Underwater Acoustic Sparse Aperture SystemPerformance: Using Transmitter Channel State

Information for Multipath & Interference RejectionLisa J. Burton,† Andrew Puryear,∗ Pierre F.J. Lermusiaux,† and Vincent W.S. Chan∗

†Multidisciplinary Simulation, Estimation, and Assimilation Systems (MSEAS)Massachusetts Institute of Technology, Cambridge, MA 02139∗Claude E. Shannon Communication and Network Group

Massachusetts Institute of Technology, Cambridge, MA 02139Email: puryear@mit.edu

Abstract— Today’s situational awareness requirements in theundersea environment present severe challenges for acousticcommunication systems. Acoustic propagation through the oceanenvironment severely limits the capacity of existing underwatercommunication systems. Specifically, the presence of internalwaves coupled with the ocean sound channel creates a stochasticfield that introduces deep fades and significant intersymbol inter-ference (ISI) thereby limiting reliable communication to low datarates. In this paper we present a communication architecture thatoptimally predistorts the acoustic wave via spatial modulationand detects the acoustic wave with optimal spatial recombinationto maximize reliable information throughput. This effectivelyallows the system to allocate its power to the most efficientpropagation modes while mitigating ISI. Channel state informa-tion is available to the transmitter through low rate feedback.New results include the asymptotic distribution of singular valuesfor a large number of apertures. Further, we present spatialmodulation at the transmitter and spatial recombination at thereceiver that asymptotically minimize bit error rate (BER). Weshow that, in many applications, the number of apertures can bemade large enough so that asymptotic results approximate finiteresults well. Additionally, we show that the interference noisepower is reduced proportional to the inverse of the number ofreceive apertures. Finally, we calculate the asymptotic BER forthe sparse aperture acoustic system.

I. INTRODUCTION

The ocean is a challenging environment for transmission ofdata, with channel reverberation in both time and frequency[1] limiting data rate. Surface waves, small scale turbulenceand fluctuations of physical properties generate random signalvariation that creates deep fades and limits reliable commu-nication. Previous work [2] has established three character-istics of the underwater acoustic channel that distinguish itfrom other wireless channels: frequency-dependent propaga-tion loss, multipath, and low speed of sound (1500 m/s).

Because of frequency dependent propagation loss, differ-ent communication architectures are appropriate for differenttransmission ranges (also called link ranges). Over very shortlink ranges (under 100m), optical wireless communication isusually appropriate, providing reliable high rate communica-tion. Underwater optical systems are limited beyond 100mbecause optical energy is quickly attenuated due to absorption

Higher Data Rate

Range of Transmission (meters)

100

101

102

103

104

105

OpticalSystems

Sparse Aperture Acoustic Systems

Existing Acoustic Systems

Fig. 1. Optical wireless communication systems are extremely limited indistance. Data rate is comprised for distance of transmission.

and scattering. Optical communication in the ocean is evenmore limited in turbid or turbulent water [3].

For link ranges beyond the limit of optical communication,high frequency acoustic systems are appropriate. These sys-tems provide high data rate and reliable communication bytransmitting in the 10-30kHz band. The usable frequency bandfor high frequency, short range acoustic systems is restrictedprimarily by channel attenuation and transducer bandwidth[2]. In practice, high frequency acoustic systems are limitedbeyond 1km because of significant path loss. Thus, for longlink ranges low frequency acoustic systems are appropriate,providing reliable communication where optical or high fre-quency acoustic systems suffer from too much attenuationto be useful. The three regions (short range, intermediaterange, and long range) along with the optimal architecturefor each region is shown in Fig. 1. The region labeled opticalsystems corresponds to the short range region, sparse apertureacoustic systems corresponds to the intermediate range region(high frequency acoustics), and existing acoustic systems cor-responds to the long range region (low frequency acoustics).This paper focuses on the intermediate range: we propose anarchitecture for efficient communication in this high frequencyintermediate range region.

