Improved Multi-Channel Blind De-Convolution Algorithm for Linear Convolved Mixture DS/CDMA Signals

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1 Joural of Commucatos Vol. 9, No., October 4 Imroved ult-chael Bld De-Covoluto Algorthm for ear Covolved xture DS/CDA Sgals Hao Cheg, Na Yu, ad Ju u Telecommucato Egeerg aboratory, Chegdu Uversty Natoal Key ab. of Commucato, Uversty of Electroc Scece ad Techology, Chegdu, Cha Emal: Dr.cheghao@63.com; yu88@63.com; u_ju@uestc.edu.c Abstract I ths aer, a mroved mult-chael bld decovoluto algorthm for code dvso multle access sgals s roosed. Through mmzg the average squared crossoutut-chael-correlato, the roosed scheme acheves a better source searato erformace for the lear covolved mxture system. Ulke most of the exstg lear covolved mxture searato algorthms the lterature, the mroved algorthm ca obta the orgal users sgal eve low sgal-to-ose rato a o-cooerato scearo. Smulato results show that the roosed otmal scheme ot oly acheves sueror erformace o the bt error rate to those of the exstg oes, but also rovdes a coverget soluto uder dfferet chael codtos. Idex Terms CDA sgal, o-cooerato, bld decovoluto, gradet algorthms I. INTRODUCTION Sread sectrum sgals have bee used for the secure commucatos for several decades. Nowadays, they are also wdely used outsde the mltary doma, esecally Code Dvso ultle Access (CDA) system. They are maly used to trasmt at a low ower wthout terferece due to jammg to others users or to multath roagato. The CDA techques are useful for the secure commucato whle the recever has to kow the sequece used by the trasmtter ad recover the trasmtted data usg a correlator []. Drect sequece sread sectrum (DSSS) trasmtters use a erodcal seudo-radom sequece to modulate the base-bad sgal before trasmsso. I the cotext of sectrum survellace, the seudo-radom sequece used by the trasmtter s ukow as well as other trasmtter arameters such as durato of the sequece, symbol frequecy ad carrer frequecy. Hece, ths cotext, a DSSS sgal s very dffcult to detect ad demodulate, because t s ofte below the ose level. I ths aer, our motvato s to determe the sreadg sequece automatcally a mult-user CDA system, whereas the recever has o kowledge of the trasmtter s seudo-ose (PN) sequece. We also auscrt receved Jue, 4; revsed October, 4. Ths roject was suorted by the Natoal Natural Scece Foudato of Cha (6768) ; the Natoal Natural Scece Foudato of Cha (578); the Secalzed Research Fud for Natoal Key ab of Commucato (456345ZS9) Corresodg author emal: Dr.cheghao@63.com. do:.7/jcm reset the techque of DSSS ad exla the dffculty to recover the data a ufredly cotext. The algorthm s artcularly mortat a o-cooeratve codto of the mltary feld. Ths algorthm ca decode the local sread sectrum sequece, so as to motor, terferece, ad decher the eemy sgal. A more comlex mult-chael bld de-covoluto algorthm s requred to acheve better source searato []. We roose a method to estmate the seudo-radom sequece wthout ror kowledge about the trasmtter. Oly the durato of the seudoradom sequece s assumed to have bee estmated. (Ths ca be doe by the method roosed [3]). Ths method s based o mmzg the average squared cross outut chael correlato crtero. There are may ways to costruct a umercal algorthm based o the above crtero for bld source searato. We derve here a ew stochastc gradet algorthm, whch has these characterstcs for the otmzato of the de-mxg matrx D. Exermetal results are gve to llustrate the erformaces of the