Relay-Aided Cooperative Underwater Acoustic Communications:Selective Relaying †

Xilin Cheng∗, Rui Cao‡, Fengzhong Qu§, and Liuqing Yang∗∗ Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA

xlcheng@lamar.colostate.edu, lqyang@engr.colostate.edu‡LSI corporation, Milpitas, CA 95035, USA, ray.cao@lsi.com

§ State Key Laboratory of Fluid Power Transmission and Control,Institute of Underwater Technology and Ship Engineering, Zhejiang University, China, jimqufz@zju.edu.cn

Abstract—Underwater acoustic communications (UAC) fea-tures frequency-dependent signal attenuation, long propagationdelay, and doubly-selective fading. Thus, the design of reliableand efficient UAC protocols is very challenging. On the otherhand, cooperative relay communications, which can provide reli-able data transmission, is a very attractive technology for UAC. Inour recent work, we proposed a practical asynchronous relayingprotocol tailored for UAC: Asynchronous Amplify-and-forwardrelaying with Precoded OFDM (AsAP). This protocol resolvesboth the time synchronization difficulty and the frequency-selective issue of UAC channels. However, the AsAP protocoladopts fixed amplification and uniform power allocation amongthe source and the relays, which limits the system performance.Thus, in this paper, we design a modified AsAP system with aninstantaneous amplification scheme, and investigate the optimumpower allocation based on the criteria of maximum average SNRat the destination. Assuming statistical channel state information(CSI) is available, we analyze the average SNR at the destinationand perform the power optimization on the AsAP system. Theanalytical results suggest a selective relaying scheme, where onlythe relay with best average SNR forwards the source information.Therefore, we propose an AsAP protocol with selective relaying(SR-AsAP). Finally, the performance of the new SR-AsAP schemeis simulated, and the benefits are illustrated through comparisonwith the direct-link system and the AsAP protocol.

I. INTRODUCTION

Underwater acoustic sensor networks (UWASN) have beenattracting growing research interests in recent decades dueto emerging applications including remote control in off-shore oil fields, pollution monitoring, ocean observation, andmilitary purposes, e.g., surveillance and intrusion detection[1], [2]. Underwater acoustic communications (UAC), whichadopt acoustic waves as the information carrier, is a promisingcommunication technique for UWASN.

The particular features of acoustic signal propagation andthe underwater environment bring formidable challenges toUAC. First, the acoustic signal is characterized by low carrierfrequency and distance-dependent bandwidth. This signifi-cantly limits UAC capacity, especially at long transmissiondistances [3], [4]. Secondly, long and variant delay, causedby slow acoustic signal propagation (around 1500 m/s) andenvironmental effects, hampers accurate time synchronization

† This work is in part supported by National Science Foundation undergrant #1129043, National Nature Science Foundation of China under grant#61172105 and #61001067, and Scientific Research Fund of Zhejiang Provin-cial Education Department under #Y201122658.

and well-coordinated medium access [5], [6]. Finally, thedoubly-selective nature of UAC channels causes transmissionsto be error-prone [7]. Therefore, the design of UAC protocolsneeds to achieve reliable and efficient data transmission withrelaxed synchronization requirements.

Cooperative relay communications, which can provide re-liable data transmission with extended distances, have beenextensively studied in terrestrial wireless communications [8].As a promising solution for long-range reliable UAC, coop-erative relay communications have aroused growing interestfrom UAC researchers. Studies [6], [9], [10], [4] demonstratethat cooperative relay communications can improve channelcapacity and data reliability for UAC. In our recent work [4], apractical relay-aided UAC (RA-UAC) protocol, AsynchronousAmplifiy-and-forward (AF) relaying with Precoded OFDM(AsAP), is designed to achieve reliable data communicationsand combat time asynchronism. In this scheme, precodedOFDM resolves the frequency selectivity issue of UAC chan-nels. Additionally, the asynchronous AF relaying solves timesynchronization difficulties and facilitates relay processing,i.e., all relays amplify and forward the received signal to thedestination asynchronously without any time coordination withother relays. However, there are two design factors in AsAPthat can lead to degraded system performance. First, the relayamplification factor is fixed based on source-relay channelstatistics. Secondly, all relays are assumed to participate inthe transmission, and the transmit power is uniformly allocatedamong the source and the relays. Although these simplify thesystem implementation, the end-to-end system performance isconstrained.