For the intermediate range, refraction and reflection cancause the transmitted signal to reach the receiver by twoor more paths. This phenomenon, called multipath, causesintersymbol interference (ISI) and signal fading [4], thus pre-

Burton, L.J., A. Puryear, P.F.J. Lermusiaux, and V.W.S. Chan, 2009. Underwater Acoustic Sparse Aperture System Performance: UsingTransmitter Channel State Information for Multipath & Interference Rejection. Proc. IEEE Oceans Conference (Bremen, Germany, 10-14 May 2009)

Receive Plane

z=Z ρ'

Transmit Plane

z=0 ρ

Submarine

Research Vessel

Fig. 2. One application of short range acoustic communication is data transferbetween a submarine and a research vessel on the sea surface. Here, a transmitplane array of apertures on a submarine communicates to a receive plane ona research vessel.

senting significant challenges in signal transmission. Multipathgenerally is strongest for horizontal channels [2]. Reflectionsand refraction that cause multipath are particularly problematicwhen the channel includes a boundary (either air-sea or sea-sea floor) or is within the “deep” sound channel created by thebackground sound speed profile. Current work reports the useof equalizers or vector receivers to mitigate multipath inducedISI [5]. The architecture proposed in this paper uses knowledgeof the instantaneous stochastic Green’s function, so calledchannel state information (CSI), to reject interference causedby multipath. More specifically, the proposed architecture usesCSI to optimally predistort the transmitted waveform whileusing coherent detection with optimal spatial recombination tomitigate multipath interference, thereby minimizing bit errorrate (BER). This architecture consists of one transmit and onereceive plane array with sparse apertures, meaning apertures ineach plane are placed at least one coherence length apart andthat each aperture diameter is smaller than a coherence length.The link range of this architecture is primarily limited by theneed for up-to-date CSI at the transmitter, which is delayedat least by the amount of time required to measure and feedback the information. As a result, the maximum transmissiondistance is a function of the duration of time the channelimpulse response is approximately constant, or coherencetime, and the speed of sound in sea water. Experimentshave shown coherence times of about one second for mid-to-high frequencies [6]. Thus, this architecture is limited toa link range of less than 1km by the low speed of soundand coherence time. This architecture can be applied forany transmitter-receiver plane orientation, including verticalas in submarines or underwater vehicles communicating withreceivers at the sea surface (Fig. 2).

Many emerging applications in the ocean that require high-speed, reliable wireless communication have the potential tobenefit from the proposed architecture. An example of suchan application is a real-time ocean observation system, whichis used more and more frequently to adaptively and optimallysample physical features of the ocean. Today, the number ofautonomous ocean sensing vehicles used in semi-coordinated

Fig. 3. Example of adaptive sampling requiring vehicles/surface-craft/shorecommunications. Illustrated are sampling paths generated for fixed 2-Dobjective fields using Mixed-Integer Programming. The background maps areproportional to error standard deviation. Grey dots are starting points for theAUVs and white dots are the computed optimal termination points. (a) Optimalpath of two vehicles and (b) three vehicles [15]. The question answered is“assuming the error field remains constant for the next day, on which path doI send my AUVs?.”

operations is often larger than three and this number is rapidlyincreasing [7], [8], [9], [10]. For intelligent sensing with thesevehicles, reliable high rate communications are needed. Pre-dictions with uncertainties and data assimilation [11], [12] canbe utilized to plan the sensing expected to be most useful. Suchplanning of optimal vehicle paths is referred to here as pathplanning. When path planning is also a function of the data tobe sampled, the term adaptive sampling (Fig. I) is used [13].For intelligent coordination among vehicles, information mustbe exchanged on multiple levels. Two-way communicationsover varied distances and bit rates are required among vehicles,from vehicles to ships or other surface crafts, and/or fromvehicles to shore and command and control.

The new acoustic communication systems discussed in thispaper would enhance such communications and would enablemore intelligent and coordinated autonomous operations. Theschemes recently developed for autonomous ocean sensing viaadaptive sampling, path planning and onboard routing wouldreadily benefit. Such schemes include: coordinated adaptivesampling with non-linear predictions of error reductions [13],[14]; Mixed Integer Linear Programming (MILP) for pathplanning [15]; nonlinear path planning using genetic algo-rithms [16]; dynamic programming and onboard routing forpath planning [17]; command and control of assets over theweb, directly from model instructions [18]; and strategiesfor fastest transit and routing [19]. Applications abound ina variety of fields including ocean science and engineering;ocean energy; ecosystem-based management; coastal moni-toring, undersea surveillance, homeland security and harborprotection; and multi-scale climate monitoring and prediction.

