CURRENTLY, the third generation (3G) mobile communication

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1 Optial Perforae for DS-CDMA Systes with Hard Deisio Parallel Iterferee Caellatio eo va der Hofstad, Marte J Klo Abstrat We study a ultiuser detetio syste usig ode divisio ultiple aess CDMA We show that applyig ultistage hard deisio parallel iterferee aellatio HD-PIC sigifiatly iproves perforae opared to the athed filter syste I ultistage HD-PIC, estiates of the iterferig sigals due to other users are used iteratively to iprove owledge of the desired sigal We use large deviatio theory to show that the bit-error probability BEP is expoetially sall ad ivestigate the expoetial rate of the BEP after several stages of HD-PIC We propose to use the expoetial rate of the BEP as a easure of perforae, rather tha taig the sigal-to-oise ratio, whih is ot reliable i ultiuser detetio odels We show that the expoetial rate of the BEP reais fixed after a fiite uber of stages, resultig i a optial hard deisio syste Whe the uber of users beoes large, the expoetial rate of the BEP overges to log / / We provide ituitio oerig the uber of stages eessary to obtai this asyptoti expoetial rate Fially, we give Cheroff bouds o the BEP s These estiates show that the BEP s are quite sall as log as = o/ log whe the uber of stages of HD-PIC is fixed, ad eve expoetially sall whe = O for the optial HD-PIC syste Keywords Code divisio ultiple aess, hard-deisio parallel iterferee aellatio, large deviatios, expoetial rate, perforae easures, optial hard-deisio syste I Itrodutio CUENTLY, the third geeratio 3G obile ouiatio syste is beig itrodued This syste is based o ode divisio ultiple aess CDMA The perforae of CDMA is aily liited by iterferee fro other users: the ultiple aess iterferee Partiularly, the suseptibility to the ear-far situatio, whih a sigifiatly redue apaity i the absee of good power otrol, is a proble Therefore, there exists a great iterest i tehiques whih iprove the apaity of CDMA reeivers see [5], [], [6] ad the referees therei Iitial researh o ulti-user reeivers for CDMA has deostrated the potetial iproveets i apaity ad earfar resistae The best ow tehique is a iu lielihood sequee estiator [9], whih obtais joitly optiu deisios for all users usig iu lielihood detetio Ufortuately, this tehique is of high oplexity A straightforward tehique is alled iterferee aellatio, see [6], Chapter, [], [9], [8] ad the referees therei The idea is that we try to ael the Faulty of Iforatio Tehology ad Systes, Delft Uiversity of Tehology, Meelweg, 68 CD Delft, the Netherlads, tel : , fax: , eail: rwvaderhofstad@itstudelftl, jlo@itstudelftl iterferee due to the other users Iterferee aellatio, espeially hard-deisio parallel iterferee aellatio HD-PIC, is the ost proisig ad the ost pratial tehique for upli reeivers see [6] This is beause i WCDMA proposals all user sigals are deodulated oheretly, whih aes ipleetatio effiiet Apart fro that, as we will see i this paper, HD-PIC sigifiatly iproves perforae I the literature, ot uh attetio has bee give to obtai rigorous aalytial results for HD-PIC systes The papers that exist o this subjet fous o approxiatig the sigal-to-oise ratio SN, for exaple [] ad [0] However, usig the SN as a easure of perforae by substitutig the SN i the Gaussia error futio ipliitly relies o Gaussia assuptios For PIC systes, this assuptio is false ad it leads to iorret results as we will argue i Setio II-E We use large deviatio theory [3], [8] to obtai aalytial results for the bit-error probability BEP More preisely, rather tha alulatig the SN, we alulate the expoetial rate of the BEP We will first explai the essee of this rate Suppose p is a sequee of probabilities ad p 0 as Ofte, this deay is expoetially i, so that we ivestigate the expoetial rate defied as I = log p For fiite, but ot too sall, we a read this as p e I Thus, the probability p is aily haraterized by its expoetial rate We stress that the expoetial rate of the BEP is ot a probability It turs out that the BEP for athed filter MF ad HD- PIC systes have the sae struture as p This aes it possible to use the expoetial rate as a easure for the perforae For the MF odel, the SN ad the expoetial rate are asyptotially equivalet, eaig that whe the uber of users is suffiietly large, both substitutig the SN ito the Gaussia distributio ad the use of the expoetial rate iply that the BEP has the sae asyptoti for This is show i Setio II-E I [7], a MF odel is ivestigated usig large deviatio tehiques However, the ephasis lies o saplig tehiques For a oe-stage softdeisio PIC odel, results have bee obtaied i [6] I that paper, large deviatio tehiques are used to prove properties of the expoetial rate ad the BEP I [7] ad

2 [], various results have bee prove for the HD-PIC syste I this paper, we start by repeatig results of oe-stage ad ulti-stage HD-PIC of [7] ad [] We exted these results to values of the proessig gai ad the uber of users that are oparable usig Cheroff bouds The ai ovelty of this paper is that we obtai results for systes with ifiite stages of HD-PIC, the so-alled optial HD-PIC syste Oe of the striig results is that after fiitely ay iteratios, the expoetial rate does ot irease ay further I other words, applyig ore stages of iterferee aellatio does ot iprove perforae ayore Aother striig result is that the obtaied expoetial rate is stritly positive, regardless of the uber of users We will give bouds o this expoetial rate Furtherore, we give ituitio o the uber of stages eessary to obtai this expoetial rate II Syste ad bagroud I this setio, we explai the syste, ad give previous results oerig HD-PIC A Syste odel We first itrodue a atheatial odel for CDMA systes We defie the data sigal b t of the th user as b t = b t/t, for, where b =, b,, b 0, b, {, + Z ad where for x, x deotes the largest iteger saller tha or equal to x For eah user,, we have a odig sequee a =, a,, a 0, a, {, + Z ad we put a t, the retagular spreadig sigal, geerated by a =, ie, a t = a t/t, where T = T/, for soe iteger The variable is ofte alled the proessig gai The trasitted oded sigal of the th user is the s t = P b ta t osω t,, where P is the power of the th user ad ω the arrier frequey The total trasitted sigal is give by rt = s j t 3 j= I pratie, the sigals do ot eed to be syhroized, ie, it is ot eessary that all users trasit usig the sae tie grid However, for tehial reasos we do assue so I [7], a asyhroous MF syste is ivestigated, usig large deviatios tehiques Oe iportat result i that paper is that the asyhroous syste has uh siilarities, opared to the syhroous syste I fat, the expoetial rate for the syhroous ad asyhroous syste are the sae For the ore advaed HD-PIC syste, this idea arries over, so that the results ivestigated i this paper tur out ot to differ fro the results for a asyhroous syste Ideed, we expet that the asyhroous odel has the sae expoetial rate of the BEP as the syhroized odel Therefore, the syhroiity assuptio is without loss of geerality We assue that there is o additive white Gaussia oise AWGN Thus, we are dealig with a perfet hael However, we believe that ay of our results reai true whe there is little AWGN To retrieve the data bit b 0, the sigal rt is ultiplied by a t os ω t ad the averaged over [0, T ] For sipliity, we pi ω T = πf, where f N to get f [6], Eq 3- T T 0 P = b 0 + rta t osω t dt j= Pj b j0 a ji a i As is see fro, the deoded sigal osists of the desired bit ad iterferee due to the other users I the ideal situatio the vetors a 0,, a, ad a j0,, a j,, j, would be orthogoal, so that i= a jia i = 0 I pratie, the a-sequees are geerated by a rado uber geerator To odel the pseudo-rado sequee a, let A i, =,,,, i =,,,, be a array of idepedet ad idetially distributed iid rado variables with distributio i=0 PA = + = PA = = / 5 Assuig the odig sequees to be rado, we odel the sigal of as P b 0 + j= Pj b j0 A ji A i, i= where we have replaed i = 0,, by i =,,, for otatioal oveiee I pratie, the a-sequees are ot hose as iid sequees ather, they are arefully hose to have good orrelatio properties Exaples are Gold sequees [] or Kasai sequees [0] However, it is oo i the literature to use rado sequees, so that a detailed aalysis is possible Better perforae a be ahieved for wellhose deteriisti odes We let ˆb 0 be the estiator for b 0 give by sgr { P b 0 + j= Pj b j0 A ji A i, where, for x, the radoized sig-futio is defied as +, x > 0, sgr x = U, x = 0, 6, x < 0 i=