In this paper, we design a modified AsAP protocol whichadopts adaptive amplification based on instantaneous source-relay channel state information (CSI). To further improveperformance, maximum ratio combining (MRC) is used toform the decision statistics from the direct-link and relay-linksignals. In addition, we investigate the optimal transmit powerallocation of the modified AsAP protocol. As UAC channelsare fast-varying and have slow signal propagation speed,transmitter-side instantaneous CSI is not available. Thus, weformulate a power optimization problem, which maximizesthe end-to-end average signal-to-noise ratio (SNR), based onthe assumption that complete statistical CSI is known at thesource. Analytical results indicate that the optimum solution

978-1-4577-2091-8/12/$26.00 ©2011 IEEE

is to allocate all relay transmit power to the relay with themaximum average SNR. This corresponds to the selectiverelaying scheme. According to this observation, we proposea selective relaying AsAP protocol (SR-AsAP) particularlydesigned for UAC. To verify the benefits of the SR-AsAPprotocol, we simulate the error performance and compare itwith the direct-link system and the AsAP system [4].

The rest of the paper is organized as follows. The under-water acoustic channel model is introduced in Section II. Themodified AsAP protocol is presented in Section III. Then thepower optimization is studied in Section IV. After that, the SR-AsAP protocol is designed and its performance is simulatedin Section V. The summarizing remarks are given in SectionVI.Notation: Superscripts (·)T and (·)H stand for the transposeand the Hermitian transpose, respectively; IN is the N × Nidentity matrix; 0M×N denotes the all-zero matrix of size M×N ; FN represents the N ×N DFT matrix with the (m,n)thentry being 1√

Ne−j 2π

N (m−1)(n−1); diag(h) is a diagonal matrixwith h on its diagonal; ‖ · ‖ represents the Frobenius norm;1M denotes the all-one column vector of size M × 1.

II. UAC CHANNEL MODEL

Compared to terrestrial wireless channels, UAC channelsexhibit different features: frequency-dependent signal attenu-ation and multipath channel with sparse delay taps [3], [11].

A. Signal Attenuation Model

Underwater acoustic signal attenuation A is dependent onboth distance D and frequency f :

A(D, f) = Dka(f)D, (1)

where k is the path loss exponent, which reflects the geometryof acoustic signal propagation. In this paper, k = 1.5 is usedfor practical spreading. The frequency dependency is capturedby a(f), which is given by Thorp’s formula [12]:

10loga(f) =0.11f2

1+f2+

44f2

4100+f+2.75·10−4f2+0.003.(2)

In Eqs. (1) and (2), distance D is in km, and frequency f isin kHz.

B. Multipath Channel Model

Due to the slow propagation of underwater acoustic waves,signals reflected from the sea surface and bottom arrive atthe receiver with distinct delays. This results in a sparsemultipath channel. Here we consider the discrete-time signalmodel of UAC channels. A UAC channel of length L isrepresented as h = [h0, · · · , hL−1]

T , within which only Lnz

taps are nonzero. Each nonzero tap hl, l ∈ {l0, · · · , lLnz−1}corresponds to one arrival, and has the independent complexGaussian distribution with zero mean and variance σ2

l , i.e.,hl ∼ CN (0, σ2

l ). The variance is computed as:

σ2l =

Γ2l

A(dl, fc), (3)

s...

r1

rR

d

x GLCP FH

Source

Hsr +

ωr

F Ar FH

Relay

Hrd +

ωd

yd

Fig. 1. System topology for the AsAP protocol

where A(dl, fc) is the attenuation computed in Eq. (1), dl isthe propagation distance of the lth arrival, which is determinedby the number of reflections with the channel geometry de-scribed in [13], and fc is the carrier frequency. Γl = (1/

√2)rl

is the refection loss of acoustic waves at the surface and thebottom, where rl is the number of reflections for path l.