Other applications that could potentially benefit from theproposed architecture include submarines communicating withships at the sea surface and underwater vehicles communi-cating with other vehicles within a fleet. Additionally, longdistance communication can be accomplished via multihop-ping, as multiple hops over several short distances allowtransmission at a higher rate than its single-hop counterpart

1

4

2

6

5

3

N

ρ'

ρ'

ρ ρ

Fig. 4. In an undersea network, several underwater vehicles communicate toall or some of the other underwater vehicles. Thus, they act both as receiveρ′ and transmit ρ planes.

[2]. Generally, this architecture can also be applied to anunderwater network (Fig. 4) with transmission between eachvehicle and a central source, or between each pair of vehicles,so that each vehicle acts both as a receiver and transmitter.

In Sec. II, we describe the system geometry and discussthe applicability of the architecture in the ocean environment.Additionally, we describe the optimal modulation, detection,and channel measurement scheme in terms of the singularvalue decomposition (SVD) of the instantaneous stochasticGreen’s function. In Sec. III, we derive asymptotic distributionof the system singular values and present a detailed discussionof the relevance of the theory to a wide range of real-worldapplications. Next, we show that the detected interferencepower decreases proportional to the inverse of the number ofreceivers. Using the theories describing asymptotic singularvalue distribution and interference power as a function ofreceivers, we derive an analytic closed-form expression foroptimal system performance in terms of BER. We also provethe modulation and detection scheme provided in Sec. IIachieves the optimal performance as the number of receiveapertures approaches infinity. Further, we show very goodagreement between a Monte Carlo simulation and theoreticalresults. This is followed by a conclusion in Sec. IV.

II. SYSTEM DESCRIPTION

The system geometry consists of a transmit plane ρ withntx apertures and a receive plane ρ′ with nrx apertures.For this paper, we use the term transmit aperture to meantransmit acoustic emitter and receive aperture to mean areceive hydrophone. We define a pair of ancillary variables,nmin = min(ntx, nrx) and nmax = max(ntx, nrx), whicharise often in the following analysis. The transmit and receiveplane are assumed to be parallel with an origin along acommon axis. Location and orientation of the system in theopen ocean is kept general without restriction on distance fromsea surface or floor. We assume the apertures in each planeare placed at least one coherence length apart, ensuring thentxnrx links are uncorrelated. For a propagation distance of800m and a signal between 15-35kHz, experimental data hassuggested a coherence length of 35cm [1]. Thus requiring the

Receive Plane

ρ'

Transmit Plane

ρ

Fig. 5. Sparse aperture system geometry: Transmit and receive plane(consisting of apertures) at some depth below the sea surface. Orientationof the system under the water is not assumed to be horizontal.

apertures to be a coherence length apart does not create aspatially prohibitive plane array. Additionally, we assume eachaperture diameter is smaller than a coherence length. We referto this geometry as a sparse aperture system (Fig. 5).

Coherence length depends on the conditions and physicalprocesses of the ocean. Strong turbulence decreases the co-herence length, as two points close in space are not stronglyrelated.

We assume a coherent narrowband wave, with center wave-length λ, is transmitted from the ρ plane, propagates over zmeters, and is detected at the receive plane ρ′. The field isdetected coherently using either a heterodyne or homodynereceiver. Also, we assume the channel state is fully knownby the transmitter. We will discuss methods of estimating thechannel state at the end of this subsection.

We model the multipath interference as being a finitenumber of discrete rays. As such, the received field can bemodeled as:

y[n] =α(d)

√nmax

H(d)x[n] + w[n]+

α(1)

√nmax

H(1)x[n−∆1] + ...+α(m)

√nmax

H(m)x[n−∆m]

