On the Relationship Between Capacity and Distance in an Underwater Acoustic Communication Channel

Transcription

1 On the Reationship Between Capacity and Distance in an Underwater Acoustic Communication Channe Miica Stojanovic Massachusetts Institute of Technoogy ABSTRACT Path oss of an underwater acoustic communication channe depends not ony on the transmission distance, but aso on the signa frequency. As a resut, the usefu bandwidth depends on the transmission distance, a feature that distinguishes an underwater acoustic system from a terrestria radio one. This fact infuences the design of an acoustic network: a greater information throughput is avaiabe if messages are reayed over mutipe short hops instead of being transmitted directy over one ong hop. We asses the bandwidth dependency on the distance using an anaytica method that takes into account physica modes of acoustic propagation oss and ambient noise. A simpe, singe-path timeinvariant mode is considered as a first step. To assess the fundamenta bandwidth imitation, we take an information-theoretic approach and define the bandwidth corresponding to optima signa energy aocation one that maximizes the channe capacity subject to the constraint that the transmission power is finite. Numerica evauation quantifies the bandwidth and the channe capacity, as we as the transmission power needed to achieve a prespecified SNR threshod, as functions of distance. These resuts ead to cosed-form approximations, which may become usefu toos in the design and anaysis of acoustic networks. Categories and Subject Descriptors: A.1. [Genera]: Introductory and survey; H.1.1. [Information systems]: Systems and information theory. Genera Terms: Theory. Keywords Underwater acoustic communications, underwater acoustic networks, acoustic channe capacity. 1. INTRODUCTION With the avaiabiity of high speed acoustic communication techniques, the maturing of underwater vehices, and the advances in sensor technoogy, integration of point-to-point communication inks into autonomous underwater networks has been steadiy gaining interest over the past years, both from the research viewpoint Permission to make digita or hard copies of a or part of this work for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. To copy otherwise, to repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. WUWNet 6, September 25, 26, Los Angees, Caifornia, USA. Copyright 26 ACM /6/9...$5.. [1], and that of the design and depoyment of first experimenta networks [2]. It is envisioned at this time that some of the immediate appications of acoustic networking technoogy wi incude coaborative missions of mutipe autonomous vehices, and the depoyment of ad hoc underwater sensor networks. The design of such systems is the subject of on-going research. One of the questions that arise naturay at this time is what are the fundamenta capabiities of underwater networks in supporting mutipe nodes that wish to communicate to (or through) each other over an acoustic channe. Whie research has been extremey active on assessing the capacity of wireess radio networks (e.g., [3]) no simiar anayses have been reported for underwater acoustic networks. The few avaiabe anayses focus on the acoustic channe capacity. For exampe, [4] uses a time-invariant channe mode with additive Gaussian noise that may or may not be white, whie [5] uses a Rayeigh fading mode, with additive white Gaussian noise (AWGN). Neither of these anayses addresses the capacity dependence on distance. Underwater acoustic communication channes are characterized by a path oss that depends not ony on the distance between the transmitter and receiver, as it is the case in many other wireess channes, but aso on the signa frequency. The signa frequency determines the absorption oss which occurs because of the transfer of acoustic energy into heat. This fact impies the dependence of acoustic bandwidth on the communication distance. The resuting bandwidth imitation is a fundamenta one, as it is determined by the physics of acoustic propagation, and not by the constraints of transducer devices. The absorption oss increases with frequency as we as with distance, eventuay imposing a imit on the avaiabe bandwidth within the practica constraints of finite transmission power. Consequenty, a shorter communication ink offers more bandwidth than a onger one in an underwater acoustic system. For exampe, transmission over 1 km can be performed in one hop, using a bandwidth of 1 khz, or by reaying the information over 1 hops, each of which is 1 km ong, but offers a bandwidth on the