Underwater Acoustic Communications: Design Considerations on the Physical Layer

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1 Underwater Acoustic Communications: Design Considerations on the Physical Layer Milica Stojanovic Massachusetts Institute of Technology Cambridge, MA 239 Abstract Acoustic roagation is characterized by three major factors: attenuation that deends on the signal frequency, multiath roagation, and low seed of sound (5 m/s). The channel has a sarse imulse resonse, where each hysical ath acts as a time-varying low-ass filter, and motion introduces additional Doler sreading and shifting. Because roagation is best suorted at low frequencies, acoustic communication systems are inherently wideband. The way in which these facts influence the design of signal rocessing methods is considered for single-carrier and multi-carrier systems. Moreover, the facts that the available bandwidth and transmission ower deend heavily on the distance, and that channel latency is high, bear imortant imlications on the design of network architectures and related rotocols. I. INTRODUCTION Underwater acoustic channels are generally recognized as one of the most difficult communication media in use today. Acoustic roagation is best suorted at low frequencies, and the bandwidth available for communication is extremely limited. For examle, an acoustic system may oerate in a frequency range between and 5 khz. Although the total communication bandwidth is very low (5 khz), the system is in fact ultra-wideband, in the sense that bandwidth is not negligible with resect to the center frequency. Sound roagates underwater at a very low seed of 5 m/s, and roagation occurs over multile aths. Delay sreading over tens or even hundreds of milliseconds results in a frequencyselective signal distortion, while motion creates an extreme Doler effect. The worst roerties of radio channels oor hysical link quality of a mobile terrestrial radio channel and high latency of a satellite channel are combined in an underwater acoustic channel. As the history of underwater acoustic communications testifies, major advances in signal rocessing were made when the hysical nature of roagation was resected through roer channel modeling. Examles that illustrate this fact include combined modeling of multiath and hase distortion for equalization in single-carrier wideband systems [], a method used in a real-time acoustic modem [2]. More recently, detection of multi-carrier signals has been shown to benefit from exlicit Doler shift modeling, while sarse channel estimation, which recognizes the fact that underwater multiath is not contiguous but consists of isolated signal arrivals, is being used to imrove the erformance of both single-carrier and multi-carrier systems. In this aer, we take a tutorial overview of the channel roerties, aiming to reveal those asects of acoustic roagation that are relevant for the design of communication systems. While a comlete and accurate model of an underwater acoustic channel remains elusive, we focus on a simlified treatment that may rovide a communications engineer with sufficient detail for effective system design. In articular, we take a fresh look at modeling a oint-to-oint channel resonse, beginning from first rinciles. The aer is organized into three sections that address () attenuation and noise, (2) multiath roagation, and (3) the Doler effect. Imlications of acoustic roagation extend beyond the hysical layer, and we conclude the aer by considering their imact on the design of future underwater networks. II. ATTENUATION AND NOISE Perhas the most distinguishing roerty of acoustic channels is the fact that ath loss deends on the signal frequency. This deendence is a consequence of absortion, i.e. transfer of acoustic energy into heat. In addition to the absortion loss, signal exeriences a sreading loss which increases with distance. The overall ath loss is given by A(l, f) =(l/l r ) k a(f) l lr () where f is the signal frequency, and l is the transmission distance, taken in reference to some l r. The ath loss exonent k models the sreading loss, and its usual values are between and 2 (for cylindrical and sherical sreading, resectively). The absortion coefficient a(f) is an increasing function of frequency, which can be obtained using an emirical formula [3]. It may be interesting to note that log a(f) α + α f + α 2 f 2 for frequencies u to about 5 khz. Noise in an acoustic channel contains the ambient noise and site-secific noise. The ambient noise, which is always resent, may be modeled as Gaussian, but it is not white. Its ower sectral density decays at aroximately 8 db/decade. If we define a narrow band of frequencies of width f around some frequency f, the signal-to-noise ratio (SNR) in this band can be exressed as SNR(l, f) =S l (f)/a(l, f)n(f) (2) where S l (f) is the ower sectral density of the transmitted signal, whose ower may be adjusted according to the dis-