= y(d)[n] + y(1)[n] + ...+ y(m)[n] + w[n](1)

where x[n] ∈ Cntx is a vector representing the transmittedfield at time n, y[n] ∈ Cnrx is a vector representing thereceived field at time n, and w[n] ∈ Cntx represents additivewhite circularly symmetric complex Gaussian noise receivedat time n. α(d) ∈ R is the path attenuation for the directpath and α(k) ∈ R is the path attenuation for the kth

multipath component. The direct channel is represented byH(d) ∈ Cnrx×ntx : element h(d)

ij of H(d) is the direct pathgain from the ith transmit aperture to the jth receive aperture.The matrices H(k), k ∈ {1, ...,m} represent the multipathinterference: element h(k)

ij is the gain of the kth reflected (orrefracted) path from the ith transmit aperture to the jth receiveaperture. ∆i is the time of arrival of multipath component imeasured from the time of arrival of the direct path. The pathdelay may be positive or negative because a multipath ray can

travel at a higher speed than the direct ray. The terms of thesecond equality in (1) are defined as:

y(d)[n] =α(d)

√nmax

H(d)x[n]

y(1)[n] =α(1)

√nmax

H(1)x[n−∆1]

......

...

y(m)[n] =α(m)

√nmax

H(m)x[n−∆m]

(2)

where y(d)[n] is the received field due to the direct pathand y(k)[n] is the received field due to the kth multipathcomponent.

We have implicitly invoked Taylor’s frozen turbulence hy-pothesis [20] by assuming H(d) is time invariant. This is trueif the bit period is much less than the oceanic coherencetime, which occurs for the transmission range specified. Thenormalization (nmax)−1/2 is chosen to ensure:• additional apertures will not increase the system power

gain and• that reciprocity is satisfied.We decouple the input-output relationship of H(d) with a

singular value decomposition:

1√nmax

H(d) = UΓV† (3)

where the ith column of U is an output eigenmode, the ithrow of V is an input eigenmode, and the i, ith entry of thediagonal matrix Γ is the singular value, or diffraction gain,associated with the ith input/output eigenmode. We use (·)†as the conjugate transpose. For the context of this paper, aneigenmode is a particular spatial field distribution, or spatialmode. We will denote the diffraction gain associated with theith eigenmode as γi so that:

yi = γixi + wi (4)

where x, y, and w are related to x, y, and w through the usualSVD, such as in [21]. Note wi retains its circularly symmetriccomplex Gaussian distribution. We have not included themultipath components in (4).

Binary phase shift keying (BPSK) is a common modulationscheme for acoustic communication systems with coherentdetection. Though results developed here may easily be ap-plied to other modulation schemes (e.g. binary on-off keying,frequency shift keying, quadrature modulation, etc.), we willlimit ourselves to the discussion of BPSK systems for brevity.We will show that the optimal scheme to minimize BERusing BPSK is to allocate all power to the input eigenmodeassociated with the maximum singular value. To transmit a bitC[n] ∈ {0, 1} with power Et, the optimal transmit field issimply:

x[n] = vmax√Ete

iπC[n] (5)

where vmax is the input eigenmode associated with the largestsingular value. The associated optimal spatial recombinationscheme is:

φ[n] = Re{u†maxy[n]} (6)

where φ[n] is a sufficient statistic at time n and umax is theoutput eigenmode associated with the largest singular value.We will prove the optimality of this modulation and detectionscheme in the next section.

The channel state can be measured by transmitting animpulse from each transmitter sequentially and recording thetime variation of the received field at each receiver. Fromthis measurement, we can build up a channel transfer matrixas a function of time. We can then assign the time instancewith the largest power gain to H(d). In practice, we do notneed to calculate the multipath transfer matrices. They canbe measured, if desired, by selecting the time instances withthe m + 1 largest power gains, corresponding to the directpropagation path and m multipath components, and creatingH(d) and H(k). The need to compute the SVD and feed backsome information regarding the direct path transfer matrixH(d) limits the distance from the transmit plane to the receiveplane to be under 1000m. Time to compute the transfer matrixis small compared to the propagation time, an important con-sideration for implementation. Propagation attenuation overthis distance limits available carrier frequencies to 10kHz to30kHz.

We have presented an idealized channel measurementscheme. However, other methods are likely more feasible forimplementation. For instance, a system could establish aninitial channel state estimate via a full channel measurementand subsequently track channel state perturbations with time,thus taking advantage of current information to reduce thenumber of channel measurements. This method is effectiveprovided the tracking bandwidth is much higher than oceanictemporal coherence bandwidth. Because the apertures aresparse (transmitters and receivers in each place are at leasta coherence length apart), nrx × ntx channel measurementsare needed to fully characterize the ocean state (i.e. populateH(d)).

