Underwater Acoustic Networks: Channel Models and Network Coding based Lower Bound to Transmission Power for Multicast

Transcription

1 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER Underwater Acoustc Networks: Channel Models and Network Codng based Lower Bound to Transmsson Power for Multcast Danel E. Lucan, Student Member, IEEE, Murel Médard, Fellow, IEEE, and Mlca Stojanovc, Member, IEEE arxv: v1 [cs.it] 30 Aug 2008 Abstract The goal of ths paper s two-fold. Frst, to establsh a tractable model for the underwater acoustc channel useful for network optmzaton n terms of convexty. Second, to propose a network codng based lower bound for transmsson power n underwater acoustc networks, and compare ths bound to the performance of several network layer schemes. The underwater acoustc channel s characterzed by a path loss that depends strongly on transmsson dstance and sgnal frequency. The exact relatonshp among power, transmsson band, dstance and capacty for the Gaussan nose scenaro s a complcated one. We provde a closed-form approxmate model for 1) transmsson power and 2) optmal frequency band to use, as functons of dstance and capacty. The model s obtaned through numercal evaluaton of analytcal results that take nto account physcal models of acoustc propagaton loss and ambent nose. Network codng s appled to determne a lower bound to transmsson power for a multcast scenaro, for a varety of multcast data rates and transmsson dstances of nterest for practcal systems, explotng physcal propertes of the underwater acoustc channel. The results quantfy the performance gap n transmsson power between a varety of routng and network codng schemes and the network codng based lower bound. We llustrate results numercally for dfferent network scenaros. Index Terms Underwater Acoustc Networks, Network Codng, Lower Bound for transmsson power, mnmal transmsson power, bandwdth - dstance dependence I. INTRODUCTION Wth recent advances n acoustc communcaton technology, the nterest n study and expermental deployment of underwater networks has been growng [1]. However, underwater acoustc channels mpose many constrants that affect the desgn of wreless networks. They are characterzed by a path loss that depends on both transmsson dstance and sgnal frequency, n a far more pronounced way than a terrestral rado system. Thus, not only the transmsson power, but also the useful bandwdth depend strongly on transmsson dstance [2]. References [6] and [7] present studes of the characterstcs and desgn challenges of underwater acoustc networks. Manuscrpt receved March 8, 2008; revsed July 15, Ths work was supported n part by the Natonal Scence Foundaton under grants No , , and CNS , by ONR MURI Grant No. N , and by Unted States Department of the Navy s Space and Naval Warfare Systems Command (SPAWAR) under Contract No. N C through BAE Systems. Danel E. Lucan and Murel Médard are wth the Research Laboratory of Electroncs, Massachusetts Insttute of Technology, Cambrdge, MA (emal: {dlucan, Mlca Stojanovc s wth the ECE Department Northeastern Unversty, Boston, MA (emal: mlltsa@mt.edu) In terms of the network layer, a seres of routng schemes have been proposed for underwater networks over the recent years. References [6], [7] and [8] present surveys of dfferent routng schemes used n underwater networks. In [3] two dstrbuted routng algorthms are ntroduced for delay-nsenstve and delay-senstve applcatons. Reference [4] presents a modfcaton of the dynamc source routng protocol that adds locaton awareness and lnk qualty metrcs. In [5] a routng protocol based on local depth of the nodes s studed. Network codng was ntroduced by Ahlswede et al []. Network codng consders the nodes to have a set of functons that operate upon receved or generated data packets [9]. Work n [11] and [12] showed that lnear codes over a network are suffcent to mplement any feasble multcast connecton. Also, [12] provdes an algebrac framework for studyng ths subset of coded networks. Work n [13] presents the dea of usng lnear codes generated randomly n a network. Some practcal network codng protocols have been presented n [22] and [14]. Network codng has prevously been consdered for underwater networks, showng better performance than other routng schemes. In [15] the problem of error recovery was studed n terms of the fracton of delvered packets and total number of transmssons. In [16] routng and network codng schemes were compared based on the tme to complete the transmsson of a fxed number of packets, and the power requred to do so. Exstng results compare dfferent network schemes, but the queston remans open as to what s the gap between these schemes and the theoretcal optmum n terms of transmsson power. The objectves of ths paper are 1) to establsh a tractable model for the underwater acoustc channel that wll be useful for network optmzaton n terms of convexty, and 2) to propose a network codng based lower bound for transmsson power n underwater acoustc networks by usng those models. The bound s used to compare the performance of several network codng and routng schemes. For an underwater acoustc channel both dstance between two nodes and capacty determne transmsson power and optmal transmsson band. However, the complete model that relates these varables s complcated. Ths paper presents a smple closed-form approxmaton for transmsson power and optmal operatng frequency band as functons of dstance and capacty. Ths approxmate model stems from an nformaton theoretc analyss that takes nto account the physcs of acoustc propagaton, and colored Gaussan ambent nose. Reference [2] shows that transmsson power as functon of

2 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER dstance can be well approxmated by P(l) = pl γ. A smlar model holds for the bandwdth. The coeffcents n ths model are functons of the requred sgnal to nose rato (SNR). The present work extends ths dea of modelng the power and bandwdth as functons of dstance, but the problem s cast nto a slghtly dfferent framework. Namely, nstead of usng the SNR as a fxed desgn constrant, lnk capacty s used as the fgure of mert. The parameters of the approxmate model proposed are functons of capacty and dstance. Ths approxmate model s useful for a broad range of capactes and dstances. The complete model that relates transmsson power, transmsson band, dstance, and lnk capacty s provably convex. Snce the approxmate model s used nstead of the complete model n network optmzaton, the present work shows the operatng condtons under whch the approxmate model s convex. We assess the mnmum transmsson power requred for an underwater acoustc network. A lower bound for transmsson power s obtaned by neglectng nterference between the nodes and usng subgraph selecton [9] to establsh mnmumcost multcast connectons wth network codng. The convex cost functon for the network optmzaton s gven by the transmsson power whch depends on the dstance and a desred