Synthetic Undersea Acoustic Transmission Channels Abstract. INTRODUCTION

Transcription

1 Sythetic Udersea Acoustic Trasmissio Chaels Dale Gree*, Joseph Rice *Bethos, Ic., 49 Edgerto Dr., North Falmouth, MA 556 SPAWAR Systems Ceter, & Departmet of Physics, Naval Postgraduate School, Moterey, CA Abstract. Achievig effective through-water acoustic digital sigalig (telesoar) requires a ability to adaptively accommodate a complex ad possibly time-varyig acoustic chael. Variable combiatios of oise, iterferece, multipath, ad motio impair real-world telesoar chaels. Whe cosiderig ay oe of these factors idividually, performace degradatio may be predicted from theory. But the combiatio of these factors ca cofoud our theoretical predictive capabilities. A computer simulatio of the acoustic chael is useful for developig telesoar waveforms ad modems. The simulatio directly drives the modem receiver with a virtually propagated aalog sigal, eablig us to test the performace of the sythetic ed-to-ed telesoar lik. We have observed a close correlatio betwee simulatio-based performace ad observed performace i at-sea chaels exhibitig similar characteristics. The fact that telesoar performace is ow quite predictable i a wide variety of chaels is due, i large measure, to the use of chael simulatio. The simulatio preseted here does ot rely o physical modelig of the chael. Rather, it is based o the combiatio of theoretical multipath models (e.g., a Ricia chael) with rapidly time-varyig impulse respose fuctios, where the statistics are derived from at-sea experimets or govered by values derived from idepedet physics-based models (e.g., PC-SWAT, Bellhop, etc). Noise ad other additive iterferece are combiatios of theoretical ad stored data, ad rage rate-iduced compressio ad dilatio are icorporated. INTRODUCTION The achievemet of practical uderwater acoustic commuicatios, or telesoar, owes much to the earlier developmet of terrestrial wireless commuicatios. Ideed, the fudametal theoretical precepts of wireless commuicatios apply equally well to telesoar, ad the physical layer sigalig techiques used for telesoar would be recogized i most commuicatios textbooks. The primary differece betwee the two is the adjustmets made to accommodate to the physical chael. I particular, telesoar deals with very limited sigal badwidths, with frequecy-depedet fadig caused by multipath, with log latecies, ad with time-varyig chael impulse respose fuctios. These factors have forced the developmet of a relatively uique structure for the implemetatio of effective commuicatios i the uderwater acoustic chael. A good example of the differeces betwee telesoar ad RF wireless commuicatios is the use of power cotrol i the RF world to eable multi-access commuicatios. I the code divisio multiple access (CDMA) cell phoe system used i the U.S., a base statio maitais istataeous liks with may idividual cell phoes via a secodary chael usee by the cell phoe user. This lik is used to cotrol the trasmitted power levels of all phoes so that all primary sigals are received at the base statio with approximately equal power. This eables the use of a type of sigal which has auto- ad crosscorrelatio properties similar to those of ucorrelated, Gaussia, white oise. However, i the acoustic domai, several issues cospire to make such a etwork difficult to achieve:

2 mobile odes at some distace from a base statio may well move ito quite differet chaels by the time a power cotrol sigal could be received from the base; furthermore, the very limited badwidths ad trasducer systems curretly available have to date preveted the developmet of full-duplex sigalig. Rapid temporal variatios i the impulse respose fuctios may severely costrai the use of phase coheret sigalig but have markedly less impact o o-coheret sigalig techiques. This fudametal issue with the physics of telesoar chaels has bee addressed with some success by the use of adaptive chael equalizatio, especially via the decisio feedback equalizer (DFE) origially developed by Proakis []. However, there still are importat, uresolved issues with chael equalizers which make their performace difficult to predict. The tremedous variety of real-world telesoar chaels, combied with the very real difficulty of obtaiig sufficiet physical iformatio to fully characterize a give chael, has substatially costraied the ability of physics-based propagatio models to predict the performace of telesoar systems. Eve i those situatios i which good physical characterizatio is available, the real-time computatioal burde required for physics-based modelig of high frequecy, time-varyig chaels is cosiderable. Our approach to telesoar simulatio is to rely o statistics which describe: the (possibly) time-varyig impulse respose fuctio of the chael; the character of the iterferece (both oise ad discrete iterfereces); ad certai properties of the commuicatig platform (e.g., rage rate). These statistics may be obtaied from physicsbased propagatio models (e.g., PC-SWAT, Bellhop, etc), or they may be obtaied from pertiet at-sea experimets. Our approach is to idetify telesoar performace i statistically describable chaels, ad ot to replicate ay specific physical chael. Performace is based o metrics obtaied from real telesoar modems with aalog iput provided by a simulator. I this way we test the ed-to-ed lik as oe system. We evaluate performace i welluderstood, beig chaels, the search for worst case chael coditios which break the modem. SIGNAL DISTURBANCES The pheomea we discuss which disturb the trasmitted sigal are those which have bee observed empirically as detractig i a substatial way from modem performace. I this paper we discuss modelig of the sigal received over a complex multipath chael, represetatio of oise, modelig of discrete iterfereces ad jammers, ad costat velocity motio (rage rate). There is o particular importace of orderig i the followig descriptios. Ideed, o a specific modem, the importace of a particular chael characteristic is determied by the effect it has o modem performace: rapid temporal variatio i the impulse respose fuctio may severely costrai the use of phase coheret sigalig, but have virtually o impact o o-coheret sigalig techiques.