2 tance. For any given distance, the narrow-band SNR is thus a function of frequency, as shown in Fig.. From this figure it becomes obvious that the acoustic bandwidth deends on the transmission distance. In articular, the bandwidth and the ower needed to achieve a re-secified SNR over some distance can be aroximated as B(l) =b l β, P (l) = l ψ, where β (, ), and ψ [4]. The bandwidth is severely limited at longer distances: at km, only about a khz is available. At shorter distances, the bandwidth increases, but it will ultimately be limited by that of the transducer. The fact that bandwidth is limited imlies the need for bandwidthefficient modulation methods if more than a bs/hz is to be achieved over these channels. Another imortant observation to be made is that the acoustic bandwidth is centered at low frequencies. In fact, the acoustic bandwidth B is often on the order of its center frequency f c, which makes an acoustic communication system inherently wideband. This fact in turn bears significant imlications on the design of signal rocessing methods, as it revents one from making the narrowband assumtion (B << f c ), on which many radio communication rinciles are based. Resecting the wideband nature of the system is articularly imortant in multi-channel (array) rocessing [5] and, as we shall discuss, in synchronization for mobile acoustic systems. /AN [db] km 5km km 5km frequency [khz] Fig.. Signal-to-noise ratio in an acoustic channel deends on the frequency and distance through the factor /A(l, f)n(f). The fact that the acoustic bandwidth deends on the distance has imortant imlications on the design of underwater networks. Secifically, it makes a strong case for multihoing, since dividing the total distance between a source and destination into multile hos enables transmission at a higher bit rate over each (shorter) ho. The same fact hels to offset the delay enalty involved in relaying [6]. Since multi-hoing also ensures lower total ower consumtion, its benefits are doubled from the viewoint of energy-er-bit consumtion on an acoustic channel. III. MULTIPATH Multiath formation in the ocean is governed by two effects: sound reflection at the surface, bottom and any objects, and sound refraction in the water. The latter is a consequence of sound seed variation with deth, which is mostly evident in dee water channels. Fig.2 illustrates the two mechanisms. tx Fig. 2. rx deth c tx distance Multiath formation in shallow and dee water. The imulse resonse of an acoustic channel is influenced by the geometry of the channel and its reflection roerties, which determine the number of significant roagation aths, their relative strengths and delays. Strictly seaking, there are infinitely many signal echoes, but those that have undergone multile reflections and lost much of the energy can be discarded, leaving only a finite number of significant aths. To ut a channel model in ersective, let us denote by l the length of the -th roagation ath, with = corresonding to the first arrival (which is not necessarily the strongest). In shallow water, where sound seed can be taken as a constant c, ath lengths can be calculated using lain geometry, and ath delays can be obtained as l /c. In reference to the strongest ath, whose arrival time is estimated to some value t which determines the reference time at the receiver, we can also define relative ath delays as τ = l /c t. Surface reflection coefficient equals - under ideal conditions, while bottom reflection coefficients deend on the tye of bottom (hard, soft) and the grazing angle [7]. If we denote by Γ the cumulative reflection coefficient along the th roagation ath, and by A the roagation loss associated with this ath, then h =Γ / A reresents the gain of this ath. At this oint, we may be temted to exress the channel resonse as h(t) = h δ(t τ ) (3) as it is usually done for a radio channel [8]. However, as we have seen in Sec.II, the ath gain in an acoustic channel is a function of frequency. Hence, it is constant only for a single frequency signal, i.e. a tone. For a broadband signal, each frequency will exerience a different attenuation. Using the attenuation (), we obtain the frequency resonse of the -th ath, H (f) = Γ A(l,f) Hence, each ath of an acoustic channel acts as a lowass filter, introducing its own disersion. Path disersion is much less than the combined multiath sread of all aths, but a comlete model must account for it nonetheless. Any aroximations that may result from the general model will deend on the sectral occuancy of the signal. rx (4)

3 The overall channel resonse in the frequency domain is H(f) = H (f)e j2πfτ (5) and the corresonding imulse resonse can be exressed as h(t) = h (t τ ) (6) where h (t) is the inverse Fourier transform of H (f). To gain insight into the multiath effects, let us look at an examle of a system with transmitter and receiver laced near the bottom at a deth of 75 m and searated by 3 km. The sreading factor is taken to be k =.5, and a 3 db loss is associated with each bottom reflection. Fig.3 illustrates the results for the first six aths. Shown in the to row are the individual ath transfer functions H (f) and resonses h (t). The overall transfer function and resonse (magnitudes H(f) and h(t) ) are shown below. The total multiath sread is governed by the longest ath delay, which is on the order of tens of ms, a value that is tyically observed in exeriments. Marked by the asterisks are the values h () at delays τ. On this scale, sreading on individual aths is not obvious, but it can be clearly seen from the ath resonses above. The shae of the ath resonses may at first sight resemble a Gaussian curve. This is not exactly so, but if one were to aroximate the absortion coefficient as e αf 2, a Gaussian shae would follow. In any case, there obviously exists some cut-off frequency f which can be used to describe the bandwidth of each ath. Corresonding to this frequency is the duration /f of each ath s resonse. In the examle shown, this frequency is on the order of 5 khz. Hence, ath disersion cannot be neglected for signals of comarable bandwidth. Excet for scaling by the reflection coefficient, the resonses of individual aths are very similar in our examle, because the ath lengths do not differ much. If one were to look at a system with shorter range but same deth, the ath length difference would be more ronounced, causing