data rate va the approxmate model for each actve lnk. We show that the no-nterference assumpton n an underwater scenaro s justfed for low multcast rates, and randomly placed nodes wth nter-node dstances less than km. Fnally, the network codng based lower bound for transmsson power s used to compare dfferent routng and network codng schemes. We use some of the schemes n [16] for a concatenated relay network and extend them to a random deployment of nodes n two dmensons. Also, we use an ALOHA-lke MAC layer nstead of a TDMA scheme as n [16] snce TDMA s not scalable. Furthermore, the problem of schedulng n TDMA s NP-complete [24]. The paper s organzed as follows. In Secton II, the complete model of an underwater channel wth Gaussan nose s presented, and specal characterstcs of ths channel are hghlghted. In Secton III, the approxmate model for the underwater channel s presented. In Secton IV, convexty of the complete model s proven and condtons for convexty of the approxmate model are studed. In Secton V, the lower bound for transmsson power usng network codng and subgraph selecton s computed. Secton VI presents the schemes to be used for performance evaluaton and gves numercal results a two-dmensonal network scenaro. The gap of several routng and network codng schemes to the network codng based lower bound s determned. The last secton summarzes the conclusons of ths work and future research topcs. II. CHANNEL MODEL An underwater acoustc channel s characterzed by a path loss that depends on both dstance l n km and sgnal frequency f as A(l,f) = (l/l ref ) k a(f) l (1) where k s the spreadng factor, l ref s a reference dstance, and a(f) s the absorpton coeffcent (Fgure 1 n [2]). The spreadng factor descrbes the geometry of propagaton, e.g. k = 2 corresponds to sphercal spreadng, k = 1 to cylndrcal spreadng, and k = 1.5 to practcal spreadng. The absorpton coeffcent can be expressed n db/km usng Thorp s emprcal formula for f n khz [17], whch s an strctly ncreasng functon. The nose n an acoustc channel can be modeled through four basc sources: turbulence N t (f), shppng N s(f), waves N w(f), and thermal nose N th (f) [2]. The power spectral denstes (psd) of these nose components n db re µ Pa per Hz as functons of frequency n khz are presented n [18]. These psd s have two mportant parameters: 1) the shppng actvty s rangng from 0 to 1, for low and hgh actvty, respectvely, 2) the wnd speed w measured n m/s. Fgure 2 n [2] shows N(f) for dfferent values of s and w, and an approxmaton logn(f) = N 1 ηlog(f) for f 0 khz, where N 1 = 50 db re µ Pa and η = 18 db/dec. Assumng that the channel s Gaussan, ts capacty can be obtaned usng the waterfllng prncple [25][26]. The capacty of a pont-to-pont lnk s Z «K(l,C) C = log 2 df (2) A(l, f)n(f) B(l,C) where B(l,C) s the optmum band of operaton and K(l,C) s a constant determned to satsfy a gven constrant. The band B(l, C) could be thought of as a unon of non-overlappng ntervals, B(l,C) = [fn (l,c),f end (l,c)], where each nonoverlappng nterval has the lower-end frequency fn (l,c) and the hgher-end frequencyfend (l,c) assocated wth t. The factor 1/A(l,f)N(f) s shown n Fgure 3 n [2] for dfferent values of l. Ths fgure suggests that the optmal transmsson band changes consderably wth respect to the dstance [2]. The transmsson power assocated wth a partcular choce of (l,c) s gven by Z P(l,C) = S(l, C, f)df (3) B(l,C) where S(l,C,f) = K(l,C) A(l,f)N(f),f B(l,C) s the psd of the sgnal. The correspondng sgnal-to-nose rato s gven by: R B(l,C) SNR = S(l,C,f)A 1 (l,f)df RB(l,C) N(f)df (4) We observe that a choce of (l, C) determnes mplctly the SNR level. Hence, there s a one-to-one correspondence between the par (l, C) and the par (l, SNR). The latter parameterzaton was used n [2] to compute the transmsson power and bandwdth representaton to ensure a preset SNR, whch determnes the value of C mplctly. The present analyss focuses on usng the former parameterzaton,.e. on determnng the power and transmsson band that ensure a pre-set lnk capacty.

3 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER Frequency [khz] 1 Band at l 200m Band at l 3m Low Frequency n Band for fxed C = 0.01 kbps Hgh Frequency n Band for fxed C = 0.01 kbps Hgh Frequency n Band for Fxed SNRo = 20dB Low Frequency n Band for Fxed SNRo = 20dB dstance [km] a 1 (C) Dashed Lne: Parameter, Crcle: Approxmaton Capacty [kbps] a 2 (C) Contnuous Lne: Parameter, Square: Approxmaton Fg. 1. Hgh and low band edge frequency of the transmsson band for C = 0.01kbps, k = 1.5, s = 0.5, and w = 0m/s. Fg. 2. Parameters a 1 and a 2 for P(l,C) and approxmate model.l [0,] km, C [0,2] kbps,k = 1.5,s = 0.5 and w = 0 m/s. A. Dependence on the spreadng factor The dependence on the spreadng factor k s qute smple. Let us assume that a model for P(l,C) has been developed for a partcular value of k,.e. P(l,C,k). To determne P(l,C,k ) for k k, note that a change n k, the product A(l,f)N(f) = (l/l ref ) k a(f) l N(f) consttutes a constant scalng factor wth respect to f. Therefore, for a lnk of dstance l the term B(l, C) wll reman unchanged. Thus, f the same capacty C s requred for k and k, equaton (2) shows that the only other term that can vary s K(l, C),.e. K(l, C, k). Then, K(l,C,k ) = (l/l ref ) k k K(l,C,k). Fnally, let us use the equaton (3) to determne the relatonshp between P(l, C, k) and P(l,C,k ). Z P(l,C,k ) = K(l,C,k ) lma(f) k l N(f) df (5) Z = l k k m B(l,C) B(l,C) K(l,C,k) l k ma(f) l N(f) df (6) = l k k m P(l,C,k) (7) where l m = l/l ref to shorten the dervaton. Thus, any model generated for some parameter k has a smple extenson. Also, note that the transmsson band remans the same for any value of k. B. Interference Characterstcs The optmal transmsson band of a lnk was shown to change wth the dstance, under the assumpton that the channel s Gaussan and that the capacty of a lnk s obtaned through waterfllng. If the capacty for a lnk s low, e.g. less than 2 kbps, and the transmsson dstance s below km, the transmsson bandwdth wll also be low, and ts optmal locaton n the spectrum wll change dramatcally wth the dstance. Fgure 1 shows ths effect for C = 0.01 kbps. In ths fgure, the hgh and low band edge frequences are plotted. Ths fgure also shows the hgh and low band edge frequency f an SNR requrement of -20 db s set,.e. usng the SNR nstead of the capacty as the fxed parameter. As noted before, the constrant over the capacty s related to dfferent SNR levels dependng upon the dstance. It s clear that low values of C are related to a very low SNR value. Fgure 1 shows that f two lnks wth the same C = 0.01 kbps are establshed, one wth l 200 m and the other wth l 3 m, the optmal transmsson bands for these lnks wll not overlap; thus, they do not nterfere wth one another. Ths characterstc of the underwater channel suggests that f a network s establshed n whch the nodes are at dfferent dstances from one another, and each node has a lmted range of transmsson when the data rate requrement s very low (all vald assumptons n underwater networks), there wll be no nterference between transmssons of the varous lnks. If each lnk allocates ts band optmally, ths suggests that a form of FDMA s the optmal approach n an underwater network, where transmsson band s determned by both the dstance and the requred data rate. From a network optmzaton vew pont, the cost functon to be mnmzed s clearly separable under these assumptons, where the channel model for a lnk can be used as the cost functon for each of the separable terms. C. Numercal Evaluaton Procedure A numercal evaluaton procedure smlar to that of [2] s used to compute the value of P(l,C), ˆB(l,C) and ˆfend (l,c), for a regon of values of (l,c). The procedure starts by fxng

