IEEE JOURNAL OF OCEANIC ENGINEERING 1

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1 IEEE JOURNAL OF OCEANIC ENGINEERING 1 Iterative Per-Vector Equalizatio for Orthogoal Sigal-Divisio Multiplexig Over Time-Varyig Uderwater Acoustic Chaels Jig Ha, Member, IEEE, Sudeep Prabhakar Chepuri, Member, IEEE, Qufei Zhag, Member, IEEE, ad Geert Leus, Fellow, IEEE Abstract Orthogoal sigal-divisio multiplexig (OSDM is a promisig modulatio scheme that provides a geeralized framework to uify orthogoal frequecy-divisio multiplexig (OFDM ad sigle-carrier frequecy-domai equalizatio. By partitioig each data block ito vectors, it allows for a flexible cofiguratio to trade off resource maagemet flexibility with peak-to-average power ratio. I this paper, a OSDM system is proposed for uderwater acoustic commuicatios. The chael Doppler effect after frot-ed resamplig is modeled as a commo time-varyig phase o all propagatio paths. It leads to a special sigal distortio structure i the OSDM system, amely, itervector iterferece, which is aalogous to the itercarrier iterferece i the covetioal OFDM system. To couteract the related performace degradatio, the OSDM receiver performs iterative detectio, itegratig joit chael impulse respose ad phase estimatio, equalizatio, ad decodig i a loop. Meawhile, to avoid iversio of large matrices i chael equalizatio, frequecy-domai per-vector equalizatio is desiged, which ca sigificatly reduce the computatioal complexity. Furthermore, the performace of the proposed OSDM system is evaluated through both umerical simulatios ad a field experimet, ad its reliability over uderwater acoustic chaels is cofirmed. Idex Terms Orthogoal sigal-divisio multiplexig (OSDM, time-varyig chaels, turbo equalizatio, uderwater acoustic commuicatios. I. INTRODUCTION UNDERWATER acoustic (UWA chaels are cosidered as oe of the most challegig commuicatio media i use [1]. Specifically, UWA chaels exhibit limited available badwidth, typically of the order of 10 khz for medium-rage liks, due to the frequecy-depedet trasmissio loss. Also, UWA chaels suffer from log multipath spread ad severe Mauscript received December 28, 2016; revised August 1, 2017 ad November 7, 2017; accepted December 19, This work was supported i part by the Natioal Natural Sciece Foudatio of Chia uder Grats , , , ad , ad i part by the Fudametal Research Fuds for the Cetral Uiversities uder Grats JCQ01010, ZD0041, ad JG (Correspodig author: Jig Ha. Associate Editor: Y. Rosa Zheg. J. Ha ad Q. Zhag are with the School of Marie Sciece ad Techology, Northwester Polytechical Uiversity, Xi a , Chia ( haj@wpu.edu.c; zhagqf@wpu.edu.c. S. P. Chepuri ad G. Leus are with the Faculty of Electrical Egieerig, Mathematics ad Computer Sciece, Delft Uiversity of Techology, Delft 2826 CD, The Netherlads ( s.p.chepuri@tudelft.l; g.j.t.leus@tudelft.l. Digital Object Idetifier /JOE time variatio, usually several orders of magitude larger tha i terrestrial radio chaels, due to the low velocity of acoustic waves (omially 1500 m/s. To achieve reliable trasmissio with high badwidth efficiecy over UWA chaels, a umber of modulatio schemes ad receiver algorithms have bee ivestigated over the last three decades. Amog them, a successful phase-coheret commuicatio with sigle-carrier modulatio (SCM was demostrated i [2], where the receiver combies a adaptive time-domai equalizer (TDE with a phase-locked loop (PLL to combat timevaryig itersymbol iterferece (ISI. Although this TDE-PLL structure has bee accepted thereafter as a stadard method ad further adopted to several systems [3] [5], its performace ad complexity deped heavily o the choice of receiver parameters, such as the TDE legth ad the PLL coefficiets. This may impair robustess ad restrict practical implemetatios [6]. To cope with these problems, two low-complexity techiques, amely, orthogoal frequecy-divisio multiplexig (OFDM ad sigle-carrier frequecy-domai equalizatio (SC-FDE, have received much attetio i recet years (see [7], [8] ad referece therei. Both schemes are based o blockwise frequecy-domai processig usig discrete Fourier trasform (DFT, which allows for mitigatig the chael frequecy selectivity more efficietly. However, it is well kow that OFDM systems suffer from a large peak-to-average power ratio (PAPR ad a high sesitivity to Doppler effects [7]. O the other had, the SC-FDE system offers lower PAPR ad better Doppler tolerace, yet at the expese of a iflexible badwidth ad eergy maagemet [9], [10]. As aother promisig alterative, orthogoal sigal-divisio multiplexig (OSDM was first proposed i [11] ad [12], ad recetly applied for UWA commuicatios i [13] ad [14]. Mathematically, it is worth otig that OSDM shares a similar sigal structure with vector OFDM, which was idepedetly developed i [15]. At the trasmitter, differet from covetioal OFDM where the data block is treated as a whole ad modulated by a sigle full-legth iverse DFT (IDFT, these schemes split the data block ito segmets (termed as vectors herei ad perform several compoetwise IDFTs with legth reduced to the umber of vectors. By doig so, they attai a uified framework to trade off resource maagemet flexibility with PAPR, thus bridgig the gap betwee OFDM ad SC-FDE. As for the receiver desig, most existig studies i terrestrial IEEE. Persoal use is permitted, but republicatio/redistributio requires IEEE permissio. See stadards/publicatios/rights/idex.html for more iformatio.