3 3 with PU = = PU = + = / The rado variables U are idepedet of all other rado variables i the syste Note that the th user always aes the sae deisio whe its sigal produes a zero There are other ways to defie the sig-futio, suh as the hoie where every tie whe a zero is deteted, a ew idepedet rado U is hose Aother optio is to let the sig of 0 be equal to 0 We hoose the defiitio i 6 for tehial reasos oly We will oet ore o other hoies of sig futios i Setio III below We use the otatio i ˆb 0 to idiate that this is a tetative deisio oly We ow desribe the hard-deisio proedure The powers P are assued to be ow We estiate the data sigal s j t for t [0, T ] by reall ŝ j t = P j ˆb j0 a jt osω t The we estiate the total iterferee for the th user i rt due to the other users by reall 3 ˆr t = j= ŝ j t We use the above to fid a better estiate of the data bit b 0, deoted by ˆb 0, whih is the sgr of T = T 0 rt ˆr ta t osω tdt 7 Pj A ji A i b j0 ˆb j0 P b 0 + j= i= We are ow iterested i Pˆb 0 b 0, whih is the probability of a bit error after oe stage of iterferee aellatio We will see that this probability is ideed saller tha Pˆb 0 b 0, the probability of a bit error without aellatio This otivates a repetitio of the previous s proedure We obtai, siilar to 7, the estiates ˆb whih are the sgr of P b 0 + j= i= A ji A i Pj 0, s b j0 ˆb j0 This is alled ultistage HD-PIC Whe we have applied s steps of iterferee aellatio we spea of s-stage HD-PIC ad the orrespodig bit error probability is Pˆb s+ 0 b 0 B eforulatio of the proble We a write the probability of a bit error i a ore oveiet way Naely, beause b i =, we have P b Pj 0 + b j0 A ji A i j= = b 0 P + j= Pj i= b j0 A ji b 0 A i i= Sie A ji d = bj0 A ji we have ˆb Pˆb 0 b 0 0 = P b 0 = Psgr Z = PZ < 0 + PZ = 0, where Z, for, is defied as Z = P + We a easily boud j= Pj A ji A i i= PZ < 0 Pˆb 0 b 0 PZ 0 8 Note that the above boud is opletely idepedet of the hoie of the sig-futio i 6 The upper ad lower boud i 8 a be show to be alost equal whe 3 ad is large Siilarly, we defie for s ad, Z s = P + to obtai as i 8 PZ s j= Pj A ji A i i= [ sgr j Z s j ], 9 < 0 Pˆb s 0 b 0 PZ s 0 0 We will ow ivestigate the perforae of HD-PIC We will use as a easure of perforae the expoetial rate of a bit error, defied by H s ; = s log Pˆb b For systes with equal powers, the users are exhageable, so that the above quatity does ot deped o I that ase, we defie H s = H s ; for all We will ofte use the syste with equal powers as a referee odel We will deote this odel by the siple syste Without loss of geerality, we the tae P =, so that the fators P / disappear C Previous results for HD-PIC I [6], it is show that the expoetial rate without iterferee aellatio H, deoted there by I, for the siple syste for 3, is give by H = log + log I [7], we have obtaied a aalyti forula for the rate after oe stage of HD-PIC A differet defiitio of the sig futio is used there, but this does ot ifluee the results

4 Theore II: Whe all powers are equal where H H = i H,r, 3 r,r = sup { log hs, t, s,t with hs, t = h,r s, t equal to r r j= r r r+j e s+sj+tj+tj osh tj r I the stateet of the theore, oe should ot ofuse H,r with H ; i We see that the proble is split ito two optiizatio probles First, r represets the uber of bits that have bee estiated wrogly i the first stage For every r, we solve the large deviatio iiizatio proble The we iiize over r to obtai the rate H We a give a aalytial expressio for H,, siilar to the rate for the MF syste, that reads H, = 3 log 3 log log log For geeral r, we aot obtai a losed for expressio for H,r However, stadard uerial paages allow us to opute H,r for all ad r I Figure, H,r is show for r =,, 5 Observe that the optial r equals for 9 I probability laguage it eas that the BEP, aused by or ore bit errors i the first stage is egligible opared to the BEP aused by bit error i the first stage For = 0,, 5, r = will give the iial rate, eaig that typially a bit error i the seod stage after HD- PIC is aused by bit errors i the first stage This is further illustrated i Table I, where the optial r is give for = 3,, uber of users Fig Expoetial rates H,r for r =,, 5 idiated with,,,, respetively For / 0 a Taylor expasio of yields H = + O I fat, r {3,,9 {0,,6 {7,,5 3 {5,,8 {85,,5 5 {6,,7 6 {75,,3 7 {3,,50 8 TABLE I Optial value r for = 3,, 50 H + O 6 I [7], it is show that for the siple syste, we have that the rate has a asyptoti salig: Theore II: Fix s < ad assue that the powers are all equal The, as, H s = s s 8 + O s 7 However, these results are asyptoti oly I Figure, we see that the approxiatio beoes worse whe s ireases uber of users Fig Expoetial rates H, H 3, respetively, H ad the asyptoti behaviour,, respetively, 3 3 Here,, 8 represet the exat values, ad +,, the asyptoti values We ext desribe the extesio of the above results for the siple syste with equal powers to the ase where the powers are uequal, show i [] For the MF syste, the followig result is prove for the expoetial rate H Theore II3: For P / = P 0, H ; = P = P + O P = P Theore II3 is based o a Taylor expasio of the rate For the HD-PIC syste, it is ore ivolved to fid a asyptoti expasio, but it a still be doe