4 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER a 1 (C) Dashed Lne: Parameter, Crcle: Approxmaton a 2 (C) Contnuous Lne: Parameter, Square: Approxmaton a 1 (C) Dashed Lne: Parameter, Crcle: Approxmaton a 2 (C) Contnuous Lne: Parameter, Square: Approxmaton Capacty [kbps] Capacty [kbps] Fg. 3. Parameters a 1 and a 2 for ˆf end (l,c) and approxmate model.l [0,] km, C [0,2] kbps,k = 1.5,s = 0.5 and w = 0 m/s. Fg. 4. Parameters a 1 and a 2 for ˆB(l,C) and approxmate model.l [0,] km, C [0,2] kbps, k = 1.5,s = 0.5 and w = 0 m/s. a target value of the capacty C. Then, for each dstance l, the ntal value of K(l,C) s set to the mnmum value of the product A(l,f)N(f),.e. K(l,C) = mn f A(l,f)N(f). The frequency at whch ths occurs,.e. f 0 = argmn f A(l,f)N(f), s called the optmal frequency. After ths, K(l, C) s ncreased teratvely by a small amount, untl the target capacty value C s met. Fnally, ths procedure s repeated for each value of C n a range of nterest. At the n-th step of the procedure, when K (n) (l,c) s ncreased by a small amount, the band B (n) (l,c) s determned for that step. Ths band s defned as the range of frequences for whch the condton A(l,f)N(f) K (n) (l,c) holds. Then, the capacty C (n) s numercally determned for the current K (n) (l,c) and B (n) (l,c), usng the equaton (2). If C (n) < C, a new teraton s performed. Otherwse, the procedure stops. III. APPROXIMATE MODEL Evdently, the expressons for the complete model are qute complcated to be used n a computatonal network analyss. Also, they provde lttle nsght nto the relatonshp between power consumpton, ˆB and ˆfend, n terms of the par (l,c). Ths motvates the need for an approxmate model to represent these relatons for ranges of C and l that are of nterest to acoustc communcaton systems. The model should also provde flexblty to changng other parameters, such as the spreadng factor k, wnd speed w and shppng actvty s. As shown n Equaton (7), any approxmate model for the transmsson power generated for some parameter k has a smple extenson to any other value of k. Also, a model for the transmsson band remans the same for any value of k. By applyng the numercal procedure of the prevous secton for varous l and C and fttng the data, t s possble to obtan approxmate models for power consumpton (Eq. 8), band-edge frequency ˆf end (l,c) (Eq. ), and for the bandwdth ˆB(l,C) = ˆf end (l,c) ˆf n (l,c) (Eq. 12). Note that some mportant propertes for these parameters are kept, e.g. P(l,0) = 0. P(l,C) = l a 1(C) a 2 (C) wth (8) a 1 (C) = α 3 +α 2 C +α 1 C 2 (9) a 2 (C) = β 3 +β 2 log C +β 1 (log (C +1)) 2 ˆf end (l,c) = l a 1(C) a 2 (C) wth () a 1 (C) = α 3 +α 2 log C +α 1 (log C) 2 (11) a 2 (C) = β 3 +β 2 log C +β 1 (log C) 2 ˆB(l,C) = l a 1(C) a 2 (C) wth (12) a 1 (C) = α 4 +α 3 log C +α 2 (log C) 2 +α 1 (log C) 3 a 2 (C) = β 3 +β 2 log C +β 1 (log C) 2 (13) The transmsson power, band-edge frequency and bandwdth of transmsson band were computed for a varety of values of s, w and two ranges of nterest of the par (l,c): l (0,] km, C [0,2] kbps, and l (0,0] km, C [0,0] kbps. The models proposed ftted these cases qute well. Results are presented for k = 1.5, w = 0 and s = 0.5, for both cases. For the frst case, the α and β parameters show almost no dependence on the shppng actvty factor s, especally f the wnd speed s w > 0. Thus, the approxmate model for ths

5 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER TABLE I a 1 AND a 2 APPROXIMATION PARAMETER VALUES FORP(l,C), ˆf end (l,c) AND ˆB(l,C), WITHk = 1.5,s = 0.5 AND w = 0 M/S FOR CASE 1: l [0,] KM,C [0,2] KBPS, AND CASE 2: l [0,0] KM, C [0,0] KBPS. Case α 1 α 2 α 3 α 4 MSE β 1 β 2 β 3 MSE 1 P(l, C) e e-5 1 ˆfend (l,c) 4.795e e e-5 1 ˆB(l,C) e e e e e-7 2 P(l, C) e ˆfend (l,c) e e-5 2 ˆB(l,C) 1.696e e e case can be smplfed to consder w only as part of the model, nstead of the par (s,w). Fgures 2, 3 and 4 show parameters a 1 and a 2 for P(l,C), ˆfend (l,c), and ˆB(l,C), respectvely, for the frst case wth k = 1.5, s = 0.5 and w = 0 m/s. The values of α s and β s are shown n Table I as Case 1, for parameters a 1 and a 2, respectvely. These tables also show the mean square error (MSE) of the approxmaton wth respect to the actual parameters. A smlar result can be found for the second case wth k = 1.5, s = 0.5 and w = 0. The values of α s and β s are shown as Case 2 n Table I, for parameters a 1 and a 2, respectvely. For both ranges, the proposed models gve a very good approxmaton to the actual numercal values. Also note that for the a 1 (C) parameter of P(l,C), t s possble to use a lnear approxmaton, nstead of a quadratc model. For case 1, the values for α and β parameters n the approxmate P(l,C) model show very lttle dependence wth respect to s whle they show a greater dependency on w. Ths s not unexpected, snce the transmsson band s at a hgh frequency (between 5 and 40 KHz) where the nose psd s nfluenced more by the w (O(f 2 )) thans(o(f 3.4 )). Therefore, a further approxmaton s to dscard s and consder parameters α and β to be functons of w only. In partcular, a smple model s α = γ 3 +γ 2 log (w+1)+γ 1 (log (w+1)) 2, = 1,2,3. A smlar relaton holds for β, = 1,2,3. Table II shows γ parameters for the dfferent α s and β s. IV. CONVEXITY ANALYSIS In ths secton we prove the convexty of the complete model of P(l,C) n the entre regon of nterest,.e. postve data rates and l > 0. Then we dscuss the necessary condtons that the value of l has to fullfll to ensure convexty of the approxmate model. A. Convexty of Complete Model The convexty of the transmsson power of the complete model s stated n the followng lemma, whch s proven n the Appendx. Lemma 1 assures that P(l,C) s a convex functon wth respect to C for the ranges of nterest of C and l for the case of non-overlappng fnte bands. Lemma 1 P(l,C) s a convex, ncreasng functon wth respect to C, C > 0 and l > 0, f A(l,f)N(f) > 0 and B(l,C) = [fn (l,c),f end (l,c)], wth f n (l,c) < f end (l,c) <, and fn (l,c),f end (l,c) [fj n (l,c),fj end (l,c)], j,.e. a unon of non-overlappng fnte bands. Proof See Appendx I. Another property to be used s gven n Lemma 2. Ths lemma assures that f a lnk between a transmtter and recever j at a