FFH nd MCFH Spred-Spectrum Wireess Sensor Network Systems Bsed on the Generized Approch to Sign Processing JONGHO KIM, JAE HYUN KIM, VYACHESAV UZUKOV, WON SIK YOON, YONG DEAK KIM Digitsis, Inc., 344-, Ytp, Bundng, Seongnm 463-954 KOREA, REPUBIC OF Deprtment of Eectric nd Computer Engineering Coege of Informtion echnoogy, Ajou University Sn 5, Wonchon-dong, Pd-gu, Suwon 44-749 KOREA, REPUBIC OF Abstrct: - In this pper, the performnce of frequency-hopping spred-spectrum wireess sensor network systems bsed on the generized pproch to sign processing in the presence of noise [ 5], which empoy noncoherent reception nd trnsmission diversity, is nyzed for frequency-seective Ryeigh fding chnnes. wo different types of trnsmission diversity systems, fst frequency-hopping (FFH) [6,7] nd muticrrier frequency-hopping (MCFH)[8] wireess sensor network systems, re investigted. In order to combine received signs from trnsmit diversity chnnes, the diversity combining rue bsed on the generized pproch to sign processing is deveoped. Probbiity of error equtions re derived nd utiized to evute the performnce of two kinds of wireess sensor network systems. he effect of frequency-seective fding is so investigted in determining optimum frequency devitions of binry frequency-shift keying (BFSK) signs. he systems considered in this pper re frequency-hopping spred-spectrum (FHSS) ones with BFSK modution, noncoherent detection, nd definite diversity order. Diversity order refers to the number of hops per symbo for FFH nd the number of subbnds for MCFH wireess sensor network systems. Ech trnsmit diversity chnne is modeed s frequency-seective Ryeigh fding process nd is ssumed to be independenty fded. he mximum dey spred of ech diversity reception is ssumed smer thn one hop durtion for FFH wireess sensor network systems, which is smer thn the symbo durtion. It is so ssumed tht one symbo is trnsmitted during one hop durtion in MCFH wireess sensor network systems, nd djcent symbos in time re trnsmitted in fr distnt frequency sot such tht mutipth interference from the previous symbo is negigibe. Key-Words: - Wireess sensor network, frequency-hopping spred-spectrum, frequency-shift keying signs, generized detector, probbiity distribution function, BER performnce. Introduction Demnds for high dt rte services in wireess sensor networks hve been incresing over pst few decdes. By this reson, there is sense to use frequency-hopping spred-spectrum (FHSS) technique in wireess sensor network systems. In high dt rte wireess sensor network systems, the effects of frequency-seective fding shoud be considered due to n increse in the rtio of dey spred to symbo durtion. he effects of frequency-seective fding on FHSS system empoying orthogon BFSK signs re discussed in [9,] under the ssumption tht the frequency seprtion between two orthogon BFSK signs is rge enough for the corretion between two detector outputs to be negigibe. In prctice, it is dvntgeous to use the minimum frequency seprtion in mutipe-ccess environments to increse the number of frequency sots for given tot bndwidth []. When the minimum frequency seprtion is empoyed, the corretion between two detector outputs becuse of frequency-seective fding nd fst fding my be significnt. he use of trnsmission diversity ensures protection ginst jmming, mutipe-ccess interference, nd fding. For FHSS wireess sensor network systems, the diversity my be reized in the form of fst frequency-hopping (FHH) nd muticrrier trnsmission. FFH is convention diversity technique in FHSS wireess sensor network systems. Muticrrier