2 2 IEEE JOURNAL OF OCEANIC ENGINEERING radio commuicatios usually assume the chael to be timeivariat ad kow apriori[16] [18], which is ot valid for practical uderwater scearios. To this ed, the OSDM scheme i [13] utilizes a pilot vector dedicated to chael estimatio. Although ot takig the chael time variatio ito accout, tak test results show that OSDM outperforms covetioal OFDM ad SCM with the TDE-PLL receiver. Furthermore, by explicitly accommodatig Doppler spreads usig a basis expasio model (BEM, the Doppler-resiliet OSDM (D-OSDM scheme i [14] has the capability to achieve a reliable commuicatio over time-varyig UWA chaels. However, there are two problems arisig i the D-OSDM system. 1 At the trasmitter, zero vectors are iserted ito each trasmitted block to preserve the orthogoality of the pilot ad data vectors. This method is a extesio of the ull-subcarrier isertio scheme for the OFDM system i [19], by which chael estimatio ad data detectio ca be separated ad thus simplified. Sice the required umber of zero vectors icreases with the maximum Doppler shift [14], this system will suffer a sigificat loss i badwidth efficiecy. 2 At the receiver, the chael equalizatio i [14] is performed directly o the demodulated vectors. It requires chael matrix iversio ad icurs a complexity of O(M 3 for each vector, where M deotes the OSDM vector legth. However, it is also assumed that M is loger tha the chael delay spread to make chael estimatio easier. This system will, therefore, be computatioally expesive for UWA chaels with log delay spread. The aim of this paper is to address the above-metioed problems. The mai cotributios are detailed as follows. 1 To avoid the overhead of zero vectors itroduced for the BEM coefficiet estimatio process, a time-varyig phase model is adopted, which has bee prove to be valid i multiple systems (see, e.g., [2], [20], [21]. However, it is show that, ulike the effects of carrier fluctuatio i SCM or itercarrier iterferece (ICI i OFDM, the time variatio i the OSDM system leads to itervector iterferece (IVI. As a result, most of the existig phase estimatio ad compesatio methods caot be directly applied to OSDM. Therefore, i this paper, we leverage the slowly varyig ature of the phase, ad characterize it with fewer parameters i both frequecy ad time domais. Based o that, we further desig a alteratig least squares (ALS algorithm to perform joit chael impulse respose (CIR ad phase estimatio. 2 To alleviate the cubic complexity of chael equalizatio, the approach proposed i this paper combies soft iterferece cacellatio (SIC ad phase compesatio (PC operatios with frequecy-domai equalizatio (FDE. The motivatio behid such a desig is twofold. First, with the aid of SIC ad PC, the chael time variatio is mitigated ad the demodulated vectors ca be decoupled. Chael equalizatio is thus allowed to be imposed o each vector istead of o the etire block (or o vector groups such as i [14], by which dimesioality reductio is achieved. Secod, by exploitig the chael matrix structure, the per-vector equalizatio is performed i the frequecy domai o predistorted versios of the demodulated vectors. It avoids ivertig the chael matrix directly. Moreover, we provide a multichael FDE extesio for OSDM to collect spatial gais. Quatitatively, the complexity of the proposed FDE algorithms is about O(Mlog 2 M per vector, which is more tractable for practical applicatios. Furthermore, aother feature of the proposed OSDM system is that each data vector is ecoded idepedetly at the trasmitter. By virtue of it, the receiver ca be desiged to perform per-vector equalizatio ad decodig iteratively based o the turbo priciple. We evaluate the performace of the proposed OSDM system through both umerical simulatios ad a shallow-water field experimet, ad its reliability over timevaryig UWA chaels is cofirmed. The remaider of this paper is orgaized as follows. I Sectio II, we preset the OSDM sigal model ad the UWA chael model. I Sectio III, we describe the iterative OSDM receiver algorithm i detail, based o which further discussios o the receiver structure ad complexity are provided i Sectio IV. The umerical simulatios ad experimetal results are the preseted i Sectios V ad VI, respectively. Fially, coclusios are draw i Sectio VII. Notatio: ( stads for cojugate, ( T for traspose, ad ( H for Hermitia traspose. We reserve for the absolute value, for the Euclidea orm, ad for the Kroecker product. We use 0 M, 1 M, I M, ad e M (m to represet the M 1 all-zero vector, the M 1 all-oe vector, the M M idetity matrix, ad the mth colum of I M, respectively. F N deotes the N N uitary discrete Fourier trasform (DFT matrix, ad diag{x} deotes a diagoal matrix with x o its diagoal. Meawhile, we defie [x] as the th etry of the colum vector x, ad [X] m, as the (m, th etry of the matrix X, where all idices are startig from 0. Furthermore, [x] m : idicates the subvector of x from etry m to, ad [X] m :,p:q idicates the submatrix of X from row m to ad from colum p to q, where oly the colo is kept whe all rows or colums are icluded. A. Trasmitted Sigal II. SIGNAL MODEL Our OSDM trasmissio scheme is depicted i the upper part of Fig. 1. We cosider a trasmissio block of K a = NM a bits. Istead of beig treated as a whole such as i OFDM systems, here the block is further partitioed ito bit vectors {a } N =0 1 of legth M a, o each of which idepedet operatios icludig ecodig, iterleavig, ad mappig are performed i parallel. To be specific, the th bit vector a is ecoded usig a covolutioal ecoder to produce a coded vector b of legth M c =(M a + M t /R c, where R c (0, 1] is the codig rate ad M t 0 is the overhead itroduced by the ecoder, icludig a cyclic redudacy check (CRC code to examie data itegrity ad a termiatio sequece to reset the fial state of the ecoder. The ecoded bits are the shuffled by a radom iterleaver ad grouped ito M sets of Q bits, i.e., c =[c T,0, c T,1,...,c T,M 1 ]T, where M = M c /Q

3 HAN et al.: ITERATIVE PER-VECTOR EQUALIZATION FOR OSDM OVER TIME-VARYING UWA CHANNELS 3 Fig. 1. Block diagram of the proposed OSDM system. (a OSDM trasmitter structure. (b OSDM receiver structure. ad c,m =[c,m (0,c,m (1,...,c,m (Q 1] T {0, 1} Q. Subsequetly, each set of Q successive iterleaved bits c,m is mapped oto a 2 Q -ary complex-valued symbol from a costellatio A = { α 1, α 2,..., α 2 Q } with α i correspodig to the bit patter c i =[ c i (0, c i (1,..., c i (Q 1] T. We thus get the th symbol vector d =[d,0,d,1,...,d,m 1 ] T, where d,m = α i if c,m = c i. Note that due to the uique structure of the OSDM block, we use the two subscripts ad m i this paper to idex symbols. Alteratively, to simplify some represetatios i the followig, we also stack {d } ito a symbol block of legth K = MN ad use a sigle idexig, i.e., d =[d 0,d 1,...,d K 1 ] T. These two otatios ca be readily coverted ito each other with d M+m = d,m. Now, the OSDM modulatio ca be implemeted by a threestep procedure. First, the symbols i d are writte rowwise ito a N M matrix D with its th row filled by the th symbol vector d T. Secod, N-poit IDFTs are performed columwise to the matrix D yieldig S = F H N D. Third, the resultig matrix S is read out rowwise to obtai the basebad trasmitted sigal s =[s 0,s 1,...,s K 1 ] T. It ca be see that ulike covetioal OFDM which modulates the symbols i d oe-to-oe o K subcarriers ad geerates time-domai samples usig a K-poit IDFT, OSDM divides d ito N symbol vectors {d } of legth M ad produces the basebad sigal by compoetwise N- poit IDFTs. Therefore, i compariso with OFDM, OSDM possesses a lower PAPR due to the reductio i the IDFT size (by a factor of M, while potetially offerig frequecy diversity withi each vector. To formulate the OSDM modulatio mathematically, we defie the K K permutatio matrix I N e T M (0 I N e T M P N,M = (1. (1. I N e T M (M 1 The above modulatio process ca the be expressed as s = P H ( N,M IM F H N PN,M d = ( F H N I M d (2 where, i the first equatio, matrices P N,M, I M F H N ad P H N,M correspod to the rowwise write, N-poit IDFT ad rowwise read operatios, respectively. Note that, similar to [13] ad [14], it is assumed i this paper that M>L, where L is the maximum memory legth of the discrete-time CIR. As a result, we ca just reserve the first symbol vector d 0 as the pilot vector to facilitate the iitial chael estimatio at the receiver side. Moreover, a cyclic prefix (CP of legth K g >Lis added at the begiig of each block to elimiate iterblock iterferece (IBI, i.e., s = T cp s, where T cp =[I cp, I K ] T with I cp comprisig the last K g colums of I K. Fially, the sequece s is upcoverted to the carrier frequecy f c ad the trasmitted through a UWA chael. B. Chael Model ad Received Sigal It is kow that uder the UWA chael assumptio that path amplitudes are costat durig oe block ad a commo Doppler scale is shared amog all paths, the effect of time variatio i widebad sigals approximately reduces to a carrier frequecy offset (CFO after Doppler compesatio at the receiver via resamplig [20]. Accordigly, the basebad received sigal after CP removal ca be expressed as r k = L h l e jθ k s k l + k, k =0,...,K 1 (3 l=0 where k is the additive oise term, h l is the CIR, ad θ k =2πɛkT s stads for the phase correspodig to the postresamplig CFO ɛ with T s beig the samplig period. Moreover, the idex of s i (3 is actually take modulo-k due to the circular covolutio implemeted by the CP.

