A Recursive Approach to Compute Bit Error Rate in Underwater Channels with Multiple Paths

Transcription

1 A Recursve Approach to Compute Bt Error Rate n Underwater Channels wth Multple Paths C. Chen A. Ad Center for Wreless Communcaton and Sgnal Processng Research Dept. of Electrcal and Computer Engneerng ew Jersey Insttute of Technology ewark J 7 USA Emals: cc4@njt.edu al.ad@ njt.edu Astract In underwater acoustc channels sgnal s transmtted over several dstnct paths egenpaths from transmtter to recever due to reflectons at sea oundares where each egenpath contans a domnant specular component and a numer of scattered components. As a result an underwater acoustc channel response s the superposton of several domnant specular components and numerous scattered components. Bt error rate (BER) n multpath fadng channels has een extensvely studed n the past. However lmted research has een conducted on fadng channels wth several domnant specular components. In ths paper BER n multpath channels wth several specular components s studed. A new formula to compute the BER recursvely and effcently s derved. Then usng Jensen s nequalty one specular component Rce fadng s shown to provde the lowest possle BER. Upon usng the new BER formula and Lagrange multplers to solve a constraned optmzaton prolem t s further shown that for two domnant specular components BER acheves ts maxmum when the two components are equally weghted. More results on BER for three and four specular paths are also presented. The results shed lght on the mpact of the numer of specular paths on BER as well as the maxmum and mnmum values of BER whch are of nterest n underwater communcaton systems. Index Terms Bt error rate multpath fadng channels underwater channels acoustc communcaton constraned optmzaton wreless propagaton. I. ITRODUCTIO A real underwater acoustc channel contans several dstnct paths [] []. Besdes the drect path from the transmtter to the recever reflectons at the sea surface and ottom also create new paths. However sgnal travellng along each partcular path experences the nhomogenety and thermal mcrostructure of the ocean medum whch turn the receved sgnal nto a Ths work s supported n part y the atonal Scence Foundaton (SF) Grant CCF-89. domnant specular component and a numer of randomly scattered components. Upon the superposton of several dstnct paths an underwater acoustc channel response can e consdered to nclude several domnant specular components and many dffuse scattered components. It has een oserved through at-sea experments that an underwater acoustc channel could have two or three or even more specular components [] [4]. In the past a large volume of research has een done on the t error rate (BER) of dgtal modulatons for multpath channels and n the presence of dffuse components Raylegh fadng as well as dffuse plus one specular component Rce fadng [5]. However how the ncreasng numer of specular components may affect the BER s unknown. Therefore t s of nterest to study the BER of fadng channels wth several specular components especally n underwater acoustc systems. The proalty densty functon (PDF) of the sgnal envelope for more than one specular component s dscussed n [6]-[8]. The case of two specular components s dscussed n [6] and [7]. Two and more specular components are thoroughly studed n [8] where some seres for BER and the envelope PDF are derved wth mnmal truncaton errors. In ths paper we study ths prolem from another perspectve y addng the specular components n multple steps. Therefore a new recursve formula s derved whch has hgh computatonal effcency n calculatng the BER. ew results are derved usng Jensen s nequalty and Lagrange multplers for constraned optmzaton to determne the mpact of the numer of specular components on BER and the hghest and lowest values of BER. The rest of the paper s organzed as follows. Secton II starts wth prolem formulaton and derves the new recursve BER formula as well as ts mnma and maxma. Secton III provdes numercal results on BER n fadng channels wth multple specular components. Fnally Secton IV provdes the concludng remarks and dscusses some other possle applcatons of the derved results and formulas.

