Analysis of the Achievable Time-Frequency Diversity Gains in Coded OFDM

Transcription

1 Analyss of the Achevable Tme-Frequency Dversty Gans n Coe OFDM Anrea M. Tonello an Rccaro Bernarn DIEGM - Unversty of Une 33 Une Italy tonello@egm.unu.t Abstract We aress the problem of etermnng how to optmally explot the tme-frequency versty n a coe OFDM system. We assume to eploy a coe that generates complex coewors wth M-ary alphabet of length equal to B where s the number of tones. The coewors span B blocs over whch the fang s assume to be quasstatc but nepenent across stnct blocs. We etermne rate/complexty traeoffs for maxmum achevable versty gans an cong gans as a functon of the channel response. We g nto the benefts an savantages of eployng nterleavng an we show that t helps to maxmze versty/cong gans wth hgh probablty. Keywors Cong versty fang channels nterleavng OFDM.. Introucton Weban channels exhbt frequency selectvty an consequently severe tme sperson. When transmsson s over such channels the recever must mplement some form of equalzaton to overcome the ntersymbol nterference ntrouce by the tme-spersve channel. The equalzaton tas smplfes when transmsson s base on some form of multcarrer moulaton. Orthogonal frequency vson multplexng (OFDM) has become a popular multcarrer moulaton technque that allows for a very smple etecton scheme [9]. It s base on eployng at the transmtter se an -pont IFFT followe by the nserton of a cyclc prefx. The recever smplfes to a bloc that sregars the cyclc prefx followe by an -pont FFT. If the cyclc prefx s longer than the channel tme sperson at each FFT output we get the ata symbol transmtte on that sub-channel weghte by the channel frequency response plus thermal nose. Therefore the etector smplfes to a one-tap equalzer.e. a symbol ecson evce that requres only nowlege of the sub-channel weght. If transmsson s uncoe the frequency versty prove by the frequency selectve fang channel cannot be explote. In other wors f a sub-carrer frequency conces wth a channel null then the nformaton carre by that sub-carrer s lost. Therefore channel cong has to be eploye n orer to explot some frequency versty an grant relable transmsson. The resultng scheme s often referre to as coe OFDM (COFDM) [3][9][]. In ths paper we aress the problem of etermnng how to optmally explot the tme an frequency versty n a coe OFDM system. We assume to eploy a channel encoer before OFDM moulaton. The encoer generates complex coewors wth M-ary alphabet of length equal to B where s the art of ths wor was supporte by MIUR uner proect FIRB "Reconfgurable platforms for weban wreless communcatons" prot. RBE8RFY. number of tones. The channel s assume to be tme-varant an frequency selectve. In partcular we assume a bloc Raylegh fang moel.e. the channel s assume to be tme-nvarant over the uraton of a gven bloc but t vares n an nepenent fashon n the aacent bloc. The coewors span one or more blocs. The cyclc prefx s assume to be long enough to cope wth the channel tme sperson. The eployment of cong has the potentalty of yelng a versty an a cong gan over the uncoe OFDM system. We stuy the error rate performance an we erve esgn crtera for COFDM such that both the versty gan an the cong gan are maxmze. The analyss goes through the analyss of the parwse error probablty. In turn the parwse error probablty epens on the ran an etermnant of a generalze Vanermone matrx whose structure s a functon of the channel an the error event tself. We erve necessary an suffcent contons uner whch the generalze Vanermone matrx s full ran. ot only the ammng stance but also the error poston patterns play an mport role. It s foun that certan coes wth ammng stance equal to (wth beng the ran of the channel