trnsmission is n terntive diversity technique in FHSS wireess sensor network systems. In FHSS wireess sensor network system, diversity is obtined by chnging trnsmit frequency more thn once over one symbo durtion. he trnsmit frequency is seected from the entire trnsmit frequency bnd. In MCFH wireess sensor network system, tot frequency bnd is prtitioned into sever disjoint subbnds on which repics of the sme sign re trnsmitted simutneousy. Ech repic hops independenty in its subbnd. In FHSS wireess sensor network systems, coherent demodution for BFSK signs is retivey difficut. For these systems, the frequency-shift keying (FSK) modution with noncoherent demodution is typicy empoyed. In the present pper, BFSK modution nd noncoherent demodution re ssumed to be empoyed for both FFH nd MCFH wireess sensor network systems. Digrms of FFH nd MCFH systems re presented in more deti in []. he use of FFH technique in high dt rte wireess sensor network system my not be fesibe due to its high speed requirements. For wireess sensor network systems empoying trnsmission diversity, diversity receptions shoud be combined in some wy in the receiver. he optimum combining schemes bsed on the mximum-ikeihood criterion hve been deveoped ony for sttic nd frequency-nonseective sowy vrying chnnes [6,7,3]. For sttic chnnes with prti-bnd interference, the optimum combining is the sum of the ogrithms of zero-order modified Besse functions [7]. For sow nd frequency-nonseective Ryeigh fding chnnes, the optimum combining rue, given tht of the diversity receptions hve the sme power spectr density (PSD) of bckground noise, is squre-w equ-gin combining [3]. he optimum combining rue for frequency-seective fst vrying nd frequency -nonseective sowy vrying Ryeigh fding chnnes with the bckground noise PSD of ech diversity reception not being equ is discussed in []. In this pper, we consider the optimum combining rue for frequency-seective fst vrying nd frequency-nonseective sowy vrying Ryeigh fding chnnes bsed on the generized pproch to sign processsing in the presence of noise. Chnne Mode Consider FHSS wireess sensor network system with BFSK modution, noncoherent detection, nd diversity order. Diversity order refers to the number of hops per symbo for FFH nd the number of subbnds for MCFH wireess sensor network systems. Ech trnsmit diversity chnne is modeed s frequency-seective Ryeigh fding process nd is ssumed to be independenty fded. he mximum dey spred of ech diversity reception is ssumed smer thn one hop durtion for FFH wireess sensor network system, which is smer thn the symbo durtion. We so ssume tht one symbo is trnsmitted during one hop durtion in MCFH wireess sensor network system, nd djcent symbos in time re trnsmitted in fr distnt frequency sots such tht mutipth interference from the previous symbo is negigibe. he bsebnd equivent of the sign trnsmitted from sensor node to sink for FFH nd MCFH wireess sensor network systems cn be represented in the foowing form ( t) ( t) k p t E cos[(, k + bk d ) t + ϕ, k ] ( k ) ; FFH system () k p t E cos[(, + b ) t + ϕ, ] ( k ), MCFH system () where E is the trnsmit energy of ech diversity trnsmission, is the symbo durtion, nd is the hop durtion;, k nd ϕ, k re, respectivey, the hop frequency nd rndom phse for the -th diversity trnsmission of the k-th symbo; b k {, + } is the k-th dt symbo, nd p for t [, ] nd zero, otherwise. he frequency devition of BFSK sign is denoted by d, where is the normi- π zed frequency devition nd is the frequency seprtion between two BFSK signs. When the tot tot trnsmit energy of (t) is E, the vue of E in tot () is E nd tht of E in () is E tot. Simiry, the vue of in () is nd tht of in () is. Correspondingy, the vues of d nd woud be different for FFH nd MCFH wireess sensor network systems. he chnne mode is considered s wide-sense sttionry uncorreted scttering mode discussed in [4,5]. he ow-pss equivent impuse response of the -th diversity chnne my be represented s c τ ) A τ ) cos[ ς τ )],,,...,, (3) k k d k