3 Modelig ad Simulatig The Acoustic Chael Modelig I the followig, we first describe the receipt of a perfectly observable sigal, but we quickly recogize that empirical estimatio of chael characteristics will be costraied by the ature of the sigal probes we use. I particular, we would employ a broadbad (but bad-limited) pulse processed with a replica correlator to examie the (bad-limited) approximatio to the chael impulse respose. This specifically limits the observatio of those compoets of the impulse respose which are temporally closer tha approximately /W, with W the pulse badwidth. The impulse respose fuctio describes the (possibly time-varyig) temporal ad spectral distributio of eergy preseted to a receiver by the chael. This fuctio has bee extesively studied [,3], ad it is ot our purpose to replicate the detailed mathematical derivatios that have bee developed by others. We simply preset our versio of the received sigal as follows. The received sigal is ideally described as a collectio of N amplitude-weighted, delayed versios of the trasmitted sigal s(t) with amplitude a (t) ad delay τ : For coveiece, we assume that s(t) is aalytic (-sided spectrum). r t) = a ( t) s( t τ ). () ( I eq (), we assume the followig:. a (t) is a circular (complex) Gaussia radom variable with zero mea ad with power (variace) b = E a a (t;τ), where E a is the expectatio operator over the amplitudes for the th path.. b is costat for the duratio of a sigle trasmissio, although the complex ature of a (t) may chage durig that duratio. 3. The delays τ reflect theoretical predictios or at-sea observatios of time spread. They are costat for the duratio of a sigle trasmissio. Because the path power is costat, the time-varyig ature of the idividual path amplitude is described by the (assumed) wide sese statioary coherece fuctio of the path, ρ : ρ ( t) = E a a ( t) a ( t + t) / b () We observe that ρ, beig a coherece fuctio, has a duality i the Fourier domai which may be iterpreted as a power spectrum P (f) with itegrated power equal to b. The effective badwidth of P (f) is determied by the time differece at which ρ ( t) drops to a agreed-upo value. We defie B to be the (3-dB) badwidth of this path, or the iverse of t for which ρ ( t) =.75. We also observe that the Fourier trasform of a (t) is A (f), with the property that P (f) = E a A (f). (3) Now, we assumed that a (t) was a Gaussia fuctio, so A (f) is likewise Gaussia at every frequecy compoet.

4 As a example, cosider a coditio i which we had observed a chael with two domiat paths, with a 4 ms temporal separatio betwee them, a relative power of ad.5, respectively, ad coherece times of 5 ms ad ms, respectively. The upper trace of Figure shows a realizatio of the path weight a (t), while the lower trace shows the sample auto-coherece properties of this weight. The dashed lie i the upper plot idicates the duratio of the sigal relative to this weight. Figure shows correspodig sample results for the secod path. Note the width of the coherece fuctio for the two paths, which correspod well with the specified (a priori) coherece time. Magitude of Amplitude Spreadig Fuctio, path, Coherece time = 5 (ms) Spread Fuct. Sigal Measured Coherece Fuctio vs. Time for oe path.8 Power Time (ms) Figure. A sample temporal weightig fuctio reflectig a sigle path i a acoustic chael. the coherece time (75% cofidece level) for this path is specified to be 5 ms..5 Spreadig Fuctio, path, Coherece time = (ms) Spread Fuct. Sigal Magitude of Amplitude Measured Coherece Fuctio vs. Time for oe path.8 Power Time (ms) Figure. A sample temporal weightig fuctio reflectig a sigle path i a acoustic chael. The coherece time (75% cofidece level) for this path is specified to be ms. Accordig to Eq (), the desired sigal, s(t), is idepedetly multiplied by each of the weights, ad the products are delayed accordig to the respective path delays, ad the results summed. The upper plot of Figure 3 shows a sample waveform received over our example -path chael. A stadard sigal processig techique is to process this sigal with a matched filter, the filter beig a copy of the trasmitted waveform. The lower plot of Figure 3 shows the output of such a matched filter. It is see that this is a useful tool i

