Information-Theoretic Comparison of Channel Capacity for FDMA and DS-CDMA in a Rayleigh Fading Environment

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1 WSEAS TRANSATIONS on OMMUNIATIONS Informaton-Theoretc omparson of hannel apacty for FDMA and DS-DMA n a Raylegh Fadng Envronment PANAGIOTIS VARZAAS Department of Electroncs Technologcal Educatonal Insttute of ama 3 o m P.E.O. ama-athens, GR-35 00, ama GREEE pvarzakas@telam.gr Abstract: - In ths paper, a comparatve estmate of the channel capacty assgned to each user for frequencydvson multple-access (FDMA) and drect-sequence code-dvson multple-access (DS-DMA) schemes operatng n a Raylegh fadng channel s presented. Then, the achevable channel capacty per user (n the Shannon sense) s estmated n an average sense consderng the nherent dversty potental that both schemes provde n a Raylegh fadng channel. The non-equal user rate for the DS-DMA case s studed, consderng the rato of two neghborng user channel capactes. Hence, t s shown that, n contrast to the equal user rate, for non-equal user rate case n DS-DMA and under normalzed condtons, the average channel capacty per user, n FDMA, s hgher than that of DS-DMA and ths result s quanttatvely evaluated n terms of a defned average capacty gan factor. ey-words:- hannel capacty, code-dvson multple-access, frequency-dvson multple-access, Raylegh fadng. Introducton In nformaton theory, ([-3]), and assumng equal power users and operaton n an deal nonfadng addtve whte Gaussan nose (AWGN) channel, the Shannon capacty regon provded by a system, that utlzes FDMA or DS-DMA schemes, s the same, snce t does not matter whether the spectrum s dvded nto frequences, or codes. However, n ths case, DS-DMA s n the nformaton-theoretc sense, where dstnct codes are used for dfferent user and the recever decodes them one by one. For a tme-varant channel, as t s the case n moble rado, ts capacty,.e., the maxmum rate, at whch data can be transmtted wth arbtrarly small bt error rate (BER), can be obtaned by fndng the best dstrbuton of the transmtted sgnal power as a functon of the nstantaneous sgnal-to-nose power rato (SNR) and then averagng over the SNR dstrbuton, [4]. However, ts capacty can also be estmated n an average sense ndcatng the average best rate for the least possble errors on the average. These average capacty formulas would ndeed provde the true channel capacty, wth channel state nformaton (SI) avalable to the recever only, [5,6]. In a prevous work, [7], we compared FDMA and DS-DMA n terms of channel capacty assgned to each user, followng the method descrbed n [8]. However, we restrcted our attenton to the equal user rate case (.e. for 3 ) for the DS-DMA scheme. From a mult-user nformaton theory, ths s only a specal case of the capacty regon, whch s acheved under the respectve schemes, [9]. In contrast, here, we nvestgate the channel capacty assgned to each user, n a DS-DMA scheme, for the nonequal user rate case. Then, assumng, n DS-DMA case, a relatonshp between the neghborng user channel capactes, we derve a general expresson useful to compare, under normalzed condtons, FDMA and DS-DMA schemes operatng n a Raylegh fadng channel, n terms of channel capacty per user. The paper s organzed as follows. Secton and 3 descrbes the operaton of FDMA and DS-DMA schemes n an deal non-fadng AWGN and a Raylegh fadng channel, respectvely. The numercal results and ther comparson are gven n Secton 4. Fnal concluson are outlned n the last secton. ISSN: Issue, Volume 7, February 008