dstance l acheves a certan capacty C, another node k at dstance l < l from node, wll be able to decode the nformaton transmtted from to j. Note that the transmsson band s optmal for the lnk of dstance l. Lemma 2 C(l,B(l,C)) < C(l,B(l,C)) for l < l, f A(l,f) = (l/l ref ) k a(f) l wth k 1 and a(f) 1, f. Proof See Appendx II. B. Convexty of Approxmate Model The functon P(l, z) represents the mnmum power requred to transmt at a data rate z over a lnk of dstance l. The functon P(l, z) was proven to be a convex functon wth respect to z, usng l as a parameter (Lemma 1). However, the exact model s complcated from a computatonal vewpont. Let us determne the condtons for whch the approxmate model P(l,z) n equaton (8) s convex wth respect to z, and havng l as a fxed parameter. We study the case of z < 2 kbps. Snce the α and β parameters come from fttng the data, the only varable left to analyze s the dstance l. Note that ensurng that P(l,z) s ncreasng and convex translates nto the followng nequaltes: ln(l) a 1 (z) z ln(l) 2 a 1 (z) +ln(l) z + ln() a 2 (z) 2 ln() 2 + ln() z > 0 (14) 2 a 2 (z) z 2 + ln() a 2 (z) z 2 a 1 (z) a 2 (z) z z + 2 a 1 (z) z 2 0 (15) There s both a lnear and a quadratc constrant upon l to ensure convexty. Snce these constrants are also functons of z, the range of values of ths parameter should be consdered. From prevous results for the fttng parameters, t s possble to determne some propertes of the model for z < 2 kbps. In terms of the parameters of nterest, α 1 < 0, α 2 > 0, 2α 1 C + α 2 > 0, β 1 > 0 and β 2 > 0. Thus, for the choces of a 1 (z) and a 2 (z), the frst and second dervatves of these functons wth respect to z are a 1 (z) > 0, ä 1 (z) < 0, a 2 (z) > 0 ä 2 (z) < 0. Usng these condtons, the constrants (14) and (15) can be smplfed to ln(l) > ln() v u tln() a 2 (z) ä1(z) +max[0, a 1 (z) 2a 1 (z) 2+ (16)! ä2(z) a 1 (z) 2 + ä1(z) 2 4a 1 (z) 4] (17) a 2 (z)ä 1 (z) a 1 (z) 3

6 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER TABLE II APPROXIMATION PARAMETERS OF α AND β FORP(l,C), l [0,] KM, C [0,2] KBPS,k = 1.5,s = 0.5 γ 1 γ 2 γ 3 α e α e α β e β e β where the term under the square root s postve whch ensures real values of l. Note that for each value of z there s a mnmum value of l. Let us use the values of Case 1 n Table I to determne the (l, z) regon for whch the approxmate model s convex. For these values, f the dstance between to nodes l s at least 13 m, for any value of z <2 kbps the model wll be convex. The lmtaton to l >13 m s related to the samplng of the dstance used for computng the parameters of the approxmate model. For all practcal purposes the approxmate model P(l,z) s convex. V. LOWER BOUND TO TRANSMISSION POWER IN UNDERWATER NETWORKS The problem of achevng mnmum-energy multcast usng network codng n a wreless network has been studed prevously [9]. A wreless network, as presented n [9] can be represented through a drected hypergraph H = (ℵ, A) where ℵ s the set of nodes and A s the set of hyperarcs. A hypergraph s a generalzaton of a graph, where there are hyperarcs nstead of arcs. A hyperarc s a par (,J), where, the start node, s an element of ℵ,and J s the set of end nodes s a nonempty subset of A. Each hyperarc (,J) represents a broadcast lnk from node to nodes n the nonempty set J. Let us denote by z J the rate at whch coded packets are njected nto hyperarc (, J). If the cost functon s separable, the optmzaton problem can be expressed as follows mn X (,J) A subject to z Z z J X j J X {J (,J) A} j J θf(z J /θ) x (t) Jj, (,J) A,t T X x (t) Jj X {j (j,i) A}, I x (t) ji = δ(t) x (t) Jj 0, (,J) A,j J,t T (18) wth 8 >< R f = s, δ (t) = R f = t, (19) >: 0 otherwse where T s a non-empty set of snk termnals, a source s, a multcast rate R, and a fxed transmsson duty cycle at each lnk θ. x (t) Jj represents the flow assocated wth termnal t, sent through hyperarc (,J) and receved by node j J. In the underwater scenaro ths formulaton s used to establsh a lower bound on the transmsson power requred to acheve a multcast rate R. Assumng no nterference for transmssons n dfferent hyperarcs yelds a separable cost functon. Note that f nterference was taken nto account, the power to reach the desred data rate would ncrease. Then, the cost functon f(z J ) for each partcular hyperarc corresponds to a lnk transmsson power P(l,z J ) n order to obtan the mnmum transmsson power requred to acheve a data rate of z J, where l represents the dstance from to the farthest node j J. For the lower bound computaton, contnuous transmsson (θ = 1) s assumed. A smplfcaton of ths problem can be made under the assumpton that transmssons are omndrectonal, and consderng the fact that f a node transmts over a certan range, all nodes n that range wll be able to receve the nformaton. Ths was proven n Lemma 2. Fnally, the model for ths channel ensures that any value of z J can be acheved f enough power s used. Thus, the constrant set Z can be dropped. Although the problem for mnmum-cost multcast s well known for wreless rado networks, the cost functon presented here s dfferent because t represents the mnmum transmsson power for an hyperarc transmttng at a data rate Z, whch s gven by the power needed to transmt at capacty C = Z, wthout assumpton on technology or, more mportantly, a specfc transmsson band whch s usually the case for wreless rado networks. Thus, we are provdng a lower bound vald for any acoustc underwater network for the case of Gaussan nose. VI. PERFORMANCE COMPARISON For ths study, fve schemes are consdered. The frst scheme corresponds to the lower bound to the transmsson power usng network codng gven by solvng the problem n Secton V wth θ = 1. The second scheme corresponds to solvng the problem n Secton V for θ < 1, n order to study the effect of usng a duty cycle for lnk transmssons n underwater networks over nterference and transmsson power. The thrd scheme corresponds to usng the paths chosen by the optmal scheme but establshng a SNR requrement for the transmsson lnks wth the objectve of studyng nterference when the SNR requrement s ncreased. The lnks are consdered to transmt contnuously. The schemes (4) and (5) consder mplementatons of network codng n a rateless fashon wth the mplct acknowledgment (ACK) [16] and routng wth lnk-by-lnk ACK usng an ALOHA-lke MAC layer. Let us explan n more detal each of the schemes. 1)Network codng based lower bound to transmsson power: Transmsson power s computed by solvng the convex optmzaton problem n V and t provdes a lower bound on the optmal transmsson power for networks operatng at low data rates. For ths computaton, contnuous transmsson,.e. a duty cycle of θ = 1 s used. Ths scheme s used as the gold standard to whch the remanng schemes are compared. The no-nterference assumpton s assessed by computng the average percent of the randomly deployed networks that have at least a lnk whch suffers from severe nterference,.e. a sgnal-to-nterference rato (SIR) below 3 db.