order of 1 khz. Hence, in exchange for a more compicated system of reays, significant increase in information throughput can be obtained. At the same time, tota energy consumption wi be ower, but this is so for the radio channe as we. Before one can answer the questions of network capacity, a functiona dependence of the acoustic communication bandwidth with distance must be obtained. This is the subject of the present paper, which is organized as foows. In Sec. 2 we summarize the basics of acoustic propagation, to formuate a mode of the path oss and the ambient noise that wi be used to assess the bandwidth. In Sec. 3 we propose two definitions of the acoustic bandwidth, one a heuristic definition based

2 on the 3 db oss in the band-edge SNR and a uniform energy aocation, and the other an information-theoretic definition based on optima energy aocation for a fixed transmission power. In both cases, the tota transmission power is determined as that needed to achieve a pre-specified SNR within the given bandwidth. Sec. 4 iustrates the resuts numericay, i.e. provides quantitative measures of the bandwidth in Hz and capacity in bps, as we as the transmission power in db re µ Pa, as functions of distance. Numerica resuts ead to cosed-form approximations which provide functiona dependence of the system capacity on the transmission distance. Concusions are summarized in Sec ACOUSTIC PROPAGATION: PATH LOSS AND NOISE 2.1 Attenuation Attenuation, or path oss that occurs in an underwater acoustic channe over a distance for a signa of frequency f is given by A(, f) = k a(f) (1) where k is the spreading factor, and a(f) is the absorption coefficient. Expressed in db, the acoustic path oss is given by 1 og A(, f) =k 1 og + 1 og a(f) (2) The first term in the above summation represents the spreading oss, and the second term represents the absorption oss. The spreading factor k describes the geometry of propagation, and its commony used vaues are k =2for spherica spreading, k =1 for cyindrica spreading, and k = 1.5 for the so-caed practica spreading. (The counterpart of k in a radio channe is the path oss exponent whose vaue is usuay between 2 and 4, the former representing free-space ine-of-sight propagation, and the atter representing two-ray ground-refection mode.) The absorption coefficient can be expressed empiricay, using the Thorp s formua which gives a(f) in db/km for f in khz as [6]: 1 og a(f) =.11 f 2 1+f +44 f f f (3) This formua is generay vaid for frequencies above a few hundred Hz. For ower frequencies, the foowing formua may be used: 1 og a(f) = f 2 1+f +.11f 2 (4) 2 The absorption coefficient is shown in Fig.1. It increases rapidy with frequency, and is the major factor that imits the maxima usabe frequency for an acoustic ink of a given distance. The path oss describes the attenuation on a singe, unobstructed propagation path. If a tone of frequency f and power P is transmitted over this path, the received signa power wi be P/A(, f). If there are mutipe propagation paths, each of ength p,p =,...P 1, then the channe transfer function can be described by H(, f) = P 1 p= Γ p/ A( p,f)e j2πfτp (5) where = is the distance between the transmitter and receiver, Γ p modes additiona osses incurred on the pth path (e.g. refection oss), and τ p = p/c is the deay (c=15 m/s is the nomina speed of sound underwater). If the transmission is not directiona, such that propagation paths other than the direct one contribute to absorption coefficient [db/km] frequency [khz] Figure 1: Absorption coefficient, a(f) [db/km]. the received signa, then the received power wi be P H(, f) 2. In our treatment, we sha focus on the path oss mode that takes into account ony the basic path oss. Extensions to the mutipath propagation case are straightforward, if the attenuation A(, f) is substituted by 1/ H(, f) 2 evauated for the particuar channe geometry. 2.2 Noise The ambient noise in the ocean can be modeed using four sources: turbuence, shipping, waves, and therma noise. Most of the ambient noise sources can be described by Gaussian statistics and a continuous power spectra density (p.s.d.). The foowing empirica formuae give the p.s.d. of the four noise components in db re µ Pa per Hz as a function of frequency in khz [7]: 1 og N t(f) =17 3 og f 1 og N s(f) = 4 + 2(s.5) + 26 og f 6 og(f +.3) 1 og N w(f) =5+7.5w 1/2 +2ogf 4 og(f +.4) 1 og N th(f) = og f (6) Turbuence noise infuences ony the very ow frequency region, f<1 Hz. Noise caused by distant shipping is dominant in the frequency region 1 Hz -1 Hz, and it is modeed through the shipping