the ath resonses to differ more noticeably. Let us now consider transmission of a baseband signal u(t) modulated onto a carrier of frequency f c. The transmitted signal s(t) and the received signal r(t) are exressed as s(t) = Re{u(t)e j2πfct } (7) r(t) = Re{v(t)e j2πfct } = s(t) h(t) (8) The baseband received signal can be exressed as where v(t) =u(t) c(t) (9) c(t) =h(t)e j2πfct () is the baseband resonse of the channel, taken with resect to the frequency f c. This resonse can also be exressed as c(t) = c (t τ ) () transfer function ath transfer function x where f [Hz] x 5 f [Hz] x 5 Fig. 3. ath resonse resonse Channel resonse functions x c (t) =h (t)e j2πfcτ e j2πfct (2) is the baseband resonse of the th ath. The corresonding transfer functions are defined as C(f) = H(f + f c )= C (f)e j2πfτ C (f) = H (f + f c )e j2πfcτ (3) Note that c (t) = h (t), and, hence, the duration of c (t) equals that of the true resonse h (t). Fig.4 illustrates the baseband resonse of the direct ath, c (t), forf c =5, and 5 khz. Shown in the figure are the magnitude (solid), and the real and imaginary arts (dashed even and odd, resectively). Note that c (t) will closely resemble h (t) if f c << f. ath resonse Fig. 4. f c =5,, 5 khz f c =5,, 5 khz x 4 Baseband ath resonse for several values of the carrier frequency.

4 If the simlified channel model (3) were used, it would corresonds to a baseband discrete-ath imulse resonse c(t) = c δ(t τ ) (4) with c = h e j2πfcτ. An acoustic channel mandates a more comlicated model (), but a question naturally arises as to whether it is ossible to simlify this model to match the discrete-ath form (4). The answer to this question deends on the bandwidth occuancy of the transmitted signal. If the signal u(t) changes slowly with resect to the resonse c (t), i.e. if the signal bandwidth f is narrow enough that C (f) remains aroximately constant over f [ f/2, f/2], then the received signal can be aroximated as with v(t) = c u(t τ ) (5) c = H (f c )e j2πfcτ (6) Hence, from the viewoint of a narrowband signal, an equivalent baseband channel resonse is given in the form (4). As we have seen, an acoustic communication signal is not likely to be narrowband. Nonetheless, so long as it is bandlimited, say to some frequency region f [ B/2,B/2], we can define an equivalent baseband resonse. This is the channel resonse that encomasses only those frequencies that are relevant for the signal, i.e. C(f) =C(f)F B (f) = C (f)e j2πfτ (7) where F B (f) =for f [ B/2,B/2] and otherwise. The corresonding equivalent baseband resonse is c(t) =c(t) f B (t) = where f B (t) =Bsinc(πBt) and c (t) = +B/2 B/2 c (t τ ) (8) H (f + f c )e j2πfcτ e +j2πft df (9) Fig.5 illustrates the resonse functions of the equivalent baseband channel model for the same system arameters as before, a carrier frequency of 5 khz, and a bandwidth of 5 khz. Shown in the to row are the ath transfer functions and resonses (magnitudes C (f) and c (t) ), while C(f) and c(t) are shown below. Marked by the asterisks are the values c () at delays τ. Marked by the circles are the values c that corresond to the narrowband model (6). Note that the two are not equal in general, although in this examle they differ negligibly. The fact that they aear similar even though the signal is not narrowband is most easily exlained by looking at the transfer functions C (f). In our examle, these functions exhibit an aroximately odd symmetry around the zero frequency; hence, their integral (9) may be aroximated by that of an equivalent flat function whose value equals C () over f [ B/2,B/2]. The equivalent baseband resonse is useful for obtaining a discrete-ath channel model (one whose resonse can be reresented in terms of delta functions). This is done by alying the samling theorem to the band-limited channel resonse. One aroach is to reresent each ath resonse as c (t) = i c (i/b)sincπb(t i/b) (2) This reresentation leads to the received signal in the form v(t) = i B c (i/b)u(t i/b τ ) (2) The underlying discrete-ath equivalent baseband resonse is c (t) = i B c (i/b)δ(t i/b τ ) (22) Another aroach is to reresent the overall resonse as c(t) = i c(i/b)sincπb(t i/b) (23) The received signal can now be reresented as v(t) = i c(i/b)u(t i/b) (24) B and the underlying equivalent discrete-ath resonse is c (t) = i c(i/b)δ(t i/b) (25) B The two discrete-ath channel models (22) and (25) are not the same, although they result in the same (band-limited) received signal. The first model has non-uniformly saced coefficients (tas), which are the samles of c (t), taken every /B. Fig.6 illustrates these coefficients for our examle. Note that only a few coefficients suffice to reresent each ath. In the second model, the coefficients are uniformly saced at /B. However, because the ath delays are not exlicitly included in this model, more coefficients may be needed to cature the entire channel resonse. This fact is illustrated in Fig.7. Shown in this figure on to are the coefficients of the non-uniformly saced model, which are centered around (continuous-valued) ath delays τ, while below them are the coefficients of the uniformly-saced model. The difference can be areciated by looking at marked values c () at delays τ. As the signal bandwidth increases, so does the resolution at which the channel is reresented, and the two models become more similar. The dominant coefficients of the uniformlysaced model can then be associated with the hysical roagation aths; however, we must kee in mind that their values are not to be identified with the coefficients (6) of the narrowband resonse; on the contrary, they must be comuted over the wide range of frequencies using (9).

