Characterization and Modeling of Underwater Acoustic Communications Channels for Frequency-Shift-Keying Signals

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1 Characterization and Modeling of Underwater Acoustic Counications Channels for Frequency-Shift-Keying Signals Wen-Bin Yang and T.C. Yang Naval Research Laboratory Washington, DC 375 USA Abstract In a fading channel, bit error rate for frequencyshift-keying signals is deterined predoinantly by the envelope aplitude fading statistics of the signal. The narrowband envelope aplitude distributions are easured fro the TREX4 data (as a function of frequency) using M-sequence signals centered at 7 khz with a 5 khz bandwidth. The results do not fit the Rayleigh, Rician, Nakagai -distributions. In contrast, we find that the data are fitted well by a K-distribution. We also analyze the data in ters of long-ter and short-ter statistics. The long-ter and short-ter fading statistics are well fitted by the lognoral distribution and Rayleigh distribution respectively, choosing the average tie scale to be ~. sec. The joint probability distribution function of a lognoral and the Rayleigh distribution is approxiately the K-distribution. I. INTRODUCTION For underwater acoustic counications, the channel is characterized by (i) a long ultipath delay, which extends over any sybols causing inter-sybol interference (ISI), (ii) a high Doppler spread which iplies short channel coherence tie, and (iii) a tie-varying Doppler shift due to the relative platfor speed copared with the sound speed. This paper addresses channel characterization in the frequency doain, specifically the signal envelope aplitude statistics, which fors the basis of bit error rate predictions for MFSK signals. For M-ary frequency-shift-keying (MFSK) signals, the sybols are spread over the frequency band and odulated in both frequency and tie. To avoid ISI interference, the sybol duration (including the guard tie if appropriate) should be longer than the ultipath spread, but in practice this is often not the case. This ethod is referred to as incoherent counication, since each sybol is detected by an energy detector (for each tie-frequency grid). It is less sensitive to the channel teporal fluctuations and does not require a channel equalizer. The frequency coponents (bins) in the MFSK signaling are, in theory, orthogonal to each other, iplying that there is no leakage of the sybol energy fro one frequency channel to the other. In practice, this is not the case due to tievariant nature of the channel. Inter-frequency bin leakage can be substantial if there is significant error in the Doppler shift estiation. To iniize this effect, the frequency bin width f is often chosen to be uch larger than the uncertainty in the Doppler shift estiation. Channel characterization for MFSK odulation requires estiation of the channel spectru (the channel transfer function) as a function of frequency and tie. The bit error rate results not only fro the noise but also fro the ISI and inter- (frequency) channel interference (ICI). For bit error rate odeling/prediction in a realistic channel, the appropriate channel transfer function needs to include the effects of ISI and ICI. Bit error rate (BER) for MFSK signals depends on the envelope aplitude fading statistics as a function of frequency. Rayleigh and Rician aplitude probability distributions are two coonly assued odels for signal fading in radio frequency (RF) counications [-3]. For low frequency (e.g., < khz) sound propagation, Rayleigh and Rician statistics are associated with saturated and partially saturated schees in which the ultipaths are totally rando or partially rando. A discussion of the statistics for a narrowband signal can be found [4]. We find that neither of the above distributions holds for high frequency underwater acoustic counication signals. We deduce the channel spectru level fluctuation statistics fro data collected at sea, and provide a physics-based interpretation. Section II describes characteristics of aplitude fluctuations. Narrowband envelope aplitude distribution statistics are deduced fro data covering a wide band (4 khz) of frequencies. Section III reviews candidate fading statistical odels. Section IV deterines an appropriate odel for underwater acoustic channel. Section V provides conclusions. II. CHARACTERISTICS OF AMPLITUDE FLUCTUATIONS A. MFSK Modulation For a narrowband signal in a linear tie-variant channel, the channel transfer function can be defined by R(, t f) = H(, t f) S( f ), () where H is the tie-variant channel transfer function at frequency f, R is the received signal and S is the source aplitude. MFSK signals consist of any narrowband signals at frequencies f k, separated by f, where k =,,K, Rt (, fk) = H(, t fk) St (, fk), () where S(t,f k ) is the transitted sybol sequence in frequency bin f k at tie t, S(t,f k ) = or. Each sybol has a tie duration t = / f. Detection of sybols at the receiver is based on the sybol intensity R(t,f k ) which is heavily influenced by the channel spectral level H(t,f k ). Hence, BER /6/$. 6 IEEE