5 5 Theore II: For P / = P 0, H ; P = P + O Whe there exists a C = P /P, i P P P = P > 0, suh that, uiforly i = P P C = P P /, where the iiu rages over all {,,, equality is attaied: H ; = P P = P + O = P A siple exaple where the above oditio is satisfied is whe j, P j /P C for soe C We refer to [] for ore exaples To see that the results above iply that HD-PIC really iproves perforae, it is useful to ivestigate the siple syste For this ase, H ad H redue to H ad H It is lear that HD-PIC gives a sigifiat irease i perforae opared to the MF reeiver, sie / is uh large tha /, so that exp / is uh saller tha exp / Note the shift fro / to / I order to preserve the desired quality level, it is thus possible to osider for exaple a derease of the proessig gai by a fator For the syste with uequal powers, we see that H P ; = P ad H ; P = P The sae olusio olusio holds: iterferee aellatio sigifiatly iproves perforae The differee is that is replaed by = P /P We will ow give a heuristi explaatio of Theore II We will start by explaiig the result for s =, ad the oet o s > We will argue that H,r r + 8r 8 Ideed, usig 3, we the see that the iiu over r is obtaied whe r, yieldig H / To see 8, ote that H,r is the rate of the evet that Z < 0,, Z r+ < 0, Z < 0, ad all the other Z j > 0 for j = r +,, Whe we ow all the sigs of the Z s, we a substitute these sigs i the forula for Z This yields that Z = + r+ j= A l A jl 9 l= Note that E Z j = for all j Therefore, the evet {Z j 0 is ot a large deviatio This explais that we a show that the evets Z j > 0 for j = r +,, do ot otribute to the rate ad we a leave these evets out i the defiitio of H,r Fially, idepedee of the sets {Z < 0,, {Z r+ < 0, ad {Z < 0 is learly false for all fiite However, it turs out to be asyptotially true for large, as show i [7] This yields that r+ H,r li = log P + r+ j= log PZ < 0 l= A l A jl < 0 The first ter produes rh r/, the seod ter a be oputed to give /8r whe r is large This gives a iforal explaatio of the asyptoti result i 9, ad explais that to have oe bit error at the first stage of HD-PIC, we eed roughly / bit errors at level oe This explais that oe-stage of HD-PIC sigifiatly dereases the BEP We fially give a iforal explaatio of Theore II for s > For = σ s s σ=, let H s, deote the rate of the evet that {Z σ < 0 for all σ s, ad {Z σ > 0 for all / σ s The we have that Let r s σ H s = i H s, 0 = s σ I [7], we show that we have H s, rs + rs 8r s + + rs s 8r s s + 8r s s Note that the right had side does ot deped o the preise struture of σ s s σ=, but oly o the sizes The reaso is that all depedee betwee Z σ is differet levels is ditated by the sigs of these quatities After these sigs are substituted, the rado variables are asyptotially idepedet for large, lie i the ase for s = Miiizatio of the right had side of over r s,, rs s yields r s σ Substitutio yields 7 s σ/s D Cheroff bouds ad large depedig o The Cheroff boud a be used to boud probabilities fro above by a expoetial, ad is true regardless of, For the MF syste, the Cheroff boud is straightforward to alulate The Cheroff boud is give by Pˆb 0 b 0 e H 3 ad this holds for ay ad eall fro 6 that H /, so that Pˆb 0 b 0 e / for ay

6 6 ad For the HD-PIC syste, a siilar stateet a be derived Whe we ivestigate for exaple the ase of equal powers, we are able to prove Pˆb 0 b 0 r r= e H,r The Cheroff boud is ot tight i the sese that it approxiates the BEP with ay desired preisio Istead it gives a upper boud i a very siple for Note that if is large, the Cheroff boud is doiated by the ter whih has the sallest rate Tae for exaple = ad = 63 We a show that see Table I H, iiizes H,r Therefore we expet that ost of the su is otributed by the seod ter Substitutio leads to Pˆb 0 b = 5 0 The ai otributio learly oes fro r = 7%, the otributios fro r = 3 ad r = are also relevat % ad 5%, but the reaider of the su is egligible % I Figure 3, siulated BEP s are opared with the Cheroff boud i as a futio of It is see that the Cheroff boud ideed gives upper bouds The expoetial rate appears i this figure as the slope of the BEP ad the Cheroff boud The differee i slopes is very sall, idiatig that the large deviatio approah is ideed proisig for perforae evaluatio These exat BEP s are obtaied by a iportae saplig proedure see [] ad [3] for the details proessig gai Fig 3 BEP ad Cheroff boud The BEP for = 3, 6, 9 is ared with,,, respetively The Cheroff boud for = 3, 6, 9 is ared with +,,, respetively We ext use the Cheroff boud i order to tae large with Fro a pratial poit of view, oe always wishes to have as ay users as possible, so that the situatio where is fixed ad ay be violated Istead, we ow tae = We prove the followig result: Theore II5: Whe suh that = o log, s li s log Pˆb 0 b 0 s s 5 8 The proof is give i Setio V-B The above result iplies that whe suh that = o log, we have that Pˆb 0 b 0 is to leadig asyptotis equal to exp This should be otrasted with the behaviour that is expeted whe is of the order This is the etral liit regie where we expet that the bit error probability overges as to a o-zero ostat It pays to tae slightly saller tha istead of a ultiple of, at least whe osiderig a fiite uber of HD-PIC stages! I Setio III-B we will give a ore preise explaatio of the etral liit behaviour ad its relatio to the HD-PIC results We rear that the Cheroff boud for s = a easily be exteded to ertai ases with differet powers Ideed, P whe we assue that j j, P C, we a easily show that li P log Pˆb s 0 b 0 P 6 E Measures of perforae: sigal-to-oise ratio vs expoetial rate We ext ivestigate the relatio of the above result to the sigal-to-oise ratio SN, whih is defied as SN = E Z VarZ, where Z deotes the deisio statisti i the odel uder osideratio We osider the ase ad P = We will show that i this ase the SN is ot a good easure of perforae, ad therefore, we propose to use the expoetial rate istead The asyptoti result for H i 6 yields that the expoetial rate of the BEP is approxiately / Hee, the BEP is approxiately BEP e 7 For the MF syste, we ow that SN= /, sie E Z = ad VarZ = / To get a approxiatio for the BEP, oe ofte substitutes the SN i the well-ow Q-futio Sie see [5] e x / πx x < Qx < e x / πx, substitutig the SN yields the approxiatio BEP e SN / This approah gives the approxiatio BEP e e, whih agrees with the large deviatio result i 7