7 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER NODE 1 NODE 2 NODE 3 PACKET A PACKET A PACKET A PACKET B PACKET PACKET B PACKET CKET D 1 D Fg. 5. Medum Access Protocol for schemes (4) and (5) Tx T Fg. 6. Subgraph selecton for scheme (4). Selected subgraph wth values of z J to provde a uncast rate of 0, where dashed lnes represent unused transmsson ranges. Note that hyperarcs nclude n the same range, e.g. the possble hyperarcs wth node 1 as the starng pont are 1{2},1{2, 4},1{2, 3, 4}. 2)Network Codng wth optmal power consumpton for lnks wth fxed duty cycle: Transmsson power s computed by solvng the convex optmzaton problem n (V) for lnks wth a fxed duty cycle,.e. θ < 1. Ths value provdes a lower bound on the optmum power consumpton for networks operatng at low data rates when lnks have a partcular duty cycle. By convexty of the cost functons used, ths bound wll be hgher than for the prevous scheme. Ths scheme s used to llustrate the effect upon transmsson power and nterference when the lnks transmt at a fxed duty cycle θ < 1 by comparng ths scheme to the prevous scheme. 3)Network Codng wth SNR requrement on lnk transmsson: Ths scheme s a heurstc scheme that mposes an SNR requrement for transmssons. Usng the subgraph selected by solvng the problem n (V) for θ = 1, t computes the SIR on the dfferent lnks for a varety of SNR constrants usng the models of transmsson power and band n [2]. As n scheme (1) and (2), the percent of randomly deployed networks wth at least a lnk wth severe nterference s computed. Results of ths scheme suggest that contnuous transmsson wth a moderate SNR requrement causes severe nterference. A soluton to ths problem s to use of a duty cycle θ < 1, smlarly as n scheme (2) when there s an SNR requrement. 4)Network codng n rateless fashon wth mplct ACK: For a concatenated relay network as n Fgure 5, the path between a source node and snk node s fxed and ncludes 3 Rx t all relay nodes. If a node b s closer than node a to the collectng node, a s sad to be upstream wth respect to b, and b s sad to be downstream wth respect to a. For the concatenated relay network ths orderng s qute natural. Ths problem was studed n [16]. For a two-dmensonal scenaro, subgraph selecton [9] wth lnear and separable cost functons are used to determne the actve lnks n the network and the transmsson power requred for each lnk. The cost functon of each hyperarc s computed based on the approxmate formulas for transmsson power and bandwdth for a fxed SNR level [2]. The weght of each lnk s gven by DP(l,SNR)/B(l,SNR), where D s a constant common to all lnks related to the number of transmtted bts per burst and modulaton used. For the performance computaton usng optmal modulaton,.e. Gaussan sgnalng, the weght for each lnk n the path s DP(l,SNR)/C(l,SNR), where C(l, SNR) s the functon of capacty related to the par (l,snr). We assume that the codng s over very large number of data packets. Ths could be extended computng an error probablty based on error exponents [26]. Once the subgraph has been selected, f several lnks share the same transmttng node, ths node wll randomly choose the lnk to use. The weght of each lnk n the random choce s gven by the fracton of data rate the optmzaton problem assgned to each of these lnks. Let us consder the network n Fgure 6 as an example, where dashed lnes represent unused hyperarcs. If node 1 transmts to both node 2 and 4, but we send a rate of 90 unts through hyperarc z 12, whle we send unts through hyperarc z 12,4, then when node 1 transmts t wll do so 90 % of the tme to reach node 2 only, and % of tme usng enough power to reach nodes 2 and 4. Fnally, we have to determne whch nodes are upstream and downstream to each node n the subgraph. If the subgraph corresponds to a sngle path, the choce s clear. If there are multple paths, we use the followng heurstcs: we start by orderng the nodes startng at the transmtter and lookng at the nodes drectly connected to t n the optmal subgraph. These nodes are ordered as follows: the node assocated wth the lnk wth hgher data rate from the transmtter s consdered to be drectly downstream from the source node, the node wth the second hghest data rate s consdered to be downstream wth respect to the prevous one, and so on. In Fgure 6, 2 s drectly downstream of 1, and 4 s downstream of 2. Once all nodes connected to the transmtter (let us call ths set of nodes S) are ordered, we proceed to order the nodes connected to S by a smlar procedure as for the case of one node descrbed before. In the example, S = {2,4} and the nodes connected to t are {3,4}. If a node connected to one of the nodes n S has already been ordered, lke node 4 n the example, the lnk s dscarded keepng the prevous order of the nodes. We update S wth the nodes that were connected to S and not prevously n t, untl we reach the recever. For the network example n Fgure 6 the orderng s 1,2,4,3. For ths partcular scheme, once a relay node gets ts frst coded packet,.e. a packet formed by a random lnear combnaton of data packets, t wll transmt untl the recevng node sends a confrmaton that all the nformaton has been receved. The same happens at the source node. However, nodes eavesdrop on other transmssons. If a node receves a

8 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER coded packet from a node further downstream wth the same, or a greater number of degrees of freedom than what t has, t wll stop transmttng and update ts nformaton f necessary. Degrees of freedom n ths settng represents the number of packets that were lnearly combned to form the coded packet as n [16]. The node wll resume transmttng f an nnovatve packet,.e. a packet wth a new random lnear combnaton of data packets useful for decodng the nformaton, s receved from a node upstream. The snk node wll retransmt a coded packet wth ts degrees of freedom when a coded packet s receved. Ths strategy assumes that there s a mechansm that nforms the collectng node about the number of degrees of freedom that consttute the total message or that ths number s fxed a pror. 5)Routng usng lnk-by-lnk acknowledgement: For a concatenated relay network, the path between the source node and the snk node s fxed and ncludes all relay nodes. Ths problem was studed n [16]. For a two dmensonal scenaro, the snk and the source are chosen randomly and the shortest path s computed before startng data transmsson n uncast. The weght of each lnk s computed based on the approxmate formulas for transmsson power and bandwdth for a fxed SNR level n the same fashon as the cost functon per lnk of scheme (4). In the current scheme, every tme a node receves a packet, t wll retransmt the packet and send an acknowledgement to the prevous node. Once a packet has been acknowledged, the node can start transmttng a new data packet n ts queue. If t has no new packets to transmt, t wll only transmt f a node upstream sends new nformaton, or sends a prevous packet, n whch case the node wll acknowledge ths packet. In terms of the physcal layer, schemes (4) and (5) use both PSK modulaton, whch mples the use of a data rate n each lnk that s lower than capacty, and Gaussan sgnalng assumng that the encodng s over a large number of bts. In order to deal wth the SNR requrement, we use an approxmate model for the transmsson