Correlations in Underwater Acoustic Particle Velocity and Pressure Channels H. Guo 1, A. Abdi 1, A. Song 2, and M.Badiey 2

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1 Correlaton n Underwater Acoutc Partcle Veloct Preure Channel H. Guo A. Ad A. Song M.Bade Center for Wrele Councaton Sgnal Proceng Reearch et. of Electrcal Couter Engneerng New Jere Inttute of Technolog Newark NJ 7 USA Phcal Ocean Scence Engneerng College of Marne Earth Stude Unvert of elaware Newark E 97 USA Eal: hg5@njt.edu ad@ad.njt.edu ajong@udel.edu ade@udel.edu Atract eendng on the angle of arrval (AOA other channel charactertc dfferent te of correlaton aear aong acoutc reure acoutc artcle veloct channel. To have accurate channel odel for councaton te degn a colete et of correlaton are derved n th aer for a vector enor arra. In addton to exact reult accurate et le colex aroxaton are derved a well. Baed on the exact correlaton dela read of reure artcle veloct channel are calculated a well. Keword Channel characteraton frequenc atal correlaton underwater councaton acoutc vector enor. I. INTROUCTION A vector enor can eaure non-calar coonent of the acoutc feld uch a the artcle veloct whch cannot e ened a ngle calar (reure enor. Vector enor artcle veloct channel have recentl een rooed for underwater acoutc councaton [][][5][]. In ultath channel uch a hallow water a vector enor receve the gnal through everal ath. Th ntroduce dfferent level of correlaton n an arra of vector enor. A et of le cloed-for forula whch exre reure-veloct atal frequenc correlaton n ter of the channel araeter are derved n [] [] for a vertcal vector enor arra. Exact atal frequenc correlaton for an olque vector enor arra are needed to have ore accurate nforaton for councaton te degn. For exale coherence wdth dela read can e calculated ung thee correlaton. Bac defnton of channel n a vector enor arra a tattcal odel for correlaton of an olque vector enor arra are ntroduced n Secton II. General exact exreon for varou correlaton of nteret are derved n Secton III. For a Gauan angle of arrval odel all angle read aroxate exreon are derved n Secton IV. Nuercal reult for exact aroxate correlaton alo the dela read are gven n Secton V. Fnall dcuon concludng reark are rovded n Secton VI. II. SIGNAS AN CORREATIONS IN A VECTOR SENSOR ARRAY For an olque vector enor arra a hown n Fg. n the two-denonal - (range-deth lane there one reure trantter at the far fled called Tx hown a lack crcle. We alo have two receve vector enor rereented two lack quare wth the center of arra at. Each vector enor eaure the reure a well a the coonent of the artcle veloct all n a ngle co-located ont. Th ean that there are two reure channel a well a four reure-equvalent veloct channel all eaured n Pacal ( Newton/. In Fg. reure channel are rereented traght dahed lne wherea reure-equvalent veloct channel are hown curved dahed lne. The veloct channel v v v v are defned a v v ( jρ ω jρ ω. In the aove equaton ρ the dent of the flud n kg/ j ω f the frequenc n rad/. B ultlng the veloct channel n ( wth ρ c the negatve of the acoutc edance of the flud where c the eed of ound n / we otan the reure-equvalent veloct channel where ρcv ρcv ρcv ρcv. Wth λ a the wavelength n k /λ ω /c a the wavenuer n rad/ Th work uorted n art thenatonal Scence Foundaton (NSF Grant CCF-89 Fg.. A two-eleent olque vector enor recever arra.