activity factor s, whose vaue ranges between and 1 for ow and high activity, respectivey. Surface motion, caused by wind-driven waves (w is the wind speed in m/s) is the major factor contributing to the noise in the frequency region 1 Hz - 1 khz (which is the operating region used by the majority of acoustic systems). Finay, therma noise becomes dominant for f>1 khz. The overa p.s.d. of the ambient noise, N(f) = N t(f) + N s(f) +N w(f) +N th(f), is iustrated in Fig.2, for the cases of no wind (soid) and wind at a moderate 1 m/s (dotted), with varying degrees of shipping activity in each case. The noise decays with frequency, thus imiting the usefu acoustic bandwidth from beow. It may be usefu to note that in a certain frequency region the noise p.s.d. decays ineary on the ogarithmic scae. The foowing approximation may then be usefu: 1 og N(f) N 1 η og f (7) This approximation is shown in the figure (dash-dot) with N 1 = 5 db re µ Pa and η=18 db/decade. 2.3 The AN Product and the SNR Using the attenuation A(, f) and the noise p.s.d. N(f) one can evauate the signa-to-noise ratio (SNR) observed at a receiver

3 noise p.s.d. [db re micro Pa] wind at 1 m/s wind at m/s shipping activity,.5 and 1 (bottom to top) 1/AN [db] km 5km 1km 5km f [Hz] Figure 2: Power spectra density of the ambient noise, N(f) [db re µ Pa]. The dash-dot ine shows an approximation 1 og N(f) =5 18ogf. over a distance when the transmitted signa is a tone of frequency f and power P. Not counting the directivity indices and osses other than the path oss, the narrow-band SNR is given by P/A(, f) SNR(, f) = (8) N(f) f where f is the receiver noise bandwidth (a narrow band around the frequency f). The AN product, A(, f)n(f), determines the frequency-dependent part of the SNR. The factor 1/A(, f)n(f) is iustrated in Fig.3. For each transmission distance, there ceary exists an optima frequency f o() for which the maxima narrow-band SNR is obtained. The optima frequency is potted in Fig.4 as a function of transmission distance. In practice, one may choose some transmission bandwidth around f o(), and adjust the transmission power so as to achieve the desired SNR eve. We comment more on such choices in the foowing section. 3. BANDWIDTH AND CAPACITY 3.1 A heuristic bandwidth definition A possibe definition of the system bandwidth is that of a 3 db (or some other eve) bandwidth. We define the 3 db bandwidth B 3() as that range of frequencies around f o() for which SNR(, f) >SNR(, f o())/2, i.e., for which A(, f)n(f) < 2A(, f o())n(f o()) = 2AN min(). Once the transmission bandwidth is set to some =[f min(), f max()] around f o(), the transmission power P () can be adjusted to achieve the desired narrow-band SNR eve at f o(). Aternativey, and perhaps more meaningfuy, one may set the desired transmission power in accordance with the tota SNR corresponding to the bandwidth. If we denote by S (f) the p.s.d. of the transmitted signa chosen for the distance, then the tota transmitted power is P () = S (f)df (9) frequency [khz] Figure 3: Frequency-dependent part of narrow-band SNR, 1/A(, f)n(f). Practica spreading (k =1.5) is used for the path oss A(, f). Moderate shipping activity (s =.5) and no wind (w =) are used for the noise p.s.d. N(f). and the SNR is SNR(, ) = S(f)A 1 (, f)df N(f)df (1) In this definition, the SNR depends on the transmitted signa p.s.d., and so does the tota transmission power P (). In the simpest case, the transmitted signa p.s.d. is fat, S(, f) =S for f, and esewhere. The tota transmission power is then P () = S. If it is required that the received SNR be at east equa to some pre-specified threshod SNR, then the minima transmission power can be determined from SNR and. When the 3 db bandwidth is used, the corresponding transmission power is determined as P 3() =SNR N(f)df B 3 () B 3() (11) B 3 () A 1 (, f)df Whie this definition of the acoustic system bandwidth may be intuitivey satisfying, there is nothing to guarantee its optimaity. It may be possibe to achieve a better utiization of resources through a different energy distribution across the system bandwidth. In other words, we may adjust the signa p.s.d. S (f) in accordance with the given channe and noise characteristics A(, f) and N(f) so as to optimize some performance metric. We do so in the foowing section. 3.2 Capacity-based bandwidth definition A performance metric that naturay comes to mind is the channe capacity. Assuming that the noise is Gaussian, and that the channe is time-invariant for some interva of time, the capacity can be obtained by dividing the tota bandwidth into many narrow sub-bands, and summing the individua capacities. The ith subband is centered around frequency f i,i=1, 2,...and it has width f, which is sma enough that the channe transfer function appears frequency-nonseective, i.e. the ony distortion comes from a constant attenuation factor A(, f i). The noise in this narrow