2 Report Docuentation Page For Approved OMB No Public reporting burden for the collection of inforation is estiated to average hour per response, including the tie for reviewing instructions, searching existing data sources, gathering and aintaining the data needed, and copleting and reviewing the collection of inforation. Send coents regarding this burden estiate or any other aspect of this collection of inforation, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Inforation Operations and Reports, 5 Jefferson Davis Highway, Suite 4, Arlington VA -43. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to coply with a collection of inforation if it does not display a currently valid OMB control nuber.. REPORT DATE SEP 6. REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE Characterization and Modeling of Underwater Acoustic Counications Channels for Frequency-Shift-Keying Signals 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Research Laboratory Washington, DC 375 USA 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES). SPONSOR/MONITOR S ACRONYM(S). DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unliited. SPONSOR/MONITOR S REPORT NUMBER(S) 3. SUPPLEMENTARY NOTES See also ADM6. Proceedings of the MTS/IEEE OCEANS 6 Boston Conference and Exhibition Held in Boston, Massachusetts on Septeber 5-, 6, The original docuent contains color iages. 4. ABSTRACT 5. SUBJECT TERMS 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT UU a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified 8. NUMBER OF PAGES 6 9a. NAME OF RESPONSIBLE PERSON Standard For 98 (Rev. 8-98) Prescribed by ANSI Std Z39-8

3 odeling/prediction requires knowledge of the statistical distribution of H(t,f k ), the envelope aplitude statistics. Snapshots of the channel transfer functions are estiated fro received data using pseudo-rando signals, e.g., - sequences, Ht (, fk) = Rt (, fk) / St (, fk), (3) where R(t,f k ) and S(t,f k ) are the spectral level of the received and transitted signals as a function of frequency at t = t, for the -th sybol. The M-sequence signals, have a flat spectru when averaged over any sapling periods. B. Envelope Aplitude Statistics The M-sequence data were transitted in consecutive packets, each of 5 sec duration with built in tie gaps. Total transission tie was ~ 4 inutes. The M-sequence data are processed first by reoving the transducer s frequency response fro the received data. The beginning of the M- sequence in each packet is deterined by atched filtering the data using either the probe signal before the M-sequence or the first M-sequence, a standard processing technique for counications. The M-sequence data are then Fourier transfored with a window size equal to the sybol duration, (e.g., /8 sec). The channel spectru is obtained using Eq. (3), with the received and transitted data processed in the sae way. The ean spectral level Σ(f k ) of the channel transfer function is estiated by suing the spectru level over all channel transfer function snapshots and dividing the result by the nuber of saples. We find that Σ(f k ) decreases by as uch as 5 db at the edge of the frequency band. These frequency coponents are discarded in our analysis. The Σ(f k ) varies by - db within the 4 khz bandwidth, which is attributed to the uncertainty in the transducer response curve, which was under-sapled in the original (calibration) easureents. We reove this effect by the following operation: H ' ( t, fk) = H( t, fk) / Σ ( fk), (4) new where t denotes the sybol sequence in tie. Since the data were transitted in packets, one has t = n T + j t, where n is the packet nuber, n =,, 34. T is the tie separation between packets, and j is the sybol nuber within a packet, j =,, 856. Henceforth we will drop the prie and denote the data by H ( t, fk ). The frequency coherence bandwidth can be easured by cross correlating the channel transfer functions between two different frequencies denoted by its frequency index k and k, ρ (( k k) f ) = * H ( t, f ) H( t, f ) k k k k Ht (, f ) Ht (, f ), (5) where the correlation is done for each packet (suing over j for a fixed n) and then averaged over all the packets. We find that ρ <. when k k, indicating that the channel transfer functions are uncorrelated between frequency bins. In other words, the frequency coherence bandwidth is < 8 Hz. For each frequency bin f k, we deterine the probability distribution of the envelope aplitude (or the histogra) Ht (, fk). The distributions are used to copare with soe theoretical fading statistical odels. III. FADING STATISTICAL MODELS A narrowband signal can be represent by p( f) = H( f ) e i π f, where H(f) is the coplex aplitude, H( f) = X + i Y, where X and Y are often referred to as the in-phase and quadrature coponents of the signal. BER of MFSK signals is deterined by the fading statistics of the (envelope) aplitude Z = X + Y as a function of frequency. A. Models with Gaussian Assuption Assue that both X and Y are Gaussian rando variables with probability distributions given by X ~ N( µ x ; σ x ) and Y ~ N( µ y ; σ y ). The correlation coefficient of two Gaussian rando variables is defined as ρxy E( XY) µ xµ y =. (6) σxσy The distribution of the (envelope) aplitude can be expressed in ters of µ x, µ y, σ x, σ y, ρ xy as given in Eq.