method ad show that a good estmato ca be obtaed. Ths aer s orgazed as follows. I Secto II, the fudametal models ad assumtos of IO bld decovoluto s troduced. I Secto III, we gve the DS/CDA data model ad searato crtero. The, the roosed aroach s descrbed Secto IV. Fally, exermetal results are rovded to llustrate the aroach Secto V ad a cocluso s draw. II. ODES FOR IO BIND DE-CONVOUTION [4] A. odels for IO Covoluto A mult-chael lear tme varat (TI), dscrete tme dyamcal system ca be descrbed the most geeral form as: x( k) Hs ( k - ) () where H s a m dmesoal matrx of mxg coeffcets at tme-lag (called the mulse resose at tme-lag ), ad s ( k- ) s a -dmesoal vector of source sgals wth mutually deedet comoets. It should be oted that the causalty tme doma s satsfed oly whe H for all. x ( k) s the receved sgal (sesor sgals) after covoluto mxed, ad x( k) [ x( k), x( k),, x ( )] T k s a dmesoal vector of the oututs. 4 Egeerg ad Techology Publshg 785

2 Joural of Commucatos Vol. 9, No., October 4 dscrete tme seres) trasformato the comlex frequecy doma. Wth the symbol W(z), we ca better v(k) y(k) + s(k) + H(z) uderstad the covoluto sgal tha W. x s x(k) W(z) aother exresso of x(k - ), there s o varable k W(z),so we use the symbol x stead of x(k - ). I some other alcatos, a o-causal (doubly-fte) feed-forward mult-chael de-mxg matrx (a) x(k) W y(k) W( z ) W x W (4) K where K ad are two varables of ostve tegers. The global trasfer fucto G( z ) s defed by Wm W x(k) W G( z) W( z)h( z) y(k) where H( z ) s the Z-trasformato of H, ad H s the chael mulse resose for tme doma. We ca obta form (6), H( z ) s the chael mulse resose for frequecy doma, ca also be called chael flter. Wm W H( z ) Wm I order to esure that the mxed sgal s recoverable, we ut the followg costrats o the covolute ad mxed sgal. The chael mulse resose fucto H( z ) s stable,.e., ts mulse resose satsfes the absolute summablty codto: (b) The goal of the mult-chael bld de-covoluto s to estmate source sgals usg the receved sesor sgals x(k ) oly ad certa kowledge of the source sgal dstrbutos ad statstcs. I the most geeral case, we attemt to estmate the sources by emloyg aother mult-chael, TI, dscrete-tme, stable dyamcal system (Fg. (a) ad (b)) descrbed at tme doma: (6) where z s the exresso of H at tme lag. y(k) Fg.. Illustrato of the mult-chael decovoluto models: (a) Fuctoal block dagram of the feed-forward model ad (b) Archtecture of feed-forward eural etwork (each syatc weght W s a FIR or stable IIR flter y (k ) H z W xm(k) (5) W x(k - ) x(k ) [ x (k ), x (k ),, x (k )]T acts as a dmesoal ut sgal vector whle x(k ) s a oututs (). I the ractcal alcatos, we have to mlemet the bld de-covoluto method wth a fte mulse resose (FIR) mult-chael flter wth a matrx trasfer fucto: where deotes the Eucldea orm. The chael mulse resose trasfer fucto H s a full rak o the ut crcle ( H ), that s, t has o zeros o the ut crcle. eve a low SNR. If the estmato of Tc ad f are avalable, we ca rocess the sgals the base bad ad the samlg erod ca be set to Tc. I the Secto V, we wll demostrate that s avalable. I the referece [3], the symbol durato Ts ca be estmated usg fluctuatos (3) where W(z) s the Z-trasformato of W. The Ztrasformato meas the tme doma sgal (.e., 4 Egeerg ad Techology Publshg (7) B. DS/CDA Data odel s Notatos ad Hyotheses Base o the model of Secto II, the goal of ths aer s to recover the PN sequece s k whe user s data symbol ad user s fadg coeffcet are both ukow, ad o trag samles are avalable ether. The followg assumtos wll be effect throughout the rest of the aer. We assume the arameters f, Tc ad Ts are kow advace. The more detaled estmato algorthms are rovded the lterature [3], [5], [6]. I the referece [5] ad [6], the method based o cyclo-statoarty aalyss ca estmate the ch erod Tc ad carrer frequecy f () where y(k ) [ y (k ), y (k ),, y (k )]T s a dmesoal vector of the oututs after the ovel searato algorthm, ad W s a m dmesoal coeffcet matrx at tme lag. W ca be called de-mxed matrx, too. W( z ) W x H 786