2 WSEAS TRANSATIONS on OMMUNIATIONS Operaton n a non-fadng awgn envronment It s well kwon, that channel capacty expressed by Shannon s formula establshes an upper lmt for relable nformaton transmsson over a bandlmted AWGN channel. Then, for the AWGN channel case, the capacty regon achevable by FDMA s gven by the Shannon-Hartley theorem, [0], P + R, F,F Blog () N 0B where B s the sgnal bandwdth assgned to -th user, [,,], P beng the receved sgnal power, and N 0 s the power spectral densty of the bandlmted AWGN channel. In the followed analyss, for both FDMA and DS-DMA schemes, we consder no dynamc resource among the users of the system but the equal-power and equal-bandwdth cases.e., B B s and P P s, [,,], and that all channel nputs are subjected to equal average power constrants. Then, the capacty regon for FDMA s defned by, [], P + s R B log B log ( + Γ )(), F,F s s N0Bs where R,F s the user s data rate, Γ Γ(P s /N 0 Β s ) s the receved SNR over sgnal bandwdth B s by each user, [,,]. The suffces F and refer to FDMA and DS-DMA schemes respectvely. Respectvely, the capacty regon acheved, n theory, by DS-DMA s: Ps R,, Blog Blog ( + Γss) N0B + (3) where Γ ss (P s /N 0 Β )(P s /N 0 G p Β s )(Γ /G p )(Γ/G p ) s the spread receved SNR reflectng the lowered sgnal power spectral densty due to spreadng the transmtted sgnal power P s by some factor, known as processng gan G p, defned as: B G p (4) Bs for a full coded DS-DMA system wth bandwdth B. Followng eq.(3), the sum rate, (achevable rate regon, []), R t, n DS-DMA operatng n an deal non-fadng AWGN channel, wll be equal to: R,, B Rt, log ( + Γss) (5) where,, [,..,], s the total channel capacty avalable to all users. Moreover, based on the fact that, n practce, Γ ss s well below unty (n lnear scale), eq.(5) can be approxmated by: B log ( + Γss ) (6) In the case consdered, t s assumed that both FDMA and DS-DMA schemes accommodate the same and fxed number of users that occupy the rado channel smultaneously. However, n common models for communcaton systems, a user access the channel randomly, as t gets a message to be transmtted, but the random access of users s a fundamental ssue whch s not yet satsfactorly treated n terms of nformaton-theoretc aspects. In addton, n eq.(3), t s consdered the cooperatve model for DS-DMA scheme, accordng to whch jont demodulaton and detecton of all users' sgnals can be assumed. Ths mples knowledge of all spreadng codes by the recever, and, thus multple access nterference can be assumed neglgble and gnored. As already mentoned, n nformaton theory and n an deal non-fadng AWGN channel, the requred channel capacty per user of the utlzed physcal (n FDMA) or "logcal" (n DS-DMA) channel s eventually the same,.e.:,f, (7) It must be notced that, n the DS-DMA case, we use the term "logcal channel capacty" to ndcate the exact amount of nformaton of the despread sgnal of a partcular user, whch, after spreadng, s transmtted wth no errors over the entre physcal channel. 3 Operaton n a raylegh fadng envronment In the followng, the Raylegh fadng channel s modeled as a slowly fadng, tme-nvarant and dscrete multpath channel, []. The capacty of a sngle-user flat fadng channel wth perfect SI known only to the recever has been extensvely studed, [-5]. Snce a pure FDMA scheme reduces multple-access channel to sngle-user Raylegh fadng channels, the average channel capacty has nformaton-theoretc meanng for the FDMA case. In ths case, the power of the transmtted sgnal s dstrbuted along a channel bandwdth B s that, n practce, s never greater than the coherence bandwdth B coh of the Raylegh fadng channel, meanng that no physcal dversty ISSN: Issue, Volume 7, February 008