4 optima frequency [khz] Figure 4: Optima frequency f o() is the one at which 1/A(, f)n(f) reaches its maximum. sub-band can be approximated as white, with the p.s.d. N(f i), and the resuting capacity is given by C() = [ ] f og 2 1+ S(fi)A 1 (, f i) (12) N(f i i) Maximizing the capacity with respect to S (f), subject to the constraint that the tota transmitted power P () is finite, yieds the optima energy distribution. The signa p.s.d. shoud satisfy the water-fiing principe [8]: S (f)+a(, f)n(f)=k (13) where K is a constant whose vaue is to be determined from the power P (), and it is understood that S (f). The power P () can be chosen to provide a desired SNR, SNR, simiary as before. The SNR corresponding to the optima energy distribution is given by SNR(, ) = S(f)A 1 (, f)df N(f)df = K A 1 (, f)df N(f)df 1 (14) The transmitted power is P () = S (f)df = K A(, f)n(f)df (15) If the power is determined as the minimum needed to satisfy the SNR condition SNR(, ) SNR (16) then the optima energy distribution S (f) can be obtained through the foowing numerica procedure. For each distance, we begin by finding the optima frequency f o(), and setting the initia vaue of the constant K to K () = AN min(). We then proceed iterativey, increasing K in each step by some sma amount, unti the condition (16) is met. In particuar, if K (n) denotes the current vaue of the constant K, for which the SNR is sti beow the desired threshod, then the foowing operations are performed in the n-th step: 1. Determine B (n) () as that region of frequencies for which A(, f)n(f) K (n). 2. Cacuate SNR (n) from (14) using the bandwidth B (n) () and the constant K (n). 3. Compare SNR (n) to SNR. If SNR (n) <SNR, increase K by a sma amount, and continue the procedure. For exampe, K (n+1) =(1+ɛ)K (n) was used for numerica evauation of resuts in Sec. 4, with ɛ=.1. When SNR (n) reaches (or sighty exceeds) SNR, the procedure ends. The current vaue of K (n) is set as the desired constant K, and the current vaue of the bandwidth B (n) () is set as the desired bandwidth. The optima energy distribution is { K A(, f)n(f), f S (f) =, otherwise and the tota power is obtained from (15). Finay, the channe capacity is [ ] K C() = og 2 df (17) A(, f)n(f) In comparison, the capacity (if it may be caed that) of the heuristic scheme that uses equa energy distribution across the 3 db bandwidth is [ C 3() = og 2 1+ P3()/B3() ] df (18) A(, f)n(f) B 3 () 4. NUMERICAL RESULTS The bandwidth, capacity, and transmission power were evauated through numerica integration of the expressions presented in the previous section. Resuts are presented for both the 3 db definition and the capacity-maximizing definition of bandwidth. For ack of better names, we sha refer to these two cases at the heuristic case and the optima case, respectivey. In both cases, the acoustic oss is modeed using practica spreading, k =1.5, and the noise p.s.d. is that obtained for moderate shipping activity s =.5 and wind speed w =. The SNR threshod is set to SNR =2 db. Figure 5 iustrates the resuts obtained using the 3 db bandwidth definition. The upper pot shows the bandwidth B 3() and the corresponding capacity C 3(), evauated numericay from the expression (18). The resuting bandwidth efficiency is 6.6 bps/hz. The ower pot shows the transmission power P 3(), evauated from the expression (11). For the case of optima resource aocation, we first find the transmitted signa p.s.d. for each distance and the desired threshod SNR. Fig.6 iustrates the attenuation-noise characteristic A(, f) N(f), and the optima p.s.d. S (f) obtained for =5 km. Shown together with the AN characteristic is the vaue of K for which the tota SNR reaches SNR =2 db. The points on the frequency axis where K crosses the AN characteristic mark the optima signa bandwidth for this distance and the chosen SNR threshod. The resuts obtained using the optima bandwidth definition are summarized in Fig.7. The upper pot shows the bandwidth and the corresponding capacity C(), evauated numericay from the expression (17). The resuting bandwidth efficiency is 8 bps/hz. The improvement in bandwidth efficiency owes to the optima energy-bandwidth aocation. The ower pot shows the transmission power P (), evauated from the expression (15).