For a communication system with OFDM as the physi-cal layer transmission scheme, the channel response in thefrequency domain is given as h =

√NFNPh, where N

is the subcarrier number, and P = [IL 0L×(N−L)]T is

the zero-padding matrix. The kth element of h is hk =∑L−1n=0 e

−j 2πN (k−1)nhn, k = 1, · · · , N , which is complex

gaussian distributed, i.e., hk ∼ CN (0, η) where η =∑Lnz−1n=0 E[|hln |2] =

∑Lnz−1l=0 σ2

l . η is the total energy ofall channel paths and can be computed based on knowledgeof the transmitter and receiver locations as well as the seageometry. Also, it is notable that each element of h has thesame variance.

The unique features of underwater acoustic channels havegreat impacts on the design of energy-efficient RA-UACprotocol. In the following sections, we first present the modi-fied AsAP protocol, and then investigate the optimum powerallocation among the source and the relays.

III. ASAP PROTOCOL

In AsAP [4], we consider a dual-hop relay system setup withone source node s, R relay nodes ri, i ∈ {1, . . . , R}, and onedestination node d, as shown in Fig. 1. We assume that thechannel for each link is block-stationary, different channelsare independent, and the noise is additive white complexGaussian with zero mean and variance σ2. hi,j , i, j ∈ {s, r, d}represents the channel vector between node i and j. The AsAPtransmission consists of three stages: source transmission,relay amplifying and forwarding, and destination decoding.The process of all three stages is elaborated upon in thefollowing subsections.

A. Source transmission

At the source, each generated OFDM symbol x is precodedwith grouped linear constellation precoding (GLCP) and then

broadcast to the relays and the destination. GLCP consists oftwo steps: grouping and precoding. The grouping is performedas follows. For a precoding size K , the data symbol x oflength N with elements chosen from a finite constellationset A is first divided into consecutive K blocks, each ofsize M , and then the mth (m ∈ {1, . . . ,M}) element fromeach block is selected to form a group, denoted by a vectorxm. This procedure can be mathematically represented as:xm = Ψmx, where Ψm = IN (Sm, :) is a K × N selec-tion matrix with K rows chosen from an N × N identitymatrix IN , and the indices of the K rows defined in theset Sm. For optimal grouping, each row set is chosen asSm = {m,m+M, · · · ,m+ (K − 1)M}. In the second step,each group vector xm is encoded with the precoder matrix Θof size K × K . We adopt the optimal grouping rule in [14]to design Θ, which maximizes the diversity and coding gain.Then, the coded groups are reassembled to form the precodedsymbols xs. The entire GLCP process can be presented as:

xs =

M∑m=1

ΨTmΘΨmx = Ψx, (4)

where Ψ =∑M

m=1 ΨTmΘΨm.

To overcome intersymbol interference (ISI), a cyclic prefix(CP) is inserted after taking the IFFT of xs. Then the gen-erated time-domain symbol is broadcast to the relays and thedestination with transmit power Pk

s , k ∈ {1, · · · , N} at kth

subcarrier.

B. Relay Amplifying and forwarding

After receiving the signal from the source, each relay ampli-fies the signal to compensate the S-R channel fading. Differentfrom [4], instantaneous frequency domain (FD) amplificationis chosen to enhance performance.

In FD amplification, after normal CP removal and FFT, thefrequency domain signal received by relay ri is representedas:

yri = Hs,ri

√P sxs + nri , (5)

where Hs,ri =√Ndiag(FNhs,ri) is the diagonal S-R fre-

quency domain channel matrix, P s = diag(P1s ,P2

s , · · · ,PNs )

is the source transmit power matrix with P is being the transmit

power on subcarrier i, and nri is the independent and iden-tically distributed (i.i.d.) frequency domain noise vector, i.e.,nri ∼ CN (0, σ2

riIN ).The frequency domain amplification signal is computed as:

xri = Ariyri . The amplification factor Ari is a diagonalmatrix with the amplification magnitude for each subcarrieras the diagonal entries. The amplification magnitude valuesare chosen to assure that the relaying signal power com-plies with the relay transmit power constraint. Suppose thatthe power constraint is imposed on each subcarrier k, i.e.,E|Ari(k, k)yri(k)|2 = Pk

ri . Then the amplification factor isobtained as:

Ari =

√P ri

|Hs,ri |2 P s + σ2riIN

, (6)

OFDM SymbolCP

OFDM SymbolCP

tsdt

CPt OFDMt

OFDM SymbolCP

OFDM SymbolCP

1sr dt

2sr dt

3sr dt

Fig. 2. Arrival time of signals at the destination with relay numberR = 3

where P ri = diag(P1ri ,P2

ri , · · · ,PNri ) is the transmit power

matrix at relay ri, and |Hs,ri |2 is the instantaneous channelgain, which can be obtained through channel estimation.

After amplification, the forwarding signal in the time do-main is generated by performing IFFT of xri and CP insertion.All relays forward the signals to the destination asynchronous-ly.

C. Destination decoding

Finally, the destination collects all the asynchronously ar-riving signals from the source and the relays and decodesthem. Due to the reception and processing delay at relays,it can be shown that the direct-link signal and relayed signalsare separated temporally. Suppose that the direct-link signaland the signal from relay ri reach the destination with delaytsd and tsrid respectively. According to the AsAP systemmodel, tsrid consists of S-R signal propagation time tsri ,signal reception time which is equal to the summation of CP’sduration tCP and OFDM symbol duration tOFDM, relay signalprocessing time tp, and R-D signal propagation time trid, i.e.,tsrid = tsri + tOFDM + tCP + tp + trid. As tsri + trid ≥ tsd,the following inequality tsrid − tsd ≥ tCP + tOFDM + tp canbe obtained. Figure 2 illustrates the arrival time of the signalsat the destination, and it is observed that the direct-link signaland the relayed signals are separated temporally. Thus, thedirect-link signal and the relayed signals can be combinedusing diversity techniques, and MRC is chosen in this paperto achieve the best received SNR.

After CP removal and FFT, the direct-link signal receivedat the destination is represented as:

y(1)d = Hs,d

√P sxs + n

(1)d , (7)

where Hsd =√Ndiag(FNhsd) is the diagonal S-D fre-

quency domain channel matrix, and n(1)d ∼ CN (0,Δd) is

the frequency domain noise at the destination with covariancematrix Δd = σ2

dIN . The relayed signals received at the

destination are superimposed as:

y(2)d =

R∑i=1

Hri,dyri + n(2)d

= Heq

√P sxs + neq, (8)

where Hri,d =√Ndiag(FNhs,ri) is the diagonal R-D

frequency domain channel matrix, n(2)d is the frequency do-

main noise vector at the destination, Heq is the equivalentend-to-end frequency domain channel matrix computed asHeq =

∑Ri=1 Hri,dAriHs,ri , and neq is the equivalent

frequency domain noise vector computed as neq = n(2)d +∑R

i=1 Hri,dArinri . In addition, neq is colored with covari-ance matrix Δeq = σ2

dIN +∑R

i=1 |σriHri,dAri |2.

Now, we are ready to combine y(1)d and y

(2)d using

MRC. Define yd = [(y(1)d )T (y

(2)d )T ]T , SNR matrix

γ = P sHHs,dΔ

−1d Hs,d + P sH

HeqΔ

−1eq Heq whose diag-

onal entries represent SNR at the subcarriers, and Λ =[(Hs,d

√P s)

HΔ−1d (Heq

√P s)

HΔ−1eq ]. Then the combina-

tion of y(1)d and y

(2)d is:

y∗d = Λyd

= (Hsd

√P s)

HΔ−1d y

(1)d + (Heq

√P s)

HΔ−1eq y

(2)d

= γxs+(Hsd

√P s)

HΔ−1d n

(1)d +(Heq

√P s)

HΔ−1eq neq

= γxs + n∗, (9)

where n∗ is the noise vector with covariance matrix γ, i.e.,n∗ ∼ CN (0,γ).