3 Joural of Commucatos Vol. 9, No., October 4 where G ad G are C K dmeso mxg matrces corresodg to the orgal ad the oe tme ut delayed symbols. The colum vectors of G ad G as follow: of correlato eve db (that s eough low to our method the aer). We ca adat the DS-CDA data model as follow. The receved CDA sgal r (t ) ca be wrtte the referece [7], [8]: K G aklm gl ', aklm g l ',, aklm g Kl ' l l l (3) G aklm gl '', aklm g l '',, aklm g Kl '' l l l We otce from () that our sgal model ca be regarded as a lear mxture of the delayed ad covolved sources, the artcular case whe the maxmum delay s oe tme ut. If we have kow ether the mxg matrces or symbol sequeces, the we are the framework of bld sources searato. The equato () s the geeral form of the BSS roblem. r(t ) aklmbkmsk (t mt d kl ) (t ) (8) m k l where symbols are set to K users ad receved va ath. aklm s the amltude (fadg) coeffcet of lth ath of the kth user durg the mth symbol durato, bkm s the kth user s mth data symbol. s k s the user s ch sequece: sk (t ),, t [, T ]; sk (t ), t [, T ] d kl s the delay of kth user's lth ath. The delays are dfferet for dfferet users, ad each delay s assumed to chage suffcetly slowly for most of the tme. (t ) deotes ose. The sgal s the samled by ch rate. After samlg the samles are collected to vectors of sutable dmeso. Ths deeds o the delay sread. We collect C-legth colum vectors from subsequet dscrete data samles vector rm : rm [r (mc) r (mc ) r (mc C )]T III. ADAPTIVE AGORITH We use the same basc sght ths aer, but roose a ew crtero to exlot t, whch leads to a smle ad coveet algorthm. The object fucto s defed to mmze the rsk fucto J : (9) where rm s the dscrete tme doma exresso of the r (t ) (8), ad C s the ch sequece legth. Equato (9) ca further be wrtte wth resect to the curret symbol vector ad the mmedately recedg oe: K K k l k l rm (t ) [bk, m aklm g kl ' ] [bk, m aklm g kl '' ] m () where m s a C-legth colum vectors ad t s the exresso of (t ) dscrete tme doma, ad deotes the ose vector. g kl s the early arts of the code vectors ad g kl '' s the late arts of the code vectors. g kl ' sk [C dkl ], sk [C dkl ],, sk [C ],, T g kl '',,, sk [], sk [],, sk [C dkl ], sk [C d kl ] T () where cotas d kl the vectors g kl ad g kl ''. I ths aer, we assume that the codes vectors g kl g kl are ukow. The aroach exlots drectly the bary form of the sources. The roosed algorthms yeld good results test examles wth CDA data. Due to the eural structure of the algorthms, they ca be used for trackg uroses whe the arameters are slowly chagg. Hece () could be wrtte as follows (4) J E ry y j (ry y ) j j where the varable v s the covergece coeffcets, ad varable δ s the ower factor, y ad yj are the chael s outut vectors. ry y j () s the correlato fucto of y ad yj. h() [ y () y j ()] ry y j () (5) h( k ) k where the oerator " " s the covoluto reresetato. h(k ) ad h() are the low-ass averagg flter vectors to comute a short-term estmate of the cross-correlato of outut chaels y ad yj at tme. There are a lot of umercal algorthms based o the above crtero for the o-statoary source searato roblem for the equato (4). We use a ew stochastc gradet algorthm here. For the otmzato of the demxg matrx D(), a stochastc gradet [9] udate takes the form D( ) D() () where () s the dfferetal fucto of J (), ad s the covergece coeffcet. J () D() J () q () d q () ( ) rm G bk, m Gbk, m m G bk, m m (), 4 Egeerg ad Techology Publshg (6) 787 ( ) [ ( ) ] r r y y y y j d q () j j jj (7)