III. PERFORMANCE OF SPARSE APERTURE SYSTEMS

In this section we present the performance of sparse apertureacoustic systems with wavefront control and coherent detec-tion. Specifically, we present the asymptotic singular valuedistribution, multipath rejection performance, and asymptoticBER.

A. Singular Value Distribution

The transfer function h(d)ij describing propagation through

random media can be represented as the deterministic free-space Green’s function hFSij modified by amplitude χij andphase φij perturbation terms:

h(d)ij = hFSij χije

iφij (7)

In the random field of the underwater acoustic channel, χijand φij are random variables. Assuming the entries of H(d) are

independent and identically distributed (i.i.d.), we can find theprobability density function (pdf) governing the distribution ofthe system singular values.

Theorem 3.1: Assuming the entries of H(d) are i.i.d., thepdf of diffraction gains, γ2

i , for a single ocean state, as thenumber of transmit apertures and receive apertures asymp-totically approaches infinity, converges almost surely to theMarcenko-Pastur density:

fβ(x) =

(1− β)+δ(x) +

√(x− (1−

√β)2)+ ((1 +

√β)2 − x

)+2πx

(8)

where β =min(nrx, ntx)max(nrx, ntx)

and (x)+ = max(x, 0).

Proof. A nice proof of the Marcenko Pastur density for i.i.d.matrices can be found in [22]. �

Fig. 6 shows the Marcenko Pastur density for various systemgeometries.

As stated in the theorem, the entries of H(d) must be i.i.d.for the Marcenko Pastur density to describe the singular valuedistribution. We now include a discussion that reasons thatthe entries of H(d) are in fact approximately i.i.d. for manyrealistic applications.

Because each ray traverses a different path from transmitteri to receiver j, the entries of H(d) will be uncorrelatedprovided transmit/receive apertures are separated by at leasta coherence length. It is difficult to prove independence, sowe assume that uncorrelated implies independence.

The transmission range is much larger than typical acous-tic wavelengths for this system (5-15cm), so the far fieldapproximation is valid and the difference between ampli-tude variations among paths is negligible. Additionally, thetransmission range is much larger than the distance betweentransmit and receive apertures. Fig. 7 shows the maximumarray size possible for the limiting case when the longest andshortest direct propagation paths from transmitter to receiverdiffer by 10%. Even in the most limiting case of a 100mtransmission range, the array extent is not restrictive. If allpath lengths are within 10%, we estimate these path lengths asbeing the same, so that the amplitude statistics should also bethe same for each path. Consequently, χij has approximatelyidentical statistics for all (i, j) pairs.φij is some unknown continuous distribution with zero

mean and variance σ2φ. For channels with sufficient turbulence,

σ2φ is much greater than 2π. Because the phase is purely

real, it can always be represented as a value from 0 to 2π.Because the variance is very large compared to this range,the distribution can be approximated as uniform from 0 to2π for all (i, j) pairs. Thus, φij can also be approximated asidentically distributed.

Additionally, hFSij can be absorbed into the phase term, asit is a complex number which can be represented as eiφ

F Sij so

that the transfer function now becomes hij = χijeiφ′

ij where

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

γ

f β(γ)

β=0.2β=0.5β=1

Fig. 6. Marcenko Pastur asymptotic singular value probability densityfunction.

102

103

0

200

400

600

Range of Transmission (m)

Max

Arr

ay E

xten

t (m

) (D

ashe

d Li

ne)

102

1030

500

1000

1500

Max

No.

of A

pert

ures

(S

olid

Lin

e)

Fig. 7. The maximum extent of the receive and transmit planes increaseswith propagation distance. This figure assumes a coherence length of 35cm.

φ′ij = φij + φFSij . Without loss of generality, the expressionis shown for λz = 1. Again, because of the periodicity of thecomplex exponential ei(·), this additive term does not changethe distribution and it remains uniform from 0 to 2π. Thus,hij is modeled as a function of i.i.d. random variables.

Thm. 3.1 assumes a large number of apertures. Fig. 7shows that this assumption is satisfied for physically realizablesystems. For instance, if we have a link range of 100m, wecan place up to 100 transmitters and 100 receivers and stillmeet all of the restrictions necessary for the theorem to apply.Therefore, the theory is applicable for any practical system.

