a) Wavefront grouping(b)–(d) Normalized segmentation masks

mask shapes is weak due to the normalization operation. A small constant term of 10−4

was added to the denominator of (4.19) to avoid division by zero due to numerical round-

off errors when |t− τm(θ)| À 1, causing mask magnitudes to drop off sharply outside the

region surrounding the main wavefront crests.

Although the two peaks in Figure 4.8a associated with rays that undergo a single

surface or bottom reflection are well separated from the direct path for detection purposes,

it was found that the time-domain waveforms could not be reliably segmented over much

of the array length. Under these conditions one can only realistically hope to separate the

paths that are reflected at least twice, hence a conservative classification was adopted in

Figure 4.9a. It should also be remarked that the set of weakest paths generated by the ray

tracer (surface-bottom-surface reflection) is only detected at the 9th iteration, when three

distinct parameter vectors have already been assigned to the main arrival. This poses no

problem as long as these closely-packed redundant vectors are assigned to the same spatial

channel.


Segmentation and Focusing Performance To evaluate the best possible perfor-

mance of this spatial multiplexing scheme, ray arrivals at each sensor were separated

based on the angle of incidence provided by the ray propagation code. The resulting

gain/delay values were then convolved with the transmitted pulse shape to create the

perfectly-segmented waveforms shown in Figures 4.10a,d,g. PAM pulse spectra and con-

stellations at the focus forM = 130 and 520 sensors are also shown. Grating lobes exist in

the beampatterns for M = 130 due to large intersensor separation, causing energy to be

sent in undesirable directions. In the case of spatial channels B and C one of these grating

lobes approximately coincides with a direct propagation path to the focus, and the signal

traveling through it arrives earlier and with less attenuation than the intended multiply-

reflected replica. This generates intersymbol interference and destroys the travel-time

synchronization of subchannels.

Figure 4.11 shows similar results using the empirical segmentation scheme. Perfor-

mance is virtually unaffected in channel A because the strongest arrivals are well inside

its segmentation mask and suffer little distortion. On the contrary, some degradation

occurs in channels B and C due to imperfect segmentation when wavefronts cross. There

is a considerable difference in magnitude between the refocused PAM pulse for channel A

and those of channels B and C. This is inevitable due to the very nature of the spatial

modulation approach, as the strongest eigenrays of channels B and C undergo a total of

four reflections (twice on the surface and bottom) as they travel from the source to the

mirror and then back to the focus during the reciprocal phase. Although the information-

theoretical results of Section 4.1.1 cannot be applied because the channels are not parallel,

one would intuitively expect a modest increase in capacity when all three spatial channels

are used relative to channel A alone due to the large discrepancies in SNR at the focus.

Clearly this modulation scheme would become more effective if the surface- and bottom-

reflected paths in channel A could be segmented out and assigned to B or C, as will be

done later in the presence of Doppler. But even the current wavefront grouping may be

of practical interest for reasonably large SNR if the mirror exerts some form of amplitude

control in the transmitted symbol streams {as(n)}3s=1 to improve the balance of received

power at the focus.

4.3 Coherent Communication in the Presence of Doppler

Expansion or compression of the time axis in received waveforms due to Doppler occurs

whenever a transmitter, receiver or scattering surfaces in the environment move in a way

that changes the the length of acoustic propagation paths. For a single path linking a trans-

mitter and receiver with velocity vS and vR, respectively, the time compression/stretch

factor is [125]

s =1− βR1− βS

, βS =〈vS , r̂S〉

c, βR =

〈vR, r̂R〉

c, (4.20)

4.3 Coherent Communication in the Presence of Doppler 103

(a)

−2000 −1500 −1000 −500 0 500 1000 1500 20000

50

100

150

200

250

frequency (Hz)

mag

nitu

de

M = 130M = 520

(b)

(c)

(d)

−2000 −1500 −1000 −500 0 500 1000 1500 20000

5

10

15

frequency (Hz)

mag

nitu

de

M = 130M = 520

(e)

(f)

(g)

−2000 −1500 −1000 −500 0 500 1000 1500 20000

5

10

15

frequency (Hz)

mag

nitu

de

M = 130M = 520

(h)

(i)

Figure 4.10: Ideally-segmented mirror (a) Stored pulses for spatial channel A (b) PAMpulse spectra at focus (c) Constellations at focus (d–f) Spatial channel B (g–i) Spatialchannel C


(a)

−2000 −1500 −1000 −500 0 500 1000 1500 20000

50

100

150

200

250

frequency (Hz)

mag

nitu

de

M = 130M = 520

(b)

(c)

(d)

−2000 −1500 −1000 −500 0 500 1000 1500 20000

2

4

6

8

frequency (Hz)

mag

nitu

de

M = 130M = 520

(e)

(f)

(g)

−2000 −1500 −1000 −500 0 500 1000 1500 20000

2

4

6

8

10

frequency (Hz)

mag

nitu

de

M = 130M = 520

(h)

(i)

Figure 4.11: Empirically-segmented mirror (a) Stored pulses for spatial channel A (b)PAM pulse spectra at focus (c) Constellations at focus (d–f) Spatial channel B (g–i)Spatial channel C

4.4 Time Reversal of Moving Sources Using Discrete Arrays 105

where r̂S and r̂R are the unit ray tangents at the source and receiver, pointing outward

from the source. In the case of a slowly-moving transmitter, |vS | ¿ c, and static receiver,

vR = 0, the approximate compression factor is s ≈ 1 + βS . For a passband signal x̃(t) =

Re{

x(t)ejωct}

the equivalent Doppler-distorted waveform transmitted over a single path

is

x̃(

t(1 + βS))

≈ Re{x′(t)ejωct} , x′(t) = x(t)ejνt , ν = ωcβS . (4.21)

In the case of PAM signals the effect of Doppler on the complex baseband signal can be

described as a rotation of the symbol constellation. If left uncompensated this phenomenon

can easily overcome the tracking ability of digital receivers, causing the divergence of

adaptive equalization algorithms. However, as discussed in Appendix C, these are highly-

structured time variations that can be efficiently estimated and isolated from the equalizer.

Assumption (4.21) is not always justified in applications to fast-moving AUVs, where pulse

dilation/compression can create loss of bit-timing synchronization and seriously affect even

a long equalizer. Adaptive resampling is commonly used to prevent coefficient migration in

equalizers, and a similar technique would probably be used in phase conjugation to avoid

disrupting the channel identification process. These issues related to broadband Doppler

were not addressed here, and the proposed methods in Sections 4.4 and 4.5 are therefore

restricted to low source speeds and relatively small observation intervals, where the time

scaling of baseband pulses can be neglected.

When the transmitter and receiver are linked by multiple propagation paths these can

expand at different rates, inducing distinct Doppler shifts in the various received replicas.

