TIME-FREQUENCY MODELING OF SHALLOW WATER ENVIRONMENTS: RIGID VS. FLUID SEABED

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1 TIME-FREQUECY MODELIG OF SHALLOW WATER EVIROMETS: RIGID VS. FLUID SEABED Jun Zhang, Betand Gottin, Antonia Papandeou-Suppappola and Conel Ioana Depatment o Electical Engineeing, Aizona State Univesity, Tempe, AZ, USA, GIPSA-Lab, IPG/ESIEG, Genoble, Fance ABSTRACT When the ocean seabed is consideed to be igid, the ideal waveguide model can be used to model the shallow wate envionment. Howeve, a moe ealistic ocean waveguide model teats the ocean loo as a bounday between two dieent luid media. In this pape, a equency-domain chaacteization o shallow wate envionments is poposed based on this ealistic waveguide model with a luid bounday. Fist, the time-equency chaacteistics o this model ae studied as well as the impact o the envionment paametes on the dispesive phenomena. Then a equency-domain matched ilte eceive is designed to obtain time-dispesion divesity once we sepaate the modes using waping techniques in the time-equency plane. Simulations demonstate that the new eceive design impoves the bit eo ate peomance. Index Tems Time-equency analysis, dispesive channels, time-equency mode sepaation, undewate acoustic communications, time-dispesion divesity 1. ITRODUCTIO As a waveom is tansmitted in the shallow wate envionment, it can inteact with the ocean bottom and suace, esulting in dense dispesion due to the wate medium and timevaying (TV) changes o the ocean envionment. Speciically, the shallow wate acoustic envionment is a linea TV dispesive system that shits dieent equencies by dieent amounts in time [1]. The dispesive eect intoduced by shallow wate envionments can seveely limit the peomance o undewate acoustic applications such as sona and communications. It can be shown that the popagation chaacteistics o an envionment can povide us with a speciic nonlinea unction that deines the dispesion in this envionment. Thus, the coesponding signal epesentation based on a nomal mode model can be applied to exploit the potential divesity suggested by the model when the acoustic eceive is appopiately designed to match this nonlinea unction. *This wok was suppoted by the SF CAREER Awad CCR-1342 and the Depatment o Deense Gant o. AFOSR FA In [1], a shallow wate chaacteization was consideed that matched dispesive signal tansomations on the tansmitted waveoms and was used successully o shallow wate communications to obtain time-dispesion divesity. Howeve, this chaacteization was only applicable to signals with vey high bandwidth as it assumed that the tansmitted waveom was an impulse. In [2], we pesented a genealized chaacteization that was applicable to a lage class o signals by using the nomalmodel in [3] assuming peect waveguide conditions (homogeneous luid laye with a sot top and igid bottom). In this pape, we extend ou chaacteization when these conditions ae not peect by ollowing the waveguide model with pessue-elease suace and luid boundaies in [4]. Using the new model, we popose a new time-equency eceive design which opeates on individual modes o the multi-component eceived signal instead o on the whole signal. This upgaded eceive design o an undewate communication system is expected to achieve bette bit eo ate (BER) peomance o time-dispesion divesity than the one obtained in [2]. The pape is oganized as ollows. In Section 2, we omulate the Pekeis model o shallow wate envionments. In Section 3, we pesent the time-equency (TF) chaacteistics o this model, and we investigate the impact o the envionment paametes on the dispesion eect. In Section 4, we use a TF mode sepaation technique to sepaate each component o the eceived signal and then design a new eceive stuctue with a coesponding optimal detecto. In Section 5, numeical esults o BER peomance illustate ou impoved peomance in tems o divesity ode. 2. MODELIG FOR SHALLOW WATER EVIROMET: PEKERIS WAVEGUIDE The nomal-mode model teats the ocean as a waveguide with plane, paallel boundaies, epesenting the acoustic ield in the ocean medium as a sum o nomal modes. We ist discuss the nomal-mode model o shallow wate envionments with pessue-elease suace and a luid seabed ollowing [4, 5, 3]. A simple waveguide model o the ocean is shown in Fig. 1 using the coodinate system (, z), whee Medium I, II and

2 D z z Tansmitte Medium I: ai Medium II: ocean Density ρ Sound speed c Medium III: Seabed Density ρb Sound speed cb z (, z) Receive Fig. 1. Waveguide model with point souce in Medium II at =, z = z, D m deep. This coesponds to a pessue elease suace with a luid seabed envionment [4]. III coespond to ai, ocean wate and seabed, espectively. An omnidiectional point souce with signal spectum X() is located in Medium II at = and z = z. The ocean is D m deep. We conside the sound speed in Medium II as a constant c m/s and density ρ kg/m 3, and in Medium III as a constant c B m/s and density ρ B kg/m 3. The ocean suace (at z = ) is modeled ealistically as an ideal pessue elease bounday and the ocean bottom (at z = D) is modeled as a bounday between two dieent luid