STATISTICAL MODELING OF A SHALLOW WATER ACOUSTIC COMMUNICATION CHANNEL Parastoo Qarabaqi a, Milica Stojanovic b a qarabaqi@ece.neu.edu b millitsa@ece.neu.edu Parastoo Qarabaqi Northeastern University, 230 Egan Building Boston, Massachusetts 02115 USA Fax: (617) 373-8970 Email: qarabaqi@ece.neu.edu Abstract: A mathematical model for the impulse response of a time-varying shallow water acoustic channel is proposed. The channel is modeled as a superposition of multiple propagation paths, whose lengths and relative delays are calculated from the channel geometry. Each path is characterized by a frequency-dependent path loss, and an additional random time-variation, expressed as a multiplicative distortion. Experimental signals collected during a 2008 test in the Narragansett Bay off the coast of North America are used to assess the statistical properties of the channel. Ricean distribution, conditioned on a timevarying mean, is found to be a good match for the path gain. Measurements of the average received power over short time intervals are made to assess the channel coherence, and to test the possibility of developing a feedback-based channel state prediction for power control or adaptive modulation. Keywords: Underwater acoustic communications, statistical channel modelling, channel state prediction. This work was supported by the ONR MURI Grant #N00014-07-1-0738.
1. INTRODUCTION High rate acoustic communication systems that are in use today use adaptive receiver algorithms to track the time-variation of the channel. While much progress has been made on this front, none of the existing systems exploit the knowledge of channel statistics. This is simply due to the fact that such knowledge is scarce, and there are no widely accepted models for the acoustic channel fading. However, efforts are underway to address this problem, e.g. [1]-[5]. These studies rely on experimental data collected in localized environments, and suggest different analytical models that fit the experimental measurements: while some authors find Rayleigh fading to provide a good match for their measurements [2], [3], others find Ricean fading to provide a better fit [1], [5]. Most studies consider short-term statistics, while some consider long-term statistics as well. In [3], long-term variations of a narrowband channel are modelled by a log-normal distribution, similarly as in radio channels. In this paper, we first model the time-invariant response of a wideband acoustic channel as a superposition of multiple propagation paths, where each path is approximated by a filter of an identical shape, but a different gain. The time-varying channel model is then obtained by associating a random multiplicative gain with each propagation path. This model is based on experimental observations. The measurements indicate a time-varying local average and an additional, more rapidly varying random component in each path gain. The latter constitutes fast fading which is caused by channel variations within a window of stationarity and is found to be well-matched with a Rayleigh distribution. Conditioned on the time-varying mean, this leads to an overall Ricean fading on each path. The time-varying mean causes random variations in the average (short-term) received power, which we use as a figure of merit to assess the overall coherence of the channel. Estimates of the Doppler power spectrum suggest coherence times on the order of a second. We conclude by discussing methods for predicting the large-scale channel variation, which may be useful in the design of feedback-based power control and adaptive modulation methods. The paper is organized as follows. The deterministic model is outlined in Sec.2 and the experimental measurements are described in Sec.3. Sec.4 is devoted to statistical channel characterization and the issues of channel state prediction. Sec.5 summarizes the conclusions and outlines the future work. 2. CHANNEL MODEL: TIME-INVARIANT CASE We begin by establishing a time-invariant acoustic channel model that takes into account the physical laws of propagation, namely the fact that the path loss of an acoustic channel depends not only on the transmission distance, but on the signal frequency as well. 2.1 Single Path Attenuation The acoustic path loss experienced by a signal of frequency travelling over distance is given by, (1) where is a constant scaling factor, is the spreading factor whose value is normally between 1 and 2 (for cylindrical and spherical spreading, respectively), and is the
absorption coefficient which is often expressed in db per kilometer using the Thorp's empirical formula [5] as 0.11 44 2.75 10 0.003 where is in khz. Using this value, we obtain the absorption coefficient in as 10 where is in Hz. Unless otherwise stated, we will assume in what follows that the distance is given in meters and the frequency in Hertz. (2) 2.2 Multipath Propagation In an acoustic channel, propagation occurs over multiple paths, as illustrated in Fig.1. The -th propagation path, denoted by, 0,1, acts as a low-pass filter, whose transfer function can be modelled as [7] Γ, (3) where Γ is the cumulative reflection coefficient encountered over surface and bottom reflections along the -th path. In particular, we model the surface as ideal, with a reflection coefficient 1, while each bottom reflection is modelled by a coefficient, which is a function of the grazing angle, and under idealized plane-wave propagation conditions, has a solution of the form [8]: sin cos sin cos, cos 1, where, are the density and the speed of sound in water ( 1000 and 1500 nominally), and, are the density and the speed of sound in bottom. We will use the values 1800 and 1300 to calculate the cumulative reflection coefficient of the -th path as Γ where is the grazing angle associated with the -th propagation path. Note that this angle is the same for all reflections occurring along a given path for the geometry of Fig.1. The path length, the angle, and the number of surface and bottom reflections, can be calculated from the system geometry. (4) p=3 p=1 h p=0 l 0 p=2 θ 1 h T θ 3 h R Fig. 1: Geometry of a shallow water channel is used to calculate path lengths and angles of arrival.
