Capacity analysis of underwater acoustic MIMOcommunications

Pierre-Jean BOUVET and Alain LOUSSERTUnderwater Acoustics Lab

ISEN BrestC.S. 42807, 29228 BREST cedex 2 - FRANCE

Email: pierre-jean.bouvet@isen.fr

Abstract—First introduced in the field of radio communica-tions, Multi-Input Multi-Output (MIMO) principle consists oftransmitting digital data from Nt transmitters to Nr receiverswithin the same frequency band. During the last decade, the-oretical works and real-life experiments in wireless local areaand cellular networks have demonstrated and confirmed thatMIMO was a breakthrough in digital communications. Recently,MIMO principle has been applied to underwater acoustic com-munications (UAC) with promising results. However, only fewwork have been already done on the expected gain of MIMOover underwater channel. Main objective of our paper is on theone hand to quantify theoretically the MIMO gain in underwateracoustic channel by using Shannon capacity analysis and on theother hand to give guidelines to optimize MIMO underwatersystem with respect to capacity maximization.

I. INTRODUCTION

Initially introduced during world war II for militaryneeds, underwater communications have today a growingnumber of civil and commercial applications such as remotecontrol in off-shore oil industry, pollutions monitoring inenvironmental systems, collection of scientific data recordedat ocean-bottom stations, communication between diversor underwater vehicles and mapping of the ocean floorfor detection of objects as well as for the discovery ofnew resources. As electromagnetic waves cannot propagateover long distances in seawater, underwater acoustic is thecore enabling technology for such applications. Howeverunderwater acoustic channel characteristics such as fading,extended multi-path bandwidth limitations or reverberationpose significant challenges to development of effective UACsystems [1] [2] [3].

On the other front, data rate required for UAC applicationsis continuously growing with the introduction of highquality images, real-time video as well as the deployment ofautonomous underwater networks such as ad-hoc deployablesensor networks or autonomous fleets of cooperatingunmanned undersea vehicles (UUV) [4]. Since underwateracoustic channel is very selective in frequency, transmissionthroughput could not be increased by simply increasingtransmission bandwidth. We can find a similar paradigm inwireless communication for which there is a huge requirementin high speed communication whereas usable bandwidth isrestricted by frequency allocation law. In radio frequency

field, this bandwidth limitation has been overcome by theintroduction of MIMO techniques which provides substantialgain in data rate while keeping the same transmissionbandwidth [5] [6] [7] [8]. In the past years, the contributionof MIMO to UAC systems has been deeply investigatedvia spatial modulation [9] or multi-carrier modulation [10].Initial simulation and experimental results demonstrate largegain over conventional single input single output (SISO) butthese results are strongly linked to the chosen modulationscheme and receiver algorithm as well as underwater channelenvironment.

Coming from the information theory, the channel capacityrepresents the tightest upper bound on the amount ofinformation that can be reliably transmitted over acommunications channel. Thus the channel capacity ofa given channel is the limiting information rate (in unitsof information per unit time) that can be achieved witharbitrarily small error probability regardless of the receiveralgorithm [11]. One of the questions that arises naturally iswhat is the maximum rate that a MIMO underwater acousticchannel can theoretically supported and how large is theMIMO contribution. While research has been extremely activeon assessing capacity of wireless radio transmission, onlyfew analysis have been reported mostly focusing on SISOchannel [12] [13]. Within the few available reference dealingwith MIMO underwater acoustic capacity, [14] providescapacity estimation using simulated and experimental databut under the assumption transmitter has a knowledge of thechannel transfer function which is not really feasible in UAC.Authors of [15] carry out capacity results both for SISO andMIMO channel but using a broadband propagation model thatdoes not reflect frequency selectivity brought by multi-pathpropagation. Intention of our paper is to provide a fulltheoretical analysis of the MIMO underwater capacity underthe assumption of shallow water acoustic (SWA) channelby taking into account both underwater propagation andmulti-path bandwidth limitation in order to give an accurateapproximation of maximum theoretical data rate expectedfrom a real-life MIMO UAC system.

Paper is organized as follows : section II describes theunderwater acoustic channel and how the latter affects a

978-1-4244-5222-4/10/$26.00 ©2010 IEEE

transmitted signal. In section III, we introduce a mathematicalmodel of SWA channel based on ray-theory. In section IV,we start from theoretical expression of the MIMO capacity inorder to derive an expression of MIMO SWA capacity. Finallysection V provides extended results covering several MIMOconfiguration and channel parameters.