5 evaluatig the gross characteristics of the chael: the delay betwee the two paths is accurate, ad the relative powers are reasoably close to the specificatio. 3 Simulated -Path Received PRN Sigal Amplitude Time (ms) 4 x 6 Liear Estimate of Impulse Respose Power Time (ms) Figure 3. A PRN waveform received over a simulated -path chael (upper plot). The lower plot shows a classic matched filter output whe the etire waveform is used as the filter. We ow cosider the limitatios we face i measurig ad estimatig these chael characteristics. I particular, the assumptio implicit i eq (), that the received compoets of r(t) are observable caot be justified or measured because the chael impulse respose will be estimated via the correlatio fuctio of a bad-limited waveform. The correlatio fuctio will have a temporal resolutio approximately equal to the iverse of the waveform badwidth, W. Thus, if several paths deliver the waveform with a temporal separatio less tha this resolutio, the estimatio will reflect a correlatio amog these arrivals. Furthermore, i a typical aalysis system, the temporal samplig iterval δt will be cosiderably less tha /W, so the iformatio cotaied i each temporal sample of the correlatio is itself correlated with adjacet samples. Thus we itroduce yet aother correlatio fuctio κ,m which describes the similarity betwee the th ad m th measured path compoets. κ,m = E τ a a m (4) Our fifth assumptio is thus 4. κ,m =, if -m δt > /W Simulatio At this poit we cosider oly the sampled received sigal, with a sample rate fs ad a sample iterval of δt = /fs. The trasmitted sigal is of duratio T, so there are K = T/δt samples. Thus, each path weight a (pδt) is represeted by K samples, with p K, ad the Fourier trasform A (kδf) is represeted by K frequecy samples, with k K, with frequecy spacig δf = fs/k. We choose to igore the expectatio operator, ad work with idividual realizatios, obtaiig thereby estimates a ~ (pδt) ad A ~ (kδf). The itetio is to fill the idividual spectral bis of A ~ (kδf) with idepedet, zero-mea,

6 complex Gaussia radom umbers. We use uit variace Gaussia radom umbers to fill the spectral bis lyig uder a widow fuctio, with the resultig spectrum ormalized such that the itegrated power is b. We use a Haig widow, with 3-dB power levels positioed at f ad f. We ow compute a realizatio of a ~ (pδt) as the iverse Fourier trasform of A ~ (kδf). We have ow computed all a ~ (pδt) as idepedet radom realizatios. That is, every possible delay τ reflects a idepedet path arrival. However, eq (4) prescribes the requisite path-to-path temporal correlatio. We approximate this correlatio via applicatio of a square root filter i the delay domai: a ~ (pδt) = (-(κ,- ) / ) a ~ (pδt) + (κ,- ) / a ~ (pδt) (5) The correlatio fuctio κ is empirically determied. Commets ad Observatios We obtai a path-depedet, time-varyig temporal weightig fuctio for every possible delay time via eq (5). Each weight is of the same duratio (or greater) as is the sampled sigal, s(pδt). Followig eq (), therefore, we perform the multiplicatio a ~ (pδt)s(pδt). For all delay times τ we delay the product ad add it to all previous products. The result is the received sigal, r(qδt), q (K+max(τ /δt)). Estimatio of Chael Characteristics Although the process described above for simulatig a received waveform applies to ay bad-limited trasmitted waveform, for purposes of estimatio of chael characteristics, we use a pseudo-radom (PRN) waveform for s(t). A properly costructed PRN sigal admits to separatio ito K compoets, each of which is effectively idepedet of all others. Each compoet may thus be used as a sub-correlator which is substatially immue to the presece of other compoets i the received waveform. We ca take advatage of this feature by performig a correlatio (matched filter) of the etire received waveform with each of the compoets. Because the temporal relatioships of all compoets are kow, the output of the sub-correlators ca be arraged i a waterfall display, as show i Figure 4. Here we observe the variatios across time of the power measured at ay give lag time. It is clear that this presetatio reveals much more of the time-varyig ature of the example chael tha does the classical matched filter. Ideed, although the presetatio i Figure 4 shows the magitude-squared (power) of the sub-correlator outputs, the uderlyig complex outputs cotai all of the iformatio required to costruct our chael model.

