in "Systems and Contol: heoy and Applications", published by WSES, 2, pp. 257-262, ISBN 96-852-4 Spead Spectum Codes Identification by Neual Netwoks Céline BOUDER, Gilles BUREL Laboatoie d Electonique et Systèmes de élécommunications (UMR CNRS 666) Univesité de Betagne Occidentale 6 avenue Le Gogeu, BP 89 29285 BRES cedex, FRANCE Celine.Boude@univ-best.f, Gilles.Buel@univ-best.f Abstact: - In the context of spectum suveillance, a method to ecove the code of diect sequence spead spectum signal is pesented, wheeas the eceive has no knowledge of the tansmitte s speading sequence. he appoach is based on an atificial neual netwok which is foced to model the eceived signal. Expeimental esults show that the method povides a good estimation, even when the signal powe is below the noise powe. Keywods: - Spead Spectum Communications, Spectum Suveillance, Identification, Pseudo-Random sequences, Atificial Neual Netwoks. Intoduction Although spead spectum communications wee initially developed fo militay applications, they ae now widely used fo commecial ones, especially fo code division multiple access (CDMA), o global positioning systems (GPS) []. hey ae mainly used to tansmit at low powe without intefeence due to jamming, to othes uses o to multipath popagation. he spead spectum techniques ae useful fo secue tansmissions, because the eceive has to know the sequence used by the tansmitte to ecove the tansmitted data, using a coelato [2, 3, 4]. Ou pupose is to automatically detemine the speading sequence, wheeas the eceive has no knowledge of the tansmitte s pseudo-noise (PN) code. In the next section, we pesent the technique of diect sequence spead spectum (DS-SS) and we explain the difficulty to ecove the data in an unfiendly context. hen, we intoduce ou method, which uses atificial neual netwoks to solve the poblem. Finally, section 4 shows expeimental esults in vaious configuations. 2 DS-SS technique In ode to spead the signal powe ove a boadband channel, fa in excess of the minimum bandwidth necessay to tansmit the data, the diect sequence spead spectum (DS-SS) technique consists in multiplying the infomation signal with a peiodic pseudo-noise sequence. 2. A simple model Let us note b(t) the infomation beaing signal bt () = bpt n ( nb) () whee b n =± with equal pobability and pt () is a ectangula pulse of duation b. Let us note y, the PN sequence of length P : y= y, y, L, y P (2) he tansmitted signal $y n is the poduct of both wavefoms. Let us conside a diect sequence spead spectum system without noise : $yn = bny (3) We assume the eceive knows this sequence and can despead the signal using a coelato : y$ n, y = byy n, = bn yy, = bp n (4) accoding to the popeties of PN sequences [5], the data infomation is then ecoveed. Howeve it becomes moe challenging when the eceive does not know exactly the code used by the tansmitte. Let us note ~ y a sequence simila to y, but not exactly the same. hen using a coelato with ~ y, we get : y$, ~ y = b y, ~ y = b y, ~ y (5) n n n
in "Systems and Contol: heoy and Applications", published by WSES, 2, pp. 257-262, ISBN 96-852-4 accoding to the popeties of PN sequences, yy, ~ is low [5] and then we do not ecove the data infomation. 2.2 A ealistic model ypically diect sequence spead spectum systems use binay o quadatue phase shift keying (BPSK o QPSK) data modulation. Usually the PN sequence is a binay maximal length sequence o a Gold sequence [4]. Sometimes complex signatue sequences ae used. It has been shown [6], that using complex codes povides an impovement of 3 db (in compaison with binay Gold sequences) against uses intefeence. Hee we conside a PSK data modulation, spead by a complex signatue sequence. he baseband eceive signal at the output of the eceiving filte can be witten as : st () = aht k ( ks) + nt () (6) k= whee ht () is the combined impulse esponse of the channel and the speading code : P ht () = c pt ( m) m= m c (7) and pt () = ( e g c)() t (8) P is the length of the speading sequence. { cm, m= L P } is the speading sequence. a k is the symbol numbe k. c is the chip peiod. s is the symbol peiod ( s = Pc). ct ()is the channel filte (that modelises the channel echoes). et () is the tansmitting filte. gt () is the eceiving filte. nt ()is the noise at the output of the eceiving filte. he baseband channel noise is assumed to be white, gaussian and centeed. An inteesting method to estimate ht ()is poposed in [7]. It takes pofit of blind identification techniques available fo multiple FIR channels. Good esults wee obtained. he method implicitly assume that each symbol a k has been pecisely located in time. his is a stong equiement, since no method is known to pefom time localization of the symbols without knowing the sequence. In this pape, we popose an appoach that does not equie knowledge of symbols times. It only needs pevious estimate of the symbol peiod. he method is based on atificial neual netwoks techniques. 