In turn, this differential Doppler effect causes nonuniform coefficient rotation in adap-

tive equalizers, impairing their stability and tracking performance even when the mean

Doppler shift is compensated by a phase-locked loop. Generalized receiver architectures

capable of handling multiple path delays and Doppler shifts were studied in [26, 27]. In-

terestingly, time reversal also provides a means for handling differential Doppler in such a

way that a uniformly-moving source perceives a Doppler-free time-reversed transmission.

This technique will be developed in Section 4.4.1.

4.4 Time Reversal of Moving Sources Using Discrete Arrays

The results of Section 2.3.1 show that waveform regeneration in an ideal time-reversal

cavity occurs along the time-reversed trajectory of a moving source. Focusing is thus

preserved in the presence of this particular type of non-stationarity as long as the envi-

ronment itself remains reasonably stable throughout the process. Based on these results,

similar properties will now be explored in discrete arrays, extending the static analysis of

previous sections.

Let νm,p denote the Doppler shift in the p-th ray tube received at the m-th mirror

transducer. Convolution of the rotated baseband PAM signal (3.6), (4.21) with the path


impulse response gm,p(t) yields the contribution at the receiver

ym,p(t) =

∫

gm,p(τ)x(t− τ)ejνm,p(t−τ) dτ =

∑

k

a(k)

∫

gm,p(τ)ejνm,p(t−τ)q(t− τ − kTb) dτ

=∑

k

a(k)ejνm,pkTbhm,p(t− kTb) ,

(4.22)

with

hm,p(t) = ejνm,pt

∫

gm,p(τ)e−jνm,pτq(t− τ) dτ = q(t)ejνm,pt ∗ gm,p(t) . (4.23)

From the form of (4.22)–(4.23) it is seen that the contribution of each ray can still be

regarded as a PAM sequence with distorted pulse shape hm,p(t), but with a constellation

a′(k) = a(k)ejνm,pkTb that rotates from symbol to symbol. It can also be interpreted as a

PAM signal with fixed constellation and signaling pulses that rotate by ejνm,pTb from one

symbol interval to the next.

Using the coherent communication protocol of Section 3.1.2 in the static case requires

that the distorted replicas hm(t) =∑

p hm,p(t) be available at the mirror, either through

direct measurement of channel probes, or by pulse estimation from received packets. Pulse

shapes become time variant in the presence of Doppler and obviously cannot be estimated

from instantaneous snapshots, thus rendering single-pulse probe signals useless. Due to the

particular structure of Doppler-induced variations implied by (4.22) it should be possible

to decouple the estimation of pulse components along the delay and Doppler axis, hence

reducing the number of parameters to be determined. Taking advantage of this property,

which stems from the assumption that path impulse responses remain approximately static

even as the source moves, would be somewhat restrictive, and as shown in Section 2.3.1

is not really required for wave focusing. For increased robustness, and in keeping with

the philosophy outlined in Section 2.2 for tackling coarsely-modeled environments, this

simplification was not adopted. Instead, pulses are estimated as generic functions in

delay-Doppler domain.

As in Section 3.1.2, suppose that there are P path contributions for each sensor and

the mirror simply records the received signals ym(t) =∑P

p=1 ym,p(t), conjugates and plays

them back in reverse. Then the signal at the focus is

z(t) =M∑

m=1

gm(t) ∗ y∗m(−t) =∑

k

a∗(−k)∑

1≤m≤M1≤p≤P

ejνm,pkTb[

gm(t) ∗ h∗m,p(−t)]

t−kTb. (4.24)

To gain insight into the properties of (4.24), it will be assumed that the Doppler shifts for

a given path are approximately equal in all array sensors, νm,p ≈ ν̄p, so that

z(t) =∑

k

a∗(−k)P∑

p=1

ejν̄pkTbM∑

m=1

[

gm(t) ∗ h∗m,p(−t)]

t−kTb. (4.25)

4.4 Time Reversal of Moving Sources Using Discrete Arrays 107

Using (4.23) and q(t) = q∗(−t) the innermost summation in (4.25) can be written as

M∑

m=1

gm(t) ∗ h∗m,p(−t) ≈ q∗(−t)ejν̄pt ∗

(

M∑

m=1

gm(t) ∗ g∗m,p(−t))

≈ Cpejν̄ptq(t) . (4.26)

The last equality in (4.26) is a consequence of the spatial filtering properties of time-

reversal mirrors. Similarly to (4.11) the sum of impulse response convolutions∑

m gm(t) ∗

g∗m,p(−t) can be interpreted as the output of a broadband beamformer matched to the pa-

rameters of the p-th wavefront to a superposition of P wavefronts that make up {gm(t)}Mm=1.

When the spatial selectivity is high the dominant term is∑

m gm,p(t)∗g∗m,p(−t), and under

convolution with q(t)ejν̄pt it can be approximated as Cpδ(t). Substituting (4.26) in (4.25)

yields

z(t) =∑

k

a∗(−k)P∑

p=1

ejν̄pkTbCpejν̄p(t−kTb)q(t− kTb)

=(

P∑

p=1

Cpejν̄pt

)

·∑

k

a∗(−k)q(t− kTb) .

(4.27)

This PAM signal, which does not suffer from delay dispersion, is an instantaneous sum of

P paths, each having its own Doppler shift. Large-scale envelope variations in (4.27) are

caused by time-varying interference among these contributions.

According to (4.27), a static observer at the focus perceives the same Doppler shifts

induced at the TRM by a source with constant velocity v. If, however, the observer were

moving uniformly along −v, the original shifts would be precisely compensated in its

reference frame, and the (time-reversed) transmitted PAM waveform would be recovered.

The previous derivation shows that, similarly to the ideal time-reversal cavity of Section

2.3.1, a discrete mirror is able to focus a multipath-free signal along the time-reversed

trajectory of a moving source.

Not only is the assumption ejνm,pkTb = ejν̄kTb violated for long observation periods, but

the source moves over a large enough distance such that the medium impulse responses

gm,p(t) cannot be considered time-invariant. Extrapolating from the results of Section

2.3.1 it seems likely that even under those circumstances the reciprocal field generated by a

discrete mirror would focus on the time-reversed source trajectory, but no analytical results

were obtained to support this claim. This issue is not a major concern, as time-reversal

for communications will typically occur over short periods during which the displacement

of a slowly-moving source will be small relative to the dimensions of the focal spot.

4.4.1 Doppler Compensation

Inverting the velocity v in the short period that separates the forward and reciprocal

transmissions seems quite unrealistic in practice, as it would require an extremely agile

unit at the focus. Even if it were feasible, such a strategy would entail a highly inefficient

control effort. By contrast, the moving receiver observes a multipath-free signal with twice


the original Doppler shift on each propagation path if the velocity is kept constant. In

order to simplify the receiver, it would be desirable to synthetically change the Doppler

shifts by processing at the TRM, while retaining the focusing information contained in

the received signals during the forward transmission. That is the goal of the Doppler

compensation technique developed in this section.