media. The nomal mode model is given by the solution o this ocean waveguide poblem, which is detemined by the envionment paametes and satisies all bounday conditions, including the bounday condition at the souce. Ate a detailed deivation in [4] and [5], the eceived signal spectum excited by X() at location (, z) is given by the Pekeis waveguide model: Y P ekeis () = X() C n ()Θ n (). (1) n= Without the assumption o ideal waveguide condition as in [2], the nth mode is chaacteized by 1 Θ n () = k n () e jkn(), whee k n () = 2π c 2 n2 c 2 4D ρ B c ρd π 1 c2 c 2 B is the wave numbe o the nth mode and p is the lagest mode numbe. The paamete C n () = A 2 n()sin(k zn ()z )sin(k zn ()z) is a unction o equency whee A n () = [ ( 1 2 D sin(2k ) zn()d) ρ 2k zn () and k zn () nπ D ρ ρ 2 B tan(k zn ()D)sin 2 (k zn ()D) k zn () 1 + ρ B c ρd 1 2π 1 c2 c 2 B 1. ] TIME-FREQUECY CHARACTERISTICS OF SHALLOW WATER EVIROMET MODEL The TF chaacteistic o the acoustic signal is detemined by the modal goup velocity (MGV) g n (), due to the popagating delay o the equency in the nth mode is detemined g n( ) by τ n ( ) =. The modal goup velocity is shown in Fig. 2. Fom Fig. 2, we notice that, o the waveguide model with luid seabed, the MGV appoaches c when the equency appoaches ininity, and appoaches c B when the equency appoaches the cuto equency o this mode. Modal goup velocity (m/s) c B cuto equency mode 1 cuto equency mode 2 MGV o mode 1 MGV o mode 2 MGV o mode c cuto equency mode 3 Fig. 2. Modal goup velocity (MGV) unction. We analyzed uthe the TF chaacteistics o the eceived signal using the spectogam TF epesentation (TFR) and investigated the eect o changing the paametes o the envionment. Fig. 3(a) epesents the spectogam o thee modes with envionment paametes: D = m and = 15 km. Fig. 3 uses the same set o paametes except that = 3 km. Fig. 3 shows each mode as a dispesive cuve in the TF plane. We note that the cuves stop at the cuto equency with no asymptotical behavio to the cuto equency. Fig. 3 shows that longe ange between the eceive and tansmitte causes moe dispesion in the eceived signal. Fig. 4 shows that with lage ocean depth, the modal cuto equency is lowe, and with a smalle depth, the dispesive eect is moe visible at highe equencies. (2)

3 2 2 2 e 1 (t) e (t) (a) R=m D=m R=3m D=m Fig. 3. Spectogams with envionment paametes (a) = 15 km, D = m, and = 3 km, D = m Fig. 5. (a) Spectogam o eceived signal with the IF cuve sepaatos, sepaated TF modes. R=m D=3m (a) R=m D=m Fig. 4. Spectogams with envionment paametes (a) = 15 km, D = 3 m, and = 15 km, D = m. 4. SYSTEM DESIG FOR COMMUICATIO DIVERSITY Accoding to the Pekeis waveguide in (1), the eceived signal spectum is the supeposition o multiple nomal modes. Consideing each mode as a communication subchannel, the time-dispesive channel can be intepeted as a supeposition o subchannels, and using thei TF chaacteistics, we can sepaate each subchannel and design the coesponding eceive to exploit time-dispesion divesity Sepaation o nomal modes Fig. 3 illustates that each mode o the eceived signal appeas as a dispesive cuve in the TF plane, which makes the sepaation o the modes possible. In this pape, we use a two-step TF mode sepaation technique based on wapings to sepaate the modes. The two steps ae descibed as ollows. Duing the ist step, we design p 1 instantaneous equency (IF) cuve sepaatos in the TF plane, epesented as e n (t), n =,, p 2. An IF cuve sepaato is a cuve situated between two successive modes in the TF plane as shown in Fig. 5(a) o the example in Fig. 3. In this poblem, all envionmental paametes ae assumed to be known, so that the TF stuctues o the eceived signal ae also known using the model in (1). Knowing the TF stuctues, we appopiately set M TF points (t i, i ), i = 1, 2,..., M in the middle o the space between two successive modes. The M points between the nth mode and the (n + 1)th mode will constitute the IF cuve sepaato e n (t). In the next step, the IF cuves will be pocessed with the eceived signal. Thus, the sepaation lines must have the same duation as the eceived signal. To ensue that, all the IF cuves begin at the stating point o the TF stuctue and ae extended with a staight hoizontal line until the end o the TF window. Duing the second step, we pocess the dieent modes in the TF plane using a waping technique [6, 7] based on the use o the local hamonic convolution opeato [6]. Assume that the eceived noisy signal in (1) is (t) with Fouie tansom (FT) R() = Y P ekeis () + W (), whee W () is additive white Gaussian noise. We then obtain the waping unctions om the IF cuve sepaatos as ζ n (t) = 1 t c e n(u)du, whee c >, and compute the coesponding genealized FT o (t) as [8] M (λ) = (t) dζ n(t) e j2πλ ζn(t) dt, (3) dt whee is the domain o time o the waping unction ζ n (t), and λ is eal and unitless. To obtain the ist mode, we compute the invese genealized FT ove the ange (, c / ] as ollows: (t) = c / M (λ)e j2πλζ (t) dλ, (4) whee is a nomalization equency. Then we subtact the ist mode om the eceived signal, i.e, ˆ(t) = (t) (t), and the emaining modes ae contained in ˆ(t).