() of [4] in the context of a propagation odel. It can be shown that the (envelope) aplitude distribution so obtained is very general - the only assuptions are that the in-phase and quadrature coponents (X and Y) are Gaussian rando variables. One finds that when the two Gaussian rando variables are uncorrelated ( ρ xy = ), and σ x = σ y, the envelope aplitude distribution reduces to the Rician distribution for non-zero µ x, µ y and the Rayleigh distribution when µ x = µ y =. B. Models without Gaussian Assuption If the in-phase and quadrature coponents (X and Y) are not Gaussian rando variables, there are several odels used to characterize the fading channel including Nakagai - distribution [5] and non-rayleigh statistics [6]. Nakagai -distribution is odeled for RF counications channel and is defined as below. p ( ) z / z z = z e Ω, z Γ( ) Ω, (7) where Ω is defined as its second oent and the paraeter is defined as the ratio of oents, called the fading figure. The Nakagai distribution contains the Rayleigh distribution as a special case when =. It can have fewer deep fades than

4 the Rayleigh distribution when / <, and ore deep fades than the Rayleigh distribution when >. For non-rayleigh fading statistics, K-distribution is one of popular odels to characterize reverberant edia. The K- distribution is given by [6] υ 4 z z pz( z) = Kυ, z, (8) αγ( υ) α α where υ is a shape paraeter, α is a scale paraeter, Kυ is the odified Bessel function of the second kind, of order Γ (υ is the Gaa function. A special case of υ, and ) the K-distribution, as υ and αυ = σ reains constant, is a Rayleigh distribution. Experiental data are analyzed next to identify which odel is appropriate for the underwater acoustic counication channel. IV. FADING MODELS USING EXPERIMENTAL DATA TREX4 experient was conducted by the Naval Research Laboratory in April 4, which took place in the coast of New Jersey. Figure shows a sound speed profile based upon easureent at the site. Acoustic counication data were transitted fro a fixed source to a fixed receiver array at the range of 3.4 k. Water depth in the experiental area is about 7 eters. The source and receivers were located at about 35 eters depth. The vertical array has an aperture of approxiately eters, and contains 8 hydrophones with nonunifor spacing. The data presented below are fro a single receiver; we observe little difference between the receivers. The data have a high signal-to-noise ratio (SNR) 3 db. Depth [] TREX4: 4/3/4 :37:4(UTC) N W Sound Speed [/s] Figure. Sound speed profile in TREX4 experient An M-sequence signal with a bandwidth of 5 khz centered at 7 khz was used to characterize the underwater counication channel. Each transitted packet lasted approxiately.7 sec and contained 53 M-sequences. A total of 34 packets, extended over a period of an hour and containing 7 M-sequences, were analyzed. The aplitude statistics are plotted in Fig. for different values of f k. We find that the probability distributions are very siilar (within the statistical error) suggesting that the envelope aplitudes at different frequencies (within the band) have independent and identical distributions (iid). [In Fig., three frequencies bins have a slightly different distribution than the rest of the frequencies bins. This difference could be easily caused by a sall nuber of events in the high tail distribution (due to coherent interference between the signal and noise) that would shift the probability distribution to what is shown Aplitude Fading Statistics 3 Figure. Aplitude fading statistics in all frequency bins Assuing an iid property, we can include envelope aplitudes of all frequencies (within the band) to obtain ore statistical saples. The resulting statistical distribution is fit to the candidate distributions, whose paraeters are estiated using the st and nd oents of the experiental data. The statistics of the experiental data is plotted in Fig. 3(a) (for the 8Hz frequency bin data) and is copared with the Rician/Rayleigh distributions and the distribution using Mikhalevsky s odel. (The distribution of the Mikhalevsky s odel turns out to be close to the Rayleigh distribution given the easured first and second oents of the data.) We repeat the above analysis using a different signaling design, by varying the frequency bin size f fro 8Hz to 3Hz, and to 5Hz. The resulting envelope aplitude distributions are plotted in Figs. 3(b) and 3(c) to copare with the odeled probability distributions. These plots show that the odels have a poor fit with the data. They suggest that the aplitude statistics for high frequency underwater counication signals are neither Rician nor Rayleigh distribution, nor the ore general distribution derived assuing that the in-phase and quadrature coponents are Gaussian rando variables. Fro Figs. 3(a)-3(c), one notes that the easureent data do not fit the Nakagai odel either, despite the fact that the Nakagai -distribution can provide ore deep fades than a Rayleigh distribution. In contrast, the easureent data see to fit the K-distribution.