To simplify the notation, a time-variant pulse shape hm(t, τ) is defined such that the

signals received at the mirror are written as

ym(t) =∑

k

a(k)hm(kTb, t− kTb) , (4.28)

hm(t, τ) =P∑

p=1

ejνm,pthm,p(τ) . (4.29)

Given the exponential nature of time variations in t, and taking into account (4.23) and

the multipath structure of Section 3.1 for an invariant medium, the corresponding delay-

Doppler spread function, defined in (E.2), becomes sparse in both its arguments

Um(τ, ν) = Ft

{

hm(t, τ)}

=P∑

p=1

δ(ν − νm,p)hm,p(τ)

=P∑

p=1

δ(ν − νm,p)[

fm,pδ(τ − τm,p) ∗ q(τ)ejνm,pτ ∗ ψm,p(τ)

]

.

(4.30)

The fact that Um(τ, ν) basically consists of a series of well-spaced peaks that are associated

with the various paths can be used to drastically reduce the number of parameters needed

for channel estimation and tracking [26].

As shown by (4.27), simple time reversal of received signals leads to residual rotation

of path contributions at the focus. From

y∗m(−t) =∑

k

a∗(−k)P∑

p=1

ejνm,pkTbh∗m,p(−(t− kTb)) =∑

k

a∗(−k)h∗m(−kTb,−(t− kTb)) ,

(4.31)

it is seen that the delay-Doppler spread function U ∗m(−τ, ν) = Ft

{

h∗m(−t,−τ)}

is implicitly

used in that case. Suppose now that Um(τ, ν) is estimated at the mirror and inverted in

ν before generating the signal

xm(t) =∑

k

a∗(−k)1

2π

∫ ∞

−∞U∗m(−(t− kTb),−ν)e

jνkTb dν

=∑

k

a∗(−k)

[

1

2π

∫ ∞

−∞Um(−(t− kTb), ν)e

jνkTb dν

]∗

=∑

k

a∗(−k)P∑

p=1

e−jνm,pkTbh∗m,p(−(t− kTb)) .

(4.32)

4.5 Wavefront Detection and Segmentation with a Moving Source 109

The approximation (4.26) can still be invoked to write the modified focused signal as

z(t) =∑

m

gm(t) ∗ xm(t) ≈∑

k

a∗(−k)P∑

p=1

e−jν̄pkTb

M∑

m=1

[

gm(t) ∗ h∗m,p(−t)]

t−kTb

≈∑

k

a∗(−k)q(t− kTb)P∑

p=1

Cpejν̄p(t−2kTb) .

(4.33)

But the original PAM pulse q(t) has an effective time span of only a few symbol intervals,

hence each k, p term in (4.33) will only take on significant values in a time window (k ±

∆)Tb. Under all plausible conditions |ν̄pTb∆| ¿ 1, and ejν̄p(t−2kTb) ≈ e−jν̄pkTb ≈ e−jν̄pt.

Then (4.33) coincides with (4.27), except that the Doppler shifts are inverted

z(t) =(

P∑

p=1

Cpe−jν̄pt

)

·∑

k

a∗(−k)q(t− kTb) . (4.34)

If the receiver keeps moving with velocity v during the reciprocal transmission phase, then

it will observe a multipath-free and Doppler-free signal.

4.5 Wavefront Detection and Segmentation with a Moving

Source

The method developed in Section 4.2.1 for detecting wavefronts in a static scenario will now

be extended to the time-variant case by replacing {hm(τ)}Mm=1 with pulse delay-Doppler

spread functions {Um(τ, ν)}Mm=1. According to the simplified model of Section 4.4 an

impinging propagation path is characterized by a certain delay and Doppler shift at each

mirror sensor. Wavefronts generate unique one-dimensional signatures across depth-delay-

Doppler space, much as they did in the depth-delay plane for a static source. In principle,

it would be possible to detect wavefronts by defining expanded parameter vectors θ to

reflect tentative source locations and velocities, sampling the estimated spread functions

along the corresponding trajectories

u(θ) =[

U1(τ1(θ), ν1(θ)) . . . UM (τM (θ), νM (θ))]T, (4.35)

and identifying wavefronts as the peaks of the cost function J(θ) = |u(θ)|2. This approach

was not taken, as it would lead to a significant increase in complexity due to the larger

dimensionality of the parameter space, and also because efficient computational procedures

such as the one described in Section 4.2.1 for plane waves cannot be used. Instead,

a projection technique was developed to break down this multidimensional search into

simpler subproblems.

Under the assumption that the source moves sufficiently slow so as to remain almost

still throughout the whole observation period, the delay signatures of wavefronts — unlike

actual path attenuations, which are subject to Doppler-induced phase rotation — are

negligibly disturbed relative to the time-invariant scenario examined previously. Even


PSfrag replacementsv

r̂S1

r̂S2

〈v, r̂S1〉 > 0

〈v, r̂S2〉 < 0

Figure 4.12: Departure angle and Doppler disparity in crossing wavefronts

in the latter case successful incoherent detection/segmentation requires low overlap of

wavefronts, such that it is very likely that only a single one will contribute for any given

m, τ where there is significant energy in hm(τ). In the presence of Doppler this means that

the evolution of hm(t, τ) through the t axis for fixed τ is typically governed by a single

exponential term ejνm,pt. Moreover, (4.23) shows that hm,p(τ) ≈ q(τ) ∗ gm,p(τ) because

phase variations due to the exponential term are small in the effective delay window where

q(τ) is nonzero. Hence |hm(t, τ)|2 is very close to |hm(τ)|2 in the time-invariant case for

any t, and the incoherent wavefront detection approach can still be used.

Rather than choosing an arbitrary time t to compute the cost function, it seems prefer-

able to average the squared magnitude of pulse shapes over several snapshots. Parseval’s

relation provides an equivalent expression in terms of Um(τ, ν)

∫

|hm(t, τ)|2 dt =1

2π

∫

|Um(τ, ν)|2 dν . (4.36)

Projecting |Um(τ, ν)|2 onto the depth-delay plane therefore produces essentially the same

results of the static case, and an identical wavefront detection algorithm can be used.