2 II. DERIVATIO OF THE RECURSIVE FORMULA The receved sgnal n a multpath fadng channel wth specular components and a dffuse component can e represented y [6] [8] () Rexp( jθ ) aexp( jφ ) + A exp( jφ ) where R and Θ are the receved ampltude and phase j a and Φ are the ampltude and phase of the -th specular component and A and Φ are the ampltude and phase of the dffuse component. The phases Φ... are ndependent and dentcally dstruted unform random varales over [ π ) whereas a s are constant and A has a Raylegh dstruton wth average power Ω E[ A]. Let us defne the superposton of specular components as Bexp( jψ ) aexp( jφ). Condtoned on B the PDF of R s the Rce pdf [8] f ( r ) ( Ω r)exp( Ω ( r + )) I ( Ω r) () RB where I (.) s the zero-th order modfed Bessel functon. Let U R and V B whch upon susttuton nto () results n the followng uncondtonal PDF for U f ( u) E [ Ω exp( Ω ( u+ V )) I ( Ω V u)] () U V where E s expectaton. The BER expresson of several modulatons n addtve whte Gaussan nose s an exponental functon of U [5]. Here we consder nary dfferental phase shft keyng whose BER condtoned on U s gven y P( U).5exp( U) where s the sgnal to nose rato (SR) per t. Usng () average BER wth specular components P Pu f ( udu ) can e wrtten as U where V n the exponental n (4) s replaced y E[ V] Ω a to otan the rght-hand sde of the nequalty n (5). The lower ound n (5) holds for any and can e acheved for. Ths s ecause accordng to eq. (8.) n [5] BER n Rce fadng s exactly the same as the rght-hand sde of (5). Ths completes our proof that for a fxed total power of specular components Ω Rce fadng results n the lowest possle BER. B. Recursve formula for BER Wthout loss of generalty and to smplfy the notaton let Φ. For the sum of two vectors (specular components) results n B exp( jψ ) a + a exp( jφ ). Usng the cosne formula n the trangle wth sdes B a and a we otan B a + a + aa cosφ. Wth V B and V a ths can e wrtten as V V+ a + a VcosΦ. When the thrd vector a exp( jφ ) s added we can smlarly wrte V V + a + a V cos( Φ Ψ ) condtoned on V and Ψ. By addng vectors one y one we otan the recurson V V + a + a V cos( Φ Ψ )... wth Ψ and condtoned on V.and Ψ. Ths approach s used n [] to derve the ampltude pdf recursvely whereas we use t n a dfferent way to otan new BER results y computng the BER recursvely. By susttutng V V + a + a V cos( Φ Ψ ) nto (4) we otan the followng BER condtoned on V Ψ and Φ as shown n (6). P( V Ψ Φ ) ( Ω ( V + a + a V cos( Φ Ψ )) exp. Ω + (6) u+ V P EV exp u I Vu du Ω Ω Ω V EV exp ( Ω Ω + where the ntegral n (4) s analytcally solved usng Mathematca. A. Lowest possle BER Because ( Ω > the term exp( ( Ω V) n (4) s a convex functon n V. Therefore Jensen s nequalty [9] results n (4) P.5( Ω exp ( Ω Ω (5) Based on the ntegral dentty π I( x) ( π) exp( xcos( β ξ)) dβ eq. (6) can e averaged wth respect to Φ whch s unformly dstruted over [ π ). Snce the resultng expresson depends only on V the recursve average BER can e ultmately wrtten as (7) where V. Eq. (7) ncludes Rce BER eq. (8.) n [5] as a specal case for. When snce V a s a constant eq. (7) results n (8). P exp ( V + a) Ω + EV ( Ω a I V Ω +... (7)