where A τ ) s re independent nd identicy distributed Ryeigh rndom processes nd ς τ ) s re independent nd identicy distributed uniform rndom processes within the imits of the interv [,π ]. he utocorretion function of the wide-sense sttionry uncorreted scttering chnne is given s [4] R c ( t; τ, τ ).5M[ c τ ) c( t + t; τ )] R c ( t; τ ) δ ( τ τ ), (4) where M [K] is the men nd denotes compex conjugte opertion. Since the chnne response for ech diversity trnsmission is ssumed independent nd identicy distributed, the utocorretion of ech chnne is the sme for, by this reson the subscript is dropped in (4). If we et t in R c ( t; τ ), the resuting utocorretion function R c ( ; τ ) is mutipth intensity profie, nd denoted s I c (τ ). Assuming tht the mutipth intensity profie is time invrint, the utocorretion function R c ( t; τ ) my be represented in the foowing form ( t; τ ) I c ( τ ) ψ c ( t), (5) where ψ ( t) is the utocorretion function in the c t vribe normized by I (τ ) for τ [4]. c ow-pss equivent dditive Gussin noise process with PSD N. We ssume tht dt symbo b is either + or with equ probbiity. Without oss of generity, it is ssumed tht dt symbo b is + herefter. Ech diversity reception is demoduted by noncoherent generized detector [3]. 3 Performnce Anysis FFH nd MCFH system bock digrms were presented in [Figs. nd, ]. After down converting nd dehopping, the bsebnd equivent of the received sign over the first symbo durtion my be presented s x( t) mx E p A τ ) cos[ b t + θ τ )] dτ ( t ) + n ( t), t [, ] for FFH wireess sensor network system nd x( t) mx E A τ ) cos[ b t + θ τ )] dτ + n ( t), t [, ] (7) for MCFH wireess sensor network system. Here θ t ; τ ) ς τ ) + ϕ, (8) (, nd mx is the mximum dey spred of ech diversity chnne. he noise n (t) is represented s d d (6) Figure. he noncoherent generized detector. he noncoherent generized detector consists of two brnches foowed by n enveope detector, s shown in Fig.. We ssume tht the generized receiver is time synchronous to the first rriving sign, i.e., τ. Rec the min functioning principes of the generized detector [3].he received sign must pss through preiminry fiter (PF). he effective frequency bndwidth of the PF coincides by vue with tht of the trnsmitted sign. hus, the sign with dt symbo b + psses through the PF + ony, nd the sign with dt symbo b psses through the PF ony. he ddition fiter (AF) is formed in pre wy to the PF + nd PF with the purpose to form reference smpe with priori informtion no sign for genertion of jointy sufficient sttistics of the men nd vrince of the ikeihood function. For simpicity of nysis, we ssume tht the mpitude-frequency response of the AF is nogous to
the mpitude-frequency responses of the PF + nd PF over the whoe rnge of prmeters, but it is detuned in the resonnt frequency retive to the PF + nd PF for the purpose of providing uncorreted processes t the outputs of the PF + or PF nd AF. he detuning vue must be more thn the effective frequency bndwidth of the trnsmitted signs with dt symbo b + or b so tht the processes t the outputs of the PF+ or PF nd AF wi be uncorreted. In sttic environments, when the dt symbo b +is trnsmitted, the processes t outputs of the PF + nd PF cn be thought s uncorreted. However, in fding environments, this is not true, since mutipth sign components nd sign vrition over one hop durtion my destruct orthogonity of BFSK signs. By this reson, in gener cse, we my think tht the processes t outputs of the PF + nd PF re correted. king into ccount the min functioning principes of the generized detector discussed in more deti in [3], the two generized detector chnne outputs of the -th diversity reception re denoted, respectivey, by Z, nd Z,, nd my be expressed in the foowing compex form mx E jθ τ ) Z, A τ ) e dtdτ Z τ + mx αe, Ac τ ) τ α [ η ( t) ξ ( t)] dt ; (9) E mx j[ dt+ θ τ )] A τ ) e τ e j[ t+ θ τ )] d dtdτ ζ ( t) dtdτ + [ η ( t) ζ ( t)] dt, () where