7 IR Vs. Time & Lag Power (db) Time (ms) Lag Time (ms) Figure 4. A waterfall display of the output of multiple sub-correlators, each a compoet of the trasmitted waveform, each of which is used to process the etire received waveform. Modelig ad Simulatig Rage Rate Rage rate is simply costat relative speed betwee a source ad a receiver. The effect of rage rate o a waveform is to cause a dilatio or cotractio i the temporal duratio of the waveform. With a broadbad waveform, the upper frequecies will shift more tha will the lower frequecies, hece causig a cotractio or dilatio i frequecy. It is oly with a toal waveform that cotractio or dilatio is equivalet to the so-called Doppler shift. We cosider a arbitrary waveform x(t;f ;f ; φ ) which is a fuctio of time (t), a start frequecy (f ), a stop frequecy (f ), ad a trajectory φ which defies the time-frequecy path by which the waveform migrates from f to f. Time is described such that t T. A waveform subject to rage rate may be modeled as: x α = x(αt;βf ;βf ; φ ) (6) where α is a costat multiplier, with α, ad β is a fuctio of α, both to be defied. The Fourier Trasform of x(t) is X(f),of x α (t) it is X α (f), ad it ca be show that X α (f) = X(f/α) (7) hece β = /α. A useful descriptio of α arises from cosideratios of costat velocity motio: α ( + v/c) (8) where v is the relative velocity betwee a source ad a receiver, ad c is the propagatio speed i the trasmissio chael. It is see that β (-v/c). I the followig we demostrate a simple method for imposig compressio/dilatio o arbitrary (passbad) waveforms. Let r(pδt) be ay passbad waveform with a carrier frequecy of f, ad a evelope fuctio a(pδt). There are K samples take over T

8 secods. For coveiece, we assume that r(pδt) is aalytic, with basebad represetatio: rˆ (αpδt) = a(αpδt)exp(iπf αpδt), p (K-)α (9) From eq (8) we have our compressio/dilatio factor α. For every time value of (pδt) there is a correspodig value of (αpδt) which reflects the compressed/dilated time equivalet. Note that, i eq (9), i the absece of rage rate we would have a exact basebad replica of the trasmitted sigal. We ow ivestigate a iterpolative approach to imposig rage rate effects o the basebad formulatio of the trasmitted waveform to give it the requisite characteristics of the received, distorted sigal. First, we must iterpolate the waveform evelope a(pδt) by a suitable amout to obtai a(δt ). Our experiece has bee that a factor of M = 8 is appropriate. That is, there are ow MK- samples, each sample take at Mfs samples per secod, with a ew sample iterval of δt = δt/m. To compute the distorted evelope required by eq (9), we compute a idex for which mi = [ -αpδt/δt ], () ŷ (αpδt) = a( δt ) () Fially, recogizig the residual effect of speed o the carrier frequecy, f, we multiply by the complex expoetial to obtai the fial received sigal rˆ (αpδt) = ŷ (αpδt)exp(iπf pδt (α- )) () Figure 5 shows spectral comparisos betwee a trasmitted LFM sigal, a aalytical represetatio of the sigal received from a kt source, ad the sigal received with kts, as produced by the umerical process. Note the offsets i frequecy caused by the rage rate, which is the same offset for the model ad for the umerical techiques. Theoretical & Numerical Impositio of Rage Rate Distortio Tx Sigal --> - Theory kts --> -4-6 Numerical kts --> Level (db) Frequecy (Hz) Figure 5. Compariso of the Power spectra of the origial trasmitted LFM (upper trace), ad the received sigal perturbed by kts of rage rate. The middle trace reflects a theoretical model, while the third trace reflects a umerical impositio. They are virtually idetical. The vertical offsets i the plots are icluded oly to clarify the presetatio.