2 we fnall otan jk jk. Each vector enor n Fg. rovde three outut gnal. For exale Rx generate one reure gnal r two reure-equvalent veloct gnal r r eaured n the drecton reectvel. Slar to [] [] the receved ra at the vector enor arra of th aer are hown n Fg. A two-denonal roagaton of lane wave n the - (range-deth lane aued n a te-nvarant envronent wth a the water deth. The vector enor arra centered at. Vector enor located at / ( / where a vector enor at / + ( /. All the angle are eaured wth reect to the otve drecton of counterclockwe. We odel the rough ea otto t urface a collecton of catterer reectvel uch that >> >>. In th aer we ue (. (. to ndcate all the otto urface coonent angle varale etc. In Fg. the -th otto catterer rereented S... wherea S denote the -th urface catterer.... Ra cattered fro the otto the urface toward the vector enor are hown old lne ra toward the center of arra are hown dahed lne. The ra cattered fro S ht Rx Rx at the AOA reectvel. The traveled dtance are laeled reectvel. Slarl the cattered ra fro S nge Rx Rx at the AOA reectvel wth a the traveled dtance hown n Fg.. The ule reone of the reure channel ( τ ( τ aed on the dela the AOA the correondng Fourer tranfor P ( f P ( f are gven n equaton (-(9 of [] for uchannel Tx Rx Tx Rx reectvel. The reure-equvalent channel of nteret n the dela-ace frequenc-ace doan can otaned artal dervatve of ( τ ( τ P ( f P ( f wth reect to horontal vertcal drecton. ( Fg.. Ra receved a two-eleent olque vector enor arra n a hallow water channel. III. EXACT CORREATION EXPRESSIONS The defnton of the reure channel frequenc-ace * correlaton gven CP ( f E[ P ( f + f P ( f ]. Here f the acng n frequenc doan are the vertcal horontal dtance etween the two vector enor reectvel * colex conjugate E the exectaton calculated over the dtruton of AOA fro the otto the urface. Slar to [] [] P ( f P ( f can e wrtten a / Λ ψ P ( f ( / a ex( j ex( jk [ co( + n( ]ex( jωτ / / / (( / a ex( jψ + Λ ex( jk [ co( + n( ]ex( jωτ / Λ ψ P ( f ( / a ex( j / / ex( jk[ co( + n( ]ex( jωτ / + / / (( / a ex( jψ + Λ ex( jk[ co( + n( ]ex( jωτ / + / where ω f ued to lf the notaton. The freq-ace correlaton of the reure channel can e hown to e C ( f w ( ex[ jk (co + co / ] Λ ex[ jk(n n ]ex[ jk(n + n / ] (5 ex[ jω( T T ]ex[ j ωt ] d w ( ex[ jk (co + co / ] + ( Λ ex[ jk(n n ] ex[ jk(n + n / ]. ex[ jω( T T ]ex[ j ωt ] d In (5 are the AOA of ra cong fro otto urface toward the vector enor arra center reectvel. co co n n rereent the correondng cone ne value of otto urface AOA for vector enor Rx Rx reectvel. The can e exreed n ter of n ( + ( / / ( n ( ( / / n ( ( / / n ( + ( / / co [( cot( ( / ] / co [( cot( + ( / ] / co [ cot( + ( / ] / co [ cot( ( / ] / where cot(. co(./n(.. Moreover are dtance fro the ea otto urface catterer to Rx ( ( (7 (8 (9

3 Rx reectvel. The are gven n (-( elow are functon of. T T n (5 are travel te fro otto urface catter to the Rx Rx reectvel can e wrtten a the functon of T T T T. ( c c c c We defne w ( w ( n (5 a the roalt dent functon (PF of the AOA of the wave cong fro the ea otto t urface reectvel uch that < < < <. Λ n (5 the ower rato etween the cattered coonent fro otto urface. Preure channel correlaton of two ecal cae vertcal horontal arra can e otaned ettng n (5 reectvel. The reult are gven n (5 for a vertcal arra ( for a horontal arra. Wth n (5 we otan the reure channel frequenc correlaton functon. Partcle veloct channel correlaton can e calculated takng dervatve of the reure channel correlaton functon n (5 wth reect to or. For exale dfferentaton of CP ( f wth reect to gve the correlaton etween the reure gnal at Rx the -veloct gnal at Rx. ue to the ace ltaton artcle veloct related channel correlaton are not reented here. IV. APPROXIMATE CORREATIONS FOR SMA ANGE SPREAS Exact correlaton exreon nclude ntegral over AOA whch are te conung to coute. For all angle read under certan condton uch a all