For notational convenience, denote by θr the subset of wavefront parameters related

to the source position, and by θv those related to its velocity. Once the (projected) wave-

front parameters θr are known, the same masks {ϕm(τ ;θr)}Mm=1 of Section 4.2.2 can be

created to segment Um(τ, ν) into {Um,s(τ, ν)}Ss=1 according to (4.19). As in the static case,

wavefront crossings cannot be resolved by this method, which leads to some performance

degradation relative to ideal segmentation. When Doppler is present, however, the ad-

ditional dimension ν may actually be used to overcome this limitation because crossing

wavefronts in the plane (m, τ) can be fully disjoint in (m, τ, ν). In the short-range shallow-

water scenarios of interest the departure angles of rays associated with crossing pairs of

wavefronts are sufficiently well separated in space, so that a moderate vertical compo-

nent in the source velocity vector will induce resolvable Doppler shifts (Figure 4.12). A

model-based approach can then be used to improve the segmentation by searching in the

parameter space of source velocity vectors for the one which produces the best match

to the wavefront trajectories along the Doppler axis given the range/bearing information

previously derived from the projection of |Um(τ, ν)|2 onto the depth-delay plane. The

original multidimensional search for wavefronts in depth-delay-Doppler space is thus de-

coupled into lower-dimensional subproblems. Specifically, if θr is the vector of position


parameters extracted during one iteration of the search algorithm using (4.36), then a

two-dimensional slice of Um(τ, ν) is obtained as

Um(ν;θr) =[

U1(τ1(θr), ν) . . . UM (τM (θr), ν)]T, (4.37)

The same incoherent detection procedure of Section 4.2.2 can now be applied to Um(ν;θr)

by discretizing the range of feasible velocity parameters θv, generating one-dimensional

slices through Um(ν;θr) as

u(θr,θv) =[

U1(ν1(θv);θr) . . . UM (νM (θv);θr)]T

=[

U1(τ1(θr), ν1(θv)) . . . UM (τM (θr), νM (θv))]T,

(4.38)

and detecting the velocity parameters as peaks of the cost function

J(θr,θv) = |u(θr,θv)|2 . (4.39)

Wavefront patterns in Um(ν;θr) are much simpler than in the delay-Doppler projection

(4.36), consisting mainly of a single (partial) signature. These trajectories are approxi-

mately linear except for the direct arrival, and their variation over depth is mild, satisfying

the assumption of Section 4.4 {νm,p}Mm=1 ≈ ν̄p. As in the static case, fast evaluation of

J(θr,θv) using the FFT is therefore possible. It is also worthwhile noting that the tra-

jectory of any crossing wavefront is essentially 1D, and its intersection with the 2D slicing

surface(

m, τm(θr))

∀ν is thus confined to a small point-like region in Um(ν;θr) that has

negligible impact on the cost function.

From θr, θv masks can be generated as

ϕm(τ, ν;θr,θv) = ϕ(1)m (τ ;θr)ϕ

(2)m (ν;θv) , (4.40)

where both terms on the right hand side are Gaussian-like, as in (4.18), possibly with

different variances chosen a priori. Once these masks have been grouped together, the

segmented delay-Doppler spread functions {Um,s(τ, ν)}Ss=1 are obtained similarly to (4.19).

Simplified Segmentation Accurate detection of wavefronts is not of crucial importance

under the high SNR conditions required for time reversal as long as the ensuing segmen-

tation step is not compromised. In particular, estimating the source velocity vector is not

really required, as only enough information needs to be obtained to discriminate between

upward and downward departure angles in crossing wavefronts. To illustrate this point,

Figure 4.13 shows the impinging paths on a dense mirror as 3D points (m, τm,p, νm,p) ∀m,p

for a moving source with nonzero vertical velocity, as well as their two-dimensional projec-

tions (amplitude information is absent). It can be verified that the signatures of reflected

wavefronts are approximately linear and nearly parallel to the delay-depth plane, such

that crossing pairs in Figure 4.13b are fully disjoint when viewed in 4.13c. Under these

conditions only a minor performance loss is incurred if all pairs of wavefronts are sepa-

rated by a single plane parallel to the depth axis that bissects the projection onto (τ, ν),


delayDoppler

dept

h

(a)

delay

dept

h

delay

Dop

pler

Doppler

dept

h

(b) (c) (d)

Figure 4.13: Coarse segmentation of wavefronts (a) Depth-delay-Doppler representation(b) Depth-delay projection (c) Delay-Doppler projection (c) Depth-Doppler projection

as depicted in Figure 4.13c. The bissecting line could be estimated in practice from

Vm(τ, ν) =

M∑

m=1

|Um(τ, ν)|2 (4.41)

using image processing techniques, but since this is not a central issue its orientation will

be assumed known.

Simplified segmentation masks are generated for given θr by zeroing out the contribu-

tions from one of the half-spaces on either side of the separating plane, i.e.,

ϕm(τ, ν;θr) = ϕ(1)m (τ ;θr)

(

ν − ν0(τ))±

, (4.42)

where ν0(τ) denotes the Doppler coordinate of the plane at delay τ , (·)+ is the step

function defined in (4.4) and (x)−∆= (−x)+. The choice of which step function is used

in (4.42) depends on whether the wavefront energy is located to the right or left of the

plane, as estimated from the slice (4.37) for each detected θr. To this effect the cost

function J(θr,θv) only needs to be evaluated for wavefronts with constant ν, or on a


delay

Doppler

depth 2D slice

Detected wavefrontSeparating plane

PSfrag replacements

Um(ν;θr)

Separating plane

Doppler

Detectedwavefront

1D slice

Depth

Crossing wavefrontPSfrag replacements

u(θr,θv)

(a) (b)

Figure 4.14: Wavefront detection (a) Two-dimensional slicing of delay-Doppler spreadfunction (b) One-dimensional reslicing of Um(ν;θr)

reduced and coarse grid of θv. If Doppler disparity is insufficient for reliable classification

and separation of wavefronts — the main arrival being an obvious example —, then it is

preferable to eliminate the ν dependence in that particular mask and use ϕm(τ, ν;θr) =

ϕ(1)m (τ ;θr).

Figure 4.14 illustrates the 2D and 1D slicing operations discussed above. For ease

of visualization the delay-Doppler spread function has been replaced by the path arrival

structure of Figure 4.13, which can be interpreted as a thresholding operation on |Um(τ, ν)|.

4.5.1 Simulation Results

Returning to the simulated scenario of Section 4.2.3, the source is now allowed to move

with velocity vector v = [√22

√22 ]T ms−1, leading to differential Doppler shifts between

wavefronts with the same number of bounces of up to about 1.5 Hz. Figure 4.15 shows

the ideal delay-Doppler spread function, calculated according to (4.23), (4.29), (E.2), and

discretized along the delay and Doppler axes with steps ∆τ = Tb/4 = 0.125 ms and

∆ν/2π = 0.1 Hz. Doppler shifts at the mirror were computed from ray departure angles

using (4.20) and (4.21). For completeness, Figures 4.15c–d represent the various paths

computed by the ray tracer between the source and array locations, projected onto the

depth-delay and delay-Doppler planes2.