5 ath transfer function transfer function.5.5 x f [Hz] x ath resonse resonse x resonse coefficients (magnitude) resonse coefficients (magnitude) Fig. 7. Equivalent discrete-ath models with non-uniform (to) and uniform (bottom) ta sacing f [Hz] Fig. 5. ath coefficients Equivalent baseband channel resonse functions x Fig. 6. Path coefficients c (i/b), i =, ±,... A. Single-carrier systems Multiath disersion of an underwater acoustic channel creates a frequency-selective signal distortion that must be equalized at the receiver. The fact that the multiath sread may be on the order of tens of ms or more imlies that the inter-symbol interference (ISI) in a single-carrier broadband system may san tens or even hundreds of symbol intervals, a situation very different from that tyically found in radio systems, where ISI may involve a few symbols only. Due to the length of ISI, maximum likelihood sequence detection is often abandoned in favor of comutationally feasible equalization methods. Nonetheless, long adative equalizers may still be needed, which have considerable comlexity, noise enhancement, and sensitivity to tracking constants. This roblem can be alleviated by realizing that the equalizer coefficients are a function of the channel resonse, which is sarse. Namely, although the total delay san is large, only a few coefficients may suffice to reresent the channel resonse. Channel modeling thus becomes an imortant asect of equalization, and sarsing has been investigated for decision-feedback equalization [9], [], and, more recently, for turbo equalization []. So far, sarse channel estimation has been considered in the context of the uniformly saced model. It remains for the future to tell whether additional gains can be extracted (at a reasonable cost in comlexity) by considering the non-uniform model. In ursuing these gains, fine details of signal design must be areciated. In articular, the greater the bandwidth (and the symbol rate), the grater will be the ISI san, but the resolution will imrove both in delay and in time. The former will enable more efficient sarsing (more similarity between the two discrete-ath models), while the latter will enable faster adatation (more frequent observations of the time-varying channel). B. Multi-carrier systems A different aroach to overcoming the frequency selective distortion is through multi-carrier modulation. Orthogonal frequency division multilexing (OFDM) for acoustic channels has recently gained interest as the first exeriments have shown successful high-rate transmission [3], [4]. Future efforts will undoubtedly rely on channel modeling, and this task should be aroached keeing in mind the broadband nature of acoustic systems. Towards this goal, let us cast our current channel model into the framework of a broadband OFDM signal. Let us denote the signal transmitted on the k-th subcarrier of frequency f k = f + k f as s k (t) =Re{d k g(t)e j2πf kt } = Re{u k (t)e j2πft } (26) where g(t) is a unit-amlitude ulse of duration T =/ f and d k is the signal (data symbol). The broadband signal,

6 consisting of K carriers occuying a bandwidth B = K f is s(t) = K k= s k (t) =Re{u(t)e j2πft } (27) Let us assume that f is indeed small enough that H (f) H (f k ) for f [f k f/2,f k + f/2]. Given the fact that the OFDM symbol duration T has to be greater than the multiath sread of the channel if the system is going to make efficient use of bandwidth, we have that f will be on the order of Hz (or less) for a multiath sread of about ms. For this f, the narrowband assumtion is justified. Then, we have that the received signal is r(t) = K k= H (f k )s k (t τ )=Re{v(t)e j2πft } (28) It may again be interesting to comare this model to the simler case (3) which would give the received signal as r(t) = h s(t τ ) (29) For the two models to be equal, the coefficients H (f k ) would need to be indeendent of k, i.e. the ath transfer functions would need to be flat over the entire bandwidth B. However, as we have seen from Fig.3, this is not the case in a broadband acoustic system. Equivalently in baseband, we have that [ ] K v(t) = d k H (f k )e j2πf kτ g(t τ ) e j2πk ft k= (3) Demodulation is erformed so as to ensure that all the received signal comonents have been taken into account, yielding ˆτ+T ɛ y k = v(t)e j2πk ft dt = H(f k )d k (3) T ˆτ ɛ where ˆτ is the estimate of τ, ε is a safety margin, and T = T + T g, where T g ˆτ P ˆτ +2ε is a guard interval that is sufficient to accommodate for the delay sreading. The signal at the outut of the demodulator is thus reresented in the usual form, y k = H k d k, but the frequency-deendence remains in the terms H k = H(f k )= H (f k )e j2πf kτ (32) Channel estimation for an OFDM system can be erformed in the frequency domain (direct estimation of the coefficients H k ) or in the time domain. The later targets estimation of the channel resonse h(t), or, more recisely, the baseband equivalent taken with resect to f, b(t) = h(t)e j2πft. Namely, we can define the discrete Fourier air H k = K l= b l e j2πkl/k (33) where the time-domain coefficients b