correlaton matrx) have pathologc behavor an o not acheve full versty when use n a conuncton wth OFDM moulaton. We also conser the eployment of tme-frequency nterleavng. The ea s wely apple however the approach s often heurstc. Instea our analyss helps unerstanng the effect of the eployment of nterleavng n terms of cong an versty gans. It s foun that nterleavng may ecrease the versty explotaton of the coe. owever t maes t unlely. In certan contons nterleavng can preserve the versty gan of the coe but t s benefcal snce t s capable of ncreasng the cong gan.. Coe OFDM We conser a coe OFDM system archtecture as escrbe n what follows. An nformaton bt ata bloc s channel encoe nto an M-ary complex coewor a of length B. The coewor s splt nto B sub-wors of length { a ()} l =... l =... B. Each sub-wor s OFDM moulate by applyng an -pont IDFT whose outputs are n n () () = x l = a l e. () A cyclc prefx of length µ symbols s ae to cope wth the tme sperson ntrouce by the channel. We emphasze that a ( l ) s the encoe complex ata symbol transmtte on subchannel over bloc l at rate /T wth T = ( + µ ) T. It belongs for nstance to the M-QAM constellaton. roc. of Internatonal Symposum on Wreless ersonal Multmea Communcatons 3 WMC 3 Yoosua Kanagawa Japan October 9-3 ISS: WMC Steerng Boar V-93

2 n After /S converson the symbols xnt ( + lt ) = x( l) are transmtte over a frequency selectve fang channel. In our baseban screte tme moel we assume a channel mpulse response gntlt ( ; ) = α ( ) ( ) lδ nt pt = wth p {... µ } beng the µ tap elays. ote that the channel can be sparse. The channel tap gans are assume to be complex Gaussan wth zero mean (Raylegh fang) an normalze power E [ α ] =. The channel taps are tme-nvarant over the uraton of an OFDM symbol however they may vary over the followng bloc n an nepenent fashon. Ths moel correspons to the well nown bloc fang moel [4]. The recever comprses conventonal OFDM emoulaton followe by maxmum lelhoo channel econg. After havng sregare the cyclc prefx the -pont DFT outputs rea z () l = a () l () l + w () l = p () l = α () l e () for =... where w () l s a sequence of... Gaussan ranom varables wth zero mean an varance. Uner the AWG assumpton the optmal maxmum lelhoo channel =... ecoer eces n favor of the coewor a ˆ = { aˆ ( l)} l=... B that maxmzes the Euclean stance metrc B = z ( l) ( l) aˆ ( l). l= = 3. arwse Error robablty The parwse error probablty contone on the channel state nformaton s ( aa) ˆ ( aa) ˆ 4 ( a aˆ ) = Q e. (3) The rght most nequalty follows from the conventonal Chernoff boun an the square parwse error stance between the transmtte coewor an the coewor we ece n favor of s efne as B ( ˆ) = () ε () l= = aa l l (4) wth ε () l = a () l aˆ () l. From () we can wrte that B ( p ) * n pm () ˆ = αm() αn() ε () l= m n= = T α ( l) = [ α ( l)... α ( l)] aa l l e l. (5) If we use matrx notaton wth an G () l beng the matrx whose elements are ( pn pm) nm () ε () = G l = e l we can rewrte (5) as a normal quaratc form [5][8] B B () ˆ = () ()() = l= l= m= aa α l G l α l λ β. (6) In (6) λ m=... are the egenvalues of the matrx R() l G () l wth R() l = E[ α() l α ()] l beng the channel taps correlaton matrx (assume to be postve efnte) whle β m l s a sequence of... Gaussan ranom varables wth zero mean an unt varance. ote that n our moel the channel tap gans are tme-nvarant over a sngle OFDM symbol but change nepenently n the aacent bloc. Therefore we can rop the epenency of the nex l n the channel taps correlaton matrx. Uner the Raylegh fang assumpton the probablty ensty functon of the square stance can be evaluate n close-form [5][8] whch yels m n a a λ n n = n= λ ( n )! p ( a) = A e ( a) (7) wth λ = λml for = l + m an s the number of stnct egenvalues each wth multplcty m... m. Further A n are the coeffcents of the partal fracton expanson of the characterstc functon (resues) A n = ( λps). (8) ( λ ) ( )! m n m n m n m n s p= p mp s = / λ If we average over (7) we obtan the average parwse error probablty n close-form m n ( l)! λ /4 An l l+ = n= l= ( l!) ( + λ /4 ) ( a a ˆ) =. (9) Fnally wth the Unon Bounng technque we can evaluate the average bt an frame error rates. 