As a corollary to Thm. 3.1, the number of eigenmodescorresponding to non-zero singular values converges, almostsurely, to min(ntx, nrx). Additionally, the maximum singularvalue converges, almost surely, to γ2

max = (1 +√β)2.

B. Multipath Rejection

Multipath interference severely limits acoustic communi-cation systems. Previous work has focused on the use ofequalizers [5] or vector receivers [1] to mitigate the multipathinterference. The sparse aperture system effectively eliminatesmultipath induced ISI when there are many transmit andreceive apertures. The system, however, does not collect powerfrom multipath components, but only rejects it.

In this section, we show that multipath interference canbe reduced to any desired level by increasing the number oftransmitters and receivers. Further, we show that the detectedinterference power is proportional to 1/nrx. Formally, wewish to select a spatial modulation x[n] and receiver spatialrecombination scheme S to minimize the BER or, equivalently,maximize the signal to interference noise ratio:

x∗[n] = arg minx[n],S

Q

(√2E(d)

σ2w +

∑mk=1E

(k)

)

= arg maxx[n],S

(E(d)

σ2w +

∑mk=1E

(k)

) (9)

where σ2w is the energy due to the noise w[n], E(d) is average

detected signal energy and E(k) is the average detected energyin the kth multipath component. Q(·) is the q-function. Wehave assumed the waveform transmitted through a multipathchannel contributes as noise through ISI. The optimization in(9) is very general. Specifically, x∗[n] would have nulls in thedirection of multipath components if such a waveform wasoptimal. Further, the optimization will find the waveform thatpropagates most efficiently through the turbulent ocean. Thm.3.2 gives the solution to (9) for the case with many transmitand receive apertures.

Theorem 3.2: As the number of transmit and receive aper-tures grows large, the spatial field distribution that minimizesBER, as formulated in (9), is given by:

x∗[n] = a[n]vmax (10)

where we have used SVD, H(d) = UΓV†. Thus, vmax isthe input eigenvector associated with the maximum singularvalue γmax of H(d). Data is encoded by time variation ofa[n] ∈ C, which is spatially constant at a particular time. Forexample, for BPSK a[n] =

√Ete

iπC[n]. A sufficient statisticfor optimum detection is:

φ[n] = Re{u†maxy[n]} (11)

where φ[n] is a sufficient statistic at time n. The associatedoptimal probability of error is:

Perror = Q

(√2α(d)γ2

maxEt

σ2w + Et

nrx

∑mk=1 α

(k)

)(12)

Proof. To prove this theorem, we show that this schememaximizes E(d) while at the same time minimizing E(k)∀kfor a large number of transmit and receive apertures. First, we

show that (10) maximizes E(d):

E(d) = E[(Re{u†maxy(d)[n]})2]

=α(d)

nmaxE[(Re{u†maxH(d)vmaxa[n]})2]

=α(d)

nmax(Re{u†maxH(d)vmax})2Et

= α(d)γ2maxEt

(13)

Because the SVD provides a complete orthonormal basis forthe transmit and receive fields, the largest possible E(d) isγ2max. Thus we have shown that this scheme maximizes E(d).

We now show that (11) asymptotically (nrx →∞) drives theinterference power E(k) to zero:

E(k) = E[(Re{u†maxy(k)[n]})2]

=α(k)

nmaxE[(Re{u†maxH(k)vmax})2]

= α(k)E[(Re{u†maxv})2]

where we have defined v = n−1maxH

(k)vmax. So, we can write:

E(k) = α(k)E

(Re

{nrx∑i=1

u†max,ivi

})2

= α(k)E

(nrx∑i=1

Re{u†max,ivi

})2

= α(k)E[ξ2]

where umax,i is element i of umax and vi is element i ofv. We have defined ξ =

∑ntx

i=1 Re{u†max,ivi

}. The entries of

H(k) are identically distributed, independent of each other, andindependent of the entries of the direct channel matrix H(d).The entries of H(k) are independent of the entries of H(d)

because the multipath components traverse a different pathfrom the direct component and therefore experience differentfading. The central limit theorem can be applied to the sum-mation because the terms in the summation, Re

{u†max,ivi

},

are independent. As the number of receive apertures growslarge, the central limit theorem implies:

ξ ∼ N(

0,1nrx

)And as a result:

E(k) =α(k)

nrx(14)

We prove (12) by substituting (14) and (13) into (9). �In essence the receiver (11) is tuned to detect only the direct

mode. As we increase the number of receive apertures, thetuner becomes increasingly narrow. As a result multipath com-ponents become less and less likely to couple into the directmode tuned receiver. Nice intuition arises from an abstractvector space interpretation. The nrx-dimensional vector spaceis the closed set of all possible receive fields. We can viewthe received field vector as the summation of the direct field

vector, the multipath field vectors, and the noise vector. Thesufficient statistic in (11) is interpreted as an inner productbetween the received field vector and the direct field vector.The multipath field vectors have random orientation withrespect to the direct field vector. As a result, the expected innerproduct between the multipath field vectors and the direct fieldvector becomes small as the vector space dimension increases.Thus, the multipath interference is rejected.

C. Asymptotic Bit Error Rate

For the optimal modulation scheme presented in Sec. II, theturbulence average BER is:

Pr(error)

=∫ ∞

0

Q

(√2α(d)γ2

maxEt

σ2w + Et

nrx

∑mk=1 α

(k)

)f ′β(γ2

max)dγ2max

(15)

where f ′β(γ2max) is the pdf of the largest singular value.

This result is general for any sparse aperture communicationsystem, but depends on the pdf of the largest singular value,f ′β(γ2

max), which is in general unknown. However, the pdf isknown in the asymptotic case. As the number of aperturesgrows large, the pdf of the largest singular value converges,almost surely, to:

limnrx→∞

f ′β(γ2max) = δ

(γ2max −

(1 +

√β)2)

(16)

where δ(·) is the Dirac delta. Using (16) to evaluate (15)provides a closed-form expression for the probability of error:

limnrx→∞

Pr(error)

=∫ ∞

0

Q

(√2γ2max

α(d)Etσ2w

)δ

(γ2max −

(1 +

√β)2)dγ2max

= Q

(√2(

1 +√β)2 α(d)Et

σ2w

)(17)

For the asymptotic case, the interference power is completelyrejected by the receiver. As such, the interference terms E(k)

do not enter into the equation.While this result is only exact in the asymptotic case, it

provides a very good approximation for finite but large numberof apertures. The

(1 +√β)2

term is the power gain over asystem without the wavefront predistortion. This power gainterm results from the ability to allocate all of the systemtransmit power into the spatial mode with the best propagationperformance. Essentially, we select the mode with the bestconstructive interference for the particular receive aperturegeometry and ocean state.

D. Discussion

Using the monotonicity of the Q-function, it is easy to provethat the optimum system is balanced, using the same numberof transmit apertures and receive apertures. For a balanced

system, setting β = 1 yields a power gain of four. This resultis intuitively satisfying. Consider two scenarios: first, if thereare more transmit apertures than receive apertures, the systemcan form more spatial modes than it can resolve and degreesof freedom are unused. Similarly, if there are more receiveapertures than transmit apertures, the system can resolve morespatial modes than it can form and, again, degrees of freedomare unused. As a result, our intuition suggests a balancedsystem is optimal.

As the number of receive apertures becomes much largerthan the number of transmit apertures, or vice-versa, β → 0and the system performance approaches that of the systemwithout wavefront predistortion. This is also an expectedresult: as the system becomes very asymmetric, the abilityto predistort the wavefront is lost.

The gain, in terms of probability of error, of moving to adiversity system with wavefront predistortion is:

Pr(error|sparse aperture)Pr(error|no diversity)

= e−3SNR (18)

At high SNR, using the sparse aperture system provides alarge gain in BER compared to the no diversity system. Atlow SNR, the advantage of the more sophisticated system isless pronounced.

It is clear, in the asymptotic case, that the turbulence averageBER does not depend on turbulence strength. Effectively, themany, many apertures act to average out the spatial variationinduced by ocean turbulence. Turbulence strength does factorinto the system design; in stronger turbulence, apertures maybe placed closer together while in weaker turbulence, theymust be placed farther apart. Further, stronger turbulencecauses slower convergence to the Marcenko Pastur density;which means more apertures are required for (17) to be valid.