The effective SNR at subcarrier k, k ∈ 1, · · · , N , is γ(k, k),where:

γ(k, k) =

Pks

∣∣∣∣∣R∑i=1

PkriHs,ri(k, k)Hri,d(k, k)√|Hs,ri(k, k)|2Pk

s + σ2r

∣∣∣∣∣2

R∑i=1

∣∣∣∣∣∣√PkriHri,d(k, k)√|Hs,ri(k, k)|2Pk

s + σ2r

∣∣∣∣∣∣2

σ2r+σ2

d

(10)

+Pks |Hs,d(k, k)|2

σ2d

. (11)

Assume perfect CSI is known at the destination, the decisionstatistic is computed by whitening the noise as:

yd = γ−1/2y∗d

= γ−1/2γxs + γ−1/2n∗

= γ1/2xs + n. (12)

The symbol detector at the destination estimates each trans-mitted symbol according to the grouping scheme defined byΨm. The K bits in group m, m ∈ {1, . . . ,M} are decodedtogether with the optimal maximum likelihood (ML) criterion:

xm = arg minxi∈AK

‖Ψmyd −Ψmγ1/2ΨTmΘxi‖. (13)

IV. OPTIMAL POWER ALLOCATION

In [4], all relays are assumed to participate in the com-munication, and the transmit power is uniformly allocatedamong the source and the relays, i.e., P ri = P s = PIN , i ∈{1, · · · , R}. Although it is verified that the AsAP protocolwith uniform power allocation outperforms single-hop com-munication, the performance can be further improved throughtransmit power optimization.

The UAC channel is fast-varying and has slow signalpropagation speed. From a practical point of view, we assumethat only statistical CSI is available at the transmitter side,i.e., E[|Hs,ri(k, k)|2] = ηks,ri , E[|Hri,d(k, k)|2] = ηkri,d, andE[Hs,d(k, k)] = ηks,d. Due to the mathematical intractabilityof closed-form error performance of AsAP systems, we chooseto optimize the average received SNR at the destination, whichis closely related to the end-to-end error performance.

The average received SNR can be obtained by averaging theinstantaneous SNR in Eq. (10). Due to the complexity of thefirst term in Eq. (10), the closed-form E[γ(k, k)] is difficultto obtain. We will evaluate the following approximate SNRγ(k, k) instead:

γ(k, k) =

E

[Pk

s

∣∣∣∣∣∣R∑i=1

√PkriHs,ri(k, k)Hri,d(k, k)√|Hs,ri(k, k)|2Pk

s + σ2r

∣∣∣∣∣∣2 ]

E

[ R∑i=1

∣∣∣∣∣∣√

PkriHri,d(k, k)√|Hs,ri(k, k)|2Pk

s + σ2r

∣∣∣∣∣∣2

σ2r + σ2

d

]

+PksE|Hs,d(k, k)|2

σ2d

. (14)

The approximation has been shown to be tight in [15].According to the analysis in subsection II-B, each subcarrier

has the same variance, i.e., ηks,ri = ηs,ri , ηkri,d

= ηri,d, ηks,d =

ηs,d, which renders the same optimal power allocation onall subcarriers. Thus, index k is removed for presentationbrevity. With γ(k, k), Pk

ri , Pks , Hs,ri(k, k), Hri,d(k, k) and

Hs,d(k, k) replaced by γ, Pri , Ps, hs,ri , hri,d and hs,d,respectively, we obtain:

γ =

E

[Ps

∣∣∣∣∣R∑i=1

√Prihs,rihri,d√|hs,ri |2Ps + σ2r

∣∣∣∣∣2 ]

E

[ R∑i=1

∣∣∣∣∣√Prihri,d√|hs,ri |2Ps+σ2

r

∣∣∣∣∣2

σ2r+σ2

d

]+PsE|hs,d|2σ2d

(15)

=

Ps

R∑i=1

Priηri,dE[|hs,ri |2

Ps|hs,ri |2 + σ2ri

]