In practice Um(τ, ν) would be estimated for each sensor by computing the time-

frequency crosscorrelation (E.15) between the received waveform and a known reference

signal transmitted at the start of data packets. This operation actually yields a two-

dimensional convolution between the desired delay-Doppler spread function and the au-

tocorrelation of the transmitted waveform [115, 26], placing fundamental limits on the

2Figure 4.13 was based on the same scenario and provides a clearer picture of the arrival structure

across the array, but numerical values were omitted in the plots to avoid obscuring the discussion.


(a) (b)

(c) (d)

Figure 4.15: Discretized delay-Doppler spread function (a) Depth-delay projection (b)Delay-Doppler projection (c)–(d) Path parameters computed by the ray tracer

time-frequency resolution that can be achieved. The case of binary PAM signals with

nearly constant magnitude generated by maximal-length sequences is especially relevant

in coherent communications, and some properties of that family of waveforms are given

in Section E.2. Their autocorrelation has a single main peak in the time-frequency plane

whose width is inversely proportional to the total sequence duration in the Doppler axis

and to the effective signal bandwidth in the delay axis. It is thus possible to concentrate

the autocorrelation around the origin by using a long PAM sequence formed by many short

symbols, and thereby achieve arbitrary resolution in both dimensions simultaneously when

estimating Um(τ, ν).

As all the wavefronts in Figure 4.15 are disjoint in depth-delay-Doppler space, they

can be resolved by a suitably designed pseudo-random reference signal. If the symbol

interval and pulse shape of the PAM preamble are identical to those used in the remainder

of the data packet, then the finest delay resolution is approximately Tb. That value is

appropriate to the underwater environment under consideration, where differential delays


between rays are typically much larger than the signaling interval. On the other hand,

a Doppler resolution of 0.1 Hz would impose a minimum duration of about 10 s for the

reference signal, which is somewhat larger than the values commonly used in practical

communication systems [100, 50, 40], but not unreasonably so. Such fine Doppler precision

is actually not needed in this case, and a somewhat shorter preamble could therefore be

used.

It should also be remarked that, for continuous signals with effective (two-sided) band-

width smaller than 2πFi and maximum duration To, the channel input-output relation

can be represented by the discrete-time convolution (E.12) involving samples of the delay-

Doppler spread function

Um(k, l) =1

FiToUm

( k

Fi,2lπ

To

)

. (4.43)

Figures 4.15a–b, which represent(∑

l |Um(k, l)|2)1/2

and(∑

m |Um(k, l)|2)1/2

, respectively,

were obtained for Fi = 4/Tb and To = 10 s. This is an appropriate choice for the kind

of PAM reference signals envisaged here that respects the sampling issues discussed in

Section E.1.1.

A PAM preamble has the advantage of allowing simultaneous equalizer training and

estimation of the delay-Doppler spread function at the mirror. The latter would also be

required as part of the signal processing chain in a receiver architecture capable of handling

strongly time-variant channels [26, 27], but the Doppler resolution can be coarser than the

one needed for wavefront segmentation, and achievable with a shorter reference signal.

Note that the segmented spread function is only needed during the reciprocal phase, after

the incoming packet has been fully received and processed. In fact, the only reason why

the estimation of Um(τ, ν) should be based on the preamble alone is to ensure that the

underlying signal has known deterministic autocorrelation properties. That argument is

not compelling because similar behavior can be imprinted with very high probability on

PAM waveforms containing actual random data through the use of simple source coding

techniques. Assuming that the receiver is able to decode data with low error probability,

it is then possible to use an approach similar to the one of Section 3.3 for bidirectional

communication and compute the spread function for segmentation over a period of several

seconds, corresponding to a fraction of the full packet duration. Naturally, the estimation

of spread functions for equalization purposes at the mirror must still be based on a known

training sequence.

Doppler Compensation According to the plain time-reversal strategy of Section 4.4,

PAM sequences were generated at the mirror with time-varying pulse shapes described by

U∗m(−τ, ν)

xm(t) =∑

k

a(k)1

2π

∫ ∞

−∞U∗m(−(t− kTb), ν)e

jνkTb dν . (4.44)


(a)

(b)

(c)

Figure 4.16: Performance of plain TRM at moving receiver (a) Delay-Doppler spreadfunction (b) Constellation magnitude (c) 2-PSK constellation

In practice, these signals are generated in discrete time from the sampled delay-Doppler

spread function. Similarly to (E.13) this yields

xm(n) =∑

k

a(k)1

N

N−1∑

l=0

U∗m(kL− n, l)ej2πLN

lk , (4.45)

where L is the oversampling factor and it is assumed that N = FiTo is an integer.

As in Section 4.2.3, long and dense arrays will be used to approximate an ideal contin-

uous mirror, avoiding the problems associated with grating lobes due to spatial undersam-

pling. In an actual implementation these issues would be addressed by reducing the array

length and spacing the sensors nonuniformly. Figure 4.16 shows the delay-Doppler spread

function at the focus for a mirror with M = 520 evenly-distributed elements spanning the

water column. In agreeement with (4.27), it can be seen that multipath has been virtually

eliminated and the resulting pulse has negligible delay dispersion. Notice that the delay-

Doppler spread function was calculated in the reference frame of the moving source/focus,

whose velocity vector is assumed to remain constant. Accordingly, the Doppler shifts for

the P replicas are located at twice the frequency values that were considered in (4.27).

Also shown in Figure 4.16 is the 2-PSK constellation at the moving receiver after root

raised-cosine filtering of the PAM signal. In addition to phase rotation, there is some

residual magnitude modulation that results from the time-varying interference pattern of

arrivals with different Doppler shifts over a period of 1 s.

Figure 4.17 shows similar results when the Doppler compensation procedure of Section

4.4.1 is used, which simply amounts to inversion of the Doppler frequency in (4.45)

xm(n) =∑

k

a(k)1

N

N−1∑

l=0

U∗m(kL− n,N − l)ej2πLN

lk . (4.46)


(a)

(b)

(c)

Figure 4.17: Performance of Doppler-compensated mirror at moving receiver (a) Delay-Doppler spread function (b) Constellation magnitude (c) 2-PSK constellation

Similarly to Figure 4.16 there is negligible multipath, but now compression has also been

achieved in the Doppler axis, resulting in an impulse-like spread function centered at (0, 0).

The constellation confirms that both intersymbol interference and time-varying magnitude

distortion are weak, thus rendering the PAM signal easily decodable with simple receiver

structures.