2.3 Channel Response Given the impulse response of each propagation path, the resulting multipath response is obtained as τ where is the propagation delay associated with the -th path. In general, each path is characterized by an impulse response of a different shape, and this fact prevents us from obtaining a tractable, simple channel model. To explore simplified versions, let us express the transfer function so as to include the dependence on the reference path, which we take as, the direct path: Γ (6) / The frequency-dependence that distinguishes the -th path from the reference path is embodied in the term in the above expression. If this term could be approximated as a constant, one could model all the paths by an impulse response of the same shape, and a different gain. The absorption coefficient has a value very close to 1 for a broad range of acoustic communication frequencies. This fact may justify an approximation of the form We examine the viability of such an approximation in Fig. 2. Shown in the figure are the absorption coefficient, and the factor 1 for several values of the path length difference. This result indicates that the approximation (7) may indeed be valid, especially for small path length differences. The smallest path length difference shown, 15 m, corresponds to the relative path delay of 10 ms, a value that is within the multipath spread of the majority of shallow water channels. Note also that it suffices to judge the validity of approximation only within the frequency range occupied by a given system. Hence, we propose an approximation of the channel transfer function in the form, Γ / (8) where the constant may be taken as the absorption factor at some frequency within the operational bandwidth, e.g. the center frequency. Moreover, a rough approximation 1 should also be fine for a system operating in the range up to a few tens of khz. (5) (7) Fig. 2: Left: absorption coefficient in. Right: Verifying the approximation (7).
The above approximation yields a simplified channel, whose impulse response is given by τ Corresponding to this model is an equivalent baseband channel, whose impulse response with respect to a center frequency is given by where are the equivalent baseband path gains, and is the reference path response in baseband. To illustrate the model, we will focus on the geometry of the experimental channel. (9) (10) 3. EXPERIMENTAL CHANNEL An acoustic communications experiment called RACE 08 (R stands for Rescheduled) was conducted by the Woods Hole Oceanographic Institution in March 2008, in the Narragansett Bay, near the coast of Rhode Island, USA. The transmitter and receiver were deployed at 4 m and 2 m above the sea-floor respectively, and separated by 400 m in 10 m deep water. The channel probing signal was a length 4095 binary sequence, BPSK modulated into the carrier of frequency 13 khz, repeatedly transmitted at a rate 1 10kbps within about one minute. Transmit filtering was performed to compensate for the non-ideal transducer characteristic, resulting in an approximately flat overall transfer function within the signalling bandwidth 8-18 khz. The entire one-minute long signal was transmitted every two hours during several days of the experiment. Fig. 3 shows the time-invariant channel model (10) and the ensemble of estimated channel responses. Clearly, the channel is highly time-varying, but nonetheless, there is a large degree of similarity between the actual channel and the model. We thus propose a time-varying model of the form, where is a randomly time-varying path gain. In general, the path delays will also be time-varying; however, for the present experiment, where both the transmitter and the receiver were fixed on the sea-floor, no variation was observed in the path delays. Hence, we will assume that. (11) Fig. 3: Left: impulse response of the time-invariant channel model (10). Right: ensemble of impulse response magnitudes estimated from a 40 second long recording using the RLS algorithm.