3 Estimation of the speading sequence o ecove data infomation, we have to estimate ht (), without knowing the tansmitte s PN sequence. In this section we explain ou method, which is based on atificial neual netwoks. 3. heoetical analysis he tansmitted signal is the same as peviously defined. he symbol peiod s is assumed to be known, it can be estimated using cyclostationaity analysis [7]. he eceived signal is sampled, and we will note e the sampling peiod. We assume that e is such that s = Me whee M is an intege. Let us note st ()the M-dimensional vecto below : st () = st (),( st+ ), L,( st+ ) (9) [ e s e ] whee ht () and nt () ae defined in the same way. Fom the signal samples, we can ceate a matix S with M ows and N columns, whee N is the numbe of tempoal windows of duation s in the signal used fo estimation : M M L M S = s() t s( t+ s) L s( t + ( N ) s) () M M L M Let us note t = ms + t, whee t < s Fom equation (6) we can wite : st () = aht k ( + ( m k ) s) + nt () () k = st () = am kht ( + ks) + nt () (2) k = Let us note h ( k t ) the vecto below : hk( t) = [ h( t + ks), L, h( t + ( k + ) s e)] Hence we can wite : st () = am khk( t ) + nt () (3) k Since the time extension of ht ()is limited, the sum has been limited to values of k fo which h ( k t ) is not null. In the sequel, we assume fo claity, that
in "Systems and Contol: heoy and Applications", published by WSES, 2, pp. 257-262, ISBN 96-852-4 ht, s. Hence, st () can be witten as follow : st () = ah m ( t) + am+ h ( t ) + nt () (4) whee h( t) is the M-dimensional vecto containing the end of the speading wavefom (fo a duation s t ) followed by zeos (fo duation t ). h( t) = [ ht ( ), ht ( + e), L, h ( s e),, L, ]. h ( t) is the M-dimensional vecto containing zeos (fo a duation s t ) followed by the beginning of the speading wavefom (fo duation y. h ( t) = [, L,, h( ), h( e), L, h( t e)]. Hence, we can wite the matix S as follow : S h a =. + h. a + n( t) (5) whee am = [ am, am+, L, am+ N ] h( t) and h ( t) ae othogonal, and the noise is uncoelated with the signal. Hence the subspace spanned by h( t) and h ( t) can be identified by a thee layes neual netwok, whose hidden laye includes two neuons [8, 9, ]. In fact we estimate ht ()thanks to the second laye of weights. () fo t outside the inteval [ ] 3.2 Desciption of the atificial neual netwok We ceate a feedfowad netwok with thee layes : a laye of the inputs, a hidden laye of two sigmoid neuons with hypebolic tangent nonlineaities and an output laye of linea neuons. As the tansmitted signal is complex, a neual netwok algoithm has been genealized to neual netwok with complex weights [9]. he netwok s inputs ae the columns of the matix S, and the desied outputs ae the same data as the inputs. he weights ae adjusted accoding to a backpopagation algoithm [], which minimizes the mean squae eo between the netwok outputs and the desied ones. Contay to classical use of neual netwoks, the useful infomation is not the outputs of the netwok, but the weights. In fact we ecove the speading sequence in the second laye of weights. hat is the eason why thee is not pevious tain, but a taining at each expeiment. Hence we impose a condition to the two vectos coesponding to the second laye of weights. he constaint does not allow the vectos to have enegy in the same time at the same place. In this way, adding the two vectos gives us the speading sequence used by the tansmitte. 3.3 Evaluation of the esults As the weights ae complex, we ecove the speading code with a phase indetemination. It is not a poblem, because in any tansmission system symbols phase is always indeteminate on the eceive side. Anyway, in ou application, it can be useful to nomalize the phase fo esults intepetation. he phase is calculated accoding to the expected sequence, to visualize the esults, as stated below : let us note V the speading code found with the neual netwok, we visualize V $ such as $ V V = Re{ Vz} *. H, whee z = 2, with H the tue V sequence. 4 Illustative esults In many spead spectum tansmission systems, the speading code is eal when the channel effects ae omitted, then we intoduce seveal esults with eal sequences, teated with neual netwoks, the weights of which ae fist eal and then complex. hen we study a tansmission system, whee the code and the netwok s weights ae complex. o complete ou wok, we povide some esults accoding to the signal to noise atio (SNR) to the numbe of tempoal windows N of duation s and accoding to the length of the speading sequence. 4. Real sequence 4.. With eal weights he studied PN sequence is a binay Gold code of length P = 3, and we conside a BPSK data modulation. he channel adds white, gaussian, centeed, and eal noise. he SNR is -5 db (the signal powe is less than the noise powe). Fig. shows the code used by the tansmitte.