Segmented Mirror The depth-delay projection of Um(τ, ν) in Figure 4.15a is very

similar to the set of pulse shapes in Figure 4.6, and the first step of the wavefront detection

algorithm of Section 4.5 produces virtually the same parameter vectors θr shown in Figure

4.8. A bissecting plane such as the one represented in Figures 4.13b and 4.14 was then

defined, and simplified segmentation masks generated according to (4.42). Wavefronts

were classified in an ad hoc manner as in the static scenario, but now different groups were

formed because it is possible to separate the direct arrival from the surface and bottom

reflections, thus enabling a more even distribution of acoustic energy among the spatial

channels. The direct arrival wavefront intersects the separating plane, as it normally

should, and therefore the Doppler dependence of its associated mask was dropped. Figure

4.18 shows the ideally and empirically-segmented spread functions, projected onto the

depth-delay plane, for the three strongest wavefronts. The same width parameter of the

static scenario, b2 = 50(Tb/L)2, was used for Gaussian masks in this plane. Comparing

these results with the static case it can be seen that the interference between the surface-

and bottom-reflected wavefronts has been completely eliminated. Though not shown here,

the same is true for the surface-bottom and bottom-surface paths. Some undesirable

interaction remains between the direct path and surface reflection, which nearly overlap

in the depth-delay projection across the upper 20 m. To ensure suitable orthogonality

between spatial channels it would be advantageous to apply the full 3D segmentation


(a) (b) (c)

(d) (e) (f)

Figure 4.18: Depth-delay projection of segmented spread functions (a)–(c) Ideally seg-mented direct path, surface reflection and bottom reflection (d)–(f) Empirical segmenta-tion

algorithm of Section 4.5 whenever these two wavefronts are assigned to different groups.

Before presenting the results for empirical segmentation, ideal separation of wave-

fronts is considered for benchmarking purposes. Figures 4.19 and 4.20 show the spread

functions and 2-PSK constellations at the focus for the three perfectly-segmented wave-

fronts considered above with and without Doppler compensation, respectively. The results

are qualitatively similar to those of the nonsegmented mirror, with Doppler-compensated

channels exhibiting a more concentrated spread function along the Doppler dimension that

induces slower magnitude modulation of the 2-PSK constellation. In both cases low delay

spread is achieved in all spatial channels. It should also be noted that the assumption

νm,p ≈ ν̄p used in the analysis of time reversal for moving sources would imply spread

functions at the focus sharply located at 2ν̄p, which is clearly not the case in Figure 4.20.

Naturally, the deviation is more significant for the direct arrival, in which the incoming

Doppler shifts vary over a wider range across the array.

The results for empirical segmentation are provided in Figures 4.21 and 4.22, where

it is seen that the degradation relative to ideal segmentation is less important than in

the static case due to the elimination of interference from crossing wavefronts. The most

obvious impairment occurs in Figure 4.21f, where the scattered constellation reflects some

residual interaction with the direct arrival in the segmented spread function.

Finally, Figure 4.23 illustrates the performance of an ideally-segmented mirror with

M = 130 uniformly-spaced sensors using Doppler compensation. Increased constellation

scattering is clearly visible in the surface- and bottom-reflected spatial channels, but the


(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 4.19: Performance of ideally-segmented mirror with Doppler compensation at mov-ing receiver (a) Delay-Doppler spread function of direct path contribution (b) Constellationmagnitude (c) 2-PSK constellation (d)–(f) Surface reflection (g)–(i) Bottom reflection


(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 4.20: Performance of ideally-segmented mirror without Doppler compensation atmoving receiver (a) Delay-Doppler spread function of direct path contribution (b) Con-stellation magnitude (c) 2-PSK constellation (d)–(f) Surface reflection (g)–(i) Bottomreflection


(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 4.21: Performance of empirically-segmented mirror with Doppler compensation atmoving receiver (a) Delay-Doppler spread function of direct path contribution (b) Con-stellation magnitude (c) 2-PSK constellation (d)–(f) Surface reflection (g)–(i) Bottomreflection


(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 4.22: Performance of empirically-segmented mirror without Doppler compensa-tion at moving receiver (a) Delay-Doppler spread function of direct path contribution (b)Constellation magnitude (c) 2-PSK constellation (d)–(f) Surface reflection (g)–(i) Bottomreflection


(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 4.23: Performance of ideally-segmented mirror with Doppler compensation andM = 130 transducers at moving receiver (a) Delay-Doppler spread function of direct pathcontribution (b) Constellation magnitude (c) 2-PSK constellation (d)–(f) Surface reflection(g)–(i) Bottom reflection


effect is less pronounced than in the static case of Figure 4.10. This is actually quite natural

if one remembers that signals undergo at least two reflections when transmitted through

channels B and C in Figure 4.10, but only a single one in the non-direct channels of Figure

4.23. Beampatterns corresponding to multiply-reflected paths are steered further away

from array broadside, and grating lobes which contribute to multipath have a stronger

impact at the focus.

4.6 Summary and Discussion

The basic approach for time-reversed coherent communication developed in Chapter 3

was extended to explicitly incorporate spatial modulation concepts, with an aim towards

improving the efficiency in channel use by simultaneously transmitting multiple signals

through the ocean. The motivation for this was drawn from recent developments in space-

time coding for wireless communications, which predict great improvements in capacity

for multiple-antenna systems operating over Rayleigh flat-fading channels. The proposed

methods, however, do not focus on capacity and coding aspects for MIMO systems, but

rather concentrate on single-receiver (MISO) configurations.

The theoretical benefits of using multiple independent communication channels operat-

ing in parallel were first discussed in a general context. An eigendecomposition method for

MIMO transfer functions (deterministic case) or autocorrelation matrices (stochastic case)

that creates such parallel channels in the ocean by exploiting the spatial dimension was

then briefly described. The directivity patterns associated with the transmit and receive

broadband beamformers in this multiplexing scheme often have an intuitive interpretation

in terms of propagation paths, which suggests that incorporating the underlying physics

into spatial modulation methods may also make sense from a communications perspective.

Other approaches for exploting spatial diversity were also briefly discussed, such as

linear transmit precoding methods that convert a MISO Rayleigh flat-fading channel into

an equivalent SISO Gaussian channel at the receiver, where reliable demodulation can be

easily achieved. Although these methods have no optimality claims and, depending on

channel modeling hypotheses, may not even lead to an increase in (theoretical) capacity,

they are of considerable practical interest, and help to reduce the overall probability of

error when coupled with well-known transmitter/receiver blocks. The spatial modulation

method developed in this chapter can be understood as an intermediate layer that turns

a severely-reverberating medium into a few equivalent paths with small delay dispersion,

over which some of the above-mentioned diversity techniques for flat-fading channels can

be adapted.