7 7 For the siple syste with oe-stage of HD-PIC, Theore II that H /, yieldig BEP e 8 However, for a syste with oe stage of HD-PIC, usig the SN results i { BEP exp e, 9 whih is show below The latter value is far too sall opared to the true asyptotis of the BEP i 8, whih learly idiates that the Gaussia approxiatio usig the SN is o good The reasoig above a be adapted for other odels, suh as ultistage HD-PIC or the optial HD-PIC syste desribed below Therefore, usig the SN together with a Gaussia approxiatio for suh ultiuser detetio systes leads to faulty approxiatios for the BEP To prove the upper boud 9, observe that E ˆb = + Pˆb b E A A ˆb b, 30 whe is large opared to sie Pˆb b 0 Moreover, VarZ E Z Pˆb b, where for the latter boud we use that A ij = Fro the Cheroff boud see 3, ad the fat that H see 6, we ed up with This yields that so that VarZ e SN e, { e SN / exp e III The optial syste I this setio, we show that after a fiite uber of stages of HD-PIC, the expoetial rate of the BEP reais fixed The we study properties of this optial hard deisio syste, suh as its expoetial rate, ad Cheroff bouds for the BEP s The results show that the BEP is expoetially sall wheever δ for soe δ > 0 suffiietly sall, whih is quite rearable A Optial hard deisios We first list soe easy osequees for the syste with uequal powers First of all, we defie the worst ase rate of a bit error to be H s = i H s = ; 3 Hee, H s is the expoetial rate of the bit error probability of that user that has largest bit error probability Theore III: a H s is ootoe i s b There exists a s +, suh that H s = H s all s s The above shows that we a spea of the syste for s = s as the optial syste For equal powers, we ow that the rate for all users does ot iprove for s s For uequal powers, we a thi of this syste as havig the optial worse ase rate, i the sese that the rate of the bit error iiized over all users is optial We will use the defiitio of the sig-futio i 6 However, i the proof, we will oet o other defiitios of the sig-futio where a siilar result a be show We will ow prove the above theore Proof: First of all, s H s is o-dereasig Ideed, to have a bit error, oe of the iterferig users has to have a bit error i the previous stage For the siple syste, we have exhageability of users, so that the probability of a bit error at stage s is saller tha that of stage s ad the desired stateet follows The above stateet eve proves that the probability of a bit error is o-dereasig This siple arguet fails to hold whe the powers are uequal However, i this ase, we a show i preisely the sae way that the ial bit error is o-dereasig The we tae suh that the bit error i stage s is ial for that user We a just repeat the arguet give above, ad see that we eed to have a bit error fro aother user i the previous stage However, the latter probability is at ost the ial probability of a bit error By the ootoiity above H = li s H s exists To see why H with fixed is reahed i a fiite uber of stages s ad s +, defie s σ for = { : sgr Z σ 3 The σ s is the set of bit errors at stage s ad level σ, ie, the set of idies j for whih sgr j Z σ j < 0 whe we efore that sgr Z s < 0 We observe that eessarily s σ = s σ, for soe σ < σ + The Z σ + Z σ + for all ad thus s σ + = s σ + It follows that s σ + = s σ + ad thus s σ+iσ σ = s σ for all σ σ < σ, for all i N {0 By 6, all stages beyod σ are deteried by the stages,, σ ad do ot otribute to the rate Clearly soe users have bit errors for a stage s > σ Thus, the seario desribed above is oe seario to fore a bit error for a user at stage s However, H s is the worst-ase rate over all searios ad all users Therefore, H s H s s Sie H is odereasig, eessarily H s = H s for all s σ Sie s σ, the desired stateet follows The stateet of Theore III a be exteded to other defiitios of the sig-futio

8 8 For exaple, whe we let the sig of 0 to be 0, the we a opy the origial proof, apart fro the fat that we let s σ σ;0 be the set of values where Z = 0, ad s σ; the set where Z σ < 0 The, whe s 3 +, we ust s σ ;i = s σ ;i, for i = 0, ad soe σ < σ 3 + This agai iplies a periodi seario For the syste with equal powers, we are also allowed to use a sig-futio that assues values ± idepedetly every tie it is used, rather tha fixig it per user Ideed, the liit deteriig H exists, so we a reah the liit just by usig the odd I this ase, we aot have Z σ = 0, exept whe σ = Therefore, every user draws at ost oe tie a rado sig-futio, whih aes the stateet equivalet to the stateet with our origial defiitio of sgr For uequal powers this arguet ufortuately does ot hold However, Q is dese i, so that for alost all powers P,, P ad for all itegers, Z σ 0 This explais that the probles arisig fro the defiitio of the sig of zero are soewhat aadei We will ext ivestigate this optial syste eall that beause of the hard deisios, the deisios at stage σ oly deped o the deisios at stage σ For illustratio purposes, we will first ivestigate the syste with users We will use that whe user has o bit error at stage σ, user eessarily has o bit error at stage σ + We deote a bit error by ad o bit error by i Figure All possible five searios exept the trivial seario where o errors at all are ade are show user stage 3 Fig Possible searios for the HD-PIC odel with users It is lear that withi 3 stages, the set where bit errors are ade beoes periodi We will all this a periodi seario This is soewhat saller tha + A periodi seario has a big advatage over o-periodi searios As log as the seario is ot yet periodi, speifyig whih users have bit errors at a ertai stage results i a derease of the BEP Ideed, users do ot ted to have a bit error; it is ore liely to estiate a bit orretly However, i a periodi seario, this is ot true ayore For exaple, to get a bit error for user at stage s = 00, it is suffiiet to speify the positios of bit errors i stages, ad 3 the first ad last seario i Figure will do Fro that poit owards, the bit errors are deteried by the bit errors i the first three stages Two essetially differet searios are haraterizig the behaviour of the optial syste for = The first oe is the so-alled disjoit seario, where at every stage user has a bit error ad user ot, or vie versa The other seario, whih we will all the overlappig seario is the seario where at every stage both user ad user have a bit error Note that for both the disjoit as the overlappig seario, the periodi behaviour is i at stage For both searios, we a alulate the expoetial rate The iiu of the two expoetial rates idiates whih seario typially is observed Whe 3, we exted those searios i the followig way For every r, at stage, 3, 5, bit errors are ade for users i soe set, with = r At stage,, 6,, bit errors are ade for users i the set with = r Whe =, we spea of the disjoit seario Wheever =, we will all it the overlappig seario All other searios are alled partly overlappig The stateet of Theore IIIb is that after at ost + stages, the set where bit-errors are ade is periodi after a ertai stage We expet that the searios that we typially observe are periodi already at stage Ideed, speifyig the bit errors at the iitial o-periodi stages ae the BEP saller, while for the periodi stages we do ot eed to speify the positios of bit errors, as these are deteried by the first few stages The loger the uber of stages where the behaviour is ot yet periodi, the saller the BEP ad thus this behaviour is less typial This suggests that the searios that we typially observe are either the disjoit, the partly overlappig, or the overlappig seario The overlappig seario has period, while the other searios have period, so strethig the above heuristi eve further, it is teptig to assue that the overlappig seario gives the sallest expoetial rate, ad is therefore typial However, this is ot true whe is sall see also below We observe the followig pheoea The partly overlappig seario is ever optial It sees that both extrees the overlappig ad disjoit seario do a better job For sall, the disjoit seario is optial The reaso is quite siple For r fixed, i the disjoit seario, a user at stage has otributio fro r oise ters, whereas for the overlappig seario the user has otributio fro oly r ters the user does ot iterfere with its ow sigal For higher, however, the overlappig seario is optial For the ase, it is ipliitly prove i the proof of Theore IIIa that the overlappig seario is ideed optial, ad the partly overlappig seario is ever optial 3 For, also r, but uh slower tha The uerial results idiate that r / To illustrate ad 3, the optial r is show for = 000 i Table II Also, it is idiated whether the disjoit or the overlappig seario is optial I Figure 5, the expoetial rates H s are give, together H s for s =,, 3 The results for s = 3 are obtaied usig siilar tehiques as i Theore II However, we have ot stated the result for s = 3 i this paper The