II. CHANNEL CHARACTERISTICS

When compared to electromagnetic propagation through theatmosphere, underwater acoustic propagation is characterizedby significant frequency dependent perturbations and relativelyslow speed of propagation. Transmission loss and noise are theprincipal factors determining functional point of a underwatercommunication system, however the slow speed of wavepropagation leads to time-varying multipath phenomenon thatalso influences system design and generally imposes severelimitations on the system performance.

A. Transmission loss

The attenuation mechanisms that affects underwateracoustic signal can be viewed as sum of three terms : thespreading loss, absorption loss and reflection loss.

The spreading losses are due to the expansion of the fixedamount of transmitted energy over a larger area as the signalpropagates away from its source. It’s admitted that the energydecays at a rate of l−k where l is the distance and k thespreading factor describing the geometry of propagation (itscommonly used values are k = 2 for spherical spreading,k = 1 for cylindrical spreading and k = 1.5 for the so-calledpractical spreading).

The second mechanism of signal loss, called absorption loss,results from the conversion of the energy of the propagatingwave into heat. In case of under-sea acoustic transmission,the absorption loss is strongly linked to the wave frequency,such that the signal energy decay due to absorption lossis proportional to exp(−α(f)l) where α(f), referred as theabsorption coefficient, is an increasing function of frequency.The absorption loss can be expressed empirically using theThorp’s formula which gives α(f) in dB/km for a givenfrequency f in kHz [16] [17]:

10 log α(f) = 0.11f2

1 + f2+44

f2

4100 + f2+2.75·10−4f2+0.003

(1)This formula is valid for f > 400 kHz. For lower frequencies,the formula becomes:

10 log α(f) = 0.002 + 0.11f2

1 + f2+ 0.011f2 (2)

By hitting the sea-surface, sea-bottom or another under-seaobject, the sound wave is partially or totally reflecteddepending on the wave frequency, the sound speed and theobstacle type. The reflection loss for path p is denoted as Γp

and will be described more in depth is section III.

As a result, the total path loss that occurs in a under-seaacoustic channel is equal to Γp/

√A(l, f) where A(l, f) is

the sum of spreading and absorption loss:

10 log A(l, f) = k · 10 log l + l · 10 log α(f) (3)

B. Noises

There are several important natural sources of ambientnoise in the ocean at frequencies of interest for acousticcommunications. Usually four sources are considered: turbu-lence, shipping, waves and thermal noise.These four noisecomponents can be modeled by a colored Gaussian noise withfollowing empirical power spectral density (p.s.d.) given in dBre μ per Hz as a function of frequency in kHz [18] [13]:

10 log Nt(f) = 17 − 30 log f

10 log Ns(f) = 40 + 20(s − 0.5) + 26 log f − 60 log(f + 0.03)

10 log Nw(f) = 50 + 7.5w1/2 + 20 log f − 40 log(f + 0.4)10 log Nth(f) = −15 + 20 log f

where s is the shipping activity whose value ranges between0 and 1 for low and high activity respectively whereas w isthe wind speed expressed in m/s. The overall p.s.d. of theambient noise is noted N(f) and expressed as the sum of thefour above mentioned noise components:

N(f) = Nt(f) + Ns(f) + Nw(f) + Nth(f) (4)

This strong frequency dependency of the ambient noise is oneof major factor considered when selecting frequency bands forunderwater acoustics transmission.

C. Multi-paths propagation

In most environments and at the frequency of interestfor communications signals, the underwater channel resultsin multiple propagation paths from each source to receiver.In the first place, the multi-paths spread depends in thethe transmission link configuration designated as horizontalor vertical. While vertical links are subjected to little timedispersion, horizontal channels may exhibit a very longdelay spread. Mechanism of multi-paths formation are alsostrongly linked to the ocean depth: in case of deep water,multi-paths are formed by ray bending which occurs becausethe rays of sound tend to reach region of lower propagationspeed whereas in case of shallow water environment (suchas in the littoral region or the region of continental shelves),multi-path mechanism consists of reflections on surface-bottom bounces in addition to a possible direct path. Thedefinition of shallow and deep water is not a strict one, butit is generally assumed that shallow water corresponds to awater depth less than 100 m. In the following we will focuson horizontal link within SWA channel which presents a verychallenging communications scenario regarding the multiplepropagation path effects. Moreover we assumes a constantwater temperature within the whole water column whichimplies a constant sound speed (isovelocity).