3. Chernoff Boun an Desgn Crtera Averagng over the strbuton of β (exponental wth Raylegh fang) the rght most term n (3) yels B r λml E λ S ( a a ˆ) + l= m= 4 4 λml ES () B where r = r l = l an r l equals the number of non-zero egenvalues of RG ( l). Therefore the COFDM system can acheve a versty gan (slope of the error rate curve n the log scale) of r an a cong gan (shft of the error rate curve) of ( / E λ ) λml m l S wth ES = E[ a ( l) ]. The esgn of the coe shoul maxmze the versty an cong gans [4][6]. ote that f the channel s flat there s no frequency versty avalable. owever maxmum tme versty explotaton can be obtane wth cong across B blocs f the coe has mnmum tme ammng stance T (number of non-zero TE ( l) = ε ( l) for l =... = B ) equal to B [4]. On the other han f the channel s quas-statc e.g. when cong s lmte to a sngle bloc the only source of versty s the frequency versty. In the remaner of ths paper we mostly focus on the quas-statc channel case. 4. Quas-Statc Frequency Selectve Fang Channel et us assume a quas-statc frequency selectve channel wth full ran correlaton matrx = ran( R ) = <. Wthout loss of generalty we can assume cong over a sngle OFDM symbol (bloc) an set l =. Then the (frequency) versty gan s r = ran( G ). The CODFM fully explots the channel versty only when ran( G ) =. In such a case ( a a) ˆ.5(4 ) R G () an consequently the maxmzaton of the cong gan calls for V-94

3 the maxmzaton of the etermnant of the matrx G. If we assume an error sequence of length we can explctly wrte G as follows: G = F ΕF Ε = ag [ ε... ε ] p p p ω e... ω p p ω ω () F = ω = e p p p ω ω... ω where the error postons are assume to be < <... < < an the tap elays are assume to be p < p <... < p <. * et Ε = ΕΕ then G = ( ΕF) ΕF. It follows that ran( G) = ran( ΕF) = ran( F) snce Ε s full ran equal to. If we efne the relatve error stances δ = =... an the relatve channel elays τ = p p =... we can wrte F = FFF τ τ ag [... ] ag [ p p F = ω ω F = ω... ω ] (3)... τδ τδ ω... ω F = τδ τδ ω... ω Therefore the ran of F conces wth the ran of F. s obtane by eletng some rows an some columns from an Vanermone matrx W []. From the above results we can state the followng two propostons. roposton. In quas-statc Raylegh fang to acheve a frequency versty gan of (full versty) the ammng weght of the error sequence has to be at least. Consequently to acheve full frequency versty the mnmum frequency ammng stance F (number of nonzero FE ( ) = ε ( l = ) for =... ) of the coe has to be at least. roposton. The maxmum achevable versty wth a coe havng mnmum ammng stance s mn{ }. Therefore n quas-statc frequency selectve fang the versty gan s etermne not only by the frequency ammng weght of the error events but also by the error patterns (postons). The shape of the constellaton s rrelevant wth respect to the coe versty. In the followng we state several necessary an/or suffcent contons uner whch ran( G) = ran( F) = ran( F ) = r. roposton 3. All error sequences of length acheve full versty.e. ran( G ) =. roof. When the length of the error sequence s the matrx F s obtane from a Vanermone matrx W where a number of rows has been elete. Snce we assume p < < p < the matrx F s full ran.... roposton 4. All error sequences of length one o not acheve any versty beneft. Consequently uncoe OFDM