Lastly, as the total aperture size increases for a singleaperture system, the power gain saturates as the aperture sizeapproaches the coherence length. We have shown that thesparse aperture system, however, does not saturate with totalaperture size. Indeed, the number of apertures used is onlylimited by form factor constraints, not ocean constraints.

E. Simulation

A Monte-Carlo simulation was performed to validate thetheoretical results presented in (17). In the simulation, weassumed that the instantaneous ocean state was available at thetransmitter. For a single ocean state, an equiprobable binarysource was encoded according to (5), transmitted through thesimulated ocean channel, detected coherently, and the numberof raw bit errors recorded. The channel was assumed to haveone multipath component with no attenuation relative to thedirect path (α(k) = 1). This process was repeated many timeswith independent realizations of the ocean to arrive at theturbulence average BER presented in Fig. 8. In the figure, weshow theory and simulation versus SNR, or α(d)Et/σ

2w. The

number of transmitters was 100, 100, 200 and the numberof receivers was 100, 50, 100 giving β = 1, β = 0.5, andβ = 0.1.

−20 −15 −10 −5 0 5 10

10−4

10−3

10−2

10−1

100

α(d)Et/σ2

Pro

babi

lity

of E

rror

Theory (β’ = 1)Simulation (β’ = 1)Theory (β’ = 0.5)Simulation (β’ = 0.5)Theory (β’ = 0.1)Simulation (β’ = 0.1)

Fig. 8. BER vs SNR - A comparison of Monte-Carlo simulation and theoryfor binary phase shift keying.

From the figure, we see very good agreement betweentheory and simulation. As we stated earlier, the theory providesan approximate solution to any system with a large but finitenumber of apertures. Here, we see the approximation is veryclose to the theory.

IV. CONCLUSION

High-frequency acoustic transmission over the ocean chan-nel has the potential to provide intermediate range communi-cation links for a myriad of applications. Such communication,however, is prone to long (up to 1s) and deep (10-20dB) fadesand is susceptible to multipath induced ISI. In this paper,we have shown that a sparse aperture transmitter and re-ceiver system with wavefront predistortion provides protectionagainst fading and multipath interference in addition to betterperformance (average BER) compared with a single aperturesystem.

We showed that using receiver channel state information isfeasible for intermediate range acoustic communication links(up to 1km). Further we showed, for this link, that the numberof sparse apertures can practically be made large enough sothat asymptotic results approximate finite results well. Wepresented spatial modulation at the transmitter and spatial re-combination at the receiver that asymptotically minimize BER.This optimal scheme in fact transmits the spatially modulatedwaveform that propagates with the smallest diffraction losswhile, at the same time, reduces multipath interference to anylevel desired. Additionally, we showed that the interferencenoise power is reduced proportional to 1/nrx. Finally, wecalculated the asymptotic BER for the sparse aperture acousticsystem.

For many current and emerging applications, the data rateprovided by acoustic communication is less than desired. Assuch, future work might explore increasing the data rate bymodulating multiple spatial modes simultaneously. Codingacross the multiple spatial modes can provide a capacity

achieving architecture. The derivation of the channel capacitywill likely follow the wireless communication derivation forMIMO capacity found by [23]. We expect the additionalcomplexity required to implement a system that communicatesover multiple spatial modes to be justified by requirements ofemerging systems.

Other work could include an analysis of the effect of vehiclevelocity on link range. As vehicle velocities increase, thelink range will decrease because the observed channel statechanges more quickly than the coherence time. Additionally, inmost applications, system requirements on BER will determinethe number of transmitters and receivers. However, in theabsence of this constraint, establishing an optimal number oftransmitters/receivers in terms of BER per unit cost remainsan open research question.

ACKNOWLEDGMENT

Lisa Burton is very grateful to the National Science Foun-dation (NSF) for her Graduate Research Fellowship. LisaBurton and Pierre Lermusiaux also thank the Office of NavalResearch for research support under grants N00014-08-1-0680 (AWACS) and N00014-08-1-0680 (PLUS-INP) to theMassachusetts Institute of Technology. Andrew Puryear andVincent Chan would like to thank the Defense AdvancedResearch Projects Agency (DARPA) under the Technologyfor Agile Coherent Transmission Architecture (TACOTA) pro-gram and the Office of Navel Research (ONR) under the FreeSpace Heterodyne Communications program. The authors alsothank Sophie Strike.

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