4 To obtain the seconde mode, we apply the above pocedue to ˆ(t) using the IF cuve sepaato e 1 (t). Repeating the above pocedue o each IF cuve sepaato, we can then sepaate each mode om the eceived signal. Fo the example in Fig. 5, we use the mode sepaation technique on the eceived signal excited by the waveom X() = o P = 3 modes. The IF cuve sepaatos ae shown in Fig. 5(a) and the sepaated components ae shown in Fig Receive stuctue Unde the Pekeis waveguide assumption, the model in (1) can be seen as an inheent equency domain tanse unction. Howeve, due to the distotion and signal popagation uncetainty in ealistic shallow wate envionments, a moe suitable model o andom signal embedded in andom noise consists o R()=Y () +W ()=X() D n C n ()Θ n ()+W (), n= (5) whee R() is the spectum o the eceived signal, D n is a andom pocess incopoating amplitude and phase vaiability o the souce, and W () is additive white Gaussian noise with vaiance σw 2. We popose a ilte bank eceive scheme based on this signal model which can exploit the equency domain dispesion divesity existing in the nomal modes. The eceived signal spectum R() in (5) is ist pocessed as descibed in Section 4.1 to sepaate each mode, and then the outputs ae combined in the minimum-pobability-o-eo sense to obtain the tansmitted inomation symbol b. This eceive stuctue is illustated in Fig. 6. We design the tansmission waveom as X() = o whee is the cuto equency o the ist mode, and then we employ the coesponding equency domain matched ilte to each sepaated mode component o the eceived signal. The nth mode component obtained om the sepaation pocessing is expessed as R n () bx()d n C n ()Θ n () + W n (), (6) whee W n () is non-white Gaussian noise, with vaiance σw 2 n, that esulted om W () when the nth mode is sepaated. We ist deine the eceive matched waveom U n () = X()C n ()Θ n () o the nth mode. Then, as each mode is sepaated, we can peom individual matched ilteing on the n th mode by D n U n (); this can avoid inteeence with the othe modes. The output o the n th matched ilte can be expessed as Z n = R n (), D n U n () = R n ()Dn Un ()d. (7) Matched iteing TF mode sepaation R ( ) R1 ( ) R p 1 ( ) DU ( ) ( ) DU D U ( ) 1 1 R ( ) Matched iteing Combination Σ Estimated Symbol... Fig. 6. Receive stuctue scheme. Matched iteing p 1 p 1 Using the expession o R n () in (6), we can ewite (7) as b D n 2 U n ()Un ()d+ W n ()Dn Un ()d. Z n As the output o this ilte bank is given by the concatenation [ Z... Z p 1], the combination in the minimumpobability-o-eo sense is deined as the aithmetical mean o the ilteing outputs. Thus, the estimation o the tansmitted inomation symbol can be expessed as ˆb = R. (8) n= whee R( ) is the opeato that takes the eal pat. Assuming an antipodal symbol tansmission, i.e, b = +1 o b = 1, and that the pobabilities o tansmitting +1 and 1 ae equal, the minimum eo pobability detecto can be expessed as: Z n decide b=+1 i ˆb >, decide b= 1 i ˆb <. 5. PERFORMACE AALYSIS As the modes ae sepaated, each mode can be consideed as a subchannel, and the aoementioned matched ilte eceive can avoid coss-coelation inteeences between dieent modes o the eceived signal. Hence, this matched ilte eceive can exploit the potential divesity existing among the dieent modes. In this section, we ist deive the theoetical pobability o eo o the aoementioned eceive, and then we obtain the BER esults om numeical simulations. We also compae the peomance o this eceive with the peomance o a eceive that pocesses the complete eceived signal.