5 Fading Statistics (8Hz bin) Nakagai (=.65) TREX4 Data Rayleigh Unity Power K-distribution (ν=.9) Figure 3(a). Aplitude fading statistics at 8 Hz bin size Nakagai (=.56) Fading Statistics (3Hz) TREX4 Data Rayleigh Unity Power K-distribution (ν=.85) 3 Figure 3(b). Aplitude fading statistics at 3 Hz bin size Nakagai (=.767) Fading Statistics (5Hz) TREX4 Data Rayleigh Unity Power K-distribution (ν=3.7) 3 Figure 3(c). Aplitude fading statistics at 5 Hz bin size The question of interest is what is the underlying echanis for signal fluctuations (between low and high frequencies) that lead to the Rayleigh/Rician odel on the one hand and the K- distribution odel on the other hand. Recall that the Mikhalevsky s odel assues that the rando variables X and Y follow stationary Gaussian statistics. This assuption sees to be valid for low frequency signal propagation, but perhaps not appropriate for high frequency signal propagation. High frequency signals ay follow quasistationary statistics that involve two tie scales associated with long-ter fading and short-ter fading []. Over a short tie scale, the high frequency signal is heavily influenced by the icro-fine structures (e.g., turbulence) in the ocean. The signal aplitude fluctuation follows a short-ter statistics. Over a long tie scale, the aplitude fluctuations of the signal will likely be doinated by the fine-structure perturbations of the ocean, assuing that the rapid fluctuations induced by the icro-structures have been averaged out. The signal aplitude fluctuations follow a long-ter statistics, which ay be different fro the short-ter statistics. (At low frequencies, the turbulence has no effect on the signal, hence there is only the long ter statistics.) To obtain the long-ter statistics, we will introduce an average tie scale T. Long-ter statistics are obtained by averaging the signal over the tie period of T, such that the short-ter signal fluctuation has been averaged out. That is, the long-ter fluctuation statistics, H( fk, T n), can be obtained by averaging the signal intensity H( fk, t ) at a fixed frequency f k over a period of T. The aplitude, which is the square root of the average intensity, yields a distribution as shown in Fig. 4 for T =. sec. It is well fitted by a lognoral distribution. The short-ter distribution is obtained fro individual snapshots. The snapshot data are noralized by the ean aplitude for each period of T, that reflects the long ter fluctuations; i.e., reoving the effect of long ter fluctuations, H ( fk, τ ) T = H( f, )/ (, ) n k t H fk Tn. (9) The noralized data yields a distribution shown in Fig. 5 for T =. sec. One finds that the data are fitted by the Rayleigh distribution. At the short tie scale, the cause of the signal fluctuation is turbulence or other icro-fine structure disturbances. The fluctuation is fully saturated and hence is Rayleigh distribution. At the long tie, the signal fluctuation is predoinantly due to internal waves or other fine-structure disturbances. The fluctuation is partially saturated and is well described by a lognoral distribution. At the sybol level, the sybol aplitude envelope statistics follows a joint probability distribution, deterined by the short-ter probability distribution function conditioned on the aplitude distributions dictated by the long-ter probability distribution function. The K-distribution is a ixture of Gaa and Rayleigh distributions. It has been proven that lognoral and Gaa distributions are close approxiates of each other [8-9]. Consequently, one finds that

6 K-distribution is nuerically close approxiations of a ixture of lognoral and Rayleigh distributions []. Next, we evaluate the long-ter and short-ter statistics based on the goodness of fit easure of the root-eansquared error (RMSE), also known as a fit standard error. The RMSE is defined as below. RMSE = MSE = n ( F i S i ) L () i= where L=n- indicates the nuber of independent pieces of inforation involving the n data points and paraeters of the prospective probability distribution. S i denotes the sapled statistical function (either the probability distribution or the cuulative probability distribution function) based on data at aplitude x i, i =,,n. F i denotes the sapled statistics at aplitude x i based on the statistical odel. We shall evaluate Eq. () for different values of T. For each value of T, we obtain the short-ter statistics and long-ter statistics fro the data. We fit the short-ter and long-ter envelope fluctuation data with the Rayleigh and long-noral distribution