l are related to the samles of the baseband resonse b(t) taken at intervals /B. Time domain channel estimation enables ilot-based detection of OFDM signals, but it can also be used to imlement channel sarsing, which in turn will lead to an imroved erformance. Work is in rogress on this issue for acoustic channels [5]. Similarly as with single-carrier systems, it remains to be seen whether exlicit estimation of ath gains and delays can rovide further gains in erformance. C. Time variability There are two sources of the channel s time variability: inherent changes in the roagation medium, and those that occur because of the transmitter/receiver motion. Inherent changes range from those that occur on very large time scales that do not affect the instantaneous level of a communication signal (e.g., monthly changes in the temerature) to those that occur on short time scales and affect the signal. Prominent among the latter are the changes induced by the surface waves, which effectively cause the dislacement of the reflection oint, resulting in both scattering of the signal, and Doler sreading due to the changing ath length. It is out of the scoe of the resent treatment to summarize what is known about statistical characterization of these aarently random changes in the channel resonse. Suffice it to say that unlike in a radio channel, where a number of models for both the robability distribution (e.g., Rayleigh fading) and the ower sectral density of the fading rocess (e.g., the Jakes model) are well acceted and even standardized, there is no consensus on statistical characterization of acoustic communication channels. Exerimental results suggest that some channels may just as well be characterized as deterministic, while others seem to exhibit Rice or Rayleigh fading [2]. Channel coherence times below ms have been observed, but not often. For a general-urose design, one may consider worst case coherence times on the order of seconds. In the absence of good statistical models for simulation, exerimental demonstration of candidate communication schemes remains a de facto standard. IV. THE DOPPLER EFFECT Motion of the transmitter or receiver contributes additionally to the changes in the channel resonse. This occurs through the Doler effect which causes frequency shifting as well as additional frequency sreading. The magnitude of the Doler effect is roortional to the ratio a = v/c of the relative transmitter/receiver velocity to the seed of sound. Because the seed of sound is very low as comared to the seed of electromagnetic waves, motion-induced Doler distortion of an acoustic signal can be extreme. The only comarable situation in radio communications occurs in the Low Earth Orbiting satellite systems, where the relative velocity of satellites flying overhead is extremely high. (The channel there, however, is not nearly as disersive.) Autonomous underwater vehicles (AUVs) move at seeds that are on the order of few m/s, but even without intentional motion, underwater instruments are subject to drifting with waves, currents and tides, which may occur at comarable velocities. In other words, there is always

7 some motion resent in the system, and a communication system has to be designed taking this fact into account. The major imlication of the motion-induced distortion is on the design of synchronization algorithms. As the transmitter and receiver move relative to each other, the distance between them changes, and so does the signal delay. As a consequence, the leading edge of a transmitted signal may exerience one delay, while the trailing edge will exerience another. This situation is illustrated in Fig.8. Focusing on a single ath, and neglecting the ath disersion, let us look at a single ulse g(t) modulated onto a carrier of frequency f c. For a constant velocity v, the received signal is s (t + t) =s(t + t l(t ) vt ) (34) c where l(t ) is the distance traveled by the signal arriving at t. Setting this time as the reference at the receiver, we have that r(t) =s (t + t), i.e., r(t) =s(t+at τ) =Re{g(t+at τ)e j2πfc(t+at τ) } (35) where τ = l(t )/c t. With resect to the center frequency f c, the baseband received signal is f(t) =e j2πfcτ g(t + at τ)e j2πafct (36) Not counting the hase 2πf c τ, this signal is distorted in two ways: first, it is scaled in time by (+a), so that a transmitted ulse of duration T is observed at the receiver as having duration T/( + a). Equivalently, its bandwidth B is observed as (+a)b. Second, a frequency offset af c is introduced. The first tye of distortion accounts for motion-induced Doler sreading, while the second accounts for Doler shifting. g(t) T g(t+at- ) t f c B af c f synchronization. The error made in doing so is only / of a bit er, bits. Hence, a simlified model can be adoted using an aroximation g(t + at) g(t). In contrast to this situation, a stationary acoustic system may exerience unintentional motion at.5 m/s ( knot), which would account for a = 3 4. For an AUV moving at several m/s (submarines can move at much greater velocities), the factor a will be on the order