oes not prove any versty gan. F F F roof. We ust nee to notce that an error sequence of length one generates the followng Toepltz-ermtan matrx ( p p ) ( p p ) G = ε Toepltz{ ω... ω }. (4) Such a matrx has ran equal to one so that RG has also ran one wth the only non-zero egenvalue λ ( ) = trace{ RG }. The fact that the egenvalue s a functon of the error event poston translates nto sub-channels wth fferent error rates whenever the channel taps are correlate [8]. roposton 5. If the channel has ran the versty orer equals the (frequency) ammng weght of the error sequence. In other wors the mnmum ammng stance of the coe etermnes the versty orer. roof. It can be prove smlarly to roposton 3. roposton 6. If the relatve channel elays or the relatve error stances are consecutvely orere respectvely as τ = = 3... an δ = = 3... the versty gan s mn{ }. In the former case the channel s non-sparse n the latter the error event s non-sparse. If the channel s non-sparse the versty gan s etermne by the mnmum ammng stance of the coe an f t s larger or equal to then full versty s acheve. roof. The matrx F s a rectangular Vanermone matrx wth ran mn{ }. A more general suffcent conton for F to be full ran s state n the followng proposton. We have foun by numercal nspecton that the conton s not necessary. roposton 7. et a {... } be relatvely prme wth.e. GCD{ a } = then assumng the versty gan equals f max { aδ mo } < (5) =... whle assumng the versty gan equals f max{ aτ mo } <. (6) =... roof. We prove (5). Smlarly we can prove (6). et us assume. Then F s not full ran ff there exsts a τδ non-zero -tuple { λ } such that λω = =.... Ths s true ff there exsts a polynomal x ( ) = λ x δ that s τ zero for x = ω =.... Consequently t can be wrtten as the prouct of two polynomals x ( ) Qx ( ) ( x τ = ω ). = The rght most polynomal has egree whle x ( ) has egree Deg{ ( x)} max{ δ }. Therefore a necessary conton for F to =... have ran lower than s that max { δ }. =... Consequently a suffcent conton for F to have full ran s that > max { δ }. Ths proves (5) for a =. =... ow let a {... } such that GCD{ a } =. Then a has the nverse.e. t exsts a {... } such that a aa mo =. et us tae S( y) = ( y ) then f ( τ ω ) = a τ aa τ τ for all =... also S( ω ) = ( ω ) = ( ω ) = for all =.... Therefore a necessary conton for the ran to be lower than s that max { aδ mo }. Consequently (5) =... V-95

4 s a suffcent conton for F to have full ran. roposton 8. et ε be an error sequence of length that acheves full versty then every error sequence ε of length that has ε as sub-sequence acheves full versty. Further G > G where G s the etermnant of G corresponng to ε. It can be compute recursvely as follows: - ( G+ = G + ε + f+ Gf + ) (7) wth the ntal etermnant equal to ε = G = F. (8) roof. et us conser = + then we can wrte + G = F ΕF G+ = [ F f+ ] ag{ Ε ε }[ F f+ ] (9) + = F ΕF+ ε f+ f+ wth f + beng an approprate vector that s assocate to the error sequence ε = [ + ε ε + ]. It follows that the etermnant of G + s + - G+ = F ΕF ( + ε f+ ( F ΕF) f+ ) () + - = G ( + ε f+ Gf+ ) > G. ote that the frst equalty () hols snce G s assume to be full ran. The last nequalty hols snce G s postve efnte. Fnally G = F Ε F = F Ε whch proves (8) whle (7) s obtane recursvely from (). roposton 9. If the channel s non-sparse G has etermnant () G 4sn ( ( )) m n m m = ε ω ω = ε n m m= n m= m= n m= n> m n> m roof. In such hypothess F s a Vanermone matrx of sze an () follows from (8). 