R∑i=1

Priηri,dE[1

Ps|hs,ri |2+σ2r

]σ2r+σ2

d

+Psηs,dσ2d

.(16)

The optimal Ps and Pri are hard to calculate by optimiz-ing γ directly, however, the closed-form suboptimal solutioncan be found by optimizing the upper bound of γ. Basedon Jensen’s inequality, E[

|hs,r|2Ps|hs,r|2+σ2

ri

] <E|hs,r|2

PsE|hs,r|2+σ2ri

=

ηs,r

Psηs,r+σ2ri

and E[ 1Ps|hs,r|2+σ2

ri

] > 1PsE|hs,r|2+σ2

ri

=1

Psηs,r+σ2ri

. Thus the upper bound γu is given as:

γu =

Ps

R∑i=1

Pri

ηri,dηs,riPsηs,ri + σ2

ri

R∑i=1

Priηri,dPsηs,ri + σ2

r

σ2r + σ2

d

+Psηs,dσ2d

. (17)

γu can be optimized in two steps:

1) Optimization of power allocation among relays Pri , i ∈{1, · · · , R} with fixed Ps, namely, maximizing the firstpart of γu under the constraint

∑Ri=1 Pri = Ptot−Ps =

Ptot.2) Optimization of power allocation between the source and

relay(s), Ps and Ptot, subject to Ps + Ptot = Ptot.

In the first step, we begin by reexpressing the first part ofγu in matrix format as:

Ps

R∑i=1

Pri

ηri,dηs,riPsηs,ri + σ2

ri

R∑i=1

Priηri,dPsηs,ri + σ2

r

σ2r + σ2

d

=αTΛα

αT (Ξ+σ2d

‖α‖2 IR)α, (18)

where α = [√Pr1 ,

√Pr2 , · · · ,√PrR ]

T , α = α/‖α‖,

Λ = diag[Psηr1,dηs,r1

Psηs,r1+σ2r1

,Psηr2,dηs,r2

Psηs,r2+σ2r2

, · · · , PsηrR,dηs,rR

Psηs,rR+σ2

rR

], and

Ξ = diag[ηr1,dσ

2r1

Psηs,r1+σ2r1

,ηr2,dσ

2r2

Psηs,r2+σ2r2

, · · · , ηrR,dσ2rR

Psηs,rR+σ2

rR

].

Furthermore, by defining

u = c(Ξ+

σ2d

Ptot

IR

)1/2

α, (19)

where c is chosen such that ‖u‖2 = 1, the optimizationproblem can be represented as:

maxα:‖α‖2≤ Ptot

αTΛα

αT (Ξ +σ2d

‖α‖2 IR)α

= max˜α:‖ ˜α‖2≤1

αTΛα

αT (Ξ +σ2d

PtotIR)α

= maxu:‖u‖2≤1

uTDu. (20)

As D is a diagonal matrix:

D =(Ξ+

σ2d

PtotIR

)−1/2

Λ(Ξ+

σ2d

Ptot

IR

)−1/2

= diag( γs,r1γr1,d

γs,r1+γr1,d+1, · · · , γs,rRγrR,d

γs,rR+γrR,d+1

), (21)

where γs,ri =Psηs,ri

σriand γri,d =

Ptotηri,d

σd. The maximum

of uTDu corresponds to the maximum diagonal element ofD. Suppose that

i∗ = argmaxi

γs,riγri,d

γs,ri + γri,d + 1(22)

is the position that has the maximum diagonal element. Then,u will be a vector with 1 in the i∗-th entry and 0’s elsewhere.

Notice that(Ξ +

σ2d

Ptot

)1/2

is a diagonal matrix in Eq. (19),we have α = u. Thus,

α =

√Ptotu. (23)

This result indicates that with only statistical CSI the optimalcooperative strategy is to allocate all relay transmit power tothe relay with the maximum average SNR.