In a digital communication system, multi-paths propagationcauses inter-symbol interference (ISI) that have to be carefullytreated at the receiver side in order to guarantee acceptableperformance [19]. As result, the design of underwater com-munication systems is accompanied by the use of propagationmodel for predicting the multi-path configuration. Ray theoryand the theory of normal modes provide basis for suchpropagation modeling [1]. At high frequencies, ray tracing isan appropriate model and is commonly used to determine thecoarse multi-paths structure of the channel. In the followingwe will consider medium range transmission (≤ 5 km) whichusually requires high frequency and justifies the use of raymodel as the basis of our channel propagation model.

D. Time-variation

Each propagation path may be also characterized by arandom fluctuation component due to the scattering onmoving sea-surface (waves or bubbles). Motion of thereflection points results in the Doppler spreading of thereflected signals leading to time-varying impulse response ofthe channel. One can note that time-varying effect is presentwithin the underwater channel regardless an eventual motionof transmitter or receiver. For simplicity reason, we’ll assumein our study a calm sea-surface such that arrivals of surfacescattered paths in the impulse response is both fairly stableand localized in delay.

III. MIMO CHANNEL MODEL

In this section we first develop a mathematical model forsignal propagation through a SWA channel based on raytracing model derived from ocean physics. Then we provide afilter-based representation of the channel and finally we extendthis model to MIMO architecture.

A. Propagation model based on ray tracing

Fig. 1. Channel representation

In the ray propagation model, sound energy isconceptualized as propagating along rays, which arestraight lines in the case of fluid medium with constant soundspeed (isovelocity) [20]. These rays are partially reflected andrefracted when they encounter a discontinuity in sound speed.We model the SWA channel as a wave guide consisting of anisovelocity layer (represented by the seawater) between twoisovelocity half spaces: air and seabed [21]. The isovelocityassumption is justified as shallow water channel are usually

well mixed and have relatively small increase in pressureover the depth of the water column.

Let L be the transmission range, D the water depth, zt andrespectively zr be the depth of the transmitter and the receiverrespectively. The distance traveled by the sound along variousrays can be computed using geometrical approach. Let dsb

be the distance along an upward originating eigenrays withs surface reflections and b bottom reflections. The distancealong a direct ray d00 is simply equal to:

d00 =√

R2 + (zt − zr)2 (5)

In the general case, if 0 ≤ s − b ≤ 1, the distance becomes

dsb =√

R2 + (2bD + zt − (−1)s−bzr)2 (6)

Inversely, if 0 ≤ b − s ≤ 1, we have:

dsb =√

R2 + (2bD − zt + (−1)b−szr)2 (7)

Attenuation coefficient due to reflection on surface denotedΓ+ is relatively small in magnitude since the impedancemismatch between the seawater and air. If the sea is calm,reflection coefficient tends to perfect reflection value 1. Ifthe sea surface is rough (due to waves), a small loss will beincurred for every surface interaction. This loss is modeledby a constant coefficient LSS .

On the seabed boundary, the coefficient reflection dependson the impedance variation from water to seabed. Such co-efficient estimation is given in [21] using Rayleigh reflexionlaw:

Γ− =

∣∣∣∣∣ρ1ρ cosθ −

√( c

c1)2 − sin2θ

ρ1ρ cosθ +

√( c

c1)2 − sin2θ

∣∣∣∣∣ (8)

where ρ and ρ1 are density of water and sea-bed respectively,c and c1 are sound speed in water and sea-bed respectivelywhereas θ is the incidence angle of reflected wave that can becomputed from the ray dsb as follows if 0 ≤ s − b ≤ 1 [21]:

θsb = tan−1

(L

2bD + zt − (−1)s−bzr

)(9)

and if 0 ≤ b − s ≤ 1:

θsb = tan−1

(L

2bD − zt + (−1)b−szr

)(10)

Additional reflection losses due to rough or absorbing seabottom is modeled by a constant coefficient LSB . Finally thetotal reflection loss for a path with s surface and b bottomreflections is equal to:

Γsb =(Γ+ · LSS

)s · (Γ− · LSB

)b(11)

The arrival time of each ray τsb is easily computed from thedistance along the ray and sound speed:

τsb =dsb

c(12)

B. FIR tap representation

By traveling from transmitter to receiver, each ray followsa path of length dsb with arrival time τsb and an attenuationof Γsb/

√A(dsb, f). The SWA channel is modeled by taking

account all possible paths and result in an Finite ImpulseResponse (FIR) filter with following transfer function [21]:

H(f) =1√

A(d00, f)e−j2πfτ00

++∞∑s=1

s∑b=s−1

Γsb√A(dsb, f)

e−j2πfτsb

++∞∑b=1

b∑s=b−1

Γsb√A(dsb, f)

e−j2πfτsb

For practical computation, we’ll assume that number of reflec-tions on surface and sea-bottom is finite and equal to smax andbmax respectively. Total number of paths is then equal to Pwith:

P = 1 + 2smax + 2bmax (13)

The channel transfer function can be rewritten as a closed-from expression:

H(f) =P−1∑p=0

Γp√A(lp, f)

e−j2πfτp (14)

where Γp , A(dp) and τp are the total reflection loss, themedium attenuation, and the arrival time of path p respectively.As an example, figure 2 and 3 represents the channel impulseresponse (CIR) and frequency response obtained of the mod-eled SWA channel with parameters described in section V.The spectrum response exhibits a slow decrease in frequencyexplained by the frequency dependency of A(l, f).

−2 −1 0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 10−5 Channel Impulse Response

t [ms]

mag

nitu

de

Fig. 2. Example of Channel Impulse Response (CIR) of SWA channel

C. Extension to MIMO

We consider now a MIMO underwater transmission archi-tecture with Nt transmit hydrophones and Nr receive hy-drophones. On the transmit and receive sides, each hydrophonepair has a vertical and constant separation of Δzt and Δrt

−10 −5 0 5 10−100

−95

−90

−85

−80

−75

−70

−65

−60

−55

−50

−45Channel Frequency Response

freq [KHz]

|H(f

)| [d

B]

Fig. 3. Example of frequency response of SWA channel

respectively. The transmitter and receiver depths zt and zr

correspond to the middle of each array as shown in figure 4.The MIMO array is placed in vertical direction in order tomaximize delay spread difference between each sub-channelsand thus minimize spatial correlation.

Fig. 4. MIMO SWA Channel representation

The MIMO channel is modeled by Nt × Nr sub-channels,where each sub-channel corresponds to the SWA channelmodel described in previous section. As a result, the transferfunction of sub-channel linking hydrophone m ∈ [1, Nt] tohydrophone n ∈ [1, Nr] denoted Hmn(f) is simply derivedfrom equation (14) as follows:

Hmn(f) =P−1∑p=0

Γp√A(dp,mn, f)

e−j2πfτp,mn (15)

where dp,mn is the distance along the p-th path of sub-channelmn and τp,mn = dp,mn/c. For simplicity, we assume eachsub-channel to experience the same path number P . In order totake into account geometry of the MIMO array, each distancedp,mn is computed by replacing in (5), (6), (7), (9) and (10)zt and zr by ztm and zrn respectively:

zt → ztm = zt − (Nt + 1 − 2m)Δzt

2(16)

zr → zrn = zr − (Nr + 1 − 2n)Δzr

2(17)

Finally the whole MIMO channel is represented by the fol-

lowing Nr × Nt channel matrix:

H(f) =

⎡⎢⎣

H11(f) . . . HNt1(f)...

. . ....

H1Nr(f) . . . HNtNr

(f)

⎤⎥⎦

Nr×Nt

(18)

IV. CAPACITY COMPUTATION

The channel capacity represents the maximum amountof information that can be reliably transmitted over acommunications channel [11]. This quantity is defined asthe maximization of the average mutual information I(x,y)between input signal x and output signal y of the channelover the choice of the distribution of x.

In the following, we derive the capacity formula of SWAMIMO channel by taking into account all characteristicsdeveloped in section II. In order to evaluate relatively the SWAMIMO capacity, we also remind some useful capacity boundsboth for SISO and MIMO channel.

A. SWA MIMO Channel

In case of deterministic MIMO, E. Telatar derived thegeneral expression of the MIMO capacity [5]:

C = log2

(INt

+SNR

NtHRssHH

)[bits/s/Hz] (19)

where INtis the Nt ×Nt identity matrix, SNR is the signal-

to-noise ratio, H is MIMO matrix, Rss is the covariancematrix of input signal and (.)H denotes the transposeconjugate operand. As demonstrated by C. E. Shannon inthe SISO case, E. Telatar shown that mutual information ismaximized by choosing input signal vector s as independentand zero mean Gaussian samples [5].