5 B [khz] and C [kbps] A(,f)N(f) [db] =5km 1 K f [khz] AN S(f)/Smax Smax=125 db P [db re micro Pa] f [khz] Figure 6: Finding the optima p.s.d of the transmitted signa for transmission distance =5km: upper pot shows A(, f)n(f) and the constant eve K for which the received SNR equas SNR =2 db; ower pot shows the resuting p.s.d. S (f). Figure 5: Bandwidth and capacity (upper pot) and transmission power (ower pot) needed to achieve SNR =2 db. Equa energy distribution and the 3 db bandwidth definition are used. Circes indicate resuts of numerica integration; soid curves represent cosed-form approximations. Whie there is no cosed-form soution for the system bandwidth as a function of distance, a coser examination of the numerica resuts reveas that the bandwidth decays amost ineary with distance on a ogarithmic scae. A simiar observation can be made for the capacity. The power increases with distance, aso foowing a inear trend on the ogarithmic scae. Such a trend is observed for both the heuristic and the optima bandwidth definition. Hence, the foowing approximations are proposed: ˆB 3() =b 3 β 3, Ĉ 3() =c 3 γ 3, ˆP3() =p 3 π 3 ˆ =b o βo, Ĉ() =c o γo, ˆP () =po πo (19) where the coefficients b, c, p, and the exponents β, γ, π are positive constants that can be determined by curve-fitting. Least-squares approximation by a first-order poynomia on a ogarithmic scae provided the vaues of these parameters that are isted in Tabe 1. These vaues are vaid for the SNR threshod of 2 db. The coefficients are given in db reative to 1 khz, 1 kbps, and 1 µpa, for the bandwidth, capacity, and power, respectivey, whie the exponents are given in db per km. The soid curves in Figs. 5 and 7 represent the cosed-form approximations, with the circes indicating actua vaues obtained through numerica integration. Ceary, there is a very good agreement between the numerica resuts and the approximate cosedform soutions. Hence, the cosed-form expressions offer an efficient way of estimating the system resources (avaiabe bandwidth and capacity, required power) for a given distance. They may thus prove to be a usefu too in the design and anaysis of underwa- ter acoustic networks, where it might be cumbersome to evauate numericay the ink capacities and powers for every different topoogy. heuristic 3 db definition optima definition band- b 3=14.39 db re khz b o=19.76 db re khz width β 3= -.55 db re khz/km β o=-.59 db re khz/km capacity c 3=22.68 db re kbps c o=28.76 db re kbps γ 3= -.55 db re kbps/km γ o=-.59 db re kbps/km power p 3=16.78 db re µpa p o= db re µpa π 3=2.22 db re µpa/km π o=2.7 db re µpa/km Tabe 1: Parameters of the cosed-form approximations for bandwidth, capacity, and transmission power at SNR = 2dB. The resuts of Figs. 5 and 7 correspond to the SNR threshod of 2 db. For a different SNR threshod, different vaues of the bandwidth, capacity, and transmission power are obtained. The effect of varying SNR is summarized in Fig.8. Shown in the figure is the bandwidth efficiency, i.e. the ratio between the system capacity and bandwidth, C()/ in bps/hz, for severa vaues of transmission distance, = 5, 15, 25, km. The capacitymaximizing definition of bandwidth is used, and the system parameters are evauated for SNR between -15 db and 45 db. The first pot (top) provides a reationship between the bandwidth efficiency and the transmission power. The bandwidth efficiency increases with transmission power, foowing a simiar pattern for various distances. The second pot iustrates the bandwidth efficiency as a function of SNR. Athough one might expect the C/B curves to coapse into a singe curve, this is not the case, except at ow SNR. At a moderate SNR around 1 db, the C/B curves start to diverge sighty, showing a greater bandwidth efficiency for a greater distance. However, with a further increase in the