3 WSEAS TRANSATIONS on OMMUNIATIONS potental s offered, [7]. Hence, assumng that each user of the FDMA system experences the same Raylegh fadng condtons, the average channel capacty, [,..,], avalable to each user, ndcatng the average best rate for error-free transmsson, s gven n [] as: Γ B log ee E s Γ (8) where.ndcates average value, Γ γ s the average receved SNR over sgnal bandwdth B s by each user, [,..,], E[ x] s the exponental ntegral, [6], and the new suffce R used refers to the Raylegh fadng channel. It s mportant to notce, that n the followed analyss, we assume that each user s data rate RR,, n a Raylegh fadng channel, s restrcted by the average channel capacty avalable from the physcal channel of bandwdth B s so that R, (9) At ths pont t must noted that snce the communcaton channel s tme-varyng, s the average channel capacty per user, acheved by a FDMA system, and not the maxmum channel capacty acheved over such a channel, [6]. In [7], for the DS-DMA scheme operatng n a Raylegh fadng channel, we examned the case where all users nformaton rates are the same. In partcular, we assumed that all users share equally the capacty of the entre physcal channel, R of bandwdth B. However, ths s only a specal case of the capacty regon of all {R } -tuples, whch s acheved, [9,5]. Then, n ths work, we consder the case where the user average channel capactes are as follows: -, n,, n (0), R, R where n>. Thus, s the mnmum average user channel capacty and the rato of two neghborng user channel capactes s n. learly, the sum of average user channel capactes equals to the average capacty avalable from the entre physcal bandwdth B n a Raylegh fadng channel. Thus, we can wrte that: () ombnng eqs (0) and () and usng the sum of geometrc progresson to terms, [6], we obtan: n () n or equvalently, n n n (3) n for [,..,]. onsderng a DS-DMA system wth bandwdth B greater than the coherence bandwdth B coh of the Raylegh fadng channel, the wde physcal channel wll appear to be frequencyselectve to the transmtted sgnals and the maxmum number of uncorrelated resolvable paths ("nherent dversty branches") wll be gven by, []: B [B Tm ] + + B (4) coh where T m s the maxmum delay spread or total multpath spread of the fadng channel (assumed known or measurable) and [.] returns the largest nteger less than, or equal to, ts argument. Although the number of resolvable paths may be a random number, t s bounded by eq.(). Thus, eq.() ndcates the maxmum number of branches of a physcal frequency dversty scheme that conssts of the transmtted sgnal frequency elements whch exceed B coh and fade ndependently to each other. In general, the multpath-ntensty profle (MIP) n a Raylegh fadng channel s exponental, but, here, MIP s assumed fnte, dscrete and constant, so that the "resolvable" path model can be consdered to have equal path strengths on the average. In a conventonal maxmal-rato combnng (MR) RAE recever, the output s decson varable s dentcal to the decson varable whch corresponds to the output of a - branch space dversty maxmal-rato combnng technque, wth branches [7]. It s well-known that MR shows the best performance n a fadng envronment, [7]. onsequently, the maxmalrato coherently combnng recepton of DS-DMA spread sgnals, acheved by the consdered RAE recever, s equvalent to a -branch space dversty maxmal-rato combnng technque. Therefore, the probablty densty functon (p.d.f.) of the combned nstantaneous SNR γ m,t of the spread sgnal over the bandwdth B, wth no correlaton among the branches, wll follow the Erlang dstrbuton, [7],.e., ( ) ( ) γm,t γ m,t (γm,t ) exp ( )! Γm,t Γm,t p (5) ISSN: Issue, Volume 7, February 008