The wavefront segmentation approach supporting spatial modulation is based on the

observation that a time-reversal mirror ideally beamforms signals along the same directions

of incoming rays. Whenever the channel impulse responses between a source and the array

transducers are sparse, it may be possible to detect and extract the information that is

needed to beamform along individual paths, or groups of partially-overlapping paths in

4.6 Summary and Discussion 125

time and space. Equally important is the fact that the signals transmitted along the various

paths are transparently delayed at the mirror so as to ensure simultaneous arrivals. By

sending different signals along these paths, low intra-path and inter-path delay dispersion

is ensured, so that from an abstract input-output perspective the receiver at the focus

experiences a MISO channel with relatively flat frequency response.

Formally, the goal of segmentation is similar to that of iterative focusing in free space,

as described in [89]. In iterative focusing an initial acoustic pulse is reflected by various

scatterers, generating a set of wavefronts that are recorded at the mirror. As these sig-

nals are repeatedly transmitted and their echos re-recorded, energy is gradually directed

towards the strongest scatterer, until the insonification of the remaining ones becomes neg-

ligible and a single reflected wave is observed. These steady-state signals can be related to

the strongest eigenvalue/eigenvector pair of the round-trip transfer matrix containing the

response from any transducer to every other one in the mirror. In fact, it can readily be

noted that the outlined procedure is simply a form of the well-known power method for

eigenvalue computation. As the remaining nonzero eigenvalues are associated with weaker

scatterers, an adaptation of this iterative method could possibly be used to extract the

corresponding eigenvectors. Iterative focusing has also been used in ocean experiments,

but the emphasis of [98] is on assessing the concentration of energy, rather than studying

the evolution of wavefronts at the mirror.

Even if an iterative wavefront separation procedure for ocean waveguides could be

devised, the whole process would likely involve large delays that cannot be tolerated in

acoustic links. For this reason, a one-step segmentation method was proposed in this

chapter, even though it necessarily suffers some degradation when wavefronts cross. The

approach conforms to the assumptions regarding the reliability of propagation models

outlined in Sections 1.6 and 2.2, namely that the shape of wavefronts can be reasonably well

predicted, but not the evolution of complex gains across the array. Wavefronts are detected

simply by accumulating the energy in measured or estimated received pulse shapes over a

discrete set of possible wavefront parameters, and then applying a threshold. Beamforming

data is extracted directly from the mixture of wavefronts in the time domain using masks,

without ever attempting to parametrize the amplitudes.

In principle, the proposed incoherent wavefront detection method would require search-

ing over a parameter space containing the source position, and possibly some environmen-

tal features. To reduce the dimensionality of this parameter space, or at least to limit

the volume that must be searched, it would be useful for the mirror to have a reason-

ably accurate estimate of the source location. Conceivably, this type of side information

could be readily available at the mirror in a scenario such as the one depicted in Figure

1.3 when the source is fixed, or less trivially when positioning systems are available for

mobile nodes. In the latter case, it would be necessary for the mobile unit to convey its

position to the mirror by some means after self-localization. This could involve embedding

that information as part of each mirror-bound packet, or transmitting it separately in a

short packet using a low-rate and easily decodable modulation format. Alternatively, it


is possible to parametrize wavefronts without explicit reference to the source position in

typical operating conditions where their shapes are approximately planar across the array.

In that case, source range and depth are replaced by direction of arrival and delay for

every wavefront, that is, all coupling constraints imposed by the geometry of the medium

are discarded, and wavefronts are treated as fully independent entities. To simplify the

detection phase, this was the approach used in the simulations.

Simulation results for wavefront detection are quite satisfactory, as all energy peaks

corresponding to the propagation paths are detected with minor redundancy (i.e., when

two or more parameter vectors are associated with a single physical path). Upon detection

of a wavefront, a Gaussian mask is applied to extract beamforming data and to remove

it from the depth-delay plane before recomputing the accumulated energy. The width of

the mask is currently chosen a priori from the expected temporal support of stochastic

components, but it should be simple to develop an estimate for this parameter based on

the sharpness of detected peaks in the energy function.

In spite of the care that was taken in developing a sound framework for spatial modula-

tion that is intimately linked to the physical properties of sound propagation in the ocean,

the feasibility of wavefront detection and segmentation using real data remains untested.

Various authors have reported channel estimates obtained with sparse arrays where a

strong correlation in impulse response magnitudes is visible across the array. However, it

is usually not possible to clearly discern the pattern of ascending and descending wave-

fronts that was assumed in this work because only a small fraction of the water column is

sampled.

Mask-based segmentation under static conditions is intrinsically a coarse operation,

and can only produce acceptable results when wavefronts propagating in the ocean (or,

equivalently, represented in the depth-delay plane at the mirror) are widely spaced. In the

simulated environment this implies that the direct path cannot be separated from those

undergoing a single surface or bottom reflection, regardless of whether the array has enough

spatial resolution to actually resolve the corresponding beams in transmit mode. As these

wavefronts concentrate a large fraction of the total acoustic energy impinging upon the

array, grouping them together creates a large power disparity between this spatial channel

and the remaining ones, which are built from multiply-reflected paths. In turn, this will

be reflected into significant differences in terms of SNR and error probability at the focus,

which may limit the improvement in effective throughput that can be achieved by this

spatial modulation scheme in practice. Even so, the ability to independently control the

energy content in multiply-reflected paths may be useful for channel stabilization, as these

tend to exhibit stronger fluctuations [8]. If desired, the segmentation scheme allows these

paths to be “turned off”, exchanging potentially useful multipath energy for improved

stability at the receiver.

Currently, the assignment of detected wavefronts to spatial channels is not carried out

automatically. Establishing a metric that reflects the proximity of wavefronts in delay-

Doppler space and developing an assignment algorithm are topics for future research.


A major drawback of wavefront segmentation stems from the stringent directivity re-

quirements that must be imposed to ensure that paths excited by grating lobes in the

beampattern of any spatial channel are greatly attenuated. Otherwise, it would be pos-

sible for one of these spurious transmitted replicas to follow an almost direct path to

the focus, arriving there with lower attenuation than the desired one, and destroying the

delay synchronization that is crucial for transparent ISI compensation. As revealed by

simulations, nearly half-wavelength intersensor separation may be needed to avoid severe

spatial aliasing, and this translates into a very large number of required transducers for

a uniform mirror spanning the water column. A nonsegmented mirror may use far fewer

sensors because focusing is mostly accomplished by the direct path, whose grating lobes

will always beamform multiply-reflected replicas except for very large intersensor separa-

tion. Assessing the reduction in sensor count through nonuniform spacing techniques is

undoubtedly one of the issues that must be addressed if this spatial modulation scheme

is ever to become practically feasible, even if one takes into account the possibility of

endowing a fixed base station with abundant hardware resources, as suggested by Figure

1.3.

