cs.IT 2008-10-02 0

Optimization of sequences in CDMA systems: a statistical-mechanics approach

Statistical mechanics approach is useful not only in analyzing macroscopic system performance of wireless communication systems, but also in discussing design problems of wireless communication systems. In this paper, we discuss a design problem of s…

Authors: Koichiro Kitagawa, Toshiyuki Tanaka

Optimization of sequences in CDMA systems : a statist ical-mec hanics appro ac h Koic hiro Kitagaw a a , T oshiyuki T a na k a a a Gr aduate Scho ol of Informatics, Kyoto University, Jap an. Abstract Statistical mec hanics approac h is useful not only in analyzing macroscopic system p erformance of wireless comm unication systems, but also in dis- cussing design problems of wireless comm unication systems. In this paper, w e discuss a design pro blem of spreading sequence s in co de-division m ultiple- access (CDMA) systems, as an example demonstrating the usefulness of sta- tistical mec hanics approac h. W e analyze, via replic a metho d, the av erage m u- tual information betw ee n inputs and outputs of a ra ndomly-spread CDMA c hannel, and discuss the optimization problem with the a v erage m utual in- formation as a measure of optimization. It has b een sho wn tha t the av erage m utual information is maximized b y o r t hogonally-in v arian t random W elc h b ound equality (WBE) spreading sequences. Key wor ds: co de-division multiple-acces s (CD MA), replica metho d, a v erage m utual information, la r ge system limit 1. In tro duction In recen t y ears, adv ances in information a nd communic at io n tec hnologies ha v e b een demanding hig h data-rate wireless comm unications. In order to realize high data-rate wireless commu nications, bandwidth of systems should b e a s w ide as possible, whic h means that suc h systems should ha ve a large degree of freedom. Those systems are also required to b e able t o op erate Email addr esses: kitagaw a@sys. i.kyoto-u.ac.jp (Koichiro K itagaw a), tt@i.k yoto- u.ac.jp (T oshiyuki T anak a) 1 Graduate Sc ho ol of Informatics, Ky o to Univ ersity , 36-1 Y oshida Hon-mac hi, Sakyo-ku, Kyoto-shi, K yoto 606 -8501 , J apan. Pr eprint submitte d t o Elsevier Octob er 29, 2018 efficien t ly ev en in bad and unce rta in en vironmen ts. F or e xample, in urban areas, there are man y obstacles, suc h as buildings, cars, and p eople, which in teract with wireless communic atio n systems as reflecting/scattering b o d- ies, thereb y making comm unication en vironment very complex. In analyzing wireless comm unication systems, therefore, one has to r ega rd them as sys- tems with very high dimensionalit y and randomness. This is why stat istical mec hanics a pproac h is exp ected to b e useful in studying wireless comm uni- cation systems. In this pap er, w e consider a problem arising f rom considerations of m ultiple- access c hannels. Ty pically , a wirele ss comm unication syste m has to accom- mo date m ultiple users simu lta neously . In suc h a system, signals coming from differen t users in terfere with eac h o ther. How to mitigate suc h m ultiple-access in terference (MAI) is one of the most imp ort an t problems in wireless commu- nications. Co de-division m ultiple-access (CDMA) [1, 2] provides an effectiv e sc heme to mitigate MAI, and is widely used in v arious commercial systems. In CDMA, an informat io n sy mbol o f a user is mo dulated with a spreading sequence assigned to the user. Receiv er has to estimate informatio n sym b ols based on receiv ed sequences by utilizing knowle dge of spreading sequenc es of the users. A con ve ntional choice to mitigate MAI is to use pseudorandom sequence s as the spreading sequence s. Although analysis o f such randomly-spread CDMA systems w as thought to b e difficult, it has turned out tha t r eplic a metho d , whic h is an analytical to ol deve lop ed in the research field o f statis- tical ph ysics of disordered systems (spin glasses), is ve ry useful for the anal- ysis [3, 4]. Mor e precisely , these studies ha v e reve aled tha t replica method allo ws us to ev aluate “macroscopic” p erformance of CDMA systems with indep enden t a nd iden tically-distributed (i.i.d.) rando m spreading sequences in the large-system limit, suc h as mutual informat io n b etw een inputs a nd outputs, bit erro r rate, and so o n. Since theoretical p erformance of CDMA sys tems is affected by c hoices of spreading sequenc es, de sign of spreading sequences is an imp ortant pro b- lem in CDMA. There hav e b een sev eral res earche s in whic h the problem of designing spreading sequence s is fo r mulated in terms of an optimization problem. F or example, Rupf and Massey [5] discussed optimization of spread- ing sequences with the channe l capacit y of CDMA systems as a measure of optimization. They show ed that so-called W elc h b ound equality (WBE) spreading sequences , whic h minimize the tota l squared correlation (TSC) of spreading sequences , ac hiev e the c hannel capacit y . 