4 WSEAS TRANSATIONS on OMMUNIATIONS where Γ m,,t γ m,t s the totally receved average spread SNR value n the m-th, m[,,], dversty branch from all users, s obtaned from eq.(4) and the subscrpts t and m refer to the totally receved average spread SNR from all users and to the m-th dversty branch, respectvely. Thus, an expresson for the average capacty of the channel B over all spread SNR, R levels, can be wrtten as: ( γm, t ) (! )( Γ ) γ B log +γ exp m, t d(γm, t) m, t (6) m, t 0 m, t Γ learly, ths capacty estmaton, s based on the equvalence descrbed above and ndcates the average channel capacty that appears at the recever output n the form of the average best recovered data rate from all users. However, the performance of the coherent maxmal-rato RAE recever depends on the number of the employed taps and the fadng channel estmaton. If the number of taps s less than the resolvable paths' number, the recever performance wll substantally be degraded because the power of the remanng "branches" wll appear at the recever output as self-nose power. In ths work, we consder the optmum operaton of the coherent maxmal-rato RAE recever where the number of taps employed s equal to the number of resolvable paths as gven by eq.(4). If we consder that each user's sgnal appears at the recever nput wth the same average spread sgnal-to-nose power rato, the receved average spread SNR from all users wll be equal to Γ ss, and Γ m,t can be wrtten as: Γ m,t Γ ss (7) Then, combnng eqs (6) and (), the -th user s average channel capacty of a DS-DMA system operatng n a Raylegh fadng channel can be expressed as followng: G B n p s n, R n (! ) ( Γss) (8) γ log ( +γγ ) exp dγ Γ ss 0 where the notaton of the combned nstantaneous SNR γ m,t, used n eq.(6), has been changed to γ. 4 omparson bewteen fdma and dscdma n a raylegh fadng envronment In ths secton, we proceed to the comparson of the average capacty per user offered by the consdered DS-DMA and FDMA schemes, under normalzed condtons..e., for B G p B s B F B s where B F s the totally allocated bandwdth n the FDMA system. Also t s assumed that the delay spread T m of the Raylegh fadng channel equals to 3 μsec. As already stated, n nformaton theory, when these schemes operate n an deal non-fadng AWGN channel, under the assumptons set, they exhbt exactly the same channel capacty per user. However, n a Raylegh fadng channel, the stuaton s dfferent. In order to account for the effect of a dfferent user s rates n DS-DMA scheme, we ntroduce the term "average capacty gan" per user,,, [,..,], defned as the rato G of to, as gven by eqs (8) and (6), respectvely. Hence, G, can be expressed as: G n n n Γ log ee E Γ GpBs log ss 0 (! ) ( Γ ) γ Γ ss ( +γγ ) exp dγ (9) Eq.(9) serves as a general expresson for estmatng the average capacty gan per user between the FDMA and DS-DMA schemes accommodatng the same number of smultaneous transmttng users when Raylegh fadng s present. The ntegral n the numerator of eq.(9) s calculated numercally as t can not be expressed n closed form. Then, average capacty gan G s plotted as a functon of the average receved SNR Γ (n db) for a Raylegh fadng channel n Fg. for n,, G p 4, B.5MHz, and B s 30Hz. It s easy to see that, for the non-equal user rate case, n contrast to the equal user rate case, [7], FDMA scheme provdes greater average channel capacty per user than DS-DMA under normalzed condtons, when operatng n a Raylegh fadng channel. ISSN: Issue, Volume 7, February 008

5 WSEAS TRANSATIONS on OMMUNIATIONS the -th user, accordng to eq.(9), for n, B.5MHz, B s 30Hz, G p 4 and Γ0dB. Fnally, n Fg. 3, average capacty gan G s plotted aganst the number of nherent dversty branches, gven by eq.(4), for n, 3, B s 30Hz and Γ0dB. Average capacty gan <G> Average capacty gan <G> Fg.. Average capacty gan G between FDMA and DS-DMA n a Raylegh fadng channel aganst Γ(dB), accordng to eq.(9), for n,, G p 4, B.5MHz and B s 30Hz. A smlar result can be obtaned from Fg. where average capacty gan G s plotted for the -th user for n, G p 4, B.5MHz, 30Hz, and Γ0dB. Average capacty gan <G> B s Γ (db) Fg.. Average capacty gan G between FDMA and DS-DMA n a Raylegh fadng channel for Fg.3. Average capacty gan G between FDMA and DS-DMA n a Raylegh fadng channel aganst the number of nherent branches of DS- to eq.