When compared with the eigendecomposition approach of [65] for MIMO channels, the

ratio between the number of spatial channels that can effectively be created by wavefront

segmentation and the total number of projectors/hydrophones is quite low. Firstly, note

that the goal of the spatial decomposition method of [65] is to create orthogonal spatial

channels with no regard for intersymbol interference, whereas here orthogonality is not

sought, but ISI compensation is desired to simplify the demodulation process. The two

approaches are therefore not directly comparable. Secondly, the ratio mentioned above

scales almost linearly with the number of receivers if the basic MISO method is extended

to the MIMO case, as described in Section 4.1.3 (Figure 4.4). Experimental results in

[65] have shown that eigendecomposition methods suffer some degradation due to channel

variations, such that the benefits of spatial orthogonality are lost in practice and the result-

ing performance is comparable to that of simpler spatial multiplexing methods. Though

untested, it is expected that the wavefront segmentation method will inherit the robustness

properties that have been observed in plain time-reversal experiments conducted in the

ocean. Having made these remarks, it should be acknowledged once more that a drastic

reduction in sensor count is indispensable before actual applications of spatial modulation

through wavefront segmentation can be contemplated.

Extensions of plain and segmented time reversal were presented for the case of a

uniformly-moving source. The problem is relevant in practice even for slow source motion,

as significant Doppler shifts result from a combination of high acoustic frequencies used

in underwater telemetry and relatively low sound speed in the water. Uniform motion is a

plausible assumption for mobile communication nodes, but clearly not for the fixed units

of Figure 1.3, where channel variations caused by waves and current-induced oscillations

are dominant.

As in the static case, a deterministic framework was regarded as more suitable to reflect


the coherent nature of time-reversed focusing. The techniques developed for motionless

sources were generalized in a rather straightforward way by resorting to delay-Doppler

spread functions, which retain the sparseness of impulse responses in the scenarios of

interest. Although both are conceptually similar, the additional Doppler dimension in

delay-Doppler spread functions can entail a large increase in computational complexity

if sparseness is not exploited. The situation is somewhat alleviated by the fact that

the processing algorithms used to estimate spread functions and synthesize time-variant

waveforms are amenable to parallel implementation. Though certainly relevant, the issue

of efficient parametrization was not addressed in this work. In principle, it should be

possible to adopt some of the ideas developed in [26] for selecting the effective support

region for this function as a relatively small set of points in depth-delay-Doppler space.

The proposed method for Doppler compensation simply consists of inverting the delay-

Doppler spread functions along the Doppler dimension and then generating the (time-

variant) waveforms by Fourier synthesis according to the conventional time-reversal pro-

cedure. It was proved that this operation preserves focusing, while inverting the Doppler

shifts generated during the forward transmission. If the original source keeps moving with

constant velocity, Doppler will be canceled in its reference frame, thus simplifying the de-

modulation process. A sparse wavefront structure was assumed in the analysis of Doppler

compensation, but the technique itself does not require wavefront discrimination, and can

therefore be applied in both plain and segmented mirrors.

Wavefront segmentation in the presence of Doppler can be carried out in essentially the

same way as in the static case. During the detection phase, energy in depth-delay-Doppler

spread functions is accumulated along 1D trajectories (wavefront signatures) correspond-

ing to a grid of tentative wavefront parameters. The peaks of this energy function are

searched, and beamforming data extracted from the original spread functions using masks.

As this process becomes quite complex due to the additional Doppler dimension, simpler

alternatives based on 2D projections were sought. The projection approach succeeds due

to the sparsity of incoming wavefronts and the assumption of constant Doppler shifts,

causing time-variant impulse responses to be essentially identical to the invariant case, ex-

cept for a single exponential factor at almost any given delay and depth that is irrelevant

for energy accumulation. Each wavefront projection is first detected in the depth-delay

plane, and then an orthogonal slice through the spread function is extracted to determine

its precise orientation in 3D. The approach can be used with arbitrary signatures, but

for simplicity linear wavefronts were assumed in the simulations. This approximation is

sufficiently accurate for all except the direct path, which can be segmented in a simplified

way due to the absence of wavefront crossings.

In particular cases where the difference in Doppler shifts between upward- and down-

ward-departing rays can be well resolved, it is possible to simplify the segmentation process

further, detecting only depth-delay wavefront projections and extracting all the values con-

tained in half of each orthogonal slice as beamforming data. For these Doppler shifts to

become clearly distinguishable the (near-)symmetry of the problem must be broken by as-


suming, for example, that a vertical component exists in the source velocity vector. While

not exactly unreasonable, this condition seems to be somewhat awkward, as underwater

vehicles are usually required to move at approximately constant depth, performing vertical

manoeuvres less frequently.

When compared with the static case, simulation results show that segmentation is

improved due to the Doppler disparity mentioned above. It becomes possible to separate

the direct, surface-reflected and bottom-reflected arrivals3, resulting in a more even dis-

tribution of energy among spatial channels. The difference in Doppler between these two

reflected wavefronts is only about 1 Hz in the simulated environment. Under those condi-

tions, a known preamble lasting for several seconds (possibly in the range 5–10s) would be

required to obtain sufficiently accurate estimates of delay-Doppler spread functions along

the Doppler axis. As this interval is comparable to the period of swell, some blurring of

the surface-reflected path is expected in actual ocean experiments, decreasing both the

effective distance between wavefront signatures and the segmentation accuracy. Possibly,

it would become practically unfeasible to separate these three paths, in which case the

assignment of spatial channels used for static sources would have to be adopted.

Focusing results with Doppler compensation show good compression of the delay-

Doppler spread function at the focus along both dimensions. Naturally, this means that

the moving receiver perceives each spatial channel as time-invariant and frequency-nonse-

lective, as intended. The behavior along the Doppler axis changes in the absence of Doppler

compensation, but low residual ISI is still obtained in all spatial channels. Receiver struc-

tures that can handle purely Doppler-spread, single-user, channels are developed in [26],

and could possibly be extended to the present multiuser scenario under those conditions.

However, it makes sense to complement receiver-side processing by performing Doppler

compensation at the transmitter, in order to avoid deep fades in constellation magnitudes

due to differential Doppler.

3More precisely, the bottom reflection can be clearly segmented in the simulated environment. However,

the signatures of the direct path and the surface reflection are almost overlapping in delay and Doppler

throughout the top 20 m of the water column due to the depth dependence of the particular sound-speed

profile that was used. Although they can be reasonably well separated by careful choice of segmentation

masks, the differences in attenuation lead to non-negligible spilling of direct path energy into the surface-

reflected spatial channel. Because such coupling leads to undesirable residual interference at the focus, it

may be necessary to merge these two wavefronts into a single spatial channel.

wavefront segmentation - ulisboa

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