acng etween arra eleent ueful ntegral-free aroxaton for vertcal horontal arra can e otaned ung (5 (. A. Vertcal Vector Senor Arra For << n( ung + x + ( x / when x << dtance gven n (-( can e aroxated a [( + n ( / ]/ n( [( n ( / ]/ n( [ ( n ( / ]/ n( [ + ( n ( / ]/ n(. Then uttutng (7 nto (-(9 reult n n n n ( /( n n n ( / n + n n( n + n n( T T n( / c T T n( / c T ( /( cn( T /( cn(. (7 (8 (9 ( ( B uttutng (8-( nto (5 we otan the aroxaton gven n ( elow. For all angle read o that n (. << n(. we odel AOA a Gauan dtruton for oth otto urface coonent wth ean µ µ varance σ σ reectvel w w ( ( σ ex[ ( µ ( σ ] ( ( ( σ ex[ ( µ ( σ ] (. ( Ung the charactertc functon of the Gauan varale /.e. ex( jθ x( σ ex[ x /( σ ] dx ex( σ θ / ( + (( / + ( / n ( ( ( / + ( / co( + arctan( n( n( ( ( + (( / + ( / n ( ( ( / + ( / co( + arctan( n( n( ( + (( / + ( / n ( ( / + ( / co( + arctan( n( n( ( + (( / + ( / n ( ( / + ( / co( + arctan( n( n(. ( C ( f Λ w ( ex[ jk(n n ]ex[ jk(n + n / ]ex[ jω( T T ]ex[ j ωt ] d + ( Λ w ( ex[ jk(n n ]ex[ jk(n + n / ]ex[ jω( T T ]ex[ j ωt ] d ( Λ ( ex[ (co + co / ]ex[ (n n ]ex[ ω( ]ex[ ω ] C f w jk jk j T T j T d + ( Λ w ( ex[ jk (co + co / ]ex[ jk(n n ]ex[ jω( T T ]ex[ j ] ωt. d (5 (

4 ntegral n ( can e olved whch reult n the ace-frequenc correlaton n ( for reure channel. B takng the frt dervatve of ( wth reect to cro-correlaton etween reure -veloct channel n ace frequenc can e otaned. The econd dervatve of ( rovde the ace-frequenc autocorrelaton functon of the -veloct channel. For exale the atal autocorrelaton functon of the -veloct gven n (5. B. Horontal Vector Senor Arra The dtance gven n (-( can e larl aroxated a [( ( n( co( / ]/ n( + [( ( n ( co( / ]/ n( [ + ( n( co( / ]/ n( [ ( n( co( / ]/ n(. Suttuton ( nto (-(9 reult n co( co( n( n( co( co( n( n( co( + co( co( co( + co( co( T T co( / c T T co( / c T ( /( cn( T /( cn(. ( (7 (8 (9 ( Therefore the ace-frequenc correlaton n ( for the horontal arra can e aroxated a (. For Gauan AOA ( lfe to (. fferentaton wth reect to rovde correlaton related to the -veloct channel. V. NUMERICA RESUTS Here we conder the cae where the two-eleent vector enor arra n Fg. receve gnal through two ea: one fro the otto wth ean AOA µ angle read σ the other one fro the urface wth ean AOA µ angle read σ. When the angle read are all we odel the AOA wth the Gauan PF gven n (. To anale atal frequenc correlaton of the olque vector enor arra we conder the ae araeter ued n [] []. The otto urface ean AOA are 8 degree wherea the correondng angle read are.5 degree. The center of the arra lace at 5 eter whle the water deth. The ower rato Λ.. Fg. how the atal correlaton agntude of the reure channel of the olque arra. Fg. Fg. 5 how the exact aroxate ulated reure channel atal correlaton for two ecal cae vertcal horontal arra. The ulaton reult confr the accurac of derved exact correlaton exreon. Cloe agreeent etween aroxate exact reult how the uefulne of aroxaton. Satal correlaton Horontal acng /λ 8 Vertcal acng /λ Fg.. Magntude of the reure atal correlaton of an olque vector enor arra. ω( C f w jk jk j d ( Λ ( ex[ n ( + n( ] cn( ω ( w ( ex[ jkn ( jk n( j ] d. cn( + Λ + + ( ( ω σ ( ω C ( f Λ ex jkn( µ j ex k co( µ + cn( µ cn( µ tan( µ ω σ ω + ( Λ ex jkn( µ + j ex k co( µ cn( µ cn( µ tan( µ C ( Λ [n ( µ + σ co ( µ σ k co ( µ + jσ kn( µ co ( µ ]ex jkn( µ σ k co ( µ (5 + ( Λ [n ( µ + σ co ( µ σ k co ( µ + jσ kn( µ co ( µ ]ex jk n( µ σ k co ( µ.. (