in "Systems and Contol: heoy and Applications", published by WSES, 2, pp. 257-262, ISBN 96-852-4 2.5.5 -.5.5-2 5 5 2 25 3 Fig. : ansmitte s PN code he fist weight vecto is shown on Fig. 2 he PN sequence is still a Gold code. It is the same as peviously, and we conside now a QPSK modulation, damaged with a white, gaussian, centeed and complex noise. he SNR is db. o visualize the esults, we used the technique of phase nomalization. Fig. 4 shows the estimated sequence (sum of the two weight vectos)..5.5 -.5.5 -.5.5 5 5 2 25 3 35 Fig. 4 : sum of two weight vectos.5 5 5 2 25 3 35.5.5 Fig. 2 : Fist weight vecto Fig. 3 shows the second weight vecto. In compaison with Fig., we ecove exactly the speading sequence, with a shift of ten positions left, because the eceived signal is not synchonized. 4.2 complex sequence Let us now conside a complex sequence, the eal and the complex pats of which ae a Gold sequence. he infomation beaing signal is still a QPSK modulation, and the SNR is db. In this case we have to ecove the eal and imaginay pats of the sequence. -.5 Fig. 5 epesents the eal pat of the code..5.5 5 5 2 25 3 35 Fig. 3 : second weight vecto he fist weight vecto coesponds exactly to the end of the speading sequence, wheeas the end of the second weight vecto coesponds to the opposite of the beginning of the code. Moeove we can obseve that the constaint imposed to the vectos is well espected. hee is only a poblem of sign between the vectos, to ecove the speading code, we have to add the fist one, with the opposite of the second one. It is a poblem of phase indetemination..5 -.5.5 5 5 2 25 3 35 Fig. 5 : Real pat of the code 4..2 With complex weights and Fig. 6, the imaginay pat
in "Systems and Contol: heoy and Applications", published by WSES, 2, pp. 257-262, ISBN 96-852-4.5.5 -.5.5 5 5 2 25 3 35 Fig. 6 : Imaginay pat of the code Fig. 7 and 8 epesent the esults of the neual netwok estimation (espectively eal and imaginay pats of the weights). 2.5.5 -.5.5 5 5 2 25 3 35 2.5.5 -.5 Fig. 7 : Real pat of the weights.5 5 5 2 25 3 35 Fig. 8 : Imaginay pat of the weights If we compae Fig. 5 and Fig. 7, the signs of eal pat of the weights coespond exactly with the eal pat of the speading sequence. Futhemoe the signs of the imaginay pat of the weights coespond exactly to the imaginay pat of the code (Fig. 6 and Fig. 8). 4.3 Pefomances of the method Hee we intoduce some tables summaizing the pefomances of ou method. 4.3. Influence of the numbe of windows in the studied signal We study the influence of the numbe of tempoal windows N included in the signal used to estimate the speading sequence. Fo this expeiment, the sequence length is P = 3, the modulation is a BPSK and the SNR is equal to 2 db. N 5 5 2 nb_eos 5 2 able :Influence of the numbe of windows N nb_eos is the numbe of sign eos in the ecoveed sequence. When N inceases, the esults ae impoved. 4.3.2 Influence of the sequence length and the SNR he modulation signal is a QPSK filteed at the tansmitte and the eceive sides, the speading sequence is a complex code, and we study the numbe of eos with espect to the SNR fo seveal sequence lengths. Fo ou expeiment, we conside sequences of length P = 3, 63, 27, the eal and imaginay pats of which ae diffeent Gold sequences, and we have N = 2. We assume, fo simplicity, that s = Pc. So the signal to noise atio on the coelato output can be expessed as a function of the signal to noise atio on the coelato input : SNRout = P SNRin If we expess the signal to noise atio in db : SNRout = log ( P) + SNRin Hence the eo pobability pe symbol Pe can be witten as : SNR Pe = efc 2 2 out SNR 2 efc 2 2 out t with efc( x)= 2 e 2 dt x π his shows that the pefomance of a tansmitted spead spectum signal is bette with long sequences than with shot ones.