Let be λi the eigen values of H with 0 ≤ i < r wherer = rank(H), expression (19) can be expanded as [5]:

C =r−1∑i=0

(1 +

SNR

Ntλi

)(20)

Since the right part of latest expression is homogeneous to aSISO capacity (see [11]), the MIMO capacity can be viewedas a sum of r SISO capacities. The MIMO capacity is thusmaximized when rank of H reaches its maximum valuewhich is equal to max(r) = min(Nt, Nr). This leads to thewell-known property of MIMO system: at it’s bests MIMOcapacity gain is increasing linearly with minimum numberof transmit and receive sensors [6]. And more generally,advantageous of MIMO depends on matrix H, the larger rankand eigenvalues HHH have, the more MIMO capacity wehave [7].

If the channel could be known at the transmit side (withfor example a feedback loop), capacity could be maximizedby adjusting the covariance matrix RSS of transmit signalwith water-filling techniques [5] [6]. However in UWAcommunications, the high variability and latency of the

channel makes difficult any feedback from receiver totransmitter such as channel matrix is usually assumed asunknown for the transmitter.

In the absence of channel state information at the transmitterit is reasonable to choose the transmit signal s to be whitein space and in frequency directions: Rss = INt

. The totaltransmit power Ps is thus assumed to be uniformly distributedover the bandwidth B and the Nt transmit hydrophones suchas the power profile density (p.s.d.) of signal transmitted fromhydrophone m ∈ [1, Nt] is expressed as:

S(f) =PS

B · Nt, ∀f ∈

[f0 − B

2 , f0 + B2

](21)

where f0 is center carrier frequency around which the signalis transmitted. At receive side, the total power receive fromhydrophone n ∈ [1, Nr] is equal to:

PRn =∫ f0+B/2

f0−B/2

Rn(f)df (22)

where Rn is received signal p.s.d. viewed by hydrophone n:

Rn(f) =Nt∑

m=1

|Hmn(f)|2S(f) (23)

On the other front, the total noise power is:

PN =∫ f0+B/2

f0−B/2

N(f)df (24)

As a result the the signal-to-noise ratio viewed at receivehydrophone n ∈ [1, Nr] can be expressed as:

SNRn =PRn

PN(25)

whereas the SNR p.s.d. is:

SNRn(f) =Rn(f)N(f)

(26)

Previous capacity formulas are provided under the assumptionthat MIMO channel described by matrix H is flat fading. Letnow consider a tone f within the transmit bandwidth for whichthe SWA channel can be considered flat, formula (19) can berewritten as follows:

C(f) = log2

(INt

+S(f)

Nt · N(f)H(f)H(f)H

)(27)

where N(f) is the underwater noise p.s.d. described in sec-tion II whereas H(f) is the SWA channel transfer functionintroduced in section III. The total channel capacity is obtainedby integrating C(f) over the transmit bandwidth B:

C =∫ f0+B/2

f0−B/2

C(f)df [bits/s/Hz] (28)

B. SWA SISO Channel

The Shannon capacity of the SWA SISO channel is obtainedby particularizing (27) and (28) to the single hydrophone casei.e. Nt = 1 and Nr = 1 leading to the following expression:

C0 =∫ f0+B/2

f0−B/2

log2

(1 +

S0(f)N(f)

∣∣H11(f)∣∣2)df [bits/s/Hz]

(29)where the p.s.d. of the transmitted signal becomes:

S0(f) =PS

B, ∀f ∈

[f0 − B

2 , f0 + B2

](30)

Please note that more detailed results on capacity of SWASISO channel can be found in [13].

C. Upper bounds

In the absence of channel knowledge at the transmit side,capacity of an arbitrary SISO channel is bounded by the so-called AWGN capacity obtained over a memoryless additivewhite Gaussian noise (AWGN) channel and expressed in [11]as:

C̄0 = log2

(1 +

PS

PN

)[bits/s/Hz] (31)

By using (20) and (31), one can demonstrated capacity of anarbitrary MIMO channel is bounded by:

C̄ = min(Nt, Nr) · log2

(1 +

PS

PN

)[bits/s/Hz] (32)

This result could be interpreted as the optimum MIMO schemewere equivalent to min(Nt, Nr) AWGN channels operating atseparate bandwidths.