4 Joural of Commucatos Vol. 9, No., October 4 where D( ) s the dscrete tme doma of the de-mxg matrx D ad ts dmeso ad q are the row ad colum dces gradet matrx that s desged by the umber of users, ad J ( ) s the th ste crtero fucto J (4). Note the use of the stataeous value at tme of the error fucto (5) gradet comutato. The ( q, )th elemet of the stataeous gradet matrx ca easly be show to be [] (, l) 4 r ( ) r (, l)... q y y y xq [ ry ( ) ] (, ) y ry x l q (8) The above matrx exresso for the stochastc gradet udate yelds a straghtforward comutato method oce the short-tme correlatos are avalable. We ow derve effcet recursve udates form for the averagg flter. For the comutatoal effcecy, we select a frst-order IIR averagg flter wth mulse resose where uk ( ) h ( ) a u( k) a (9) s the ut ste fucto, ad a s a costat a. Wth ths form, the elemets of ( a) ryy ca easly be udated recursvely accordg to r ( ) ar ( ) ( a) y ( ) y ( ) y y y y, r ( ) ar ( ) ( a) y ( ) x ( ) yx yx, () r ( ) ad r ( ) are the correlato fuctos yy betwee y yx y ad y x. Ths comletes the smle recursve algorthm for o statoary bld source searato as below. Ste : Comute outut accordg to (4). Ste : Udate short-tme correlatos usg (). Ste 3: Comute searato flter gradet usg (8). Ste 4: Udate searato matrx usg (6). Ste 5: Go back to ste. I evaluatg the algorthms, two arameter matrces, G ad G are chose (radomly) as the covoluted mxg matrces. Both sources are zero-mea ad have ut varaces. A Gaussa vector ose rocess s added to the observato wth ose covarace I. The Sgal-to-Nose Rato (SNR) s the defed as where SNR log ( db) () s a covarace factor, ad I s a ut matrx I () I ths secto, we evaluate the erformace of the adatve algorthm by some heurstc argumets ad smulato results. B. Results Aalyss As the CDA sgal exhbt cyclo-statoary, we utlze the sectral correlato method the referece [] to get the carrer frequecy f ad the ch erod T c. Fg. shows the measured sectral correlato fucto magtude for the -users CDA system. Form Fg., the frst eak value coordate aear at 4 the frequecy Hz, accordg the defto of cyclc-correlato, we ca get the carrer frequecy at 4 f Hz, the carrer frequecy f has bee estmated. The secod eak value aear at the both sdes of each ma eak, ad the coordates s 9 Hz (that s ot dstct Fg. ) ad Hz, so we ca obta the ch erod T c s. We ca rocess the aborgal sgals the base-bad ad set the samlg erod tot c. Usg the fluctuatos of correlato method, we obta the Ts.3 s (aberracy ear uo.3%). So the PN code legth ca be kow as C T T 3. s c IV. PERFOANCE EVAUATION AND SIUATION A. Smulato Parameter Setu The mmzg average squared cross-outut-chaelcorrelato algorthm s demostrated here va comuter smulato, for a DS/CDA sgal receved the resece of varous levels of whte Gaussa backgroud ose. I the exermet, we cosdered a -users CDA system. The source sgal was geerated by the comuter ad the sgal arameters were set as follow: carrer frequecy f 5, ch erod T /, symbol erod Ts 3/ (PN code legth C 3 ), users umber K, ath, SNR (the SNR s hgher tha assumto ad assumto ). c Fg.. Sectral correlato fucto magtude for the users CDA system. For the am of observato, we oly dslay 8 ots Fg.. 4 Egeerg ad Techology Publshg 788