9 9 r r -30 d 337-0o 3-73 d 5-99 o d o o o o o o o o o o d eas disjoit seario is optial, o eas overlappig seario is optial TABLE II Expeted optial seario for optial HD-PIC odel rate H s is i fat the rate orrespodig to the disjoit seario for r = or r = For = 3, it is see that H 3 = H 3 = H 3 3, so that s 3 = For 9, we see H < H = H 3, so that s = We see that oe stage of HD-PIC gives a iproveet i expoetial rate However, addig oe ore stage does ot result i ay iproveet For 0, -stage HD-PIC gives a iproveet over oe stage HD-PIC The uerial result show that there is a blo seario i fat the disjoit seario with r = with the sae rate Therefore s = 3 i this ase For 3 50, we expet s =, eve though we aot alulate H s = s = s =3 s = uber of users Fig 5 The expoetial rates H s ad H s for s =,, 3 We ow tur to a lower boud of the expoetial rate for the optial syste Clearly, 7 has o eaig for s, but assuig 7 ad the ootoiity of s H s it follows for the siple syste that for all ɛ > 0, ɛ H whe Thus, if H overges to 0 as, it does so slower tha ay power of / The theore below states that the expoetial rate of the optial syste reais stritly positive as, ad we idetify the liitig rate uder a ertai assuptio o the powers P,, P that we will defie ow We will assue that P P hold, where the oditios P P are defied by P There exists a δ > 0 s t {j : P j [δ, /δ], P li P <, P 3 P 0, P P i Here P = P, P i = i P The ai result i this setio i the followig theore Theore III: a For the geeral syste, for all s >, H s log For the syste with equal powers the boud is obtaied for s > b If the powers are suh that P P hold, the li H s = log 3 We stress that the ase of equal powers is overed i Theore IIIb I Figure 6, a upper ad lower boud is give of the expoetial rate H s for equal powers We expet that the upper boud is i fat equal to H s, but we la a proof uber of users Fig 6 Upper ad lower bouds for the expoetial rates H s The proof of the above result will be give i the ext setios We will start with bouds o oet geeratig futios i Setio IV These bouds will be used i the proof of Theore III i Setio V B Cheroff bouds ad large depedig o I Setio V-B, we prove the followig Cheroff boud for the optial syste: Theore III3: Whe ad for s > Pˆb s 0 b 0 8 e I, 35 where I = log Whe the powers are equal, the sae result holds whe s > The above result shows that the liit of ad a be tae siultaeously, istead of the order oo i large deviatio theory first, ad subsequetly Moreover, Theore III3 shows that whe = δ with δ < I/3 log, that the the BEP is expoetially sall

10 0 Sie for = O, we have etral liit behaviour, the above Cheroff boud iplies that applyig HD-PIC suffiietly ofte eables us to redue the BEP fro O for the MF syste, to expoetially sall for the optial HD syste We ow give a ore extesive heuristi explaatio for this effet We tae = δ for soe ostat δ > 0 We assue that the powers are equal, ad for sipliity tae P = I this ase, we have that by the etral liit theore, Z N, δ Hee, the BEP is approxiately ɛ = Q/ δ Therefore, we expet ɛ bit errors i the first stage s = Now, give that there are ɛ bit errors i the first stage, we have that Z N, ɛ Therefore, the BEP is approxiately ɛ = Q/ɛ, ad we expet ɛ bit errors i the seod stage s = epeatig the above arguet, we fid that, with ɛ s = Q/ɛ s, we expet approxiately ɛ s bit errors i the s th stage The above is probably quite aurate whe s is fiite ad, but ertaily ot whe s teds to ifiity with However, fro Theore II obied with Theore III, we do ow that s whe Therefore, the above arguet fails to hold for the optial syste, ad we see that the optial HD-PIC syste has a uh saller BEP tha the BEP after ay fiite uber of stages C O the uber of stages to optiality I this setio, we will ivestigate the uber of stages eessary to reah the asyptoti optial rate log We will prove the followig theore I its stateet, we will reall that P = P, ad we will use the otatio p = i P Theore III: For the geeral syste, for all 0 < ɛ < log ad for all s suh that s ɛ log P p, H s log ɛ 36 We ow that I = log is the asyptotially optial rate for the syste where all the powers are equal Note that whe all powers are oparable eg, whe P j j, P < C for soe ostat C < uiforly i, that the ɛ log P p ɛ log Theore III states that whe we apply at least that ay stages of HD-PIC, that the the liitig rate will be asyptotially bouded fro below by I This suggests that s grows roughly lie log It is a iterestig proble to deterie how s grows ore preisely It follows fro Theore III together with Theore II that s However, it would be of pratial iportae to ow the preise rate, or eve a upper boud o s We have the followig ojeture: Cojeture III5: For the geeral syste, li sup Cojeture III5 says that log s = 37 log log s log faster tha ay sall power of log aot grow or derease Theore III, together with Theore III, suggests that s ɛ log for every ɛ 0 We believe that i fat a logarithi uber of stages is required to obtai the optial HD-PIC syste However, we have o proof for this belief This belief stes fro the fat that we expet the proof of Theore II to hold for soe s that ted to ifiity with suffiietly slowly More preisely, we expet that the strategy desribed i Setio II-C reai true for as log as s log ɛ for ay ɛ > 0 The reaso is that i, there is a essetial hage whe s log opared to s = Olog, i the sese that for the forer r σ s, r s σ+ whereas for the latter, the above overges to a ostat Whe this overgee is towards a ostat, we aot expet to be a good approxiatio Moreover, ote that whe s = log ɛ, the substitutio of s ito the right had-side of 7 gives s 8 e log /s = log ɛ e log ɛ 0 8 for all positive ɛ This is learly far away fro the rate of the optial HD-PIC syste i Theore III Therefore, we believe s log ɛ for all ɛ > 0, whih explais Cojeture III5 IV Bouds o oet geeratig futios I this setio, we will give sharp bouds o ertai oet geeratig futios that will prove to be essetial i the aalysis of the optial HD-PIC syste We defie where S = Pi X i, 38 i= PX i = ± = We will also use the otatio P = i= P i The ai result of this setio is Propositio IV: a For all N, 0 t < P b For all P t P ad all s, P s P t E [e ss+ t e S ], 39 P t E [e t S ] P t 0 Note that if is large, the S P Z, where Z has a stadard oral distributio The bouds i 39-0 show that the oet geeratig futio of S ad S, for the appropriate rages of the variables t ad s are at least bouded fro above by the oet geeratig futios of P Z ad P Z, e P s / ad / P t, respetively