(9), for n, 3, DMA, accordng B 30Hz and Γ0dB. s As t can be seen as the number of dversty branches ncreases, n a DS-DMA scheme, average capacty gan s decreased followng the respectve decrease of channel capacty acheved by the corresponded DS-DMA scheme. 5 oncluson In ths paper, we consdered FDMA and DS- DMA schemes from a mult-user nformaton theoretc pont of vew. In partcular, we derved a general expresson for the user channel capacty comparson between the prevous multple-access schemes when channel capacty (n the Shannon sense) s estmated n an average sense. It was shown that for dfferent user channel capactes, n DS-DMA case, and when operatng n a Raylegh fadng channel, FDMA provdes sgnfcantly greater channel capacty per user that DS-DMA, under normalzed condtons. However, for the equal users nformaton rate case, the stuaton s altered. It has been shown, n a prevous work, [6], that channel capacty per user, n DS-DMA ISSN: Issue, Volume 7, February 008

6 WSEAS TRANSATIONS on OMMUNIATIONS scheme operatng n a Raylegh fadng channel, exceeds that provded by FDMA. Thus, the equal and non-equal user-rate cases, n a DS-DMA scheme, appear to behave very dfferent. Our results are useful for the predcton of the nformaton-theoretc comparson of prevous schemes and then used as a fgure of mert. References: [] T.over and J.Thomas, Elements of Informaton Theory, New York: Wley, 99. [] P.Jung, P.W.Baer and A.Stel, Advantages of DMA and Spread Spectrum Technques over FDMA and TDMA n ellular Moble Rado Applcatons, IEEE Transactons on Vehcular Technology, vol. 4, 993 pp [3] I.E.Pountouraks and P.A.Bazana, Multchannel mult-access protocols wth recever collsson markovan analyss, WSEAS Transactons on ommuncatons, ssue 8, vol. 4, 005, pp [4] A.J.Goldsmth The apacty of Downlnk Fadng hannels wth Varable Rate and Power, IEEE Transactons on Vehcular Technology, vol. 46, no. 3, 997, pp [5].H.Ozarow, S.Shama and A.D.Wyner, Informaton theoretc consderatons for cellular moble rado, IEEE Transactons on Vehcular Technology, vol. VT-43, no., 994, pp [6] A.J.Goldsmth and P.Varaya, apacty of fadng channels wth channel sde nformaton, IEEE Transactons on Informaton Theory, vol. 43, no.6, 997, pp [7] P.Varzakas and G.S.Tombras, omparatve estmate of user capacty for FDMA and DS- DMA n moble rado, Inernatonal Journal of Electroncs, vol. 83, no., 997, pp [8] P.Varzakas and G.S.Tombras, Spectral effcency for a hybrd DS/FH DMA system n cellular moble rado, IEEE Transactons on Vehcular Technology, vol. 50, no.6, 00, pp [9] S.Verdu, The apacty Regon of the Symbol- Asynchronous Gaussan Multple-Access hannel, IEEE Trans. on Inform. Theory, vol. 35, no.4, 989, pp [0].E.Shannon, ommuncaton n the presence of nose, Proceedngs of IRE, vol.37, 949, pp.0-. rd [] J.G.Proaks, Dgtal ommuncatons, 3 Edt., New York: McGraw-Hll, 995. [] W..Y. ee, Estmate of hannel apacty n Raylegh Fadng Envronment, IEEE Transactons on Vehcular Technology, vol. 39, no. 3, 990, pp [3] B.S.Tsybakov, On the capacty of channels wth a large number of paths, Radoteknca. Elektronca, vol. 4, 959, pp [4] T.Ercson, A Gaussan channel wth slow fadng, IEEE Transactons on Informaton Theory, vol. IT-6, 970, pp [5] E.Bgler, J.Proaks and S.Shama (Shtz), Fadng hannels: Informaton-Theoretc and ommuncatons Aspects, IEEE Trans. on Inform. Theory, vol. 44, no.6, 998, pp [6] I.S.Gradsteyn and I.M.Ryzhk, Table of Integrals, Seres, and Products, Academc Press, p.97, 980. [7] M..Smon and M.S.Aloun: Dgtal ommuncaton over Fadng hannels: A Unfed Approach to Performance Analyss, New York: John Wl ey, 000. ISSN: Issue, Volume 7, February 008