6 =5, 15, 25,...75 km B [khz] and C [kbps] 1 2 C/B [bps/hz] P [db] =5,15,25,...75 km P [db re micro Pa] 14 C/B [bps/hz] 5 equivaent AWGN SNRo [db] Figure 7: Bandwidth and capacity (upper pot) and transmission power (ower pot) needed to achieve SNR =2 db. Capacity-maximizing energy distribution and the corresponding optima bandwidth definition are used. Circes indicate resuts of numerica integration; soid curves represent cosedform approximations. C/B [bps/hz] 1 1 =5,15,25,...75 km equivaent AWGN SNR, the curves cross each other, yieding higher bandwidth efficiency to shorter distances. As a benchmark, the pot aso shows the bandwidth efficiency of an equivaent AWGN channe, ( ) C =og B 2 (1 + SNR ) (2) AW GN We observe that the bandwidth efficiency of an acoustic Gaussian channe tends to that of an equivaent AWGN channe at ow SNR regardess of the distance, but then deviates from it as the SNR increases. For the considered mode of a time-invariant singe-path acoustic channe, the bandwidth efficiency is greater than that of an equivaent AWGN channe in the SNR range between about 1 db and 3 db, but fas beow it as the SNR further increases. It may aso be interesting to present the bandwidth efficiency as a function of the bit SNR, a figure of merit commony used in the study of communication systems. For a channe corrupted by the AWGN, the bit SNR is the ratio of the bit energy E b to the noise p.s.d. N. The noise in the acoustic channe is not white, but one can define the p.s.d. of an equivaent white noise as N () = 1 N(f)df (21) The dependence of the equivaent noise p.s.d. on the distance is caused by that of the bandwidth. The received bit energy is E b() = 1 S (f)a 1 (, f)df (22) C() Eb/No [db] Figure 8: Bandwidth efficiency as a function of transmission power, SNR and equivaent E b/n. Hence, we define the equivaent bit SNR as E b = SNR(, ) (23) N C() It may be interesting to note that athough both the bit energy and the equivaent white noise p.s.d. depend on the distance, their ratio does not. The third pot of Fig.8 shows the bandwidth efficiency as a function of the equivaent bit SNR E b/n. As a caibration benchmark, the pot aso shows the bandwidth efficiency of the equivaent AWGN channe, which obeys the reationship E b N = 2(C/B)AW GN 1 (C/B) AW GN (24) This pot presents the same resuts as the second one, but perhaps in a more famiiar framework, which ceary shows the Shannon s imit.

7 5. CONCLUSIONS It is we known that the frequency-dependency of the acoustic path oss imposes a bandwidth imitation on an underwater communication system, such that a greater bandwidth is avaiabe for a shorter transmission distance. This fact has a significant impication on the design of an acoustic network: if a greater bandwidth is avaiabe for a shorter distance, then the tota network throughput can be increased by pacing reay nodes between the informationgenerating ones. In designing a network, one wi thus inevitaby ask how many reays to use, where to pace them, and what is the overa throughput improvement; or, more generay, what is the optima resource aocation and what is the network capacity. To answer these questions, ink capacity must be known as a function of distance. This paper offers an insight into the reationship between an acoustic ink capacity and distance. As a first approximation, a simpe mode of a time-invariant acoustic channe was considered, taking into account a physica mode of acoustic path oss and the ambient noise. The bandwidth, capacity, and transmission power needed to achieve a pre-specified SNR were evauated anayticay as functions of distance. Numerica resuts were shown to admit simpe cosed-form approximations. These semi-anaytica soutions provide the needed functiona dependence between the acoustic ink capacity and transmission distance. The basic principes used in this paper can be appied to more accurate acoustic channe modes that take into account both mutipath propagation and time-variabiity. Future research shoud focus on using these resuts to assess the capacity of muti-hop acoustic systems. 7. REFERENCES [1] I.Akyidiz, D.Pompii and T. Meodia, Underwater acoustic sensor networks: Research chaenges, Ad Hoc Networks Journa, Esevier, March 25, vo. 3, Issue 3, pp [2] J.A.Rice, SeaWeb acoustic communication and navigation networks, in Proc. Internationa Conference on Underwater Acoustic Measurements, Juy 25. [3] S.Toumpis and A.Godsmith, Capacity regions for wireess ad hoc networks, IEEE Trans. Wireess Commun., vo.2, pp , Juy 23. [4] H.M.Kwon and T.Birdsa, Channe capacity in bits per Joue, IEEE J.Oceanic Eng., vo.11, No.1, pp.97-99, Jan [5] H.Leinhos, Capacity cacuations for rapidy fading communications channes, IEEE J.Oceanic Eng., vo.21, No.2, pp , Apr [6] L.Berkhovskikh and Y.Lysanov, Fundamentas of Ocean Acoustics New York: Springer, [7] R.Coates, Underwater Acoustic Systems, New York: Wiey, [8] J.G.Proakis, Digita Communications, New York: Mc-Graw Hi, ACKNOWLEDGMENT This materia is based upon work supported by the Nationa Science Foundation under Grant No