4 4 IEEE JOURNAL OF OCEANIC ENGINEERING I this paper, we further elimiate the sigle-frequecy restrictio imposed o the time-varyig phase, ad cosider {θ k } K 1 k=0 as a determiistic sequece with slow variatio to accommodate other effects of chael time variatio that caot be aggregated ito a commo Doppler scale, such as driftig of the platforms, scatterig i the medium ad slight Doppler spread amog differet paths, etc. Note that a similar chael model is also adopted for sigle-carrier UWA commuicatios, where a symbolwise PLL [2] ad a groupwise correctio [21] have bee employed for trackig θ k. However, these methods are ot applicable to blockwise modulatios such as OSDM. To estimate ad compesate θ k i this case, we first establish the model of the received sigal distorted by the time-varyig phase below. As show i the lower part of Fig. 1, the OSDM demodulatio is based o compoetwise N-poit DFTs, which reverses the modulatio process at the trasmitter. Here, we also adopt the previously metioed sigle idexig ad defie the received sigal block as r =[r 0,r 1,...,r K 1 ] T. Specifically, the iterleaver first writes the samples of r rowwise ito a N M matrix R. The, N-poit DFTs are performed columwise to the matrix R yieldig X = F N R. Fially, the matrix X is read out rowwise to obtai the demodulated block x =[x 0,x 1,...,x K 1 ] T. Aalogous to (2, the OSDM demodulatio process ca be expressed as x = P H N,M (I M F N P N,M r = (F N I M r. (4 Now, from (2 (4, the iput output relatioship of the timevaryig OSDM system ca be writte i the matrix-vector form as x =(F N I M Θ H ( F H N I M d + z (5 where H is the K K circulat chael matrix with first colum equal to the CIR vector h =[h 0,h 1,...h L ] T appeded by K L 1 zeros, Θ = diag{[e jθ 0,e jθ 1,...,e jθ K 1 ]} is the time-varyig phase matrix, ad z =[z 0,z 1,...,z K 1 ] T is the oise term. To separate the time-varyig ad time-ivariat chael effects, we reformulate the sigal model (5 ito x = GHd + z. (6 It ca be derived that H = (F N I M H ( F H N I M = H 0 H 1... H N 1 where H, =0,...,N 1,istheM M chael submatrix correspodig to the th symbol vector, which has the form (7 H = Λ H M F H M H F M Λ M (8 with Λ 2 π 2 π M = diag{[1,e j K j,...,e K (M 1 ]} referred to as the frequecy shiftig submatrix, ad H = diag {[H,H N +,...,H (M 1N + ]} as the decimated frequecy respose (DFR submatrix, sice H k = L l=0 h le j(2π/klk for k =0,...,K 1. Meawhile, it ca also be show i (6 that G = (F N I M Θ ( F H N I M = G 0 G N 1... G 1 G 1 G 0... G G N 1 G N 2... G 0 where G i, i =0,...,N 1, is the phase submatrix correspodig to the ith frequecy sample, which has the form 1 N (9 G i = diag {g i } (10 with g i =[g i,0,g i,1,...,g i,m 1 ] T ad its etries g i,m = N 1 =0 ej(θ M + m 2πi/N for m =0,...,M 1. Itiseasy to verify that G N i = G i. The proof of (7 (10 ca be foud i Appedix A. After demodulatio, the legth-k blocks x ad z i (6 are divided ito N vectors, i.e., x =[x] M:M+M 1 ad z = [z] M:M+M 1 for =0,...,N 1. It ca be see from (7 ad (9 that for time-ivariat chaels where θ k =0for all k, wehaveg = I K, ad thus the detectio of the N symbol vectors i the OSDM system ca be decoupled as x = H d + z, =0,...,N 1. (11 For this case, maximum likelihood (ML ad liear receivers have already bee proposed i the literature (see [18] ad refereces therei. However, for the time-varyig case where G is ot diagoal, iterferece amog symbol vectors may arise accordigly, i.e., x = G 0 H d + i 0 G i H i d i + z, =0,...,N 1. (12 Note that all idices i (12 are take modulo-n for otatioal simplicity. It ca be see that o the right-had side of (12, the first term models the ISI withi oe vector, the secod term represets the IVI, ad {G i } capture the phase distortio due to the chael time variatio. I [22], it is assumed that the chael frequecy respose is apriorikow or quasi-static over cosecutive blocks to compesate phase oise. This assumptio is usually iappropriate for rapidly time-varyig UWA chaels where chael estimatio is required o a block-by-block basis. Therefore, we propose a iterative algorithm for OSDM detectio i this paper, which will be described i Sectio III. III. ITERATIVE OSDM DETECTION The structure of the iterative OSDM detectio is illustrated i Fig. 2, which cosists of the followig two processig modes. 1 Pilot-based iterferece-igored preprocessig mode: This mode is activated at the iteratio β =0, where iitial chael estimatio ad equalizatio are performed without explicit IVI cacellatio. The resultig CIR ad

5 HAN et al.: ITERATIVE PER-VECTOR EQUALIZATION FOR OSDM OVER TIME-VARYING UWA CHANNELS 5 Fig. 2. Structure of the iterative OSDM detectio. symbol estimates are provided as iitial values for the followig iteratios. 2 Decisio-directed iterferece cacellatio mode: The receiver switches to this mode for the iteratios β>0, where the time-varyig UWA chael is recostructed via joit CIR ad phase estimatio, ad the low-complexity per-vector equalizatio with SIC ad PC is performed i the frequecy domai to mitigate IVI ad ISI. Furthermore, the turbo priciple is applied based o the exchage of soft iformatio with the decoder to improve the OSDM system performace iteratively. The above algorithm cotiues util the CRCs of all detected symbol vectors are matched successfully, or a prespecified umber of iteratios β max have elapsed. We ext preset the detailed descriptios of several key modules. A. Iterferece-Igored Preprocessig At the iitial iteratio β =0, the residual Doppler effect after frot-ed resamplig is igored. We, thus, adopt a estimate of g i correspodig to the zero-valued phase, i.e., ĝ (0 i = δ i 1 M (13 where δ i is the Kroecker delta. As a result, the demodulated sigal vector is ow reduced to (11, where z cotais the oise plus residual iter- ad itravector iterfereces. We further assume a moderate sigal-to-iterferece-plusoise ratio i this case, ad defie a M M diagoal frequecy-domai symbol matrix D = diag{f M Λ M d } ad a M (L +1 matrix Γ with etries [Γ ] m,l = e j(2π/k(mn+l. Sice oly the pilot vector d 0 is available at this poit, based o (11 ad the assumptio M>L, the iitial CIR estimate ĥ(0 =[ĥ(0 0, ĥ(0 1,...ĥ(0 L ]T ca be obtaied i the least squares (LS sese as ĥ (0 = 1 M ΓH 0 D 1 0 F M x 0. (14 Therefore, we have the correspodig estimates of the th DFR ad chael submatrices { = diag Γ ĥ (0} (15 ˆ H (0 Ĥ (0 = Λ H M F H (0 M ˆ H F M Λ M (16 ad the arrive at the iitial symbol estimates ˆd (0 (Ĥ(0 1x =, =1,...,N 1. (17 B. Joit CIR ad Phase Estimatio For iteratios β>0, the residual Doppler effect after froted resamplig is take ito accout by modelig it as a timevaryig phase ad assumig it chages slowly withi oe block, which is ofte the case for UWA chaels with fixed or smoothly movig trasceivers. Moreover, for the OSDM sigal model i (12, this time-varyig phase ca be further simplified as follows. 1 I the frequecy domai: It is reasoable to assume that the Doppler spread of the time-varyig phase is bouded. We ca thus reduce the umber of phase submatrices i the model, i.e., G i = 0 M M, I<i<N I (18 where I is the Doppler spa parameter. 2 I the time domai: A subvector-fadig model ca be assumed, i.e., the time-varyig phase is approximately costat over J = M/ M symbols, where M ad J are itegers deotig the umber ad legth of the quasi-static subvectors, respectively. As such, we ca reformulate the phase submatrix i (10 as G i = diag {g i } I J, I i I (19 where g i =[g i,0,g i,1,...,g i, M 1 ] T. Uder the above assumptios, the demodulated sigal vector i (12 ca be rewritte i the form I x = G i H i d i + z (20 i= I which, ulike the sigal model used i the iitial iteratio, icorporates the time-varyig chael effects explicitly. 1 Iterative Estimatio: The CIR ad phase ca ow be joitly estimated by solvig mi N 1 h,{g i } =0 x I G i H i d i 2. (21 i= I However, two issues should be observed here. First, there exists a scalig ambiguity betwee the estimates of h ad {g i }.

