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
86 Wavefront Segmentation
... 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-
4.1 Spatial Modulation in Underwater Channels 87
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
88 Wavefront Segmentation
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
4.1 Spatial Modulation in Underwater Channels 89
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
90 Wavefront Segmentation
... 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.
4.1 Spatial Modulation in Underwater Channels 91
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
92 Wavefront Segmentation
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
94 Wavefront Segmentation
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 Wavefront Segmentation 95
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.
96 Wavefront Segmentation
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.
4.2 Wavefront Segmentation 97
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
98 Wavefront Segmentation
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-
4.2 Wavefront Segmentation 99
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
100 Wavefront Segmentation
(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
4.2 Wavefront Segmentation 101
(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.
102 Wavefront Segmentation
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
104 Wavefront Segmentation
(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
106 Wavefront Segmentation
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
108 Wavefront Segmentation
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
110 Wavefront Segmentation
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
4.5 Wavefront Detection and Segmentation with a Moving Source 111
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),
112 Wavefront Segmentation
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
4.5 Wavefront Detection and Segmentation with a Moving Source 113
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.
114 Wavefront Segmentation
(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
4.5 Wavefront Detection and Segmentation with a Moving Source 115
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)
116 Wavefront Segmentation
(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)
4.5 Wavefront Detection and Segmentation with a Moving Source 117
(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
118 Wavefront 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
4.5 Wavefront Detection and Segmentation with a Moving Source 119
(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
120 Wavefront Segmentation
(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
4.5 Wavefront Detection and Segmentation with a Moving Source 121
(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
122 Wavefront Segmentation
(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
4.5 Wavefront Detection and Segmentation with a Moving Source 123
(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
124 Wavefront Segmentation
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
126 Wavefront Segmentation
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.
4.6 Summary and Discussion 127
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
128 Wavefront Segmentation
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-
4.6 Summary and Discussion 129
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.
130 Wavefront Segmentation