5. Dstance an Complexty Constrants Any bloc coe (an also any fnte length trells coes) of rate R bts/symbol wth coewors consstng of B blocs of length symbols that belong to an M-ary alphabet satsfes the Sngleton Bouns [4]: T + B ( R/log M) F + (Rlog M). Therefore the versty capablty of the coe s not nfluence by the constellaton shape. owever for a gven rate to ncrease the coe versty the sze of the constellaton has to ncrease. In terms of complexty a trells coe can acheve versty only f ts constrant length s at least. If the coe has rate R then to acheve versty ts complexty (number of states n the trells) has ( ) to be at least R [6]. Therefore the complexty ncreases exponentally wth the versty orer explotaton capablty. 6. Interleavng Interleavng s often eploye at the output of the channel encoer an before the OFDM moulator [3][7][]. Interleavng can be performe ether n frequency or n tme or ontly n tme an frequency. The most nterestng case s when both cong an nterleavng are eploye across one sngle OFDM symbol or equvalently across a number of OFDM symbols over whch the channel remans statc. In such a case the only source of versty s the frequency versty prove by the spersve channel. As we have shown n the prevous secton the versty gan s a functon not only of the ammng stance of the coe but t also epens on the poston of the errors wthn the coewor. When we eploy an nterleaver the pattern of the errors s clearly nfluence. In orer to nvestgate the effect we assume to eploy a unform nterleaver. The nterleaver maps a gven error pattern of ammng weght F an length n all ts possble stnct permutatons wth equal probablty. The same analyss tool was use n [] although n our case there s no coe concatenaton. et ε be an error sequence of ammng weght F. Then we can evaluate the average parwse error probablty compute over all possble nterleavers I : F I ( ) F ( ) ( ˆ ε ) =! ( ˆ ε ˆ ε =I( ε )). () 6. Effect of Interleavng on the Dversty Gan Assumng a quas-statc channel the nterleaver has the effect of changng the error patterns (postons). If the channel s not sparse then the nterleaver has no effect on versty (see roposton 6). On the contrary f the channel s sparse then a gven full versty error event may be mappe by the nterleaver nto a lower versty error event or vce versa nto a hgher versty error event. Overall the effect of nterleaver s to average out the versty beneft an allow for smpler although heurstc coe esgn. That s t maes the probablty of the coe to acheve full versty hgh. 6. Effect of Interleavng on the Cong Gan Agan we assume a quas-statc channel. If we assume the channel to be non-sparse nterleavng oes not affect versty however t changes the error patterns such that the cong gan may change. As an example f we pc a maxmum versty BSK coe the mnmum cong gan s acheve n corresponence to the error sequence ε wth ammng weght that mnmzes (). Further the average parwse error probablty can be boune as E S R 4sn ( n m) (3) I nm = n> m where I s the set of all possble combnatons of the error poston nces n the -tuple = [... ] wth <. 7. Examples 7. Two-Taps Channel et us conser a -ray channel moel wth uncorrelate channel taps.e. R = ag{[ ]} p = p. Then p ε ε e = = p ε ε = = RG = (4) e an compute n close form the etermnant an the egenvalues 4 RG = λλ = ε ε sn p( ) (5) > V-96

5 λ = ( + ) ε ± ( + ) ε / (6) 6 ε ε sn ( p( ) / ) > ow for all error sequences for whch p ( )mo = the versty orer equals one an the only non-zero egenvalue s λ = ( + ) ε. For nstance let s pc a geometrcally unform coe that has a sngle mnmum ammng stance coewor that s equal to two. The encoer s followe by unform nterleavng. If / p s nteger an f the stance among the two errors s equal to τ = / p the error event oes not acheve full versty. In the left plot of Fg. we show the probablty to acheve full versty as a functon of the tap elay p. It has been compute assumng all stnct pars wth equal probablty. In the rght plot of Fg. we show the strbuton of the normalze cong gan assumng BSK mappng. As the fgures show the probablty of achevng full versty ncreases when the number of tones (nterleaver length) ncreases. The cong gan strbuton s a functon of the number of tones (nterleaver length). 