6 6 IEEE JOURNAL OF OCEANIC ENGINEERING Fig. 3. Structure of the per-vector equalizatio scheme. Secod, the optimizatio problem give by (21 is actually biliear ad thus ocovex. To avoid the ambiguity ad fid a suboptimal solutio, we desig a ALS algorithm i this paper, which decouples the joit estimatio ito two LS problems ad updates the estimates of h ad {g i } i a iterative way. The details of the ALS algorithm are preseted i Appedix B. 2 Iitializatio ad Termiatio: Sice the receiver is switched to the decisio-directed iterferece cacellatio mode after the iitial turbo iteratio β =0, we defie the iput symbol vectors at the βth iteratio as d (β = { d, =0 or N (β v d (β, N r (β (22 where N v (β ad N r (β are the idex sets of the successfully decoded ad the remaiig symbol vectors up to the βth iteratio, respectively, satisfyig N v (β N r (β = {1, 2,...,N 1}, ad } are the soft symbol vectors fed back from the decoder { d (β (see Sectio III-D for more iformatio. To solve the optimizatio problem i (21, we use the decisios {d (β } istead of the true symbol vectors. The iitial values of the ALS algorithm are set as the CIR estimate ĥ(0 ad the phase estimates {ĝ (0 i }. If the chael time variatio is ot severe, we ca expect the solutio of (21 to be i a eighborhood of ĥ(0 ad {ĝ (0 i }, ad the ALS algorithm ca attai it i a moderate umber of steps. Therefore, whe the ALS algorithm is termiated, the fial chael estimates ĥ (β ad {ĝ (β i } for the βth iteratio are geerated. The, similar to (15 ad (16, the estimates of the DFR ad chael submatrices ca be updated as ˆ H (β Ĥ (β = diag (β {Γ ĥ } {[Ĥ(β ]} = diag, Ĥ(β N +,...,Ĥ(β (M 1N + (23 = Λ H M F H (β M ˆ H F M Λ M, =0,...,N 1 (24 while the estimates of the phase submatrices are { = diag Ĝ (β i ĝ (β i C. Per-Vector Equalizatio } I J, i = I,...,I. (25 To achieve reliable OSDM trasmissio i the presece of ISI ad IVI [cf., (20], we preset a per-vector equalizatio scheme, whose structure is depicted i Fig. 3. Here, the precedig SIC ad PC modules are utilized to mitigate IVI ad phase distortio, ad the low-complexity frequecy-domai equalizatio processig follows to combat the ISI caused by the symbols withi the same vector. More details of this scheme are explaied i the followig. 1 Soft Iterferece Cacellatio ad Phase Compesatio: At the βth iteratio, give the soft decisios {d (β } i (22, as well as the estimates of the chael ad phase submatrices {Ĥ (β } ad {Ĝ (β i } i (24 ad (25, the phase distortio ad IVI ca be explicitly recostructed ad removed from the th sigal vector, which yields x (β (Ĝ(β 1 = 0 Here, x (β x 0< i I Ĝ (β i Ĥ (β i d(β i = H d + z (β. (26 is the Doppler-compesated sigal vector which is subsequetly used as iput for the equalizer. z (β cotais the additive oise ad residual iterferece. 2 Frequecy-Domai Equalizatio: Both the zero-forcig (ZF ad miimum mea-square error (MMSE criteria ca be i (26. Sice they have a similar structure, we here focus oly o the liear ZF equalizer for simplicity, ad the performace evaluatios of the MMSE equalizer are provided i Sectio V. Mathematically, the liear ZF equalizatio of the OSDM system is equivalet to employed to equalize x (β ˆd (β = (Ĥ(β 1x (β (27 where ˆd (β is the estimate of the th symbol vector. Furthermore, we otice from (24 that (Ĥ(β 1 = Λ H M F H M ( 1 ˆ H (β F M Λ M. (28 This meas that, istead of computig the iverse of Ĥ (β directly as i [13] ad [14], which has a high complexity of O(M 3, it is favorable to exploit the matrix structure ad perform equalizatio i the frequecy domai. To this ed, we defie W (β =(ˆ H (β 1 as the coefficiet matrix of the frequecy-domai equalizer with [ W (β ] m,m = 1 Ĥ (β mn+, m =0,...,M 1 (29 by which the ZF equalizatio i (27 ca be rewritte as ˆd (β = Λ H M F H M W (β F M Λ M x (β. (30

7 HAN et al.: ITERATIVE PER-VECTOR EQUALIZATION FOR OSDM OVER TIME-VARYING UWA CHANNELS 7 Moreover, recall that (Ĥ (0 1 i (17 ca be factorized similarly as (28; therefore, frequecy-domai equalizatio is actually also utilized i the preprocessig step for iitial symbol vector estimatio. 3 Multichael Combiig: It is well kow that multichael combiig at the receiver collects spatial diversity gais ad thus has better resiliece agaist deep chael fadig [23], [24]. We ow cosider a OSDM system with P receive elemets. I this case, the previous CIR ad phase estimatio step is performed elemetwise, while equalizatio ad multichael combiig are carried out o a vector-by-vector basis. We defie H p,, x (β p,, ad z (β p, as the th chael submatrix, Doppler-compesated sigal vector, ad oise vector at the pth receive elemet, respectively. By stackig the sigal vectors of all P chaels together, i.e., x (β 1,. x (β P, = H 1,. H P, d + z (β 1,. z (β P, (31 the estimate of the th symbol vector obtaied by multichael combiig ca be expressed as Ĥ (β H 1, Ĥ (β 1 1, Ĥ (β H 1, x (β 1, ˆd (β =.... (32 Ĥ (β P, Ĥ (β P, Ĥ (β P, x (β P, Likewise, let us defie the estimate of the chael frequecy respose correspodig to the pth elemet at the βth iteratio as {Ĥ(β 1 p,k }K k=0. We ca the readily obtai the diagoal coefficiet matrix W p,, (β whose (m, mth etry has the form [ (β W p, (β ]m,m = Ĥ p,mn+ P, m =0,...,M 1 (33 Ĥ(β 2 i=1 i,mn + ad the multichael combiig i (32 ca be alteratively performed i the frequecy domai as D. Decodig ˆd (β = P p=1 Λ H M F H M W (β p,f M Λ M x (β p,. (34 After chael estimatio ad equalizatio, the resultig estimates ˆd =[ˆd (β (β (β (β,0, ˆd,1,..., ˆd,M 1 ]T are utilized to update the soft iformatio for each symbol vector N r (β.asshow i Fig. 2, a soft-iput soft-output (SISO demapper is first employed to compute the extrisic log-likelihood ratios (LLRs of the iterleaved bits. Here, we adopt the typical assumptio that ˆd (β,m = μ (β d,m + ξ (β with ξ (β Gaussia distributed with zero mea ad variace σ 2(β. The extrisic LLR of the qth bit i c,m, i.e., c,m (q, ca thus be expressed as (35 show at the bottom of the page, where q =0,...,Q 1, ad L (β (c,m (q is the apriorillr at the βth iteratio [25]. Moreover, the parameters μ (β ad σ 2(β are computed by [26], [27] μ (β = 1 M 1 ˆd (β,m M m =0 ď (β,m σ 2(β = 1 M 1 ˆd (β M 1 m =0 (β ˆd,m μ (β ď (β,m (36 2 (37 where ď(β,m = dec{,m} is the hard symbol decisio. The extrisic LLRs {L (β e (c,m (q} are the iput to a decoder implemeted by the stadard BCJR algorithm [28], which, i cojuctio with a pair of radom iterleaver ad deiterleaver, produces the a posteriori LLRs {L (β app(c,m (q}. Afterwards, based o a CRC, the successfully decoded symbol vectors are reassiged to N v, ad the remaiig sym- (β +1 bol vectors update their apriorillrs as L (β +1 (c,m (q = L (β app(c (β +1,m (q. Furthermore, the soft iformatio d = (β +1 (β +1 (β +1 [ d,0, d,1,..., d,m 1 ]T (β +1, N r, is computed by a SISO mapper, i.e., (β +1 d,m = 2 Q i=1 ( Q 1 α i q=0 1 2 (1 + (1 2 c i (q ( L (β app (c,m (q tah 2 (38 ad fed back for the ext iteratio [25]. Fially, the decoder releases bit vector decisios {ã } N =0 1 whe the turbo iteratio eds. Remark: I the OSDM detectio scheme described above, although per-vector equalizatio is used i (30 ad (34, the decodig is performed i a batch maer, i.e., o soft decisios are updated util all symbol vector estimates are obtaied. For this reaso, we refer to the scheme as parallel iterative detectio (PID. I compariso, sice ecodig at the trasmitter is coducted idepedetly for each symbol vector, the decodig ca also be performed o a vector-by-vector basis, i.e., oce oe symbol vector estimate is obtaied, its soft decisio is immediately computed ad fed back to update chael estimates ad improve IVI cacellatio for the ext symbol vector. We term this latter scheme as successive iterative detectio (SID, which ca be expected to have better performace tha PID. L (β e c i : c i (q=0 ( exp (c,m (q = l c i : c i (q=1 ( exp ˆd ( β,m μ ( β α i 2 σ 2(β ˆd ( β,m μ ( β α i 2 σ 2(β + q :q q + q :q q 1 2 c i (q 2 L (β (c,m (q 1 2 c i (q 2 L (β (c,m (q (35