5 Let us conside the matix o equency domain mode autocoelations deined as A = diag(a, A 1,..., A P 1) whee A i = U i ()Ui ()d, i =,..., P 1 and the matix o modal noise vaiance σ = diag(σ 2 W i ) o i =,..., P 1. Deining = σ 1 AQ A and m = σ 1 A E[D], whee E[ ] is the expectation opeato and Q is the covaiance matix o the coeicients vecto D = [D, D 1,..., D P 1] T, we can deive the theoetical BER as the pobability o eo given by [2]: P e = 1 π [ ] 2 1 det π sin 2 θ + I e m ( +sin 2 θ I) 1m dθ, (9) whee we denote T and, espectively, as the tanspose and conjugate tanspose opeations. The BER obtained om the simulation and the theoetical expession ae shown in Fig. 7, using the envionment paametes D = m, = 15 km, c = 1 m/s, c B = 18 m/s, ρ = kg/m 3 and ρ B = 18 kg/m 3 (as in Figues 3 and 5). The esults show the BER as a unction o the signal-to-noise atio (SR) o the tansmission equency band 5 2 Hz. We notice om Fig. 7 that the simulation esults ae wose than the theoetical esults. This is because the mode sepaation technique caused loss o inomation, especially at highe equencies, which leads to less accuate matched ilteing. Howeve, i we assume no inomation is lost in the pocedue o sepaation, i.e, the sepaation can geneate the exact modes, then the numeical simulation esults ae close to the theoetical esults. We also compaed the BER peomance o the new eceive with the peomance obtained in [2] whee the matched ilteing was applied to the whole multi-component eceived signal. Fig. 7 shows that the new eceive peomed bette accoding to the numeical simulation esults. 6. COCLUSIO We developed a equency domain chaacteization o ealistic shallow wate envionments and analyzed its dispesive chaacteistics. Following this model, we developed the appopiate eceive design based on using a waping technique o modal sepaation. This eceive stuctue aims to exploit the divesity existing in the system. Simulation esults show that BER peomance and divesity ode wee impoved by the aoementioned system design. 7. REFERECES [1] Y. Jiang and A. Papandeou-Suppappola, Discete timeequency chaacteizations o dis-pesive time-vaying BER Theoetical BER Simulation BER (exact modes) Simulation BER Simulation BER ome eceive SR (db) Fig. 7. Theoetical BER peomance o the new eceive (diamonds), BER peomance o the new eceive obtained by numeical simulation (squaes), BER peomance o the new eceive obtained by numeical simulation using exact modes (line) and BER peomance o the ome eceive (cicles). systems, IEEE Tans. Signal Pocessing, vol. 55, pp , May 7. [2] J. Zhang and A. Papandeou-Suppappola, Timeequency based waveom and eceive design o shallow wate communications, in Poc. IEEE Int. Con. on Acoust., Speech, Signal Pocessing, vol. 3, pp , Apil 7. [3] L. J. Ziomek, Fundamentals o Acoustic Field Theoy and Space-Time Signal Pocessing, CRC Pess, [4] G. V. Fisk, Ocean and Seabed Acoustics, Pentice-Hall, [5] G. B. Deane, Intenal iction and bounday conditions in lossy luid seabeds, The jounal o the Acoustical Society o Ameica, vol. 11, no. 1, pp , Januay [6] A. Jaot, C. Ioana, C. Gevaise and A. Quinquis, A class o linea time-vaying iltes based on non-unitay timewaping opeatos. Pat I: pinciples and examples, IEEE Tans. Signal Pocessing, submitted, 6. [7] A. Jaot, C. Ioana, C. Gevaise and A. Quinquis, A timeequency chaacteization amewok o signals issued om undewate dispesive envionments, in Poc. IEEE Int. Con. on Acoust., Speech, Signal Pocessing, vol. 3, pp , Apil 7. [8] A. Papandeou-Suppappola, Time-Vaying Pocessing: Tutoial on Pinciples and Pactice, in Applications in Time-Fequency Signal Pocessing, (A. Papandeou- Suppappola, ed.), Floida: CRC Pess, pp. 1 84, 2.