in "Systems and Contol: heoy and Applications", published by WSES, 2, pp. 257-262, ISBN 96-852-4 Hee ae ou esults fo diffeent sequence lengths and SNR in (db) P = 3 P = 63 P = 27 SNR -2-3 -4-5 -6-7 -9 eos 4 2 2 able 2 : Influence of sequence length eos ae the numbe of signs eos in the sequence estimated by the neual netwok. We obseve, that the esults ae impoved, when the sequence is longe. We get a gain of 3 db when we use a sequence length equals to 63 athe than 3, o a sequence length equals to 27 athe than 63, which coesponds about to : (log ( 63) log ( 3) ) and (log ( 27) log ( 63) ). he esults follow appoximately the same law as the eo pobability pe symbol. 5 Conclusion In the context of spectum suveillance, a method fo identification of a spead spectum tansmitte PN sequence has been poposed. Expeimental esults have been povided and show good estimation esults. Futhe wok will include emoval of sign o phase indecision. [6] Alex W. Lam, Faith M. Özlütük, "Pefomance Bounds fo DS/SSMA Communications with Complex Signatue Sequences", IEEE ansactions on communications, Vol. 4, No., Octobe 992, pp. 67-64. [7] Michail K. satsanis, Geogios B. Giannakis, "Blind Estimation of Diect Sequence Spead Spectum Signals in Multipath", IEEE ansactions on signal pocessing, Vol. 45, No. 5, May 997, pp. 24-252. [8] G. Buel, "Réseaux de Neuones en aitement d Images : des Modèles héoiques aux Apllications Industielles", hèse de Doctoat de l Univesité de Betagne Occidentale, Best, 99. [9] N. Rondel, "Réseaux de Neuones pou le aitement d Antenne et pou la Commande Réféencée Capteu", hèse de Doctoat de l Univesité de Betagne Occidentale, Best, 996. [] D. E. Rumelhat, G. E. Hinton, R. J. Williams, "Leaning intenal epesentations by eo backpopagation", Paallel Distibuted Pocessing, D. E. Rumelhat and J. L. Mc Clelland, chapte 8, Badfod book, MI Pess, 986. Refeences: [] D. homas Magill, Fancis D. Natali, Gwyn P. Edwads, "Spead-Spectum echnology fo Commecial Applications", Poceedings of the IEEE, Vol. 82, No. 4, Apil 994, pp. 572-584. [2] Raymond. L. Picholtz, Donald L. Schilling, Lauence B. Milstein, "heoy of Spead- Spectum Communications - A utoial", IEEE ansactions on Communications, Vol. COM- 3, No. 5, May 982, pp. 855-884. [3] Chales E. Cook, Howad S. Mash, "An intoduction to spead spectum", IEEE communications magazine, Mach 983, pp. 8-6. [4] John G. Poakis, Digital communications, hid Edition, Mac Gaw Hill Intenational Editions, 995. [5] Dilip V. Sawate, Michael B. Pusley, "Cosscoelation Popeties of Pseudoandom and Related Sequences", Poceedings of the IEEE, Vol. 68, No. 5, May 98, pp. 593-69.