8 8 IEEE JOURNAL OF OCEANIC ENGINEERING TABLE I COMPLEXITY (IN TERMS OF CMS OF THE PROPOSED OSDM RECEIVER Source Pilot-based iterferece-igored preprocessig mode (β =0 Decisio-directed iterferece cacellatio mode (β >0 Chael estimatio Complexity NM +3 2 log 2 M + M +(L +1N Chael equalizatio (per vector M log 2 M +3M Chael estimatio h & { H } 3 2 NM log 2 M +(2I +3NM +2(L +1N (per update {G i } 1 2 NMlog 2 M +(Ī2 + Ī +2NM +ΣĪ M Chael equalizatio (per vector ĪMlog 2 M +4ĪM IV. FURTHER DISCUSSIONS A. Comparisos With Other Existig Systems 1 Sigal Structure: It ca be see from (2 that the trasmitted sigal of OSDM reduces to that of the covetioal OFDM ad SC-FDE whe N = K ad N =1, respectively. Otherwise, it cosists of N superimposed symbol vectors of legth M.The OSDM modulatio may also look similar to the modulatio scheme whose trasmitted sigal cosists of M CP-free OFDM blocks of legth N, i.e., s =(I M F H N d. However, the chael equalizatio i the latter case will be much more complicated tha the per-vector equalizatio proposed here for OSDM, sice IBI will arise therei without CPs. I additio, compared with the OSDM system i [14], the proposed scheme here requires o isertio of zero vectors ad performs ecodig o a vector-byvector basis. As such, higher badwidth efficiecy ad iterative per-vector equalizatio ca be achieved. Furthermore, the parallel trasmissio property of the OSDM may be remiiscet of the multibad scheme discussed i [26] ad [29]; however, their sigal structures are fudametally differet. To be specific, the multibad scheme modulates a commo symbol stream oto N separated subbads, while the OSDM scheme allocates distict symbol vectors oto N iterleaved subbads represeted by {H } i (8. Aother widely used parallel trasmissio scheme is MIMO-OFDM (see [30] for a example. However, ulike the MIMO-OFDM system performig per-subcarrier equalizatio based o P r P t chael matrices [30, (4], where P t ad P r are the umbers of trasmit ad receive elemets, the per-vector equalizatio i the OSDM system is based o the M M chael submatrices {H }. 2 Receiver Processig: The turbo detectio processig i this paper differs from that give i [31] i aimig to iteratively mitigate IVI other tha ICI. O the other had, regardig ISI suppressio, SC-FDE (e.g., [21] is ormally performed blockwise sice the chael matrix is circulat ad ca be diagoalized by the DFT matrix F K. I cotrast, the ISI i OSDM systems is cofied withi each vector ad the circulat structure is o loger held for the chael submatrices {H }; therefore, extra frequecy (ushiftig operatios, i.e., post- (premultiplicatio by Λ M (ΛH M, are eeded for equalizig the th vector [cf., Fig. 3]. Recallig that SCM is also deemed as DFT-precoded OFDM [32], i this perspective, we ca cosider OSDM as a form of geeralized multicarrier modulatio precoded with {F M Λ M }. B. Computatioal Complexity We here focus o the algorithms of chael estimatio ad equalizatio i the proposed OSDM receiver. For each turbo iteratio, the computatioal complexity i terms of complex multiplicatios (CMs are summarized i Table I, where we defie Ī =2I +1, ad ΣĪ =(Ī3 +3Ī2 Ī/3 deotes the complexity of solvig a liear system of Ī equatios with the Gaussia elimiatio method. 1 Chael Estimatio: Clearly, compared with the computatios of (14 ad (15 at β =0, the ALS algorithm for joit chael estimatio whe β>0 ivolves a larger complexity. However, it is iterestig to ote that the overall complexity is still kept o the same order, i.e., about O(Mlog 2 M per update, sice M is usually much larger tha N ad I over UWA chaels with log delay spreads. This merit is maily attributed to two characteristics of the desiged ALS algorithm. First, the CIR estimatio ca be implemeted by M-poit IDFTs, ad o matrix iversio is eeded [cf., (56]. Secod, a divide-adcoquer strategy is used for the phase estimatio, where MĪ coefficiets are iterleaved ad partitioed ito M vectors of legth Ī to avoid ivertig a large matrix [cf., (62]. 2 Chael Equalizatio: As metioed above, we exploit the structure of the chael submatrix i (24, the iverse of which ca the be decomposed as (28 accordigly, i.e., two frequecy shifts, two DFT operatios, ad a diagoal matrix iversio. This observatio ispires us to perform chael equalizatio i the frequecy domai [cf., (30], istead of ivertig the chael submatrix directly i (27. As a result, compared with the OSDM receiver schemes i [13] ad [14], the pervector computatioal complexity of chael equalizatio is sigificatly reduced from O(M 3 to O(Mlog 2 M. Moreover, for a multichael receiver case, we ca readily verify from (34 that the complexity of chael equalizatio is kept liear with the umber of receive elemets P. V. NUMERICAL SIMULATIONS Throughout this sectio, we cosider a coded OSDM system with blocks of legth K = 1024 ad duratio T = 256 ms. Thus, the symbol samplig period is T s = T/K = 0.25 ms, ad the total sigal badwidth is BW = 1/T s = 4 khz. O each vector, the source bits are appeded by a 4-bit CRC ad a 2-bit all-zero termiatio code, i.e., we have M t =6. The resultig sequece is the ecoded usig a rate R c =1/2 covolutioal

9 HAN et al.: ITERATIVE PER-VECTOR EQUALIZATION FOR OSDM OVER TIME-VARYING UWA CHANNELS 9 Fig. 4. BER performace compariso amog OFDM, SC-FDE, ad OSDM with differet vector sizes (over time-ivariat chaels. Fig. 5. BER performace compariso of OSDM systems equipped with differet umbers of receive elemets (over time-ivariat chaels. code with polyomials (5, 7, passed through a radom iterleaver, ad mapped oto a QPSK costellatio, i.e., Q =2. Moreover, we isert a CP of legth K g =32ad use a carrier frequecy of f c = 6kHz. The simulated UWA chael has symbol-spaced paths with maximum memory legth L =20, which correspods to a multipath delay spread of τ max = 5 ms. Idepedet Rayleigh fadig taps are adopted with a expoetially decayig power delay profile, where the average power differece betwee the first ad last taps is 6 db. Furthermore, at the receiver, the time-varyig phase values {θ k k =0,...,K 1} durig each block are geerated by a update equatio θ k+1 = θ k +2πɛT s + ξ k (39 where ɛ = a Δ f c is the postresamplig CFO with a Δ beig the estimatio error of the Doppler scalig factor, ad ξ k deotes the extra phase distortio caused by other chael time variatio effects. We here model {ξ k } as i.i.d. radom variables draw from a real Gaussia distributio N (0,σ 2 ξ. Based o the above settigs, the performace of the proposed OSDM system is evaluated i three aspects as follows. A. Frequecy-Domai Equalizatio We start with evaluatig the performace of the per-vector frequecy-domai equalizatio i OSDM systems. To isolate its effect, we temporarily disable the turbo iteratio ad directly measure the ucoded bit-error rate (BER at the output of the equalizers (before decodig. Moreover, the chael is assumed to be time-ivariat for simplicity. I Fig. 4, our focus is o the OSDM systems with various vector legths M = 32, 64, 128 ad sigle-elemet receptio. Sice the coditio M>Lis met for all cases, by treatig d 0 as the pilot vector, chael estimatio i the OSDM systems ca be easily performed by usig (14. The, ZF ad MMSE equalizatio follow ad their ucoded BERs are plotted. Also, the performaces of covetioal OFDM ad SC-FDE (equivalet to OSDM with M =1 ad M = K with MMSE equaliza- tio based o perfect chael estimates are icluded to serve as bechmarks. It ca be see that whe equipped with MMSE equalizers, the OSDM system outperforms its OFDM couterpart i the high sigal-to-oise ratio (SNR regime, ad the performace advatage icreases with the vector legth M.The reaso behid this is that the OSDM system implicitly ejoys a itravector frequecy diversity, thaks to beig precoded with {F M Λ M }. As expected, i this case, SC-FDE provides a lower boud o the MMSE equalizatio performace, sice it correspods to the logest vector legth. O the other had, if ZF equalizatio is used, the BER performace degrades cosiderably due to its oise ehacemet effect caused by the ill-coditioed chael submatrices {H }, while the performace gai brought by icreasig M is ow trivial. A detailed theoretical aalysis of the diversity order ca be foud i [18]. The iferior performace of the OSDM system with ZF equalizatio ca be improved by itroducig multichael combiig at the receiver. As show i Fig. 5, for OSDM systems with fixed vector legth M =64, whe the umber of receive elemets P icreases, both MMSE ad ZF equalizatio produce lower BERs; however, the performace gap betwee them becomes much arrower. B. Joit CIR ad Phase Estimatio We ow cotiue with time-varyig chaels ad evaluate the performace of joit CIR ad phase estimatio based o the proposed ALS algorithm. Although, due to the low velocity of acoustic waves (omially 1500 m/s, the Doppler scalig factor at the receiver frot ed is typically o the order of 10 4 to 10 3, the time variatio i the received sigal ca be expected to reduce greatly after resamplig. As such, i the followig simulatios, we set the residual Doppler scalig factor a Δ to [0:0.2 :1] 10 4, ad the stadard deviatio σ ξ =2πa ξ f c T s with a ξ = [cf., (39]. Cosider a OSDM system with vector legth M = 128. The iput SNR of the OSDM system is fixed to 25 db, while the measured output SNR of MMSE equalizatio is