In the second step, after choosing the relay ri∗ , the resultingγu in Eq. (17) is given as:

γ∗u(Ps, Ptot) =

γs,ri∗γri∗ ,d

γs,ri∗ + γri∗ ,d + 1+ γs,d. (24)

Then, the optimization problem can be formulated as follows:

maxPs, Ptot

γ∗u(Ps, Ptot)

subject to Ps + Ptot = Ptot.

Referring to the solution of the similar problem in [16], theoptimal power allocation result is given as:

P∗s =

Ptotηs,ri∗ ηri∗ ,d

σ2ri∗ σ2

d+

Ptotηs,dηri∗ ,d

σ2dσ

2ri∗

+ηs,d

σ2d

g +

√1+Ptotηs,ri∗ /σ2

ri∗1+Ptotηri∗ ,d/σ2

d

ηs,ri∗ ηri∗ ,d

σ2ri∗ σ2

dg

(25)

P∗tot =

Ptotηs,ri∗ ηri∗ ,d

σ2ri∗ σ2

d− Ptotηs,dηri∗ ,d

σ2dσ

2ri∗

− ηs,d

σ2d

g +

√1+Ptotηs,ri∗ /σ2

ri∗1+Ptotηri∗ ,d/σ2

d

ηs,ri∗ ηri∗ ,d

σ2ri∗ σ2

dg

, (26)

where g =ηs,ri∗ ηri∗ ,d

σ2ri∗ σ2

d

+ηs,dηri∗ ,d

σ2dσ2d

− ηs,ri∗ ηs,d

σ2ri∗ σ2

d

. Define the S-R

power allocation ratio as αp = Ps/Ptot, then the optimal S-Rpower allocation ratio is α∗

p = P∗s /Ptot. The maximum γu in

Eq. (17) can be computed by substituting P∗s and P∗

tot intoEq. (24):

γoptu = γ∗

u(P∗s , P∗

tot). (27)

Notice that the relay selection in step 1) depends on thevalue of Ps and Ptot, while the optimal power allocationbetween Ps and Ptot also depends on the selected relay. Tosolve this problem, we adopt an exhaustive search method.Observe that the optimization result of step 1) suggests onlya single relay for any power allocation between Ps and Ptot.Thus, we can perform optimal power allocation for each relayusing Eq. (25) and (26). Then the relay which maximizes thevalue of Eq. (27) is selected.

In summary, we optimized power allocation between thesource and the relays using the upper bound of γ. The optimalpower allocation result is to choose the relay which maximizesEq. (27) to transmit while the other relays keep silent.

V. SELECTIVE RELAYING

Based on the power optimization results in Sec. IV, wepropose the following SR-AsAP protocol to enhance thesystem performance.

A. Protocol description

The SR-AsAP consists of three phases: relay selection, relaynotification, and AsAP transmission.

1) Relay selection: assuming the localization service isavailable and relay selection is performed by the sourceas follows. First, the source chooses the potential relaysri, i ∈ {1, · · · , R}. According to the simulation resultsof the optimal relay location in [4], it is suggested tochoose potential relays close to the midpoint between thesource and the destination. Secondly, ηs,d, ηs,ri , ηri,d, r ∈{1, · · · , R} are computed as in Section II-B. Finally, fora given total transmit power Ptot, the source computesthe values of P∗

s , P∗tot and γopt

u with Eqs. (25), (26), and(27) for each relay. Then the relay with the maximumvalue of γopt

u is selected.2) Relay notification: after relay selection, the source broad-

casts the index of the selected relay as well as thepower allocation information, i.e., P∗

s and P∗tot. Then,

the selected relay is activated to forward the source signalwhile other relays keep silent.

3) AsAP transmission: the source generates informationsymbols modulated by GLCP OFDM with transmit powerP s = P∗

sIN , and broadcasts them to the relay andthe destination. After receiving the data signal from thesource, the selected relay amplifies the received signalwith transmit power matrix P r = P∗

totIN . Finally,the destination combines the direct-link signal and therelayed signals using MRC and performs decoding.

B. Performance simulations

To demonstrate the performance of the proposed SR-AsAPprotocol, we obtain the bit error rate (BER) through simula-tion. By comparing with other protocols, the benefits of ourselective relaying design are verified.