α. 5 is the coefficient chrcterizing the prt of energy of the trnsmitted sign with the symbo b + t the output of the PF owing to destruction of BFSK signs orthogonity; η (t) is the zero-men Gussin noise with the vrince σ n forming t the output of the AF; ξ (t) is the zeromen Gussin noise with the vrince σ n forming t the output of the PF + ; ζ (t) is the zero-men Gussin noise with the vrince σ n forming t the output of the PF ; the reference (mode) sign j dt E e is formed t the output of MSG + nd j dt the reference (mode) sign E e is formed t the output of MSG, but for the considered cse b + the st reference sign is equ to zero due to the min functioning principes of the noncoherent generized detector [,3]. he effect of destruction of BFSK signs orthogonity is represented s the first nd second terms of (), which wi be referred to, herefter, s interference component. he second term in (9) nd the third term in () represent the bckground noise forming t the output of the generized detector. he first term in (9) nd the first nd second terms in () re zero-men compex Gussin rndom vribes. he st term in (9) nd () is zero-men but not Gussin rndom vribe. he distribution w of the st term in (9) nd () (the bckground noise of the generized detector) is discussed in more deti in [3,4]. he vrinces of Z, nd Z re given by E mx α E α Eσ n + σ τ σ mx,.5m[ Z, ( ) ], t + τ σ n τ ) dtdτ + ; (),.5M[ Z, τ mx τ t + τ τ ) cos( t)( dtdτ t + τ τ ) cos( t)( dtdτ σ 4 n +. () he corretion coefficient is given by.5m[ Z, Z, ] ρ σ σ,, mx 4 αe j t σ n [ ( ) ] t t e dt dt dτ + τ τ σ,σ, ] ) 4 ). (3) he decision-mking rues re mde bsed on pirs of noncoherent generized detector outputs, R, Z, nd R, Z, for {,,..., }. hey shoud be combined in some wy to form decision sttistics for the noncoherent generized detector. o find the optimum diversity combining rue
bsed on the mximum-ikeihood criterion, we shoud find the condition joint probbiity density function (pdf) of noncoherent generized detector outputs, R, nd R, for {,,..., } conditioned on trnsmitted dt symbo. his pdf is referred to s ikeihood function. Becuse ech diversity reception is ssumed independent of ech other, the ikeihood function for dt symbo b + cn be expressed s F r, r,..., r, r, r,..., r b ) R (,,,,,, + FR ( r,, r, b + ) F R ( r,, r, b + where ), (4) is the condition joint pdf of the non-coherent generized detector outputs for the -th diversity reception. Using procedure discussed in [], finy we cn obtin the condition pdf of, F R ( r, (, r, b R nd R, r,r, + ) σ σ ( ρ ),, σ, r, + σ, r, ρ r,r, σ, σ, ( ρ ) ) e,σ, ( ρ ) K σ, (5) where K ( ) is the zero-order modified Besse function of the second kind. Simiry, the ikeihood function for dt symbo b wi be obtined from (4) nd (5), by exchnging r,, r,, σ,, nd σ, in (5), respectivey. After strightforwrd gebric mnipution nd extrction of common terms in the og-ikeihood functions, the optimum decision rue is derived s ˆ σ + b < >, σ, ( R, R, ). σ,σ, ( ρ ) bˆ (6) Reference to (6) shows tht the decision vribe ssocited with b ˆ + is constructed s the weighted sum of squres of R, for ; the decision vribe ssocited with b ˆ is constructed in simir mnner. hese vribes re compred to estimte trnsmit symbo. Note tht the combining rue is differed from the combining rue in [7], which is deveoped for sttic chnnes. In (6), it cn be shown tht the -th weighting fctor depends on the vrinces nd corretion coefficient of noncoherent for the generized detector outputs for the -th di- versity reception. he vrince σ, is composed of sign nd bckground noise components, nd the vrince σ, is composed of interference nd bckground noise components. he numertor σ, σ, represents difference between sign power nd interference power, since the bckground noise power in σ, nd σ, is the sme. σ, σ, is the sme for, when the trnsmit