10 10 IEEE JOURNAL OF OCEANIC ENGINEERING Fig. 6. Output SNR performace of the OSDM receiver for differet chael modelig parameters I ad J. Fig. 8. BER performace compariso betwee the OSDM system proposed here ad the D-OSDM system i [14]. Fig. 7. Output SNR performace of the OSDM receiver for differet ALS iteratios γ max. Fig. 9. BER performace compariso betwee two iterative OSDM detectio schemes, i.e., PID ad SID. adopted as the performace metric. Fig. 6 shows the impact of choosig the parameters I ad J [cf., (18 ad (19] o the system performace. As a bechmark, if the IVI-igored processig is applied (which correspods to the case of I =0ad J = M = 128, the output SNR deteriorates rapidly whe a Δ deviates from 0. I cotrast, the ALS algorithm has a capability of joitly estimatig the chael ad time-varyig phase. Moreover, by icreasig I ad/or decreasig J, IVI ca be recostructed ad the cacelled with more accuracy. Higher output SNRs are thus achieved at the cost of a icrease i computatioal complexity. Furthermore, sice the ALS algorithm performs joit CIR ad phase estimatio i a iterative way, its covergece property is show i Fig. 7, where γ max deotes the umber of ALS iteratios [cf., Appedix B], ad we set I =3ad J =8. It ca be observed that while the first two iteratios yield a sigificat improvemet o the output SNR of the OSDM system, the performace gai of further iteratios becomes egligible. For practical use, oly two to three iteratios are usually sufficiet for covergece of the ALS algorithm. C. Iterative OSDM Detectio I this sectio, we further take decodig ito accout ad discuss the performace of the iterative OSDM detectio. Badwidth efficiecy ad BER performace comparisos are first coducted betwee the OSDM system proposed i this paper ad the D-OSDM system i [14]. Here, we set the residual Doppler scalig factor to a Δ = ad the vector legth to M =64. For the D-OSDM system, sice 2V zero vectors have to be iserted aroud the pilot vector ad each of the U data vector groups, where V is the maximum discrete Doppler shift, its badwidth efficiecy ca be computed (with our otatio by η D-OSDM = QM [N 2V (U + 1] MN + K g. (40 I this simulatio, we set U =1to miimize the overhead of zero vectors, ad select V =1, 2, 3, which correspod to the badwidth efficiecies η D-OSDM = 1.45, 0.97, 0.48 b/s/hz, respectively. O the other had, the badwidth efficiecy of the

11 HAN et al.: ITERATIVE PER-VECTOR EQUALIZATION FOR OSDM OVER TIME-VARYING UWA CHANNELS 11 Fig. 10. A sapshot of measured CIRs o each receiver elemet (Chael 01 through 04. OSDM system i this paper is η = M a (N 1 MN + K g. (41 For the assumed settigs, we have M a = QMR c M t =58 ad η = 0.82 b/s/hz, which lies betwee that of the D-OSDM system with V =2ad V =3. Fig. 8 demostrates the BER performace of the two OSDM systems, where our iterative OSDM detectio is embodied by the PID scheme (with ZF equalizatio for fair compariso. It is iterestig to ote that i the high-snr regime, PID has a comparable performace to the D-OSDM system with V =2eve at the iitial iteratio, i.e., β =0. Moreover, whe β 3, it outperforms the D-OSDM system with V =3. This observatio idicates that compared to isertio of zero vectors as i the D-OSDM system, it may be favorable to use the frequecy bad to perform codig, by which a lower BER ca be achieved with the aid of iterative detectio. Furthermore, Fig. 9 compares the iterative detectio performace of the PID ad SID schemes at SNR = 15 db with various Doppler scalig factors. Sice both schemes perform the same preprocessig, their performaces are idetical at the iitial iteratio. However, as the iteratios progress, SID starts to offer a performace advatage over PID because of its ability to immediately decode ad feed back the soft iformatio of the curret equalized symbol vector. VI. EXPERIMENTAL RESULTS I this sectio, we preset prelimiary uderwater field test results of the proposed OSDM system. The experimet was coducted at the Dajiagkou reservoir, Hea Provice, Chia, i Jauary The water depth was varyig from 30 to 50 m. Two ships were used as trasceiver platforms ad deployed 3 km apart. The OSDM sigal was trasmitted from a depth of about 20 m ad received by a four-elemet vertical array (cosistig of Chael 01 through 04 at the same depth with iterelemet Fig. 11. Time-varyig CIR o Chael 01 durig 30 s. spacig 0.25 m. A typical example of the measured CIRs is show i Fig. 10, where for compariso purposes, all impulse resposes have bee commoly ormalized by the strogest amplitude i that of Chael 01. It ca be see that the multipath delay spread τ max 30 ms. Moreover, Fig. 11 displays the timevaryig impulse respose of Chael 01, which is estimated by performig a trai of liear frequecy-modulated (LFM chirp correlatios. Sice o platform motio was ivolved, the experimetal chaels exhibited oly slow time variatios. Withi a observatio duratio of 30 s, the multipath structure was lagged by 1.49 ms, correspodig to a Doppler scalig factor of Over these UWA chaels, there were a total of 16 data packets trasmitted cosecutively, each of which has the structure show i Fig. 12. It comprises four OSDM blocks separated by blak itervals, with two LFM probes iserted for both sychroizatio ad Doppler estimatio purposes. The experimetal OSDM parameters were early the same as those used i Sectio V. The oly differece is that we fixed M = K g = 128 i the experimet to guaratee that: 1 the CP was log eough, i.e., T g = K g T s τ max ; ad 2 the assumptio M>τ max /T s = 120 was valid. The resultat bit rate of the OSDM system ca be computed as M a (N 1 T + T g kb/s. (42 Furthermore, at the receiver, sice the Doppler scalig factor is oly o the order of 10 5, the frot-ed resamplig is deliberately skipped, ad thus the task of compesatig for the time variatios is left to the iterative OSDM detector. We first set the chael time variatio parameters to I =2ad J =16, ad use the BER as a performace metric. The BERs of the proposed PID ad SID schemes are listed i Tables II ad III, respectively, from which similar observatios as those i the simulatios ca be readily made as follows. 1 With SIC improved via the immediate soft iformatio update of the curret symbol vector, it is reasoable to say that SID ca produce superior performace over PID,

12 12 IEEE JOURNAL OF OCEANIC ENGINEERING Fig. 12. Structure of the trasmitted OSDM packet. Fig. 13. Number of bit errors of the proposed OSDM receiver at each iteratio durig blocks (a Sigle-chael ZF equalizatio. (b Sigle-chael MMSE equalizatio. (c Two-chael ZF equalizatio. (d Four-chael ZF equalizatio. TABLE II BER RESULTS OF THE PID SCHEME TABLE III BER RESULTS OF THE SID SCHEME Num. of Iter. No. Equalizatio Chaels β =0 β =1 β =2 β =3 P =1 ZF MMSE P =2 ZF MMSE P =4 ZF MMSE Num. of Iter. No. Equalizatio Chaels β =0 β =1 β =2 β =3 P =1 ZF MMSE P =2 ZF MMSE P =4 ZF MMSE although the BERs of these two schemes are the same i some cases (e.g., whe P =2ad β =1with ZF equalizatio due to limited experimetal data. 2 By utilizig the oise variace estimated i eighborig blak itervals of duratio T b, MMSE equalizatio outperforms its ZF couterpart. Their performace gap is impressive especially whe sigle-chael processig is adopted, i.e., P =1. 3 Whatever the choice for the detectio scheme (PID or SID or equalizatio method (ZF or MMSE, the OSDM system performace improves as the umber of chaels P or iteratios β icreases. I particular, it is observed that ZF

13 HAN et al.: ITERATIVE PER-VECTOR EQUALIZATION FOR OSDM OVER TIME-VARYING UWA CHANNELS 13 Fig. 14. Output SNRs of the experimetal OSDM system for differet receiver parameters. (a (c Output SNRs correspodig to various chael modelig parameters I ad J with P =1, 2, ad 4, respectively. Iteratio parameters are here fixed at β max =3ad γ max =2. (d (f Output SNRs correspodig to various iteratio parameters β max ad γ max with P =1, 2, ad 4, respectively. Chael modelig parameters are here fixed at I =2ad J =16. ad MMSE equalizatio achieve error-free trasmissios at iteratio β =1withP =4ad 2 chaels, respectively. However, it is oteworthy that both iterative detectio schemes fail to coverge i the case of sigle-chael ZF equalizatio. Let us take a closer look to explai this pheomeo ow. I Fig. 13, we cosider four receiver settigs ad plot the umber of bit errors at each iteratio durig OSDM blocks 1 20 (i.e., over the first five packets. Meawhile, the oise variaces estimated for each block are also provided for compariso. It ca be see that there exists a correlatio betwee the system performace ad the oise variace. Roughly speakig, the OSDM receiver produces more bit errors over the duratio where the oise curve rises up. This observatio implies that the performace degradatio of sigle-elemet ZF equalizatio is maily due to the oise ehacemet effect. I cotrast, by takig ito accout the oise explicitly or itroducig spatial diversity, the performace of the OSDM receiver suffers much less from the SNR reductio with MMSE equalizatio or multichael combiig. Furthermore, we ispect the experimetal output SNRs correspodig to differet choices of the parameters I, J, β max, ad γ max. The OSDM receiver here adopts the PID scheme ad MMSE equalizatio with P =1, 2, 4 chaels. First, from Figs. 14(a (c, it ca be see that, as expected, the output SNR improves whe the Doppler spa parameter I icreases or the quasi-static subvector size J decreases. However, compared with the substatial performace gais achieved by expadig I, the output SNR shows a weak depedece o J. This is reasoable because the Doppler effects i the experimetal chaels were ot severe, ad thus oly slow variatios appear alog the diagoals of the phase submatrices {G i }. As a extreme example, whe I =0, the beefit of shorteig J becomes egligible, sice i this case the system performace is limited maily by IVI rather tha itravector time variatios. Secod, as for the iteratio parameters β max ad γ max, Figs. 14(d (f demostrate that the output SNR saturates quickly as β max icreases, while it stays almost costat whe γ max 1. This suggests that for moderate chael Doppler scales, the proposed OSDM receiver eeds oly a few iteratios to guaratee covergece, which may offer some savigs i complexity. VII. CONCLUSION OSDM is a geeralized modulatio scheme coectig OFDM ad SC-FDE. Aalogous to ICI iduced i OFDM systems, OSDM suffers from IVI over time-varyig chaels. I this paper, a OSDM system is proposed for UWA commuicatios. It does ot eed isertio of zero vectors, ad performs ecodig vectorwise at the trasmitter. Accordigly, two iterative detectio schemes, PID ad SID, are also provided at the receiver, where to couteract the effects of the UWA chael, a ALS algorithm is preseted for time-varyig chael estimatio ad a per-vector chael equalizer is desiged for IVI ad ISI cacellatio. The proposed OSDM system achieves: 1 sigificat complexity reductio o chael equalizatio by performig pervector equalizatio i the frequecy domai; ad 2 better BER performace by utilizig badwidth resources for ecodig istead of isertig zero vectors, ad with the aid of iterative detectio. The results of umerical simulatios ad a field experimet suggest that OSDM is a promisig cadidate for highrate commuicatios over time-varyig UWA chaels, offerig more flexible cofiguratios tha the covetioal OFDM ad SC-FDE.

14 14 IEEE JOURNAL OF OCEANIC ENGINEERING APPENDIX A PROOF OF (7 (10 Sice H i(5isak K circulat matrix, it ca be diagoalized by the DFT matrix, i.e., where H = F H K diag {[H 0,H 1,...,H K 1 ]} F K = F H K P N,M HP H N,MF K (43 H = H 0 H 1 Also, we ca factorize F K as where... HN 1. (44 F K = P N,M (I N F M Λ (F N I M (45 Λ = Λ 0 M Λ 1 M... Λ N 1 M. (46 The, substitutig (43 ad (45 ito the first equatio of (7, we have with H = Ξ H HΞ (47 Ξ = P H N,MF K ( F H N I M = (I N F M Λ (48 where, i the derivatio of (48, we have used the fact that P N,M is uitary, i.e., P H N,M P N,M = I K, ad the idetity of Kroecker products, i.e., (A B(C D =(AC (BD, applied to matrices with matchig dimesios. Based o (47 ad (48, we readily obtai (7 ad (8. As for the phase matrix, let us defie a N M matrix Θ =[θ 0, θ 1,...,θ M 1 ] with etries [Θ],m = e jθ M + m, ad a N N matrices Θ m = diag{θ m }, m =0,...,M 1.We ca ow have Θ 0 Θ 1 P N,M ΘP H N,M =.... (49 Moreover, it is easy to kow that Θ M 1 F N I M = P H N,M (I M F N P N,M. (50 Therefore, the matrix G i (9 ca be rewritte as G = P H N,M (I M F N P N,M ΘP H ( N,M IM F H N PN,M G 0 G1 = P H N,M. P.. N,M (51 G M 1 where G m = F N Θ m F H N for m =0,...,M 1. Sice Θ m is a diagoal matrix, G m is a circulat matrix with first colum 1 N F N θ m =[g 0,m,...,g N 1,m ] T. (52 Substitutig matrices G m ito (51, we arrive at (9 ad (10. APPENDIX B ALS ALGORITHM FOR JOINT CIR AND PHASE ESTIMATION To differetiate from the turbo iteratio idex β i Fig. 2, the iteratio idex i the ALS algorithm is here deoted by γ. With this otatio, supposig the estimates of the CIR ĥ γ ad the phase coefficiets {ĝ γ i } have bee obtaied at the γth iteratio, we ca the compute the chael submatrices {Ĥ γ }, the phase submatrices {Ĝ γ i }, ad the Doppler-compesated sigal vectors {x γ } based o (24, (25, ad (26 similarly. The ALS algorithm is developed i a iterative framework via two-step LS estimatio. Similar methods are also adopted for phase oise estimatio i OFDM systems (see, e.g., [33]. We here exted it to the proposed OSDM system. Specifically, the first LS step fixes the phase coefficiets ad updates the CIR estimate. Accordig to (26, it ca be derived that x γ = H d + z γ = Λ H M F H M D (Γ h+z γ (53 where z γ is the oise term. The, premultiplyig both sides of (53 by A = D 1 F M Λ M, we arrive at y γ = Γ h + v γ (54 where y γ = A x γ ad v γ = A z γ. Furthermore, by stackig all N sigal vectors i (54, ad defiig Γ = [Γ T 0, Γ T 1,...,Γ T N 1 ]T, y γ γ T γ T γ T =[y 0, y 1,...,y N 1 ]T, ad v γ γ T γ T γ T =[v 0, v 1,...,v N 1 ]T, it follows that y γ = Γh + v γ. (55 Sice the colums of {Γ } are mutually orthogoal, i.e., Γ H Γ = MI L+1 for ay, the LS estimate of h at the (γ +1th iteratio has the form ĥ = ( Γ H Γ 1 Γ H y γ = 1 K N 1 =0 Γ H y γ. (56 Notice that the matrix iversio is avoided i (56 ad the estimate ĥ ca be efficietly computed via M-poit IDFTs.