Fig. 4 illustrates the time-variation of the path responses, at several delays. In particular, shown in the figure are the reference tap (at delay 0), and the two prominent taps at delays 0.5 ms and 0.7 ms. Several important observations can be made from this figure. First, we notice that the reference tap is very stable, while the others are not. The apparent variation of the reference tap gain is most likely due to the estimation noise, i.e. it is an artefact of the estimation process, not an inherent property of the channel (although it is possible that both are present). Secondly, we note that the other reflections exhibit a high degree of timevariability, which is evident in two forms: a time-varying local average (plotted in solid line in the figure), and a more rapidly varying instantaneous deviation about the average (plotted in dots). The latter includes estimation noise as well as any inherent rapid fluctuation. The time-varying local average is local in the sense that it represents an average taken over a short interval of time during which channel appears stationary in the wide sense. It is important to note that different taps exhibit different rate of variation, i.e. that they cannot be characterized by the same Doppler spread. Fig. 4: Variation of the path gains (delays 0, 0.5 and 0.7 ms) over a 40 second interval of time. Shown are the tap gains (dotted) and their local average (solid). 4. STATISTICAL CHARACTERIZATION Statistical channel characterization is concerned with determining two types of functions: the probability density function (pdf) and the power spectral density (psd). In order to provide a meaningful pdf characterization, one must first ensure that the process is stationary in the strict sense. However, we have already observed that on the time scales of interest to wideband communications, the path gain process is not stationary even in the mean. We thus proceed as follows: we first subtract the time-varying local average, and then estimate the pdf of the so-obtained process. Crucial in this procedure is the selection of the time-averaging window used to obtain the local average,. Specifically, one would like to select the window size such that the resulting estimate of the pdf is consistent over time. Obviously, this window needs to be shorter for those tap gains that are varying more rapidly. The exact manner in which the averaging window is selected remains rather heuristic. In Fig. 5 we show the histogram of the path gain associated with the delay of 0.5 ms (the strong reflection). The window size has been chosen as 0.1 s, resulting in an obviously good match with a Rayleigh pdf for the path gain amplitude. Hence, we conclude that, conditioned on the mean value, this tap gain can be modelled as a Ricean fading process. The mean value, however, is itself a random process, characterized by a coherence time that is on the order of 0.1 s.
Fig.5. Histograms of the tap gain (strong reflection at delay of 0.5 ms) with the time-varying local average removed: amplitude and a Rayleigh fit (left); phase and a uniform fit (right). Similar observations can be made for the other tap gains. Again, it is important to note that each tap will be characterized by a different time-varying local average, and a different variance. Assuming a properly chosen window that reflects the coherence time of that path, the variance appears to be constant, at least over the one-minute interval under consideration. We estimate this variance as the time-average 1,, where the time index spans the observation interval (40 s) in steps of 1 0.1. At this point, we have decomposed the path gain process into the sum of two parts: a randomly time-varying mean, and a zero-mean complex Gaussian process of a given variance. Following the general radio-communications literature, one could refer to these two processes as the slow and fast fading, respectively. Each of these processes can be characterized by its pdf and psd (we already have the pdf for the fast component). In what follows, we will focus on the psd characterization of the slow component, which we find to be the more interesting one for reasons that will become apparent shortly. The notion of fast and slow is particularly meaningful from the viewpoint of designing a communication system: fast variations occur within a signal block (data packet), while slow variations may influence the average received power in a particular signal transaction. Because the fast variations are of a somewhat unpredictable nature, they must be dealt with at the receiver side. Slow variations, however, may be more predictable, and this fact can be exploited not only at the receiver side, but also at the transmitter side - if the propagation delay allows it. The time-varying mean of the path gains will lead to a time-varying average power. Judging by the result of Fig. 4, the average power can vary by as much as several db over a time interval of few seconds - long enough for a feedback mechanism to be implemented between the receiver and the transmitter. Such a feedback mechanism would allow for adaptive power control at the transmitter side. In particular, the transmitter could adjust its power so that the receiver always gets the minimum needed for a pre-specified quality of reception (within the constraints of maximal power, of course). Such a strategy would yield savings in the total transmission power consumed. Alternatively, the transmitter could take advantage of the times when the channel is in a good state by implementing an adaptive modulation scheme. Such a strategy would yield an increase in the average bit-rate, a crucial step towards approaching the channel capacity [9]. The possibility of implementing adaptive power control or adaptive modulation relies on the ability to estimate and predict the slow channel variation. Motivated by this fact and the potential gains of adaptive modulation, we focus on characterizing the time-varying average power. (A similar analysis can be conducted for individual path gains.) (12)