power is the sme nd fding process is independent nd identicy distributed for ech diversity chnne. he denomintor σ, σ, ( ρ ) represents tht the weighting fctor shoud be sm when the noise power increses, σ, nd σ, increse nd ρ decreses. o compre the performnce of FFH nd MCFH wireess sensor network systems nd to evute the effects of diversity order in typic frequency-seective fding chnnes, the PSD N of the dditive Gussin noise for ech diversity reception is ssumed to be the sme, i.e., N N, where N is the one-sided therm noise PSD. From this ssumption, the vrinces nd corretion coefficient of the noncoherent generized detector outputs given by () (3), re the sme for : σ, σ, σ σ, nd ρ ρ for {,,..., }. With this ssumption, the optimum combining rue in (6) becomes squre-w equ-gin combining, which is the sme resut s in [3], where orthogonity between BFSK signs is mintined. Bsed on (6) nd the bove ssumption, the probbiity of error for the optimy combined sign my be expressed s P e F( D b + ) dd, (7) where D is the decision vribe defined s D D, nd D R, R,. F( D b + ) is the condition pdf of D, given b +. he condition pdf F ( D b + ) my be found using (4) nd (5) with pproprite trnsformtions of rndom vribes. It cn be shown tht the decision vribe D in (7) my be viewed s speci cse of the gener qudrtic form studied in [5], where the chrcteristic function-bsed pproch is presented to obtin simpe cosed-form expression for P e. Using technique discussed in [, 5], the probbiity of error cn be obtined s,
γ Pe πj( + γ ) ( ) Γ v dv, ( v) (8) where Γ is circur contour of rdius ess thn unity tht encoses the origin, nd γ is defined s σ σ + ( σ + σ ) 4 ρ σ σ γ. σ σ + ( σ + σ ) 4 ρ σ σ (9) For, the contour integr is zero by Cuchy s theorem [6], since the integrnd in (8) is n nytic function in Γ. However, for, the contour integr shoud be ccuted using Residue theorem [6]. hus, the probbiity of error in () cn be determined by γ P e ( ), () ( + γ ) which my be expressed in n terntive form P e ( + ) + γ. () ( + γ ) It is not difficut to prove n equivence of () nd (). It shoud be noted tht when ρ, () becomes the probbiity of error eqution deveoped for frequency-nonseective sow Ryeigh fding chnne [3]. Figure. BER performnce of FFH wireess sensor Network system for vrious dey spreds ( 3,.63). D 4 Performnce Evution he BER performnce of FFH nd MCFH wireess sensor network systems is evuted using (9) nd (). he vrinces nd corretion coefficient in () (3) re required for (), nd ccuted by Monte Cro integrtion technique [7]. he utocorretion function of fding chnne in (5) is ssumed to be described by n exponenti mutipth intensity profie nd Jkes fding mode [8] µτ µ mx µ ( e e ) mx ( t; τ ) I ( D t), µ ( + µ ) e () where µ is decying fctor nd set to.5; D is the mximum Dopper spred, nd I ( x ) is the zeroorder Besse function of the first kind. Orthogon signing, i.e.,, is impied, uness expicity specified. Figures nd 3 show the performnce of FFH nd MCFH wireess sensor network systems bsed on the generized pproch to sign processing in the pre- Figure 3. BER performnce of MCFH wireess sensor network system for vrious dey spreds ( 3,.63). D sence of noise for sever vues of mximum dey spreds, when the normized mximum Dopper spred D.63. Diversity order is set to 3. he performnce of FHSS wireess sensor network systems is found to be significnty degrded in frequency-seective fding environments with dey spred. he performnce degrdtion due to dey spred is found much more severe in FFH wireess sensor network system thn in MCFH wireess sensor network systems. his fct cn be expined in the foowing mnner. he probbiity of error my be proved to be monotonicy decresing function of γ by differenttiting (9) with respect to γ. From () (3), nd (9), γ is observed to be reted to the rtio of mx to, which is defined s n effective dey spred.