5 Joural of Commucatos Vol. 9, No., October 4 Fg. 3 shows the -users CDA sgal sources wth AWGN chael (SNR=3). Wth the comarso of the results the Fg. 3 ad Fg. 5, the estmated data after searato s smlar to the sources sgal. I ths exermet, the all symbols were correctly estmated, whch shows the data are receved correctly. Accordg the assumto before, we have estmated the PN code legth that becomes very smle to seek the -users PN code. We adot tradtoal slage correlato method or matrx Ege aalyss techques to resolve the roblem. I the aer, we use the slage correlato method, ad get the -users PN code as Fg Samlg tme(ms) Fg. 3. -users CDA sgal sources wth AWGN chael Fg. 6. The -users PN codes Samlg tme(ms) Fg. 4. The -users data after covolute mxg Fg. 7. Adatato of the coeffcet the de-mxg matrx - D Fg BER (%) Samlg tme(ms) Samlg tme(ms) The outut data after searato 5 5 SNR Fg. 4 shows the -users data after covolute mxg. Two ukow GODEN sequeces are used as the sreadg codes. The outut data after searato betwee 7~8 s show Fg. 4. The utary outut bary estmated data after searato betwee 7~8 s show Fg Egeerg ad Techology Publshg Fg. 8. BER for covoluted mxture CDA sgals searato The self-adatve coeffcets the de-mxg matrx D are show Fg. 7 whe the.9. The exermet result of the bt-error-rate (BER) to SNR s show Fg

6 Joural of Commucatos Vol. 9, No., October 4 C. The Ifluece of J,, ad Secod, the least value of affects the covergece lmt of rsk fucto. The value of ower factor s coheret wth the rsk fucto covergece lmt. We test samles for 3 tmes uder the same codto, ad get the rsk fucto J over tme, as Fg.. The shae ad eergy factor's value dcate the fluece whch eergy factor has o the algorthm. Covergece factor v also exst the smlar fluece lke eergy factor as Fg.. I the algorthm, we choose the low ass flter (9). The am s to abbrevate the algorthm comlexty ad average the de-mxg outut data. Uder the same codto, we get the rsk fucto J over, as Fg.. The cocluso has bee receved that the value of s ager, the covergece rate s slower but stablzato ad s smaller; the covergece rate s faster but ustable. The rsk fucto J fluctuato wth the samle seral s show the Fg. 9. The recurso oerato after about 3 ot, the rsk fucto J s coverget. The recurso oerato after about 6 ot, the rsk fucto J s ear to. The covergece rate s accordace wth the adatato of the coeffcet the de-mxg matrx D as Fg. 7. Fg. 9. Fluctuato of rsk fucto J Fg.. Varety flter arameter fluece o the J V. CONCUSIONS Fg.. Varty ower factor fluece o the J Fg.. Varty ower factor J v fluece o the A bld source searato algorthm for the DS-CDA sgals s develoed ad demostrated ths aer. By mmzg the average squared cross-outut-chaelcorrelato, a crtero fucto J s roosed to estmate the sreadg code. Ths techology ot oly ca be used mltary for CDA sgal terceto, also ca be used a commercal at-jammg for CDA sgal. The exermet results show that the roosed algorthm s a romsg alteratve to the exstg BSS algorthms for DS/CDA sgals. Ths algorthm s also alcable to sgals wth log code, such as commercal commucato sgals, whch allow t to be mlemeted wth a reasoable comutatoal comlexty ad over reasoably short receto tervals. Ths method s also a artcular alcable estmato dowlk sgal rocessg, because the codes of the terferg users are ukow. It s caable to searate the sgals ad remove the mult-ath chael effect. We also fd the macts o de-mxg matrces about covergece factor v ad the ower factor. The ower factor exst dual effects o the de-mxg matrces. Frst, whe the value of s lager, the covergece rate s faster but u-stablzato []. From a cotrastg ot of vew, the covergece rate s slower but stable. 4 Egeerg ad Techology Publshg REFERENCES [] T. C. Yag ad W. B. Yag, ow sgal-to-ose-rato uderwater acoustc commucatos usg drect-sequece 79