15 HAN et al.: ITERATIVE PER-VECTOR EQUALIZATION FOR OSDM OVER TIME-VARYING UWA CHANNELS 15 Also, from (56, we ca obtai the correspodig chael submatrices {Ĥ }. The above chael estimate is the used to costruct the secod LS problem ad further update the phase coefficiets. We ow substitute {Ĥ } ito (20, which gives x = I i= I G i Ĥ i d i + z. (57 By extractig the mth subvector, i.e., premultiplyig both sides of (57 by E m =[I M ] mj: mj+j 1,wehave x, m = I g i, m E m Ĥ i= I i d i + z, m = B, m ḡ m + z, m (58 where x, m = E m x, z, m = E m z, ad ḡ m = [g I, m,...,g 0, m,...,g I, m ] T (59 [Ĥ γ ] +1 B, m = E m +I d +I,...,Ĥ I d I (60 for m =0,..., M 1. Likewise, by stackig all the mth subvectors together from {x }, ad defiig x m = [ x T 0, m, x T 1, m,..., x T N 1, m ]T, z m =[ z T 0, m, z T 1, m,..., z T N 1, m ]T, ad B m T T T =[B 0, m, B 1, m,...,b N 1, m ]T, it yields x m = B m ḡ m + z m. (61 Therefore, the LS estimate of ḡ m at the iteratio γ +1is ( ˆḡ m = B H 1B H m B m m x m. (62 Notice that the computatio of (62 oly ivolves iversio of a small matrix with size (2I +1 (2I +1. Furthermore, the estimate of the phase coefficiets {ĝ i } ca be readily obtaied by iterleavig, i.e., ĝ = P M,2I +1ˆḡ (63 T T,...,ĝ T where ĝ =[ĝ I 0,...,ĝ I ] T ad ˆḡ T T T =[ˆḡ 0, ˆḡ 1,...,ˆḡ M 1 ] T. The ALS algorithm termiates whe γ reaches a give umber γ max, or there is o sigificat declie i the objective fuctio i (21 as the umber of iteratios icreases. If this coditio is satisfied at the γ e th iteratio, the ALS algorithm outputs its fial estimates at the turbo iteratio β as ĥ(β = ĥ γ e ad ĝ (β i = ĝ γ e i,fori = I,...,I. ACKNOWLEDGMENT The authors would like to thak Dr. T. Ebihara for several helpful discussios o the previous works [13], [14], ad the aoymous reviewers for their commets ad suggestios that cotributed to improve the quality of this paper. REFERENCES [1] M. Stojaovic ad J. 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16 16 IEEE JOURNAL OF OCEANIC ENGINEERING [25] M. Tüchler, A. Siger, ad R. Koetter, Miimum mea squared error equalizatio usig a priori iformatio, IEEE Tras. Sigal Process., vol. 50, o. 3, pp , Mar [26] P. va Walree ad G. Leus, Robust uderwater telemetry with adaptive turbo multibad equalizatio, IEEE J. Ocea. Eg., vol. 34, o. 4, pp , Oct [27] J. W. Choi, T. Riedl, K. Kim, A. Siger, ad J. Preisig, Adaptive liear turbo equalizatio over doubly selective chaels, IEEE J. Ocea. Eg., vol. 36, o. 4, pp , Oct [28] L. Bahl, J. Cocke, F. Jeliek, ad J. Raviv, Optimal decodig of liear codes for miimizig symbol error rate, IEEE Tras. If. Theory,vol.20, o. 2, pp , Mar [29] G. Leus ad P. va Walree, Multibad OFDM for covert acoustic commuicatios, IEEE J. Sel. Areas Commu.,vol.26,o.9,pp , Dec [30] B. Li, J. Huag, S. Zhou, K. Ball, M. Stojaovic, L. Freitag, ad P. Willett, MIMO-OFDM for high-rate uderwater acoustic commuicatios, IEEE J. Ocea. Eg., vol. 34, o. 4, pp , Oct [31] J. Huag, S. Zhou, J. Huag, C. Berger, ad P. Willett, Progressive iter-carrier iterferece equalizatio for OFDM trasmissio over timevaryig uderwater acoustic chaels, IEEE J. Sel. Topics Sigal Process., vol. 5, o. 8, pp , Dec [32] N. Beveuto, R. Diis, D. Falcoer, ad S. Tomasi, Sigle carrier modulatio with oliear frequecy domai equalizatio: A idea whose time has come Agai, Proc. IEEE, vol. 98, o. 1, pp , Ja [33] Q. Zou, A. Tarighat, ad A. Sayed, Compesatio of phase oise i OFDM wireless systems, IEEE Tras. Sigal Process., vol. 55, o. 11, pp , Nov Jig Ha (M 10 received the B.Sc. degree i electrical egieerig ad the M.Sc. ad Ph.D. degrees i sigal ad iformatio processig from Northwester Polytechical Uiversity, Xi a, Chia, i 2000, 2003, ad 2008, respectively. He is curretly a Associate Professor at the School of Marie Sciece ad Techology, Northwester Polytechical Uiversity. From Jue 2015 to Jue 2016, he was a Visitig Researcher at the Faculty of Electrical Egieerig, Mathematics ad Computer Sciece, Delft Uiversity of Techology, Delft, The Netherlads. His research iterests iclude wireless commuicatios, statistical sigal processig, ad particularly their applicatios to uderwater acoustic systems. Sudeep Prabhakar Chepuri (M 16 received the M.Sc. degree (cum laude i electrical egieerig ad the Ph.D. degree (cum laude from the Delft Uiversity of Techology, Delft, The Netherlads, i 2011 ad 2016, respectively. He has held positios at Robert Bosch, Idia, durig , ad Holst Cetre/imec-l, The Netherlads, durig He is curretly with the Circuits ad Systems Group at the Faculty of Electrical Egieerig, Mathematics ad Computer Sciece, Delft Uiversity of Techology. His research iterests iclude mathematical sigal processig, statistical iferece, sesor etworks, ad wireless commuicatios. Dr. Chepuri received the Best Studet Paper Award for his publicatio at the ICASSP 2015 coferece i Australia. He is curretly a Associate Editor for the EURASIP Joural o Advaces i Sigal Processig. Qufei Zhag (M 03 received the B.Sc. degree i electrical egieerig ad the M.S. degree i sigal ad iformatio processig from Northwester Polytechical Uiversity, Xi a, Chia, i 1990 ad 1993, respectively, ad the Ph.D. degree i sigal ad iformatio processig from Xidia Uiversity, Xi a, Chia, i He is curretly a Professor at the School of Marie Sciece ad Techology, Northwester Polytechical Uiversity. His research iterests iclude spectral estimatio, array sigal processig, uderwater commuicatios, ad etworkig. Geert Leus (M 01 SM 05 F 12 received the M.Sc. ad Ph.D. degrees i electrical egieerig from the KU Leuve, Leuve, Belgium, i 1996 ad 2000, respectively. He is curretly a Atoi va Leeuwehoek Full Professor at the Faculty of Electrical Egieerig, Mathematics ad Computer Sciece, Delft Uiversity of Techology, Delft, The Netherlads. He is also a Fellow of EURASIP. His research iterests are i the broad area of sigal processig, with a specific focus o uderwater commuicatios, array processig, sesor etworks, ad graph sigal processig. Dr. Leus received the 2002 IEEE Sigal Processig Society Youg Author Best Paper Award ad the 2005 IEEE Sigal Processig Society Best Paper Award. He was a Member-at-Large of the Board of Goverors of the IEEE Sigal Processig Society, the Chair of the IEEE Sigal Processig for Commuicatios ad Networkig Techical Committee, ad the Editor-i-Chief for the EURASIP Joural o Advaces i Sigal Processig. Hewasalsoothe Editorial Boards of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ad the IEEE SIGNAL PRO- CESSING LETTERS. He is curretly a Member of the IEEE Sesor Array ad Multichael Techical Committee, a Associate Editor for the Foudatios ad Treds i Sigal Processig, ad the Editor-i-Chief for the EURASIP Sigal Processig.