3 I ( ( ) aa ). P.5( Ω exp ( Ω ( a + a ) Ω + (8) For the expectaton n (7) s over V whch depends on Φ. Therefore P.5( Ω + ) E Φ exp ( Ω + ) ( V + a) I( ( Ω a V). Followng the same approach eq. (7) for 4 ultmately results n 4 Ω + Φ Φ Ω P.5 E exp ( V a ) I( ( Ω a4 V) (9) () wth V defned efore and V cos( Φ Ψ ) ( a+ acos Φ)cosΦ + asn Φsn Φ. Equatons (8)-() are computatonally more effcent than ntegrals over products of Bessel functons (see [8] and references theren). C. Hghest and lowest BERs usng Lagrange multplers The BER n (8) s a functon of a and a. For a fxed total specular power a + a Ω BER vares as a and a change and t s of nterest to fnd out what values of a and a maxmze or mnmze the BER. Ths a constraned optmzaton prolem suject to the constrant a + a Ω. To solve ths usng Lagrange multplers [] we defne the Lagrange functon La ( a ) P + ( a + a Ω ) where s the Lagrange multpler. By settng partal dervatves of L zero we otaned three crtcal ponts. They agree wth the ponts n [] derved usng a BER expresson that s entrely dfferent from the new recursve BER n (7). The BER expresson n [] s not effcent for numercal computatons. Also t s not analytcally nvestgated there whch pont gves the maxmum BER and whch one gves the mnmum. The ponts ( a a ) ( Ω ) and ( Ω ) ndcate that the specular power s n one component only whch s Rce fadng and s shown n (5) to e the lowest BER. The thrd crtcal pont ( a a ) ( Ω / Ω /) ndcates two equal ampltude specular components. Usng the Hessan matrx [] we have shown the BER s maxmum at ths crtcal pont. Detaled dervatons are provded n Appendx. III. UMERICAL RESULTS Let K Ω Ω e the specular to dffuse power rato and μ a / a... e the -th specular power rato. Also consder unt total power.e. Ω +Ω wthout loss of generalty. Usng eq. (8) Fg. shows the BER for versus μ wth specular power rato K 5 db 5 db and db. In agreement wth the theoretcal dervaton BER s maxmum when the two specular components have equal strength that s μ and s mnmum when one specular component s much stronger than the other whch means μ or. log K5 db K5 db KdB μ a /a Fg.. Bt error rate n a fadng channel wth two specular components versus the specular power rato wth db and dfferent specular to dffuse power ratos K. log - - μ a /a μ a /a Fg.. Bt error rate n a fadng channel wth three specular components versus the specular power ratos wth db and specular to dffuse power rato K db. The BER for s plotted n Fg. versus μ and μ wth K db and db usng eq. (9). Consstent wth the analytcal fndng that mnmum BER occurs when one specular component s much stronger than the others we oserve that the mnma n Fg. are located at ( μ μ ) (..)(.)(.). The maxma appear

4 to occur when ( μ μ ) ()(.)(.). Ths agrees wth the constraned optmzaton result that maxmum BER for two specular components happens when the ampltudes are equal. lmted to underwater acoustc channels and systems. Another case where the results are of nterest s a spatally selectve system wth drectve antennas where only a lmted numer of waves from certan drectons mght e receved [6]. Smlar stuaton can occur n wdeand systems where a short delay n may contan only a small numer of multpath components []. The results could also e of nterest n vehcular wreless sensor networks [4]. log - - μ μ. μ μ μ μ μ.μ. μ.μ (db) Fg.. Bt error rate n a fadng channel wth three specular components versus SR wth dfferent specular power ratos μ and μ and the specular to dffuse power rato K db. More numercal results are shown n Fg. and 4 wth dfferent specular power ratos μ for 4. Two strong specular components provde hgher BERs and one strong specular component gves lower BERs. To study the mpact of n Fg. 5 BER s plotted for equal-strength specular components usng (8)-(). Unt power Raylegh fadng BER.5( [5] s also plotted as a reference. For a fxed total specular power as shown n (5) usng Jensen s nequalty (Rce fadng) has the lowest BER. Moreover has the hghest BER. Interestngly BER for 4 s hgher than the BER for at hgh SRs. As ncreases BER approaches the Raylegh case. IV. COCLUSIO In ths paper a new BER formula s derved for multpath channels wth specular paths. Upon usng ths formula t s proved that Rce fadng provdes the lowest BER among all possle values for. For equal-ampltude specular components the numercal results show that wth a fxed total specular power BERs le etween and curves. Moreover 4 provdes a BER hgher than specally at hgh SRs. These results are useful for system desgn and performance predcton n underwater acoustc channels contanng a drect ray as well as few surface and ottom reflected rays n early arrvals. However the fndngs and derved formulas are not log - - μ μ.μ 4. μ μ μ 4 μ μ μ 4. μ μ μ 4 μ.μ.μ (db) Fg. 4. Bt error rate n a fadng channel wth four specular components versus SR wth dfferent specular power ratos μ μ and μ 4 and the specular to dffuse power rato K db. APPEDIX THE BER COSTRAIED OPTIMIZATIO PROBLEM AD LAGRAGE MULTIPLIERS Wth two specular components BER n multpath fadng channels s gven y (8). Defne the Lagrange functon as Ω Ω + aa Ω + a a La ( a ).5( Ω e I + ( + Ω ). () Takng the partal dervatves of L wth respect to a and a respectvely results n Ω Ω + aa a Ω + ( Ω La ( a )/ a e I Ω Ω + aa a Ω + ( Ω e I + a Ω Ω + aa a Ω + ( Ω La ( a )/ a e I