D. Rayleigh bounds

In wireless communications, a usual channel model con-sists of modeling frequency fading as i.i.d. random Rayleighcoefficients. Each channel flat coefficient hmn of matrix Hare modeled as uncorrelated complex Gaussian samples withzero mean and unit variance. Channel capacity is computedby taking the expectation of instantaneous capacity over therealization of H:

C̃ = EH

[log2

(INt

+PS

NtPNHHH

)][bits/s/Hz] (33)

Such capacity is called ergodic since it’s assumed that thchannel transfer function obeys to an ergodic random process.As previously, the equivalent SISO capacity is obtained byparticularizing to the single hydrophone case:

C̃0 = Eh00

[log2

(INt

+PS

NtPN

∣∣h11|2)]

[bits/s/Hz] (34)

V. SIMULATION RESULTS

A. Simulation set-up

Channel parameters chosen for simulation works are sum-marized in Table I. These parameters are derived from MIMOexperiments made in [10]. SWA channel capacities are numeri-cally evaluated using (28) and (29) with a frequency resolutionequals to Δf = 7.6 Hz. We use a quite high frequencyresolution in order to take into account large frequency fadingsbrought by SWA channel.

TABLE ISIMULATION PARAMETERS

Letter Name Default value

F0 Carrier frequency 32 kHz

B Signal Bandwidth 12 kHz (Variable)

Fs Sampling frequency 500 kHz

Δf frequency resolution 7.6 Hz

L Range 1500 m (variable)

D Water depth 20 m (variable)

Nt Number on transmit hydrophones 2

Nr Number on receive hydrophones 2

zt Transmit array depth 9 m

Δzt Vertical separation of transmit array 0.6 m (variable)

zr Receive array depth 9 m

Δzr Vertical separation of receive array 0.6 m (variable)

k Spreading factor 1.5

c Sound speed in water 1500 m/s

ρ Water density 1023 kg/m3

c1 Sound speed in sea-bottom 1650 m/s

ρ1 Sea-bottom density 1500 kg/m3

LSS Absorption loss at sea-surface −0.5 dB

LSB Absorption loss at sea-bottom −3 dB

B. Capacity of the SWA MIMO channel

Figure 5 shows the MIMO 2 × 2 capacity (C) and SISOcapacity (C0) results of a SWA channel with the defaultparameters described in table I. As references, 2 × 2 AWGNcapacity (C̄), SISO AWGN capacity (C̄0), 2 × 2 Rayleighi.i.d. capacity (C̃) and SISO Rayleigh i.i.d. capacity (C̃0) arealso plotted in the same figure. First, we can observe the largecapacity gain brought by MIMO architecture. As an examplefor SNR = 15 dB, SWA channel capacity is increasedfrom 3.9 to 8 bits/s/Hz representing a maximum MIMOthroughput of 96 kbits/s with the chosen bandwidth of 12kHz. Thus the MIMO gain is about 103 % demonstratingthe SWA capacity is increasing linearly with min(Nt, Nr) asshown in section IV. As a result, larger could be obtained byincreasing the number of hydrophones at transmit and receiveside.

By comparing with the Rayleigh bounds, one can see thatC curve is quite close to C̃ whereas C0 is lower than C̃0.For the SISO case, capacity result confirms the limitation touse a Rayleigh fading model to represent the SWA channelas already pointed out in literature by experimental results[20]. However due to the high fading diversity brought bythe multi-sensors approach, MIMO SWA channel tends toapproach MIMO Rayleigh i.i.d. model. Finally, both SISO andMIMO SWA curves are staying quite far from AWGN boundswhich is note very surprising since SWA channel remains avery challenging and difficult communication channel.

0 5 10 15 200

1

2

3

4

5

6

7

8

9

10

Signal−to−noise ratio (SNR) [dB]

Cap

acity

(C

) [b

it/s/

Hz]

2×2 AWGN2×2 Rayleigh i.i.d. channel2×2 SWASISO AWGN channelSISO Rayleigh i.i.d. channelSISO SWA

Fig. 5. MIMO 2 × 2 and SISO capacities as function of SNR. Defaultparameters are set for SWA channel model.

C. Bandwidth and communication range

As described extensively in section II, sound propagationin water is very sensitive to the wave frequency. Thisdependency is illustrated by the fact that SNRn(f) includeson the one hand a frequency dependent part represented byquantity A(l, f)N(f) also called in the literature AN product[13] and on the other hand a frequency fading coefficientexp(−j2πfτp) . In a MIMO configuration, this dependence isillustrated in figure 6 where SNR(f) expression is plotted asfunction of frequency for increasing values of communicationrange D. As in SISO case, SNR becomes more frequencyselective by increasing the communication range leading to asmaller bandwidth usable for transmission. At the same time,figure 6 shows that operating frequencies are also decreasingfor long range value meaning that high frequencies arededicated for small or medium range communications. It isalso interesting to notice that SNR becomes smoother for highvalue of D, demonstrating that for long range, the dominantfactor is not longer the frequency fading but transmission lossrepresented by the AN product.