Fig.6 shows the time-coherence properties of the average power. The average power is obtained by filtering the instantaneous power, which is calculated from the impulse responses at times as, where ranges across the delay span of the channel response in steps of 1. (13) Fig. 6. Left: Auto-correlation and power spectral density of the average power. The psd is obtained by the Welch's method, applied to 10 second intervals with 50% overlap. The three autocorrelation curves correspond to the Welch's psd (dashed), conventional auto-correlation (solid), and the Jake's psd plotted as a reference (dotted). Right: Channel state prediction: normalized average received power (solid), delayed version (dotted) and prediction (dashed). The 3 db Doppler spread of about 0.2 Hz indicates a coherence time of 5 s. Hence, it may be possible to exploit some (albeit a small) amount of coherence within a time interval of a second or less, needed to implement a feedback between the transmitter and receiver at short range. For the geometry at hand, the two-way travel time over 400 m is about 0.5 s. The easiest way in which the transmitter can decide on the channel state is to simply assume that it is identical to the one reported by the receiver. This type of channel state estimation will obviously be sensitive to the feedback delay. An improved estimate can be obtained by predicting the channel based on the delayed feedback. To test this possibility, we look at a predictor that operates on a series of average power measurements obtained every second. The measurements, made on a logarithmic scale, are denoted by 10log. A one-step prediction is made as 1 The model parameters are updated based on the error 1 1. Fig. 6 shows the results of prediction. A predictor of order 20 is used, and its coefficients are updated by an RLS algorithm. The actual prediction obviously yields better results when the feedback delay is not negligible with respect to the channel coherence time. (14)
5. CONCLUSIONS A deterministic multipath model, based on the channel geometry and the frequencydependent path loss in a wideband acoustic system, was augmented to include a randomly time-varying gain for each propagation path. Based on the 8-18 khz experimental measurements in a 400 m long shallow water channel, the path gains were found to obey a conditional Ricean distribution, with conditioning on a time-varying local mean in subsecond stationarity intervals. Correlation properties of the average (short-term) received power were assessed via Doppler spectrum estimation, showing coherence times on the order of a second. Considerable variation in the average received power serves as a motivation for investigating methods that would predict the channel state in order to implement feedbackbased adaptive modulation and power control. We have briefly outlined a possible channel state prediction method, which demonstrates the possibility to exploit the limited channel coherence. More sophisticated prediction models, as well as a comprehensive analysis of adaptive modulation gains, are the subject of future research. At present, our results are limited to the detailed investigation of a short segment of experimental data, and future work will enlarge this scope. In particular, the probability density function of the time-varying local mean, and the large-scale coherence properties of the channel need to be assessed. Establishing a relationship between the so-obtained experimental models and the physical processes that occur in the ocean remains a challenging task. REFERENCES [1] X. Geng and A. Zielinski, An eigenpath underwater acoustic communication channel model, in Proc. IEEE Oceans Conf., 1995. [2] M.Chitre, A high-frequency warm shallow water acoustic communications channel model and measurements, J. Acoust. Soc. Am., vol. 122 (5), pp. 2580-2586, Nov. 2007. [3] W.B. Yang and T.C. Yang, High-frequency channel characterization for M-ary frequency-shiftkeying underwater acoustic communications, J. Acoust. Soc. Am., vol. 120 (5), pp. 2615-2626, Nov. 2006. [4] J. Preisig, Acoustic propagation considerations for underwater acoustic communications network development, ACM SIGMOBILE Mobile Computing and Communications Review, vol. 11, No. 4, pp. 2-10, Oct. 2007. [5] A.Radosevic, J.Proakis and M.Stojanovic, Statistical characterization and capacity of shallow water acoustic channels, in Proc. IEEE Oceans Europe Conf., 2009. [6] L.Berkhovskikh and Y.Lysanov, Fundamentals of Ocean Acoustics, Springer, 1982. [7] M.Stojanovic, Underwater acoustic communications: design considerations on the physical layer, invited paper, in Proc. IEEE/IFIP Fifth Annual Conference on Wireless On demand Network Systems and Services (WONS), Jan. 2008. [8] F.Jensen, W.Kuperman, M.Porter and H.Schmidt, Computational Ocean Acoustics, Springer Verlag, 1994. [9] A.Goldsmith and S-G.Chua, Variable-rate variable-power MQAM for fading channels, IEEE Trans. Commun., vol. 45, pp.1218-1230, Oct. 1997.