of 3, a value that cannot be ignored. In such a case, the aroximation g(t + at) g(t) cannot be justified. Non-negligible motion-induced Doler sreading thus emerges as another major factor that distinguishes an acoustic channel from the mobile radio channel, and dictates the need for exlicit delay synchronization in all but stationary systems. The aroach that has demonstrated successful erformance in single-carrier broadband acoustic systems is that of couled equalization and synchronization []. In this aroach, the hase offset, which account for the Doler shift, is estimated jointly with the equalizer coefficients (i.e. the channel resonse). The so-obtained hase estimate can also be used to comute the Doler factor, which is then used to resamle the incoming signal, thus exlicitly erforming delay synchronization by decomressing the signal in time. The entire rocedure is erformed adatively and in a closed loo. This rocedure has been imlemented in a real-time acoustic modem [2], and shown robust erformance in a variety of conditions. In multi-carrier systems, equalization is accomlished easily in the frequency domain, but the Doler effect creates a articularly severe distortion. In radio systems, where time comression/dilation can be neglected, the only distortion remaining is the frequency offset. Since the system is narrowband, the Doler shift aears as almost equal for all subcarriers. This fact greatly eases the task of synchronization, and many efficient synchronization algorithms have been develoed for OFDM radio systems. In a wideband system, the situation is quite different. Here, each frequency f k is shifted by an amount that cannot be aroximated as equal for all subcarriers. The effect is that of an accordion, as illustrated in Fig.9. This figure offers an exaggerated view, but nonetheless one that serves to illustrate the fact that the Doler effect in a wideband acoustic system causes non-uniform frequency shifting. /(+a) t f c f T/(+a) B(+a) f Fig. 8. Motion causes changes in the signal duration and frequency. The Doler factor a = v/c in an acoustic channel can be several orders of magnitude greater than in a radio channel. B=K f f The way in which these distortions affect signal detection deends on the actual value of the factor a. For comarison, let us look at a highly mobile radio system. At 6 km/h ( mh), we have that a =.5 7. This value is low enough that Doler sreading can be neglected. In other words, there is no need to account for it exlicitly in symbol Fig. 9. f k f k (+a) Motion-induced Doler shift is not uniform in a wideband system. f

8 In a mobile acoustic system, the signal must first be resamled in order to reduce the Doler factor to a reasonably low value. This can be done using an indeendently obtained estimate of the Doler factor. Since large values are at hand to begin with, there is likely to be some error that cannot be neglected. If the original signal is characterized by a Doler factor a, and resamling is erformed using an estimate a, then the resamled signal will be characterized by a residual factor a =(a a )/(+a ). Once the residual Doler shift is much smaller than the subcarrier sacing, af k << f, FFT demodulation can be erformed. In a multiath channel, there exists a ossibility that each ath will exerience a different Doler effect. To develo a channel model for this situation, one can follow the stes of Sec.III, including a residual Doler factor a for the -th ath. Time variation can also be included, and we do so by assuming that the channel transfer functions H (f) do not change much during one OFDM block, but may change from one block to another. We denote the values observed during the n-th OFDM block by H (f k,n). A similar aroximation is made for the Doler factors; namely, we assume that motion occurs at a constant velocity over one block, although it may change to a slightly different value in the next block. These assumtions are easily justified for block lengths on the order of tens of ms. Note, however, that their validity is comromised as the block length increases, and this ultimately limits the bandwidth efficiency that can be achieved using simle ost- FFT rocessing on a time-varying acoustic channel. To model the Doler effect, let us exress the signal transmitted on the k-th carrier as s k (t) =Re{ d k (n)g(t nt )e j2πk f(t nt ) e j2πft } n (37) This signal travels over the channel in which the ath length variation over an interval of time t within one block is modeled as l (t n, + t) =l(t n, ) v (n) t (38) where t n, is the time of arrival of the leading signal edge in block n, i.e. t n, = t n, + T n +a (n) = t, + T +a (i) i= (39) The velocity v (n), i.e. the Doler factor a (n) =v (n)/c, can be calculated from the channel geometry, i.e. the angle of signal arrival, and the actual motion of the transmitter, receiver and reflection oints. For the first incoming block, n =, we have that t, = l (t, )/c. Corresonding to the strongest ath is the time t, at which the receiver sets its local time to. In reference to this time, the relative initial ath delays are τ () = t, t. The Doler effect on the signal s k (t) is modeled as s k,(t n, + t) =s k (t n, + t l (t n, ) v (n) t ) (4) c for t (,T /( + a (n)). Setting t n, + t = t + t, this signal can be exressed as s k,(t + t) =s k (t + a (n)t τ (n) a (n) nt ) (4) for t (t,n t,t,n+ t ), where the relative ath delays are given by n τ (n) =(+a a (i) (n))[τ () +a (i) T ] (42) Substituting for the transmitted signal (37), the comosite received signal is obtained as r k(t) = H (f k,n)s k,(t + t) =Re{ d k (n) n g(t + a (n)t τ (n) a (n) nt nt ) e j2πk f(t+a (n)t τ(n) a (n) nt nt ) e j2πf(t+a (n)t τ(n) a (n) nt ) } (43) and, summing all the carriers, we obtain r (t) = K k= r k (t). Suose now that we have an estimate a of the dominant Doler factor. This estimate can be obtained from dedicated channel robes, or directly from the signal by estimating the duration (comression/dilation) of each block. When the system velocities remain aroximately constant over several consecutive blocks (AUVs will not change their seed significantly over a few seconds), averaging can be erformed to imrove the estimate. Resamling yields the signal K r(t) =r t ( +a )=Re{ v k (t)e j2πft } (44) i= k= that will be fed to the demodulator. Corresonding to k-th carrier of this signal is the comonent v k (t) = H (f k,n)e j2πf k(τ (n)+a (n) nt ) d k (n) n g(t + a (n)t τ (n) a (n) nt nt ) e j2πk f(t nt ) e j2πf ka (n)t (45) where a (n) = a (n) a +a (46) is the residual Doler factor of the -th ath. Assuming that each received block is now confined to its own interval of duration T, FFT demodulation can be erformed in the usual manner. Because of the residual Doler effect, however, there will still be some shifting from one subcarrier to another, which causes inter-carrier interference (ICI). The demodulated signal in the k-th subband is thus a suerosition y k (n) = y kl (n) (47) l We will assume that the the same value a is used for multile blocks, although in general a different value a (n) can be used for each block.

9 where ˆτ()+nT +T ɛ y k,l (n) = v l (t)e j2πk f(t nt ) dt (48) T ˆτ ()+nt ɛ In the resence of Doler effect, the safety margin ɛ must account for additional delay sreading. Secifically, this will accumulate to n ɛ (n) =a a (i) a τ () +a (i) T (49) i= over n blocks, which should be a small fraction of T under normal conditions (e.g., even for a (i) a on the order of 4 and, say, blocks). Then, we have that y k,l (n) =d l (n) where ρ () ρ () k,l (n) = k,l (n)h (f l,n)e j2πf k(+a )τ () e jθ() k +a (n) sincφ() k,l (n) ejφ() k,l (n) φ () k,l (n) = a (n)2πf l +(l k)2π f T +a (n) 2 (5) (5) The factor ρ () k,l (n) models the ICI which results from the residual Doler effect on the -th ath. Clearly, in order for the ICI not be destructive, we need a (n)f l << f, l. The demodulator in the k-th subband will then yield 2 (n) H (f k,n)e j2πf k(+a )τ () e jθ() k (n) +z k (n) y k (n) d k (n) (52) where any residual ICI is treated as indeendent additive noise z k (n), and the only distortion remaining is contained in the hase θ () (n), which is given by k n θ () k (n) =2πf a (i) a k +a i= (i) T (53) For a constant a,wealsohavethat θ () k (n +)=θ() k (n)+2πf k a (n)t (54) where a (n) = a (n) a +a (55) (n) If the channel geometry is such that the range is much greater than deth (which often is the case) and if the Doler effect is redominantly caused by the transmitter/receiver motion, i.e. not motion of a reflection oint, then it is reasonable to assume that the Doler factor is aroximately the same for all the aths, a (n) = a(n). The received signal then simlifies to the model [3], y k (n) d k (n)h k (n)e jθ k(n) + z k (n) (56) 2 We are neglecting amlitude scaling by +a (n). This model clearly indicates the major distortion caused by the motion, which is the time-varying hase θ k (n +)=θ k (n)+ a(n)2πf k T (57) i.e. the non-uniform Doler shift a(n)f k. Whereas the channel gain may vary in time as H k (n) = H (f k,n)e j2πf k(+a )τ () (58) it does so more slowly than the hase θ k (n), which can change by 2π a(n)f k T from one block to another. For examle, if a(n) = 4, f k =5 khz, and T =5 ms, the hase change will be 2π/. In a conventional aroach to OFDM signal detection, each block is detected indeendently. This can be done by allocating null subcarriers for frequency offset estimation, and ilot subcarriers for channel estimation. This aroach takes no advantage of coherence between adjacent blocks, and it is referred when the channel is varying raidly [4]. Another aroach is based on the model (56), (57). In this aroach, a(n) is estimated adatively and used to comute the hases for all subcarriers, without reserving any null subcarriers. A single arameter is thus needed for synchronization of all K subcarriers, which may be many (e.g. 24) in a bandwidth efficient acoustic system. This model-based aroach has shown good erformance with exerimentally recorded data [3], [5]. At this time, no attemt has been made to develo receivers that accommodate the ossibility of ath-secific Doler shifts. However, exerimental recordings of acoustic channels demonstrate that such situations