11 Proof of Propositio IVa This boud is easiest Let Z have a stadard oral distributio, the we ow that E e tz = e t / Hee, we get that [ ] E [e ss+ t S ] = E e s+ tzs [ = E osh P i s + ] tz i= We use that osht e t /, to arrive at E [e ss+ t S ] E [ i= e P i s+ tz ] = e P s E [e P s tz+tp Z ] We oplete the proof by otig that, for t P, ad by rearragig ters E [e P s tz+tp Z ] = e P s t P t, P t Proof of Propositio IVb The lai for 0 t P follows fro Propositio IVa proved above The lai for t < 0 is ore diffiult, ad we will use idutio o Defie f t = E [e t S ] The idutio hypothesis is that f t P t for all P t 0 Clearly, for = 0, the above is trivial, as both the left ad right had side are equal to We ext advae the idutio We write f t = E [e t S ] = e tp E [e t S + P ts X ] = e tp E [e t S osh P ts ] We agai use that osht e t / to arrive at f t = E [e t S ] e tp E [e t+pt S ] = e tp f t+p t To prove the lai, we first show that for P t 0, P P t + P t 0 Ideed, sie P t 0, we have 0 + P t, so that P t + P t P t P t + P t = P P t + P t = P P t + P t 0, where the last iequality follows fro P P 0 ad t 0 We therefore a substitute the idutio hypothesis for, so that it reais to show that e tp/ P P t + P t P t Sie e x + x + x / for all x 0, e tp/ = e tp Multiplyig ad rearragig ters gives P t + P t / P t + P t / P P t + P t = + t[ P P P ] +t [ P P P + P P P + P /] +t 3 [ P P P / + P P P ] +t [ P 3 P P /] = P t + P t / + P P P /3t 3 / PP 3 P t /3 = P t + P t + P t P t + P P t P t, sie + P t 0, t 0 ad + P P t 0 opletes the proof V Proofs This I this setio we will first prove Theore IIIa ad III, where we will use Propositio IV The, we will prove the Cheroff bouds i Theore II5 ad III3 i Setio V-B This proof aes use of the proof of Theore Theore IIIa Fially, we prove Theore IIIb for equal powers The proof of Theore IIIb for uequal powers satisfyig the power oditios P P will be give i the appedix A Proof of Theore IIIa ad Theore III I this setio, we will prove Theore IIIa ad Theore III siultaeously For opleteess, we will repeat the stateets For s = +, H s log 3 For every 0 < ɛ < I = log For s = ɛ logp/p, H s log ɛ I Setio III we have show that there is a optial HD- PIC syste, ad that for s +, the set of bit errors is periodi For ay set A {,,, we let P A = i A P i Whe s = +, there ust be a σ + suh that P s σ P s I fat, whe the powers are equal, we σ have that P s = P σ s, ad the above ust happe σ already whe s We fous o that level σ ad are oly iterested i the evet {P s σ P s σ Furtherore, whe s = ɛ logp/p +, there ust be a σ ɛ logp/p + suh that P s ɛp s σ σ

12 Ideed, whe this should ot be the ase after σ = ɛ logp/p stages, we have log ɛ P s ɛ σ P s σ = exp log P P 0 ɛ p Sie ɛ e ɛ, also log ɛ/ɛ, so that However, always P s σ P s exp log P P = p σ p P s σ p, so that p ɛp s, σ so ertaily after at ost s = logp/p/ɛ + stages the desired evet has ourred We fous o the level σ at whih the desired evet ours ad are oly iterested i that ourree At this poit we rear that oe we obtai for P s σ ɛp s that σ log P P s ɛp s σ σ log ɛ, we iediately obtai the result for ɛ = 0, whih is the stateet 3 We use that whe A = s σ ad B = s σ, that the for all B, we have that P Z σ = P + l= j A\{ Pj P A jl A l As the uber of ofiguratio σ s s σ= is just fiite, we obtai that H s is bouded fro below by the iiu over A ad B suh that P B ɛp A of log P j A l= log P Pj P A jl A l + P j A l= B\{j log e P B P A B t E e t Pj P A jl A l + P B 0 B 0 j, A B PjP A ja, where the last iequality is the expoetial Chebyheff s iequality for t 0 We write S A = j A Pj A j to ed up with H s i log A,B P B ɛp A E e t SASB + P B P A B t 5 We will ow boud the oet geeratig futio fro below usig Propositio IV We first write S A = S A B + S A\B, ad we use the fat that S A\B is idepedet fro S B, S B A to get that E e t SBSA equals E e t SBSB A e t SBS A\B 6 = E e t SBSB A osh t Pj S B j A\B E e t SBSB A e t 8 P A\BS B We write the right had side of 6 as E e t + t 8 P A\BS A B e t + t P A\BS B AS B\A + t 8 P A\BS B\A ad use Propositio IVa with s = t + t P A\BS B A ad t = t P A\B 0 to boud the expetatio over S A\B as PA\B P B\A t / = E e t + t 8 P A\BS A B e P A\B PA\B P B\A t / E e ts / B A PA\B P B\A t / t/+p B\A t / P A\B P B\A t / S B A PB A t, where the last iequality is valid as log as t where t = t + t P A\B = t + P B\A + P A\B t / P A\B P B\A t / P B A + P B\A t/ + P A\B t / P A\B P B\A t / ad Fro the first expressio for t it is lear that t t, so that we restrit /P A B t 0 Fro the seod expressio for t above it is straightforward to prove that t 0 We proeed by ultiplyig the two square roots to obtai E e t SASB sie tp B A t P AP B P B A, 7 P A\B P B\A + P A B P A\B + P B\A = P A P B P B A Substitutig 7 ito 5 yields that for all t P A B we have that H s is bouded fro below by i A,B log P B ɛp A tp A B t P AP B P B A P B P B A t 8 Sie P B A P B 0, substitutig t = P B H s i A,B P B ɛp A P A P B + P B A log P B A P B P B results i + P B A P B 9