power, hgh band edge frequency and bandwdth as functons of SNR smlar to the work n [19]. When PSK modulaton s used, probablty of packet error due to nose over the lnk from node to j s obtaned from the probablty of bt error by P packet Error (,j) = 1 (1 P bt error ) n, where n s the number of bts n the packet, and P bt error s computed usng the standard PSK bt error probablty. Note that nodes farther away from the transmtter have some probablty of recevng the packet correctly. For Gaussan sgnalng, the probablty of packet error due to nose s consdered to be zero for all nodes n range, and 1 for all nodes further away. In terms of the MAC layer, schemes (4) and (5) use an ALOHA-lke MAC layer. Ths ALOHA protocol consders a fxed number of bts per data packet and uses the optmal transmsson band for an SNR requrement per lnk. Thus, the duraton of the transmtted packet depends on the transmsson dstance [2]. Every node has a probablty to access the medum every T unts of tme followng a Bernoull process. Transmsson delay s consdered usng a typcal value of sound speed (1500 m/s). Fgure 5 shows an example of usng ths MAC layer for three nodes wth D 1 >> D 2. In ths Deployments wth at least one lnk wth SIR < 3 db (%) θ = 1 θ = 0.5 θ = 0.1 θ = 0.01 θ = 0.05 SNR = 20 db SNR = 5 db SNR = 0 db Number of Nodes Fg. 7. Percent of deployments n a fxed square of 5x5 km 2 wth SIR < 3 db n at least one lnk vs number of nodes deployed n that area, for the frst three schemes. For scheme 1 θ = 1, whle scheme 2 s shown wth dfferent values of θ to acheve uncast rate of R = 0.1 kbps. Performance for scheme 3 s shown for dfferent SNR values. example, when node 1 transmt a packet to node 2, ths packet also reaches node 3. Note that the duraton of the packet transmtted from node 1 to node 2 (Packet A) s large compared to the packet transmtted from node 2 to node 3 because of the relaton of dstance to bandwdth/capacty for a fxed SNR value mentoned above [2]. Once node 1 has transmtted the packet t wll try to transmt agan, and t has some probablty to start transmsson every tme slot T. Let us assume that node 2 has a data packet for node 3. Fgure 5 shows the case when the data packet transmtted from node 2 to node 3 suffers a collson at node 3 wth a new packet transmtted from node 1 to node 2. We consder that a collson at any recever causes a loss of all packets nvolved n the collson for that recever. Let us study some numercal results that correspond to a network n whch nodes are deployed randomly n a two dmensonal space. Uncast connectons of rate R are establshed,.e. the network has one transmtter, one recever chosen randomly, and, possbly, several relay nodes. The number of nodes ranges from 3 to 8. The transmsson power lower bound, as an average over random deployments, wll be compared wth transmsson power of schemes for routng and network codng. Also, a comparson between the schemes (1) and (2) n terms of nterference s presented. Note that the transmsson power lower bound s computed assumng that all nodes are wthn the transmsson range of the others,.e. full connectvty. Fgure 7 llustrates the effect of ntroducng a duty cycle θ (dashed lnes) for lnk transmsson under a random deployment n a 5 x 5 km 2 square. For θ = 1 n Fgure 7,

9 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER Network Codng wth Implct ACK Routng lnk by lnk ACK Transmsson Power (µ Pa) 8 7 PSK Gaussan Sgnalng Routng lnk by lnk ACK R 1 kbps Network Codng wth Implct ACK R 1 kbps Lower Bound R = 1 kbps Transmsson Power ( µ Pa) 8 R 0.2 kbps R 2 kbps R 1 kbps Number of Nodes Number of Nodes Fg. 8. Average transmsson power of networks deployed randomly on a 1x1 km 2 square. Network schemes operatng at SNR = db and a lower bound for transmsson power (scheme 1) s presented. Model used consders k = 1.5, s = 0.5 and w = 0 m/s. whch corresponds to scheme 1, note that less than 3 % of the random deployments cause severe nterference (SIR < 3 db) over at least one lnk. Ths corroborates the no-nterference assumpton used durng the analyss to obtan a lower bound for transmsson power. Furthermore, ths percentage seems to have lttle dependence on the number of nodes deployed n the network. When a value of θ < 1 s used, Fgure 7 shows that the percentage of deployments wth SIR < 3 db ncreases for the ntal decrements of θ, but decreases as θ becomes very small (below 1.5% for θ = 0.01). Although ths may seem counter ntutve, ntroducng a duty cycle causes the lnk to transmt at a hgher data rate when t s actve whch translates to usng more bandwdth and power to acheve that data rate n the underwater channel. Although duty cycle reduces nterference by not usng the channel contnuously, the combned effect wth the ncreased transmsson bandwdth and power causes more nterference for ntal decrements on the value of θ. Ths s a transent effect, and t has a breakng pont for a small value of θ when the probablty of havng nterference s small. As the value of θ decreases the transmsson power can be shown to ncrease. Ths s an expected effect snce the cost functon s convex and the value of θ s a constant parameter to all lnks n ths problem. Fgure 7 shows the results n contnuous lnes for the thrd scheme wth dfferent SNR requrements. The fgure presents the percentage of random deployments that have at least one lnk sufferng from severe nterference. For very low SNR, the assumpton of no-nterference s justfed. However, even for SNR = -5 db the percentage of deployments wth severe nterference for a uncast connecton ncreases dramatcally, especally when the number of nodes n the network ncreases. Fg. 9. Average transmsson power of networks deployed randomly on a 1x1 km 2 square. Network schemes operatng at SNR = db usng Gaussan sgnalng. Model used consders k = 1.5, s = 0.5 and w = 0 m/s. A smlar effect occurs when SNR = 0 db. One way to reduce nterference whle havng an SNR requrement s to use a smlar approach as s scheme (2),.e. to have a transmsson duty cycle n each of the lnks. Schemes (4) and (5) show an mplementaton usng an ALOHA MAC protocol, where every lnk has an assocated duty cycle when t has some data to transmt. Let us compare the transmsson power of scheme (4) and (5) to the lower bound usng both PSK and Gaussan sgnalng n a 1 x 1 km 2. Fgure 8 shows the average transmsson power for dfferent number of nodes n the network, both actve and nactve,.e. before determnng the shortest path or solvng the subgraph selecton problem, wth a transmsson power computed to obtan a burst SNR = db. The average data rate for the dfferent schemes s R 1 kbps. Ths fgure shows optmal sgnalng (Gaussan sgnallng) and a PSK modulaton, whch llustrates that close to 6 db n the gap between a PSK modulaton and the lower bound s due to the choce of the modulaton. Notce that the gap between the average transmsson power for Gaussan sgnalng and the lower bound of R = 1 kbps n Fgure 8 s about 11 db for scheme (4) and 13 db for scheme (5). Also, t shows that ths gap s mantaned as more nodes are deployed. Some part of the gap s related to the MAC protocol used. Another s related to the db SNR requrement whch s usually used for a practcal mplementaton. Fgure 9 compares transmsson power for dfferent data rates usng Gaussan sgnalng. The number of transmtted bts was kept constant, whle the transmsson probablty over each lnk was ncreased to acheve the desred rate. Note that transmsson power ncreases by 3 db for scheme (4) whle t ncreases by almost 5 db for scheme (5) when the data rate s