respectively and deterine the best paraeters that fit the data. Having deterined the paraeters for the best fit, we then deterine the RMSE of the fit using Eq. (). Long-ter Fading Statistics (T=. sec) with the expectation that the long-ter fluctuation should not change significantly with the tie window T as long as it is long-ter. We note that there is no theoretical basis that the distribution has to be Rayleigh. Thus a % RMSE is quite reasonable. Figure 6 plots the su of the short-ter and longter RMSE. The iniu occurs around T ~.5 sec. We find that T =. and.4 sec yield a reasonable RMSE. For the above data analysis, we use T =. sec. We have also evaluated the fit of the data with the theoretical distribution using the Kologorov-Siirnov test statistics [6]. The results are very siilar and not explicitly shown here Short-ter Fading Statistics (T=.sec) Noralized Aplitude (db) Figure 4. Long-ter fading statistics as T=. sec vs. lognoral distribution We study the short-ter envelope aplitude statistics by deterining the RMSE values of the Rayleigh statistics as a function of the tie scale T. The iniu of the test is located at T.3 sec. The RMSE is % for T between.5 and.5 sec. It indicates that the Rayleigh distribution is a good fit for the short-ter fading statistics during this tie window. When T exceeds.5 sec, the fit deteriorates significantly. We interpret this result to ean that for T>.5 sec the effects of fine-structure processes are no longer negligible and need to be included. For long-ter statistical distribution, the RMSE between the data and the Rayleigh distribution is % for T between.4 and. sec,.5% for T between.3 and.63 sec, and % for a large window of T up to.5 sec. This is consistent 3 Figure 5. Short-ter fading statistics as T=. sec vs. Rayleigh distribution V. CONCLUSIONS In this paper, we presented envelope aplitude (fading) statistics for narrowband high frequency signals over a wide band of frequencies (5-9 khz). The envelope aplitude statistics shows a non-rayleigh or a non-rician distribution behavior. The conventional odels for the envelope aplitude distributions, developed for low frequency unsaturated, partially saturated and fully saturated signal fluctuations; do not fit the high frequency aplitude statistics data. The reason is that these odels assue a fading statistics that is valid for all tie scales. Our analysis of the high frequency data indicates two tie-scale fading phenoena: long-ter versus short-ter fading. The division between the two is deterined by using RMSE test, which is about. sec tie scale for the TREX4 data. We found that the long-ter aplitude fading statistics follow a lognoral distribution and the short-ter aplitude fading statistics follows a Rayleigh distribution. The signal aplitude distribution based on the joint long-ter and shortter distributions yields a distribution nuerically close to the K-distribution, which is found to be a good fit of the high frequency data.

7 ACKNOWLEDGMENT This work is supported by the Office of Naval Research. The authors thank J. Schindall, M. McCord and P. Gendron for their effort in conducting the ACOMMS experient during the TREX 4 experient, and our NRL colleagues for aking the TREX4 experient a success. 5 Su of RMSE Errors Root Mean Squared Error (%) Average Tie Scale (x /8 sec) Figure 6. Su of RMSE errors vs. average tie scale REFERENCES [] R.J.C. Bultitude, Measureent, Characterization and Modeling of Indoor 8/9 Mhz Radio Channels for Digital Counications, IEEE Counications Magazine, June 987, pp. 5-. [] W.C.Y. Lee, Mobile Counications Engineering, McGraw-Hill Book Copany, 993. [3] T.S. Rappaport, Indoor Radio Counications for Factories of the Future, IEEE Counications Magazine, May 989, pp [4] P.N. Mikhalevsky, Envelope Statistics of Partially Saturated Processes, J. Acoustical Society of Aerica, 7(), July 98, pp [5] J.G. Proakis, Digital Counications, fourth edition, McGraw-Hill Book Copany,. [6] D..A. Abraha, Modeling Non-Rayleigh Reverberation, SR-66, SACLANT Undersea Research Center, La Spezia, Italy, 997. [7] S.O. Rice, Matheatical Analysis of Rando Noise, Bell Syste Technical Journal, Vol. 3, p., 945. [8] J.R. Clark and S. Karp, Approxiations for lognorally fading optical signals, Proc. IEEE, 58, (97). [9] N.L. Johnson and S. Kotx, Distributions in statistics: continuous univariate distributions, Wiley, New York, 97. [] A. Abdi and M. Kaveh, K distribution: an appropriate substitute for Rayleigh-lognoral distribution in fading-shadowing wireless chanels, Electronic letters, 34, (998).