5 ω( C f w jk jk j d ( Λ ( ex[ ( + co( + co( ] ( ( cn( ω + ( Λ ( ex[ ( co( co( + ]. w jk jk j d cn( co( µ ( C ( Λ ex jk + co( µ + jk j ω ( ( c n( µ σ n( µ ( ex k + n( µ k + ω ( ( cn( µ tan( µ co( µ + ( Λ ex jk co( µ jk + j ω c n( µ σ n( µ ex k n( µ + k ω. c n( µ tan( µ ( (.9.8 Exact Aroxaton Sulaton.9.8 Satal correlaton agntude /λ Fg.. Magntude of the reure atal correlaton veru / λ for a horontal arra. Satal correlaton agntude Exact Aroxaton Sulaton /λ Fg. 5. Magntude of the reure atal correlaton veru / λ for a vertcal arra. Magntude of frequenc correlaton Sulated reure correlaton Exact reure correlaton Sulated veloct- correlaton Exact veloct- correlaton Sulated veloct- correlaton Exact veloct- correlaton f (H Fg.. Magntude of frequenc correlaton n a ngle vector enor. To anale the dela read of reure veloct channel we need ther frequenc correlaton functon. Fg. how the noraled exact ulated frequenc correlaton of reure veloct channel for the ae et of araeter ntroduced at the egnnng of th ecton. et u defne the coherence wdth a the frequenc at whch the noraled correlaton ecoe.5. Then the dela read can e condered to e the nvere of th coherence wdth. Accordng to Fg. the dela read of -veloct channel aller. Th can e etter undertood coarng the ule reone of reure veloct channel n Fg.7 otaned lottng equaton ( of [] t dervatve wth reect to. Clearl the reone of channel read over a aller range of dela n th exale. To tud the act of ean AOA angle read on the dela read of reure artcle veloct channel one can refer to Fg. 8 Fg. 9. ela read are lotted n Fg. 8 veru ean AOA whle angle read are fxed uch that σ σ 5 deg. On the other h n Fg. 9 dela read are 5

6 Preure channel ule reone agntude Veloct- channel ule reone agntude Veloct- channel ule reone agntude ela read (ec Preure channel Veloct- channel Veloct- channel Angle read: σ σ 5 deg τ (econd Fg. 7. Iule reone of the reure veloct channel. lotted n ter of equal angle read where µ deg. µ 5 deg. ( deg.. In oth fgure Λ.5 aued.e. equal ower for otto urface coonent. Content wth the Fg. Fg. 7 we ee n oth fgure the dela read of the -veloct channel aller for the hallow water odel condered n th aer. Accordng to Fg. 8 dela read are alot contant when ean AOA change uch that o µ ncreae µ µ. Th ght e ecaue the angle read are fxed o the nuer of ultath coonent cong fro dfferent drecton wth dfferent dela doe not change. However when angle read ncreae a n Fg. 9 then ore ra fro new drecton wll e receved whch wll aount to a wder range of dela (travel te n channel ule reone. Th can exlan the ncreae of dela read n Fg. 9. VI. ISCUSSION AN CONCUSION In th aer we have reented a ra-aed tattcal/geoetrcal fraework for characteraton of acoutc vector enor arra correlaton n hallow water. Exact correlaton exreon for an artrar vector enor arra are derved. Ung thee exreon one can calculate the exact correlaton etween reure veloct channel n ter of eleent acng frequenc earaton angle of arrval water deth arra locaton. Ueful aroxate correlaton exreon are alo derved when angle read are all. ela read of reure veloct channel are alo tuded ung channel frequenc correlaton functon. The reult of th aer are requred for the degn erforance aeent of ngle uer [][][5] ultuer [] underwater councaton te that oerate through acoutc artcle veloct channel Mean angle of arrval: µ - µ (deg. Fg. 8. ela read of reure veloct channel veru ean angle of arrval. ela read (ec Preure channel Veloct- channel Veloct- channel Mean angle of arrval: µ deg. µ 5 deg..5 5 Angle read: σ σ (deg. Fg. 9. ela read of reure veloct channel veru angle read. REFERENCES [] A. Ad H. Guo A new coact ultchannel recever for underwater wrele councaton network acceted for ulcaton n IEEE Tran. Wrele Coun. 8. [] A. Ad H. Guo Sgnal correlaton odelng n acoutc vector enor arra acceted for ulcaton n IEEE Tran. Sgnal Proceng. 8 [] A. Ad H. Guo A correlaton odelng for vector enor arra n underwater councaton te n Proc. MTS/IEEE Ocean Queec Ct QC Canada 8. [] A. Ad H. Guo P. Sutthwan A new vector enor recever for underwater acoutc councaton n Proc. MTS/IEEE Ocean Vancouver BC Canada 7. [5] A. Song M. Bade P. Hurk A. Ad Te reveral recever for underwater acoutc councaton ung vector enor n Proc. MTS/IEEE Ocean Queec Ct QC Canada 8. [] H. Guo A. Ad Multuer underwater councaton wth ace-te lock code acoutc vector enor n Proc. MTS/IEEE Ocean Queec Ct QC Canada 8.