It cn be shown tht γ decreses with the effective dey spred, due to n increse in σ nd decrese in σ nd ρ. hus, the vue of γ is smer for FFH thn for MCFH wireess sensor network system, for given dey spred, since the effective dey spred for FFH wireess sensor network system is times rger thn tht of MCFH wireess sensor network system. In ddition, comprison between FFH nd MCFH wireess sensor network systems bsed on the generized nd Neymn Person receivers is shown in Figs. nd 3. he high superiority of the generized receiver over the Neymn Person detector is obvious. o investigte the effects of corretion between two generized detector outputs, the performnce of FFH nd MCFH wireess sensor network systems with the corretion ignored nd mx. 5 re obtined by setting ρ in (9) nd potted in Figs. nd 3. he rge differences between the corretion ignored nd not-ignored cses indicte tht the corretion shoud not be ignored. 5 Concusions he BER performnce of FFH nd MCFH wireess sensor network systems bsed on the generized pproch to sign processing in the presence of noise in frequency-seective Ryeigh fding chnnes is presented nd compred with the BER performnce of the sme systems under the use of the Neymn Person receiver. he optimum diversity combining rue bsed on the mximum-ikeihood criterion is deveoped. It is found tht the optimum combining is the weighted sum of the squres of non-coherent generized detector outputs. A weighting fctor is shown to depend on the vrinces nd corretion coefficient of noncoherent generized detector outputs for ech diversity reception. Bsed on the deveoped optimum diversity combining decision-mking rue under the use of the generized pproch to sign processing in the presence of noise, the expressions for the probbiity of error re derived nd evuted for vrious chnne conditions. It is found tht the use of the generized detector ows us to rech better BER performnce. Acknowedgment: his work ws supported in prt by prticiption within the imits of the project A Study on Wireess Sensor Networks for Medic Informtion sponsored by IIA, Kore. References: [] V. uzukov, Sign Processing in Noise: A New Methodoogy. Minsk: IEC, 998. [] V. uzukov, A new pproch to sign detection, Digit Sign Processing: A Review Journ, Vo. 8, No. 3, pp. 66 84, 998. [3] V.uzukov, Sign Detection heory. New York Springer-Verg,. [4] V.uzukov, Sign Processing Noise. Boc Rton, ondon, New York, Wshington D.C.: CRC Press,. [5] V. uzukov, Sign nd Imge Processing in Nvigtion Systems. Boc Rton, ondon, New York, Wshington D.C.: CRC Press, 4. [6]. Mier, J. ee, nd A. Kdrichu Probbiity of error nyzes of BFSK frequency-hopping system with diversity under prti-bnd jmming interference Prt III: Performnce of squre-w sef-normizing soft decision receiver, IEEE rns., Vo. COM-34, No. 7, 986, pp. 669 675. [7] G. i, Q. Wng, V. Bhrgv, nd. Mson, Mximum-ikeihood diversity combining in prti-bnd noise, IEEE rns.,vo. COM-46, No, 998, pp. 569 574. [8] E. nce nd G. Keh, A diversity scheme for phse-coherent frequency-hopping spred-spectrum system, IEEE rns., Vo. COM-45, No. 9, 997, pp. 3 9. [9] E. Gerniotis nd M. Pursey, Error probbiities for sow-frequency-hopped spred-spectrum mutipe-ccess communictions over fding chnnes, IEEE rns.,vo. COM-3, No. 5, 98, pp. 996 9. [] B. Soimn, A. Gvieux, nd A. Hiion, Error probbiity of fst frequency hopping spred spectrum with BFSK modution in seective Ryeigh nd seective Ricin fding chnnes, IEEE rns., Vo. COM-38, No., 99, pp. 33 4. [] K. Cheun nd W. Strk, Probbiity of error in frequency-hop spred-spectrum mutipe-ccess communiction systems with noncoherent reception, IEEE rns., Vo. COM-39, No. 9,99, pp. 4 4. [] O. Shin nd K. ee, Performnce comprison of FFH nd MCFH spred-spectrum systems with optimum diversity combining in frequencyseective Ryeigh fding chnnes,ieee rns. Vo. COM-49, No. 3,, pp. 49 46.
[3] J. Pierce, heoretic diversity improvement in frequency-shift keying, in Proc. IRE, Vo. 46, No. 5, 958, pp. 93 9. [4] B. Ibrhim nd A. Aghvmi, Direct sequence spred spectrum mtched fiter cquisition in frequency-seective Ryeigh fding chnnes, IEEE J. Seect. Ares Commun., Vo., No. 6, 994, pp. 885 89. [5] J. Prokis, Digit Communictions, 3 rd ed. New York: McGrw-Hi, 995. [6] E. Kreyszig, Advnced Engineering Mthemtics, 7 th ed. New York: Wiey, 993. [7] W. Press et., Numeric Recipes in C, nd ed. Cmbridge, U.K.: Cmbridge Univ. Press, 99. [8] W. Jkes, Microwve Mobie Communictions New York: Wiey, 974.