We adopt a simulation setup similar to [4]. The AsAPsystem has the S-D distance D = 1000 m. The waterdepth is 30 m. All nodes are placed underwater at the samewater depth. Thus, each node can be identified by a two-dimensional coordinate. Figure 3 illustrates the selection ofpotential relays under this setup. The coordinates of the sourceand the destination are (0, 0) and (1000, 0) respectively. Onlythe green nodes within the circle are chosen as potential relayswhile other nodes outside of the circle are not utilized. Thecircle is centered at (500, 0) with radius D/4 = 250. Binaryphase shift keying (BPSK) signaling is used. The OFDMprecoding size is chosen to be K = 8, and the OFDM symbolsize is N = 1024. The carrier frequency is set as fc = 30kHz with bandwidth B = 10 kHz. The noise variances at allreceivers are the same, i.e., σ2

d = σ2r1 = · · · = σ2

rR .In Figs. 4 and 5, we plot the BER performance of the

direct-link protocol, the AsAP protocol with uniform powerallocation, and the SR-AsAP protocol. Three relays wererandomly placed within the circle in Fig. 3. For the AsAPprotocol with uniform power allocation, all relays amplify andforward the received signal to the destination, and power isuniformly allocated among the source and the relays. For the

D

/ 4Ds d1r

2r

3r(0,0)

(500,0)

(1000,0)

Fig. 3. Potential relay selection: the green nodes within the circleare chosen as potential relays while other nodes outside of the circleare not utilized. The circle is centered at the midpoint between thesource and the destination with radius D/4

0 5 10 15 20 25 30 3510

−8

10−6

10−4

10−2

100

Transmit power/Bit (dB)

BE

R

Non−precoded Direct−linkNon−precoded, R=3, AsAPNon−precoded, R=3, SR−AsAP

Fig. 4. The uncoded BER performance of the direct-link protocol,the AsAP protocol with uniform power allocation, and the SR-AsAPprotocol. The water depth is 30 m and the number of potential relaysis R = 3.

SR-AsAP protocol, the source selects the best relay, and thepower is allocated between the source and the selected relayaccording to P∗

s and P∗tot. From the two figures, it can be

observed that the SR-AsAP protocol outperforms the direct-link protocol and the AsAP protocol with uniform powerallocation for both the precoded and uncoded relay systems.In Fig. 6, we also compare the BER performance of the SR-AsAP protocols with the optimal αp and with αp = 0.5. It canbe seen that the SR-AsAP protocol with optimal αp slightlyoutperforms the the SR-AsAP protocol with αp = 0.5.

VI. CONCLUSIONS

Cooperative communications are very beneficial to under-water acoustic communications. In this paper, we modifyour AsAP protocol and explore the optimal power allocationamong the source and the relays in order to improve thereliability and efficiency of AsAP. The optimization results

0 5 10 15 20 25

10−6

10−4

10−2

100

Transmit power/Bit (dB)

BE

R

Precoded, Direct−linkPrecoded, R=3, AsAPPrecoded, R=3, SR−AsAP

Fig. 5. The coded BER performance of the direct-link protocol,the AsAP protocol with uniform power allocation, and the SR-AsAPprotocol. The water depth is 30 m and the number of potential relaysis R = 3.

0 5 10 15 20 2510

−8

10−6

10−4

10−2

100

Transmit power/Bit (dB)

BE

R

Non−precoded, SR−AsAP, Optimal α*p

Non−precoded,SR−AsAP, αp = 0.5

Precoded, SR−AsAP, Optimal α*p

Precoded, SR−AsAP, αp = 0.5

Fig. 6. BER performance comparison of SR-AsAP with optimalpower allocation α∗

p and with power allocation ratio αp = 0.5.

show that, based on the statistical CSI, only the relay whichhas the maximum effective SNR should transmit while otherrelays keep silent. Thus, we further propose a selective-relaying AsAP protocol. Finally, the performance of our SR-AsAP protocol is simulated. The results show that the SR-AsAP protocol outperforms the AsAP scheme.

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