are ossible, including those in which the Doler shift has different sign on different aths [6]. In addressing receiver design for such channels, the model given by (52) and (54) may be a useful first ste. V. DESIGN CONSIDERATIONS FOR ACOUSTIC NETWORKS Imlications of acoustic roagation extend beyond the hysical layer, affecting all the layers of a network architecture. We have already seen that the bandwidth-distance deendence builds a strong case for underwater multi-hoing. In addition to the bandwidth, there are two major factors that influence the design of network rotocols: the transmission ower and the low seed of sound. In an acoustic system, ower required for transmitting is much greater than ower required for receiving. Transmission ower deends on the distance, and its tyical values are on the order of tens of Watts. 3 In contrast, the ower consumed by the receiver is much lower, with tyical values ranging from about mw for listening or low-comlexity detection, to no more than a few Watts required to engage a sohisticated rocessor for high-rate signal detection. In slee mode, from 3 An acoustic signal roagates as a ressure wave, whose ower is measured in Pascals (commonly, in db relative to a micro Pascal). In seawater, Watt of radiated acoustic ower creates a sound field of intensity 72 db re µpa m meter away from the source.

10 which a node can be woken on command, no more than mw may be needed. Underwater instruments are battery-owered, and, hence, it is not simly the ower, but the energy consumtion that matters. This is less of an issue for mobile systems, where the ower used for communication is a small fraction of the total ower consumed for roulsion, but it is imortant for networks of fixed bottom-mounted nodes, where the overall network lifetime is the figure of merit. One way to save the energy is by transmitting at a higher bit rate. (This is one more reason to investigate efficient rocessing methods for wideband signals.) For examle, the WHOI modem [2] has two modes of oeration: high rate at 5 kbs and low rate at 8 bs. This modem will require about 6 times less energy er bit (8 db) in the high-rate mode. The receiver s energy consumtion will also be lower, although it requires 3 W for detection of high-rate signals as oosed to 8 mw for detection of low-rate signals (the difference is about 2 db). Another way to save the energy is by minimizing the number of retransmissions. In random access networks, which are suitable for serving a varying number of users that transmit in a bursty manner, this task is made difficult by high channel latency. For examle, the basic rincile of carrier sensing multile access that a node should transmit only if it hears no on-going transmissions is comromised in an acoustic channel where the ackets roagate slowly, and the fact that none are overheard does not mean that some are not resent in the channel. Multile access with collision avoidance (MACA) has been used in the early acoustic network trials [7], and a number of variants have since been roosed [8], [9]. A different aroach has been sought through the design of coordinated sleeing schedules for underwater nodes [2], [2]. Aart from the rotocol design, it must be ket in mind that selection of ower and bit rate will influence its erformance: reducing the ower reduces the level of interference; increasing the bit rate makes the ackets shorter and reduces the chances of collision. Low seed of sound further challenges the throughut efficiency of any data link control scheme that requires automatic reeat request (ARQ), because current technology suorts only half-dulex oeration. Careful consideration of the hysical layer arameters can hel to design data ackets so as to take maximal advantage of limited resources [22]. The imlications on routing rotocols are similarly imortant. At this time, it is not certain in which direction the underwater networks will develo, as ossible alications deend on the network caabilities, which are still develoing, and the question of network caacity remains oen. Both ad hoc networks, and infrastructure-based ones can be envisioned. In either case, acoustic roagation imlies design rinciles that may be quite different from those used in radio networks (for a cellular underwater network, this issue is discussed in [23]), while the harshness of the environment dictates systems that are neither small nor easily deloyable, and certainly not inexensive or disosable [24]. ACKNOWLEDGMENT This work was suorted in art by the ONR MURI grant N and the ONR grant N REFERENCES [] M.Stojanovic, J.Catiovic and J.Proakis, Adative multichannel combining and equalization for underwater acoustic communications, Journal of the Acoustical Society of America, vol.94 (3), Set. 993, [2] L. Freitag, M. Grund, S. Singh, J. Partan, P. Koski and K. Ball, The WHOI micro-modem: An acoustic communications and navigation system for multile latforms, in Proc. IEEE Oceans Conf., 25. [3] L.Berkhovskikh and Y.Lysanov, Fundamentals of Ocean Acoustics, Sringer, 982. 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