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
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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
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
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.