7. Repetton Coe et us conser a repetton coe wth rate R = /. It generates for each nput symbol a coewor of length equal to thus the mnmum ammng stance equals. et us assume a channel wth ran. The frequency versty gan epens on how we map the coewors nto the subchannels. It certanly equals f we transmt the = K coe symbols n the natural orer across the tones. In fact n such a case roposton 6 hols. Instea f we eploy a unform nterleaver there s a non-zero probablty that the nterleave coewors generate error patterns for whch the acheve versty s less than. If the channel s not sparse then versty s acheve for all nterleaver realzatons. Further the cong gan strbuton changes resultng n possbly better average parwse error probablty. Ths s shown n Fg. where we conser a 4 taps non-sparse channel wth / R =.5 ag{[ ]} an we compute accorng to (9) the parwse error probabltes corresponng to all possble nterleaver patterns. We assume an error event wth weght or 4 wth BSK mappng. It s nterestng to note that the worst performance (lowest cong gan) s acheve by a non-sparse error pattern. As the nterleaver length (number of tones ) ncreases the spectrum broaens. In any case there exst nterleavers that sgnfcantly better the performance. 7.3 Trells Coe If we conser trells coes the performance s etermne not only by the ammng stance of the coe but also by the error postons. The constructon of trells coes has to be carre out by searchng for coes for whch F s full ran. A wely eploye approach s to use bnary convolutonal coes followe by nterleavng an symbol mappng. Accorng to our analyss the effect of nterleavng s to average out performance.e to allow for maxmum versty an cong gans wth hgh probablty. 8. Conclusons Several necessary an suffcent contons have been erve for achevng maxmum versty an cong gans n COFDM. The effect of nterleavng on versty an cong gan has been analytcally stue. References [] S. Beneetto D. Dvsalar G. Montors F. ollara Seral concatenaton of nterleave coes: performance analyss esgn an teratve econg IEEE Trans. on Info. Theory pp May 998. [] R. orn C. Johnson Matrx analyss Cambrge Unv. ress 985. [3] R. assalle M. Alar "rncples of moulaton an channel cong for gtal broacastng for moble recevers" EBU revew no. 87 pp August 987. [4] R. Knopp. umblet On cong for bloc fang channels IEEE Trans. on Info. Theory vol. 46 pp January. [5] C. Schlegel Error probablty calculaton for multbeam Raylegh channels IEEE Trans. on Comm. pp March 996. [6] V. Taroh. Seshar R. Calerban Space-tme coes for hgh ata rate wreless communcaton: performance crteron an coe constructon IEEE Trans. Info. Theory pp March 998. [7] A. Tonello Asynchronous multcarrer multple access: optmal an sub-optmal etecton an econg Bell abs Techncal Journal vol. 7 n. 3 pp. 9-7 February 3. [8] A. Tonello Exact matche flter performance boun for multtone moulaton n fang channels roc. of WMC 3 Yoosua October 3. Full paper submtte to IEEE Trans. Wreless. Comm. Aprl 3. [9] Z. Wang G. Gannas "Wreless multcarrer communcatons" IEEE Sgnal rocessng Magazne pp May. [] W. Zou Y. Wu "COFDM: An overvew" IEEE Trans. on Broacastng pp. -8 March 995. rob [ versty = ] robablty of Full Dversty o =6 * = secon ray elay p =6 ammng Weght= o Average =64 ammng Weght= o Average rob [ cong gan = abscssa ].3 o =6 * = Cong Gan Dstrbuton for p = λ λ /(4 ) Fg.. robablty of achevng full versty an cong gan strbuton. =6 ammng Weght=4 o Average 8 8 =64 ammng Weght=4 o Average SR (B) SR (B) Fg.. Spectrum of parwse error probablty wth 4 taps fang channel V-97