13 3 It is lear that this lower boud of H s is dereasig i P A ad that P A ɛ P B Therefore substitutig P A = ɛ P P B still gives a lower boud Clearly, A B P B [0, ] The above lower boud is therefore attaied at i 0 α log ɛ + α + α α = i fα 50 0 α Differetiatig fα wrt α gives that f α equals + + α ɛ + α + = α ɛ α α ɛ + α + α Hee, f α > 0 for α < + / ɛ ad f α < 0 for α > + / ɛ Therefore, the iiu of f is attaied at either α = 0 or at α = Substitutio yields that f0 = log + ɛ = log log 3 ɛ ad f = log 9 ɛ = log + log 8 Fially observe that for 0 ɛ /3, ɛ e ɛ ɛ + ɛ ɛ + ɛ/3 = ɛ/3, so that ɛ log ɛ/3 Furtherore, for 0 ɛ /, / ɛ /ɛ Substitutig this yields log ɛ, 3 ɛ log 3 8 ɛ ɛ ɛ Therefore, sie log 3 + / > log + /, H s i{f0, f = log ɛ This opletes the proof of Theore III Substitutig ɛ = 0 gives Theore IIIa B Proof of the Cheroff bouds We will start by provig Theore II5 for s = The extesio to s > will follow later, ad is a sall adaptatio of the proof for s = We have that Psgr Z < 0 PZ 0 P Z 0, r+ r r= = Z 0, i =r+ Z 0 We split the su over r i two parts: r ad r > We start with the first ter The Cheroff boud gives P Z 0, r+ = Z 0, i =r+ Z 0 e H,r We boud, usig that r r ad e H,r e H, r= r e H,r + = e log e H log + e H The first ter o the right had-side is e o, sie = o log iplies log = o Therefore, this ter is bouded fro above by e o e H = e o e +o This proves that the first ter has the right order, ad it reais to show that the other ter is a error ter I order to do this, we will first prove the followig lea Lea V: For every, P Z 0, r+ = Z 0, i =r+ Z 0 Proof: We boud, usig PA B PA, P Z 0, r+ We a opute that = Z 0, r P Z 0 = r = Z = i =r+ Z r P r = A l l= = i= 0 e r Z 0 A il, 5 Therefore, by the expoetial Chebyheff iequality, for every t 0, the probability of iterest is bouded by i E e t SrS i E e t S t, r e SrS S r t 0 t 0 where we write S = l= A l We ext boud the oet geeratig futio fro above We first use the idepedee of S r ad S S r to obtai E e t S t r e SrS S r = E e t S r t r osh S r 5 We ext use the boud osht e t / to boud the expressio above as E e t S r t osh S t t+ r E e S/ r 53

14 As log as t + t r, we a use Propositio IVb to obtai E e t S r e t SrS S r Therefore, we arrive at P Z 0, r+ = Z i t 0 exp rt rt / 0 5 0, i Z =r+ log rt rt The optial t is attaied at t = / For this hoie, we have that t + t / = / /r This justifies 5 We further observe that for t = /, we have that log rt rt = log + r Fially, observe that log + x x/ for all 0 x, whih opletes the proof of the lea We ow oplete the proof of Theore II5 for s = usig Lea V Sie e = o, the su over r satisfyig r > is bouded by r> r e r = log e e r>0 log exp log This satisfies the required boud sie log = o, so that we have Psgr Z < 0 e +o + e r +o = e +o The proof for s > is siilar, ad we poit out the differees oly We a use the proof of Lea V to show that the probability that there are at least s /s bit errors i the first stage is a error ter if > 0 is large eough Therefore, we oly have to deal with the ase where there are at ost s /s bit errors i the first stage I this ase, a easy extesio of the proof of Lea V shows that the probability that there are at least s /s bit errors i the seod stage is a error ter if > 0 is large eough Therefore, we ay also assue that there are at ost s /s bit errors i the first stage We a repeat this arguet, so that we oly have to deal with the probability that user 0 has a bit error i stage s, iterseted by the evets σ s σ s σ/s We ow a use the Cheroff boud ad show that the bioial fators are of lower order We ext tur to the proof of Theore III3, whih is quite easy whe we use results of the proof of Theore III i Setio V-A I the proof, we have used the fat that for s +, there ust be a stage σ suh that P s σ The rate of this evet is prove to be bouded fro P s σ below by I = log The uber of possible stages at whih this a happe is +, ad the uber of possibilities for σ s ad s σ are eah, sie σ s ad s σ aot be epty The above arguet leads to a overall fator of + 8 This opletes the proof, sie Pˆb s 0 b 0 P {P s P s σ σ + σ 8 i P σ s A,B:P B P A 8 e I = A, s σ = B We ote that the above proof also holds if we hoose the sig0 to be equal to ± idepedetly every tie C Proof of Theore IIIb I order to prove Theore IIIb, we have to fid a strategy that has asyptoti rate log For sipliity, we will assue that all powers are equal The proof for uequal powers is ore tehial ad is therefore deferred to Appedix A For sipliity, we assue that the powers are all equal to We ote that whe that eessarily for all σ =, σ = Hee, we have ow foud a strategy that iplies bit errors at all stages, so that the rate of this evet is a upper boud for the rate of the optial syste We fix r = ad for tehial reasos we assue r to be odd We will first ivestigate the rate log P =, = r Due to the fat that all users are exhageable, ad the rate futio of the vetor Z, Z = is ovex, we have that log P =, = r = log P r = r = Z 0, Z 0, =r+ =r+ Z 0, Z 0, where Z deotes Z where the sigs of the Z are substituted This stateet will be prove i ore detail i Appedix A

15 5 We ext ote that Z = = = = i= i= =,j= + A ji A i j= A ji A i A i 0 i= = Moreover, we have that r = Z 0 Hee, =r+ Z 0, so that we a reove the evet { =r+ Z 0 Sie PX 0, Y 0 PX Y/ 0, Y 0, we a boud the expoetial rate fro above by r log P Z r Z 0, Z 0, = = Z 0, where for r, Z Z = Z = = =r+ + A ji A i, i= j=r+ r + A ji A i i= + i= while for r +, Z = + We abbreviate { E = r + Clearly, E = E 3 = { r + { i= r,j= r =r+ j= r =r+ j= i= [ + j= r A ji A i, j= r A i A ji j= A i A ji 0, i= i= A i A ji 0, ] A i A ji 0 PE E = PE E E 3 + PE E E 3, so that, aordig to the largest-expoet-wis priiple, { li log PE E = i li log PE E E 3, li log PE E E3 We wish to show that for ad r large, log PE E = log PE E E 3 I order to do so, we show that for all r li if log PE E E3 log Furtherore, we will show that li log PE E log PE 56 ad as r, log PE = log + o 57 This iplies diretly whe r is large li log PE E E 3 = li log PE E log PE = log + o = o Ideed, whe x i{y, z ad x < z, the eessarily x = y Equatios 55, 56 ad 57 also iply the stateet i Theore IIIb as we will show ow Taig r gives li li r log PE E E 3 log The reaider of this proof therefore fouses o provig 55, 56 ad 57 We prove 55 i the followig lea, 56 i Lea V3 ad 57 i Lea V Lea V: As, li if log PE E E3 log Proof: We boud the rate of iterest fro below by li log PE 3 = li log P = li log P i= =r+ =r+ Z 0 [ r ] + A ji A i < 0 j=0 We a follow the proof of Setio V-A with A = {0,, r ad B = {r,, see eg 9 with A B = This results i log PE 3 log whe r / r r + log 3 +,