7 Joural of Commucatos Vol. 9, No., October 4 sread-sectrum sgals, Proc. OCEANS 7 Euroe, Aberdee, 7,. -6. [] J.. Yag ad G. K. Hog, Adatve multchael lear redcto based dereverberato tme-varyg room evromets, Proc. th Sgal Processg Coferece, Euroea orocco, 3,. 5. [3] G. Burel, Detecto of Sread sectrum trasmssos usg fluctuatos of correlato estmators, Proc. ISPACS,. [4] Z. Koldovsky ad P. Tchavsky, Tme-doma bld searato of audo sources o the bass of a comlete ca decomosto of a observato sace, IEEE Tras. o Audo, Seech, ad aguage Processg, vol. 9, , Feb.. [5]. K. u,. Hog, ad J. Zhag, A teratve carrer frequecy estmate algorthm based o hgh-order cyclc cumulats, Proc. IEEE CIE Iteratoal Coferece o Radar, Chegdu,, [6] K. Waheed ad F.. Salem, Bld formato-theoretc multuser detecto algorthms for DS-CDA ad WCDA dowlk systems, IEEE Tras. o Neural Networks, , Jue 5. [7]. H. m ad P. Como, Bld multlear detfcato, IEEE Tras. o Iformato Theory, vol. 6,. 6 8, Feb. 4. [8] S. Saberal ad H. Amdavar, Nolear detector desg for cdma sgals the resece of ukow terferers usg maxmum etroy method ad comarso wth SIC, IEEE Commucatos etters,. -4, Ja. 4. [9] F. Dg, G. G. u, ad X. P. u, Partally couled stochastc gradet detfcato methods for o-uformly samled systems, IEEE Tras. o Automatc Cotrol, vol. 55, , Aug.. [] S. C. Douglas ad. Guta, Scaled atural gradet algorthms for stataeous ad covolutve bld source searato, Proc. Acoustcs, Seech ad Sgal Processg 7, Arl 7, [] A. Abd, J. A. Barger, ad. Kaveh, A arametrc model for the dstrbuto of the agle of arrval ad the assocated correlato fucto ad ower sectrum at the moble stato, IEEE Tras. o Vehcular Techology, vol. 5, , ay. [] C.. Che, R. Bhata, ad R. K. Sha, ultdmesoal declusterg schemes usg Golde Rato ad Kroecker Sequeces, IEEE Tras. o Kowledge ad Data Egeerg, vol. 5, , Jue 3. Hao Cheg was bor Jagsu, Cha 976. He receved the B.S. degree aled strumet techology ad.s. degree hyscal electrocs 999 ad, resectvely, both from South East Uversty of Cha, Najg, Cha, ad the Ph.D. degree commucato egeerg from Uversty of Electroc Scece ad Techology of Cha (UESTC), Chegdu, Cha 7. He s curretly workg the telecommucato egeerg College of Chegdu Uversty, Schua, Cha. Hs curret research terests clude wreless sgal rocessg, sectral estmato, ad array theory. Na Yu was bor Schua Provce, Cha, 98. He receved the B.S. ad.s. degree electroc formato scece ad techology from the Southwest Jaotog Uversty of Cha, Chegdu. She s curretly workg the Telecommucato Egeerg College of Chegdu Uversty, Chegdu, Cha. Her research terests clude research terests clude formato securty, comuter etworks, ad moble alcato develomet. Ju u was bor Schua, Cha 975. He receved the B.S.,.S. ad Ph.D. degree electroc formato scece ad techology from the Uversty of Electroc Scece ad Techology of Cha (UESTC), Chegdu 997, ad 6 resectvely. Hs research terests clude Hs research terests clude formato securty, mage rocessg, ad comutg etwork. 4 Egeerg ad Techology Publshg 79