A second illustration is obtained by plotting in figure 7

0 5 10 15 20−200

−150

−100

−50

Frequency [kHz]

SN

R(f

)

1.5 km10 km50 km100 km

Fig. 6. SNR(f) at output of 2×2 SWA channel as a function of frequency

MIMO and SISO capacities as function of D for a fixedSNR of 15 dB. As expected both SISO and MIMO SWA

capacities decrease when D increases. This phenomenon isa classical feature that distinguishes an underwater acousticsystem from a terrestrial radio one : UAC capacity has astrong dependency in the transmission distance [13]. Howeverone can see that MIMO SWA capacity fades much rapidlythe SISO one meaning the frequencies used for MIMOtransmission have to be carefully chosen as function of thedesired range.

0 2 4 6 8 100

2

4

6

8

10

12

14

Range [km]

Cap

acity

(C

) [b

it/s/

Hz]

2×2 AWGN2×2 Rayleigh i.i.d. channel2×2 SWASISO AWGN channelSISO Rayleigh i.i.d. channelSISO SWA

Fig. 7. MIMO and SISO Capacities as function of communication range,SNR = 15 dB

D. Influence of water depth

Figure 8 provides capacity results for varying value of waterdepth D. One can see that MIMO capacity gain is slowlydecreasing as water column depth increases. The largest gainis obtained for very shallow water (≤ 30 m). This result iseasily explained by the fact that MIMO capacity is stronglylinked to the correlation between sub-channels mathematicallyrepresented by matrix determinant of HHH : the lower is thespatial correlation the larger is the MIMO gain (see sectionIV). The lowest spatial correlation is reached in case of richmulti-paths environment which happens when channel waterdepth is very shallow. However some specific combinationsof path delays lead to low correlation configurations whichexplain local maximums observed along the MIMO SWAcurve.

50 100 150 200 250 300 350 400 450 5003

4

5

6

7

8

9

10

11

12

13

14

Water depth [m]

Cap

acity

(C

) [b

it/s/

Hz]

2×2 AWGN2×2 Rayleigh i.i.d. channel2×2 SWASISO AWGN channelSISO Rayleigh i.i.d. channelSISO SWA

Fig. 8. MIMO and SISO Capacities as function of water depth, SNR = 15dB

E. Optimizing MIMO architecture

As illustrated previously, MIMO capacity is function ofthe correction between each sub-channels linking each multi-sensors. Since reflection interfaces (Sea-surface and sea-bed)are situated in the vertical direction relatively to hydrophones,the lowest spatial correlation configurations are obtained withhydrophone array oriented in vertical direction. Furthermore,an easy way to ensure uncorrelated links is to space sufficientlyeach hydrophones within the transmit or the receive arrays asshown in figure 9 where MIMO SWA capacity is plotted asfunction of Δzt and Δzr (which are supposed to be equal).One can see that perfect uncorrellation is obtained whenhydrophone separation is greater than 0.8 m. The oscillationphenomenon observed for large value of Δzt and Δzr isexplained by the phase combination of Hmn(f) which islinked to distance and leads to periodic variation of thecorrelation ratio. On the other hand it is interesting to noticethat low-spaced array leads to capacity which remains largerthan the SISO case. In fact, for the extreme value of Δ = 0,MIMO SWA capacity is 4.8 b/s/Hz whereas SISO SWAcapacity is equal to 3.93 which still represents a capacity gainof 22 %.

0 2 4 6 8 102

4

6

8

10

12

14

16

Hydrophones vertical separation [m]

Cap

acity

(C

) [b

it/s/

Hz]

2×2 AWGN2×2 Rayleigh i.i.d. channel2×2 SWASISO AWGN channelSISO Rayleigh i.i.d. channelSISO SWA

Fig. 9. MIMO and SISO Capacities as function of hydrophones verticalseparation (Δzt = Δzr), SNR = 15 dB

VI. CONCLUSIONS

The primary aim of this study was to quantify and predictgain brought by MIMO technology for underwater acousticcommunication. By using the ray-theory approach andunder the assumption of shallow water environment, thepaper provides a model of the MIMO underwater acousticchannel taking into account all the under-sea dominantperturbations. A numerical evaluation of the underwaterchannel capacity is then proposed leads to an accurateapproximation of maximum achievable throughput for agiven set of underwater communication parameters. Extensivesimulation results for several transmission parameters andchannel configuration show the expected MIMO gain issubstantial and the capacity increase is in the same order asin wireless transmission. The study demonstrated also thatsome restrictions on frequencies, communication range as

well as underwater environment have to be made in orderto ensure MIMO capacity gain. Especially, medium rangetransmission over shallow water channel seems to be well adedicated system to MIMO transmission.