10 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER 2008 Energy = Transmsson Duraton x Transmsson Power ( µ Pa ) kbps 1 kbps 2 kbps 2 kbps 1 kbps Routng lnk by lnk ACK Network Codng wth Implct ACK 0.2 kbps Number of Nodes Fg.. Energy for transmsson for networks deployed randomly on a 1x1 km 2 square. Network schemes operatng at SNR = db usng Gaussan sgnalng. Model used consders k = 1.5, s = 0.5 and w = 0 m/s. ncreased from 1 kbps to 2 kbps,.e. the gap between scheme (4) and (5) ncreases as data rate ncreases. Ths fgure shows also that the gap between schemes (4) and (5) s very low when the data rate s 0.2 kbps. Note that an ncrease n data rate s related to an ncrease n the collson probablty n the ALOHA protocol. For the same settng, Fgure shows transmsson energy for both schemes. The energy requred for transmttng at the chosen data rates remans constant n the case of network codng, whle t ncreases for routng when hgh data rates are attempted. Multple transmssons of one packet are the man cause of the ncreased energy consumpton for scheme (5), caused both by packet losses and long delays n transmttng an ACK packet gven the ALOHA MAC layer. Whle scheme (4) transmts nnovatve packets at each transmsson, scheme (5) tres to retransmt the same packet f no ACK has been receved. Consder the case of a long delay n transmttng an ACK,.e. the packet was correctly receved but the ACK s transmtted a long tme after recepton, scheme (5) can generate several transmssons of the same data packet. Ths nvolves an addtonal energy consumpton. For the same number of transmssons, scheme (4) wll transmt several nnovatve packets, whch are useful n decodng the nformaton at the recever. These results llustrate that codng, subgraph selecton and the eavesdroppng capabltes assocated wth network codng allow a better performance when the collson probablty ncreases. However, notce that when transmsson rates are low the benefts of network codng are less marked. Ths s explaned by the fact that an mplct ACK mght or mght not be receved by an upstream node. If t s not receved, the node wll keep transmtng nnovatve packets. Ths effect s partcularly evdent when Gaussan sgnalng snce only nodes n range of transmsson wll correctly receve a packet. If we use a smlar example as n fgure 5, node 1 wll not receve any mplct ACK from node 2, and t wll contnue to transmt untl nformed that all nformaton was receved at the snk node. Thus, an explct ACK procedure should be used f node deployments are lkely to produce these stuatons, especally f only one node s actvely generatng new data packets. VII. CONCLUSIONS A tractable model for the underwater acoustc channel has been establshed and used for network optmzaton. A lower bound for transmsson power n underwater acoustc networks was obtaned based on network codng, and ths bound was used to compare the performance of several network codng and routng schemes. The closed-form approxmate models for the tme-nvarant acoustc channel where shown to provde a good ft to the actual emprcal values by numercal evaluaton for dfferent ranges of dstance l and capacty C, as well as nose profles correspondng to dfferent shppng actvty factor and wnd speed. The parameters obtaned for these approxmate models can be used n the case that a dfferent spreadng factor s needed, snce the band-edge frequency ˆf end (l,c) and the bandwdth ˆB(l,C) were shown to be nvarant to the spreadng factor k, whle the power scales as P(l,C,k ) = (l/l ref ) k k P(l,C,k), where l s n km. Also, the approxmate model of P(l,C) was shown to be almost ndependent of the shppng actvty factor s whle havng a marked dependency on the wnd speed w for l < km and C < 2 kbps. Ths dependence on w s qute smooth and can be approxmated by a smple model, resultng n a complete model for the P(l,C) for a range of values (l,c) that s of nterest to a typcal underwater communcaton system. We show that the complete model s convex for all C > 0 and dstances l > 0. Snce the complete model s complcated to solve a network optmzaton problem, we present condtons on the mnmum dstance between nodes that ensure convexty of the approxmate model. Convexty of ths model allows us to use t n more complex scenaros, for example, n the framework of layerng as optmzaton decomposton [20] [21]. Ths work shows that the no-nterference assumpton used n the computaton of the lower bound to transmsson power n the underwater scenaro s justfed for low multcast rates, and randomly placed nodes wth dstances under km between each other. We present numercal results that confrm the valdty of ths assumpton, showng that less than 3% of the lnks suffer severe nterference. We show that solvng the optmzaton problem when the lnks have a fxed duty cycle for transmssons can reduce nterference f the duty cycle s low enough. However, hgh duty cycles can effectvely ncrease nterference because of the hgh dependence of the bandwdth to the transmtted data rate n an underwater channel. The network codng based lower bound was used to determne the gap of dfferent medum access protocols and network schemes for some multcast rate n underwater networks. Ths comparson s carred out for several routng

11 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER and network codng schemes. We present numercal results n uncast scenaros by deployng nodes n a two dmensonal envronment. Network codng wth mplct acknowledgements ntroduced n [16] has better performance than routng wth lnk-bylnk ACK n terms of transmsson power, especally when the probablty of collson s ncreased. The gap between routng wth lnk-by-lnk ACK and network codng wth mplct ACK n terms of transmsson power was shown to ncrease as the transmsson probablty ncreased, whch s closely related to the probablty of packet collson. However, study of a network codng scheme wth an explct ACK should be consdered to mprove performance. Also, network codng wth mplct acknowledgement compared to common rateless network codng schemes allows to save resources, e.g. memory requred n the nodes, and rate adaptaton followng a smlar analyss as n [23]. Also, there has been prevous work on network codng based Ad-Hoc protocols, such as CODECAST [22], whch could be extended to use mplct acknowledgements, and adapted to the underwater acoustc channel. Future research should consder usng error exponents [26] to compute the error probablty when Gaussan sgnalng s used. Ths provdes a more accurate estmate of transmsson power when encodng s performed over a fnte number of bts wth a data rate approachng capacty. Fnally, future research should address the ssue of scalablty,.e. study f the results are vald when the number of nodes n the network s ncreased but mantanng the same node densty. APPENDIX I PROOF OF LEMMA 1 We consder a set Ξ of bands, each band Ξ havng a fend (l,c) and f n (l,c) assocated P to t. Then, P(l, C) = K(l,C)(f end (l,c) fn (l,c)) P R f end (l,c) f n A(l,f)N(f)df and C = (l,c) P R f end (l,c) f n (l,c) log K(l,C) 2 df. Usng the Lebnz Integral A(l,f)N(f) rule, the fact that K(l,C) = A(l,f end (l,c))n(f end (l,c)) and K(l,C) = A(l,f n (l,c))n(f n (l,c)), that A(l,f)N(f) s ndependent of C, and that the dervatve X of the sum s the sum of the dervatves, P(l,C) = K(l,C) (fend(l,c) f n(l,c)). Takng the second dervatve: 2 P(l,C) 2 = X + P K(l,C) 2 K(l,C) 2 (fend(l,c) f n(l,c)) (20) «f end (l,c) f n (l,c) (21) Takng the dervatve of C wth respect to tself, usng Lebnz Integraton Rule and K(l,C) = A(l,f end (l,c))n(f end (l,c)) and K(l,C) = A(l,f n (l,c))n(f n (l,c)), then: 1 = 1 K(l,C) ln(2)k(l,c) X (f end(l,c) f n(l,c)) (22) Snce K(l,C) > 0 for any l > 0 and C > 0 by the physcs of the channel and fend (l,c) f n (l,c) > 0, C > 0,l > 0, and the bands are non/overlappng and ln(2) > 0 ths mples that K(l,C) > 0. Then P(l,C) = ln(2)k(l,c) > 0, l > 0,C > 0. Takng a second dervatve to the C expresson wth respect to tself: 2 K(l,C) P (f end (l,c) f n (l,c)) = (23) X 2 K(l,C) 2 + K(l,C) Thus, 2 P(l,C) 2 = X K(l,C) (f end(l,c) f n(l,c)) (24) «f end (l,c) f n (l,c) (25) 2 X (fend(l,c) f n(l,c)) where (fend (l,c) f n (l,c)) > 0, C > 0, fnte and non-overlappng and K(l,C) > 0. Thus, 2 P(l,C) 2 > 0 APPENDIX II PROOF OF LEMMA 2 Snce A(l,f) = (l/l ref ) k a(f) l A(l,f) and l = (k/l ref )(l/l ref ) k 1 a(f) l + (l/l ref ) k ln(a(f))a(f) l > 0 snce a(f) 1 and l ref > 0. Then, A(l,f) > A(l,f),l > l. K(l,C) Also, A(l,f)N(f) 1, f B(l, C) whch mples K(l,C) log 2 ( ) 0, f B(l,C). Let us compute the A(l,f)N(f) capacty of a lnk of dstance l when we use the optmum band and spectral densty for a lnk of dstance l and capacty C,.e. B(l,C) and S(l,C,f) = K(l,C) A(l,f)N(f),f B(l,C), respectvely. Then, C(l,B(l,C)) = R B(l,C) log 2 1+ K(l,C) A(l,f)N(f) A(l df (26),f)N(f) > R B(l,C) log 2 df = C (27) 1+ K(l,C) A(l,f)N(f) A(l,f)N(f) REFERENCES [1] Partan, J., Kurose, J., Levne, B. N., A Survey of Practcal Issues n Underwater Networks, In Proc. WUWnet 06, pp , Los Angeles, Sept [2] M.Stojanovc, On the Relatonshp Between Capacty and Dstance n an Underwater Acoustc Communcaton Channel, ACM SIGMOBILE Moble Computng and Communcatons Revew (MC2R), pp.34-43, vol.11, Issue 4, Oct [3] Pompl, D., Meloda, T., Akyldz, I. F., Routng algorthms for delaynsenstve and delay-senstve applcatons n underwater sensor networks, In Proc. MobCom 2008, pp , Los Angeles, CA, USA, Sept [4] Carlson, E.A., Beaujean, P.-P., An, E., Locaton-Aware Routng Protocol for Underwater Acoustc Networks, In Proc. OCEANS 2006, pp. 1-6, Boston, MA, USA, Sept [5] Yan, H., Sh, Z. J., Cu, J.H., DBR: Depth-Based Routng for Underwater Sensor Networks, In Proc. IFIP Networkng 08, pp. 1-13, Sngapore, May [6] E. M. Sozer, M. 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12 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 11, NOVEMBER [12] Koetter, R., Médard, M., An algebrac Approach to network codng, IEEE/ACM Trans. Netw., vol. 11, no. 5, pp , Oct [13] Ho, T., Medard, M., Koetter, R., Karger, D.R., Effros, M., Sh, J., Leong, B., A Random Lnear Network Codng Approach to Multcast, Trans. Info. Theory, vol. 52, no., pp , Oct [14] Chachulsk, S., Jennngs, M., Katt, S., Katab, D., Tradng Structure for Randomness n Wreless Opportunstc Routng, In Proc. COMM 07, pp , Kyoto, Japan, Aug [15] Z. Guo, P. Xe, J. H. Cu and B. Wang. On Applyng Network Codng to Underwater Sensor Networks, In Proc. of WUWNet 06, pp , Los Angeles, Sept [16] Lucan, D. E., Médard, M., Stojanovc, M., Network Codng Schemes for Underwater Networks: The Benefts of Implct Acknowledgement, In Proc. WUWnet 07, pp.25-32, Montreal, Quebec, Canada, Sept.2007 [17] Berkhovskkh, L., Lysanov, Y., Fundamentals of Ocean Acoustcs, New York, Sprnger, 1982 [18] Coates, R. 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A., Elements of Informaton Theory, 2nd Edton, New Jersey, Wley, 2006 [26] Gallager, R.G., Informaton Theory and Relable Communcatons, New York, Wley, 1968 Murel Médard s a Professor n the Electrcal Engneerng and Computer Scence at MIT. She was prevously an Assstant Professor n the Electrcal and Computer Engneerng Department and a member of the Coordnated Scence Laboratory at the Unversty of Illnos Urbana-Champagn. From 1995 to 1998, she was a Staff Member at MIT Lncoln Laboratory n the Optcal Communcatons and the Advanced Networkng Groups. Professor Médard receved B.S. degrees n EECS and n Mathematcs n 1989, a B.S. degree n Humantes n 1990, a M.S. degree n EE 1991, and a Sc D. degree n EE n 1995, all from the Massachusetts Insttute of Technology (MIT), Cambrdge. She s an assocate edtor for the IEEE Journal of Lghtwave Technology. She has served as an Assocate Edtor for the Optcal Communcatons and Networkng Seres of the IEEE Journal on Selected Areas n Communcatons, as an Assocate Edtor n Communcatons for the IEEE Transactons on Informaton Theory and as an Assocate Edtor for the OSA Journal of Optcal Networkng. She has served as a Guest Edtor for the IEEE Journal of Lghtwave Technology, the Jont specal ssue of the IEEE Transactons on Informaton Theory and the IEEE/ACM Transactons on Networkng on Networkng and Informaton Theory and the IEEE Transactons on Informaton Forensc and Securty: Specal Issue on Statstcal Methods for Network Securty and Forenscs. She s a member of the Board of Governors of the IEEE Informaton Theory Socety. Professor Médard s research nterests are n the areas of network codng and relable communcatons, partcularly for optcal and wreless networks. She was awarded the IEEE Leon K. Krchmayer Prze Paper Award 2002 for her paper, The Effect Upon Channel Capacty n Wreless Communcatons of Perfect and Imperfect Knowledge of the Channel, IEEE Transactons on Informaton Theory, Volume 46 Issue 3, May 2000, Pages: She was co- awarded the Best Paper Award for G. Wechenberg, V. Chan, M. Médard, Relable Archtectures for Networks Under Stress, Fourth Internatonal Workshop on the Desgn of Relable Communcaton Networks (DRCN 2003), October 2003, Banff, Alberta, Canada. She receved a NSF Career Award n 2001 and was co-wnner 2004 Harold E. Edgerton Faculty Achevement Award, establshed n 1982 to honor junor faculty members for dstncton n research, teachng and servce to the MIT communty. She was named a 2007 Glbreth Lecturer by the Natonal Academy of Engneerng. Professor Médard s a House Master at Next House and a Fellow of IEEE. Danel Lucan receved the B.S. degree n Electroncs Engneerng (summa cum laude) and the M.S. n Electroncs Engneerng (wth honors) from Unversdad Smón Bolívar, Venezuela, n 2005 and 2006, respectvely. He s currently a Ph.D. canddate at Massachusetts Insttute of Technology (MIT). Hs research nterests nclude dgtal communcatons, wreless communcatons and networks, network codng, and ther applcatons to moble rado and underwater acoustc communcatons. Mlca Stojanovc graduated from the Unversty of Belgrade, Serba, n 1988, and receved the M.S. and Ph.D. degrees n electrcal engneerng from Northeastern Unversty, Boston, MA, n 1991 and After a number of years wth the Massachusetts Insttute of Technology, where she was a Prncpal Scentst, n 2008 she joned the faculty of Electrcal and Computer Engneerng Department at Northeastern Unversty. She s also a Guest Investgator at the Woods Hole Oceanographc Insttuton, and a Vstng Scentst at MIT. Her research nterests nclude dgtal communcatons theory, statstcal sgnal processng and wreless networks, and ther applcatons to moble rado and underwater acoustc communcaton systems.