5 Ω Ω + aa a Ω + ( Ω e I + a. () Ω Ω + aa I Ω + ( Ω + e (6) Solvng the two equatons aove provdes Ω ( ) Ω + aa a a I Ω + ( Ω e () whch leads to three crtcal ponts. they are: ( a a ) ( Ω ) ( Ω ) and ( Ω / Ω /). At the ponts ( a a ) ( Ω ) and ( Ω ) the specular power s allocated to a sngle component only whch s Rce fadng. As shown n (5) BERs at these two ponts are mnmum. To determne whether the pont ( a a ) ( Ω / Ω /) gves a mnmum or a maxmum BER we need to solve an equaton ased on the Hessan matrx [] where the equaton s gven y (4). The determnant on the rght hand sde of (4) needs to e computed at ( a a ) ( Ω / Ω /). From () we know at the pont ( a a ) ( Ω / Ω /) we have (5). L( a a ) / a x L( a a ) / a a a a a L( a a ) / a a L( a a ) / a x a Ω Ω + Ω I Ω + ( Ω Ω Ω + Ω I Ω + ( Ω.5e +.5 e.. (4) (5) Moreover second dervatves of L are gven n (6) and (7). Upon susttuton of (5)-(7) nto (4) the roots of (4) can e shown to e x x exp( Ω / ( Ω ) I( Ω /( Ω )/( Ω <. The negatve sgn [] ndcates the pont ( a a ) ( Ω / Ω /) corresponds to the maxmum BER. Ths agrees wth the numercal results n Fg.. La ( a )/ a a La ( a )/ a a Ω Ω + aa aa Ω + Ω + Ω + ( Ω Ω Ω + aa Ω I Ω + ( Ω.5e I I e La ( a )/ a La ( a )/ a Ω Ω + aa aa Ω + Ω + Ω + ( Ω Ω Ω + aa Ω I Ω + ( Ω.5e I I e Ω Ω + aa I Ω + ( Ω e +. REFERECES (7) [] X. Geng and A. Zelnsk An egenpath underwater acoustc communcaton channel model n Proc. IEEE Oceans Conf. San Dego CA 995 pp [] X Lurton An Introducton to Underwater Acoustcs: Prncples and Applcatons. Sprnger-Praxs. [] P. Qaraaq and M. Stojanovc Statstcal modelng of a shallow water acoustc communcaton channel n Proc. Underwater Acoustc Measurements Conf. afplon Greece 9. [4] M. Chtre A hgh-frequency warm shallow water acoustc communcatons channel model and measurements J. Acoust. Soc. Am. vol. pp [5] M. K. Smon and M. S. Aloun Dgtal Communcaton over Fadng Channels nd ed. Wley 5. [6] G. D. Durgn T. S. Rappaport and D. A. de Wolf ew analytcal models and proalty densty functons for fadng n wreless communcatons IEEE Trans. Commun. vol. 6 pp. 5. [7] J. Frolk On approprate models for characterzng hyper-raylegh fadng IEEE Trans. Wreless Commun. vol. 7 pp [8] A. Ad On the utlty of Laguerre seres for the envelope PDF n multpath fadng channels IEEE Trans. Info. Theory vol. 55 pp [9] T. M. Cover and J. A. Thomas Elements of Informaton Theory nd ed. Wley 6. [] M. Smon On the proalty densty functon of the squared envelope of a sum of random phase vectors IEEE Trans. Commun. vol. pp [] S. S. Rao Optmzaton Theory and Applcatons nd ed. Wley984. [] R. K. Bandla and A. Ad On the performance lmts n fadng channels wth artrary numer of multpath components n Proc. Conf. Inform. Sc. Syst. Prnceton J 4 pp [] A. S. Molsch Mole rado channels n Wdeand Wreless Dgtal Communcatons A. F. Molsch Ed. Upper Saddle Rver J: Prentce Hall PTR pp [4] J. Frolk A case for consderng hyper-raylegh fadng channels IEEE Trans. Wreless Commun. vol. 6 pp

6 log Raylegh (db) Fg. 5. Bt error rate n a fadng channel wth 4 specular components versus SR wth equal ampltudes K db and the correspondng Raylegh t error rate.

7