16 6 The strategy of the proof is first to haraterize the behaviour for ad the showig that r gives the desired result Lea V3: For r fixed, li log PE E Proof: We fous o r log P r + r,j=0 + r =r j=0 log PE 59 A i A ji 0, i= A i A ji 0 i= Usig Craér s Theore ad ivoig the otatio = {0,, r ad 0 + = {r,, gives that the rate above is give by if log E e tr/+ss + t S r 0 t,t 0 Sie S ad S + are idepedet, we a boud the rate 0 of iterest fro above by if e tr/ osht S r E e ts r 60 t,t 0 log It is suffiiet to prove that for all δ > 0, t δ as Ideed, sie osh x, the rate o the right had-side is bouded of 60 fro above by if e log tr/ osht S r E e ts r t δ,t 0 if e log δr/ E e ts r t 0 = δr if t log E e ts r = δr li log PE Sie δ is arbitrary sall ad r is fixed, the desired stateet the follows It is hee suffiiet to prove that uder t δ, the ifiu a ot be attaied for suffiietly large We prove this stateet by otraditio Assue that the ifiu is attaied for a t δ Sie r is odd, osht j osht e t /, so that e tr/ osht j r r e t r/ e r t r Substitutig the result above gives if e log tr/ osht j r E e ts r t δ,t 0 if e log r log osht j r E e ts r t δ,t 0 if e log r log osh δ r E e ts r t 0 = if t 0 log E ets r + r log r log osh δ Sie oshδ >, the last ter teds to for Thus, assuig t δ leads to log PE < 0, whih is a otraditio To oplete the proof, we let r Lea V: For r, log PE log + o Proof: We show li if log E r t ets r r = 0 log The first step is to show that wheever t /r, ht = E e ts r r > We the a olude that the ifiu is ot attaied, sie h0 = < ht for t /r Sie ht is a oet-geeratig futio, it is log-ovex, so that it suffies to show that h /r > Ideed, for all t /r, there exists a α 0, ], suh that αt = /r It ow follows that < hαt = h αt + α 0 αht + αh0 = αht + α, so that ht > I order to prove that h /r >, we observe that e r S r r Sr = e r D e Z, where Z d = N 0, Furtherore, sie E e α r S r r e α < for all α >, it the follows fro [5] Exaple 705, that as r, Sr E e r E e Z = e 565 > 5 Ideed, the ifiu is ot attaied Fially, usig that t = β/r for soe β [0,, we a agai use the arguet above to olude that Sr β β E e r e β E e β βz = + β Miiizig over β gives β = /, resultig i li if log E r t ets r r = log e/ = 0 log, whih is the desired result This opletes the proof of Theore IIIb i the ase of equal powers VI Colusios Ivestigated is a DS-CDMA syste with HD-PIC The odel iorporates iterferig users with uequal powers The proessig gai is deoted by ad the uber of users by Large deviatio theory is used to obtai qualitative stateets oerig the perforae of

17 7 the syste The expoetial rate has bee prove to be a good easure of quality For the MF syste, the expoetial rate is asyptotially equivalet with the well-ow sigal-to-oise ratio We ote that for the HD-PIC syste the sigal-to-oise ratio is ot a good easure, sie there is o Gaussia behaviour for large However, the rate is ot based o Gaussia assuptios ad this allows us to draw qualitative olusios oerig the behaviour of HD-PIC systes by usig the rate of the BEP We have also ivestigated the asyptoti expoetial rate of the BEP for ultistage HD-PIC as the uber of users teds to ifiity The asyptoti expoetial rate of the BEP shows that HD-PIC eeps ireasig perforae as the uber of users ireases! We have show that the expoetial rate of the BEP reais uhaged after a fiite uber of HD-PIC stages The uber of stages of HD-PIC after whih this happes is for 3 up to 9 users, ad for 0 up to users whe all powers are equal This is the first aalytial proof of this effet ofte observed i pratie We have show that the expoetial rate of the BEP reais uiforly positive as the uber of users ireases Uder ild oditios o the powers, we have idetified the liitig expoetial rate of the BEP after applyig suffiietly stages of HD-PIC as log whe the uber of users ireases The expoetial rate have bee used to give Cheroff bouds The boud is ot tight i the sese that it approxiates the BEP with ay desired preisio Istead, it gives a upper boud that is valid for all ad, aig it a robust upper boud For s fixed, the Cheroff boud is valid as log as = o log, whereas for the optial HD-PIC syste, it is valid as log as = o The aalysis i this paper aswers ay questios Nevertheless, ay other questios reai ope We try to suarize the ost iportat oes: What happes to the results whe AWGN ad uequal powers are osidered i the ase where we apply ore tha oe stage of HD-PIC? We expet that the result will be geeralized as i Theore II The proof of Theore II has bee exteded to ilude AWGN i [] Ca we ilude AWGN i the proof for the optial HD- PIC syste? We believe that we a prove upper ad lower bouds o the asyptoti expoetial rate of the BEP usig the ethods i this paper, but that these bouds are ot sharp I order to prove asyptotis, ew ideas will eed to be developed 3 It is well ow that e I is ot a good approxiatio to p whe log p I Ca we opute the seod order asyptotis, eg i the ase whe we apply oe-stage o HD-PIC? Ca we say ore about how ay iteratios of HD-PIC are eessary to obtai the optial syste? For istae, a we prove Cojeture III5? Appedix A Proof of Theore IIIb for uequal powers We reall that we have the followig oditio o the powers P,, P : P There exists a δ > 0 s t {j : P j [δ, /δ], P li P <, P 3 P 0, P P i, where P = P, P i = i P The result of the stateet is that uder these power oditios, we have that li H s = log Theore IIIa states that H s it is suffiiet to prove li H s log log, so that We will prove the theore siilar to the way we have prove the result for equal powers However, the derivatios will be ore tehial Siilarly to the ase i whih all powers are equal, we will fous o the evet { = = This speifies σ for all σ We will show the theore, usig a syetrizig arguet More preisely, we replae differet powers by the sae value I this ase, we a use exhageability, together with ovexity arguets i order to siplify the aalysis We ited to follow the strategy i Setio V-C as losely as possible We tae {j : P j [δ, /δ] ad we assue that is fixed ad odd ad is large Our startig poit is the probability P = = = P Z 0, i Z 0, Z 0, i Z 0, where Z = Z = P / + i= i= P / j= + j P / j A ji A i, 6 P / j A ji A i 6 Sie PA 0, B 0 PA B/ 0, B 0, we a boud the probability fro below by P Z Z 0, i Z Z 0, Z 0, i Z 0,