9 Modelig ad Simulatig Noise We have little to add to the decades-log developmet ad uderstadig of additive oise, other tha to describe our use of it i modem developmet. Whether we use artificially-geerated oise or experimetally recorded oise, whether it is represeted aalytically or as a real compoet, we always pass the oise through a badpass filter prior to addig it to ay sigal. The filter has approximately the same badwidth as does the sigal of iterest. Our received sigals (see eq ()) are always ormalized to have uit power (variace) ad the filtered oise is the ormalized to have a power (variace) such that the ratio of sigal power to oise power meets user-specified criteria for sigalto-oise-ratio (SNR). We ote a cosequece of this defiitio of SNR: because we maitai uit sigal power, the SNR available o ay idividual chael path is reduced as the umber of paths icreases. This is especially detrimetal for phase-coheret sigalig i which a adaptive equalizer attempts to remove multipath ifluece from a time-varyig chael: a path which is occasioally strog may at times be quite weak, ad the overall respose of the equalizer will be to treat the path sigal as oise bursts. CHANNEL SIMULATION (CHANSIM) Acomms performace is determied as much by acquisitio, frequecy aligmet, ad timig as it is by demodulatio ad decodig, ad our experiece is that these prelimiaries are ofte the more demadig part of successful commuicatios. We therefore have foud it ecessary to cosider the effect of the chael (ad platforms) o all aspects of the trasmitted sigal. I particular, we have foud it ecessary with our simulatios to geerate passbad waveforms i a aalog form ad provide that directly ito the output of the preamplifier of our modem receiver. We supply a few tes of secods of iterferece oly prior to the waveform to test the etire receiver, especially with regard to the automatic gai cotrol (AGC) ad false alarm performace. Bethos has received fudig from the US Navy via several modem-related programs datig back to 998 for the purpose of developig a simulator adequate to meet the eeds of modem developmet. This chael simulator (CHANSIM) is a MATLAB-based, GUIdrive, real-time emulatio of real-world acoustic chaels. It provides for most recogized effects of the chael, usig statistical characteristics to cotrol the realizatios, which are coverted via a digital -to-aalog coverter to a voltage sigal which ca drive the etire modem. Although the iteded use of CHANSIM is for modem developmet, ay bad-limited waveform may be used as a trasmitted sigal. As such, the simulator is valid for soar applicatios as well as telesoar applicatios. CHANSIM has may features both to characterize the chael ad to cotrol the flow of the simulatio. Table lists the settable chael ad cotrol features of CHANSIM.

10 Table. CHANSIM Chael Characteristics & Simulatio Cotrol Noise Types. bad-limited AWGN. stored oise files from experimets, appropriately badlimited & resampled to match the sigal bad 3. No oise Iterferece Types. toals, impulses. partial-bad, short-term oise 3. recorded aimal souds (sea lios, whales, etc.) 4. Exteral (stored files) waveforms (e.g., for multi-access iterferece) Impulse respose fuctio characterizatio. maual etry (via cursor/mouse). theoretical (Rayleigh, Ricia,etc.) 3. from stored statistics, especially for time-varyig chaels Sigal-to-oise ratio umber of realizatios Waveform basebadig & decimatio badwidth cotrol sample rate cotrol Exteral triggers Rage rate to +/- 4 kts Aalysis ad plottig tools. spectrogram. magitude 3. power spectrum 4. matched filter 5. plot idividual impulse respose realizatios CONCLUSIONS There are two basic portios of a waveform used for acoustic commuicatios: acquisitio/aligmet/timig compoets, ad the modulated sigal. I most practical situatios the two caot be separated whe evaluatig performace of a modem. We have developed a real-time simulator which tests a modem for most of the observed iflueces of the acoustic chael. Those iflueces are described by statistical parameters, which ca be obtaied from physics-based propagatio models, or from at-sea experimets. Although the simulatio icorporates may chael characteristics, the two addressed i this paper reflect the modelig of time-varyig impulse respose fuctios, ad the simulatio of platform rage rate as it affects a arbitrary waveform. REFERENCES [] M. Stojaovic, J.A. Catipovic, J.G. Proakis, Adaptive multi-chael combiig ad equalizatio for uderwater acoustic commuicatios, J. Acoustic. Soc. Amer., vol 94, pp.6-63, 993 [] P. Bello, Characterizatio of radomly time-variat liear chaels, IEEE Tras. Commu. Syst., vil. CS-, pp , 963 [3] R. Keedy, Fadig Dispersive Chaels, New York: Wiley, 969