With this capacity based approach, our paper gives a simplemean to provide an upper bound of the expected MIMO gainin the field of undersea acoustics and forms a useful tool todesign and optimize future MIMO underwater communicationsystem.

REFERENCES

[1] R. F. W. Coates, Underwater Acoustic Systems. Wiley, 1989.[2] M. Stojanovic, “Recent advances in high-speed underwater acoustic

communications,” IEEE J. Ocean. Eng., vol. 121, no. 2, pp. 125–136,Apr. 1996.

[3] R. S. H. Istepanian and M. Stojanovic, Underwater acoustic digitalsignal processing and communication systems. Kluwer Academic Pub.,2002.

[4] I. F. Akyildiz, D. Pompili, and T. Melodia, “Underwater acoustic sensornetworks: research challenges,” Ad Hoc Network Jorunal, vol. 3, no. 3,pp. 257–279, 2005.

[5] E. Telatar, “Capacity of multiantenna gaussian channel,” Bell Labs. Tech.Memo., Jun. 1995.

[6] G. J. Foschini, “Layered space-time architecture for wireless communi-cation in a fading environment when using multielement antennas,” BellSyst. Tech. Journal, vol. 1, pp. 41–59, Oct. 1996.

[7] D. Gesbert, M. Shafi, D. Shiu, and P. Smith, “From theory to practice:An overview of space-time coded mimo wireless systems,” IEEE J. Sel.Areas Commun., Apr. 2003.

[8] A. Paulraj, A. D. Gore, R. U. Nabar, and H. Boelcskei, “An overviewof MIMO communications: A key to gigabit wireless,” Proceedings ofthe IEEE, vol. 92, no. 2, pp. 198–218, Feb. 2004.

[9] D. B. Kilfoyle, J. C. Preisig, and A. B. Baggeroer, “Spatial modulationexperiments in the underwater acoustic channel,” IEEE J. Ocean. Eng.,vol. 30, no. 2, pp. 406–415, Apr. 2005.

[10] B. Li, J. Huang, S. Zhou, K. Ball, M. Stojanovic, L. Freitag, and P. Wil-lett, “MIMO-OFDM for high rate underwater acoustic communications,”IEEE J. Ocean. Eng., vol. 34, no. 4, pp. 634–644, 2009.

[11] C. E. Shannon, “A mathematical theory of communications,” Bell Syst.Tech. Journal, vol. 27, pp. 379–423 and 623–656, Jul. 1948.

[12] T. J. Hayward and T. C. Yang, “Underwater acoustic communicationchannel capacity: A simulation study,” in Proceedings of High FrequencyOcean Acoustics Conference, vol. 728, 2004, pp. 114–121.

[13] M. Stojanovic, “On the relationship between capacity and distance inan underwater acoustic communication channell,” ACM SIGMOBILEMobile comput. and Comm. Review (M2CR), vol. 11, pp. 34–43, Oct.2007.

[14] M. Zatman and B. Tracey, “Underwater acoustic mimo channel capac-ity,” in Proceedings of 36th Asilomar conf., vol. 2, 2002, pp. 1364–1368.

[15] T. J. Hayward and T. C. Yang, “Single- and multi-channel underwateracoustic communication channel capacity: A computational study,”Journal of the Acoustical Society of America, vol. 122, no. 3, pp. 1652–1661, Sep. 2007.

[16] W. H. Thorp, “Analytic description of the low frequency attenuationcoefficient,” Journal of the Acoustical Society of America, vol. 33, pp.334–340, 1961.

[17] F. H. Fisher and V. P. Simons, “Sound absorption in seawater,” Journalof the Acoustical Society of America, vol. 62, pp. 558–564, 1977.

[18] R. J. Urick, Ambient Noise in the Sea. Peninsula Pub, 1986.[19] J. G. Proakis, Digital Communications. Mc Graw Hill, 1995.[20] X. Geng and A. Zielinski, “An eigenpath underwater acoustic commu-

nication channel model,” in Proceedings of OCEAN’95, vol. 2, 1995,pp. 1189–1196.

[21] M. A. Chitre, “A high-frequency warm shallow water acoustic commu-nications channel model and measurements,” Journal of the AcousticalSociety of America, vol. 5, no. 122, pp. 2580–2586, 2007.