2 W e restrict ourselv es to considering the optimization problem of spreading sequence s of CDMA systems with t he m utual information b etw een inputs and outputs as the measure of optimizatio n. There ha v e b een sev eral researc hes in whic h suc h optimizatio n pro blems are discussed [5, 6, 7]. The y ha v e assumed inputs of the system to b e Gaussian distributed and discuss ed theoretical up- p er b ound (i.e., channel capacit y) o f m utual information b et w een inputs and outputs. If one wishes to consider realistic wireless comm unication systems, ho w ev er, it is import an t to study the optimization problem under the as- sumption of non-Ga ussian inputs. The ob jectiv e o f this pap er is therefore to discuss the optimization problem of spreading sequences of CDMA systems when one allows non-G aussian inputs. W e would lik e to emphasize that, un- lik e previous statistical-mec hanics studie s of CDMA systems [3, 4, 8] whose ob jectiv es are basically to analyze macroscopic system p erfo r ma nce, w e sho w in this pap er that the statistical-mec hanics approac h is also useful in dealing with design problems in wireless communication, with the optimization prob- lem of spreading sequenc es of CDMA systems as a de monstrative example. A digest ve rsion of this pap er has b een presen ted as a conference pap er [9 ]. 2. Problem W e consider t he follow ing real-v alued K - user CDMA c hannel mo del, y µ = 1 √ L K X k =1 s µk x k + σ n µ , µ = 1 , . . . , L, (1) where x k is an information sym b o l of user k . W e assume that { x k ; k = 1 , · · · , K } are i.i.d. ra ndom v ariables, and let p ( · ) b e the prior probabilit y of x k , whose mean a nd v ariance ar e assumed to be zero and one, respectiv ely . { s µk ; µ = 1 , · · · , L } is the spreading sequence of user k in the sym b o l interv al of in terest, and L denotes the spreading factor of the CDMA c hannel mo del. W e a ssume that the p o wer of the spreading sequences is normalized to one, so that P L µ =1 ( s µk / √ L ) 2 = 1 holds for k = 1 , · · · , K . W e assume additive white Gaussian noise (A W GN): n µ ∼ N (0 , 1) s o that σ 2 is the v ariance of A W GN. Let us intro duce the following notat io ns: y ≡ [ y 1 , · · · , y L ] T , n ≡ [ n 1 , · · · , n L ] T , x ≡ [ x 1 , · · · , x K ] T , and S = ( S µk ), S µk ≡ (1 / √ L ) s µk ; k = 1 , · · · , K ; µ = 1 , · · · , L . The system mo del (1) is then rewritten a s y = S x + σ n . (2) 3 One can consider a maximization problem of p er user m utual informatio n b et w een x and y with resp ect to the spreading sequences S , with the channel input x drawn from the probability distribution p ( x ) = Q k p ( x k ), C user = 1 K I ( x ; y ) | S , (3) where the notation I ( x ; y ) | S denotes the m utual information b et w een x and y whe n S is sp ecified. When K ≤ L , the mutual information C user is maxi- mized b y a ssigning to all users o rthogonal L -dimensional v ectors as spreading sequence s, regar dless o f the input distribution p ( · ). When K > L , on the other hand, spreading seque nces maximizing the m utual information C user are not trivial. When x k , k = 1 , · · · , K , are i.i.d. standard Gaussian ran- dom v ariables, it is kno wn that the WBE spreading sequences maximize t he m utual information C user [5]. WBE spreading sequences are characterized as [10, 11] S S T = β I L × L , β ≡ K L > 1 , (4) where I L × L is an L -dimensional iden tity matrix. When one assumes Gaussian inputs, spreading sequences maximizing C user ha v e b een iden tified in more general system mo dels than (2). F or exam- ple, in a system mo del where one allow s the p ow er of inputs to b e different, Visw anath a nd Anan tharam [6] show ed that the m utual information is max- imized b y assigning orthogonal spreading sequences to relativ ely high- p ow er users and s o- called generalized WBE spreading sequences to the remaining users, where the users are classified according to a certain criterion. Also, in a sys tem mo del whe re the inputs ma y arriv e async hronously , Luo et al. [7] sho w ed that t he m utual information is maximized by spreading sequences whic h can b e regarded as an extension of the ones whic h Visw anath and Anan tharam prop osed. On the other hand, when x k ’s are dra wn from a non-Gaussian dis tribu- tion, to the authors’ knowled ge, spreading sequences maximizing the m utual information (3) hav e not b een kno wn. W e analyze, via statistical mec hanics, spreading sequences maximizing the m utual information of the system with non-Gaussian inputs. Since the case with non-G aussian inputs is difficult to analyz e analytically , we resort to making sev eral assumptions. First, w e ev aluate m utual information in the large-system limit, in whic h the n um b er of users K and the spreading fa cto r L are both sen t to infinity while main- taining their ratio β = K /L constan t. Second, we assume random spreading. 4 More sp ecifically , w e a ssume that the sample correlation matrix R = S T S of r a ndom spreading sequence s S is asymptotically ort hogonally inv a rian t, that is, the proba bility la w of R and that of an orthogonal tra nsform U T RU are the same for any orthogonal matrix U in the large-system limit, and that empirical eigen v alue distribution of R con v erges to a limiting eigenv alue distribution ρ ( λ ) with a finite supp ort included in [ λ min , λ max ], in the large- system limit. Under the assumption of random spreading, w e consider the a v erage conditional mutual information in the large-system limit, C = lim K →∞ E S { C user } = lim K →∞ 1 K I ( x ; y | S ) , (5) where E S denotes exp ectation with resp ect to S , and where I ( x ; y | S ) is conditional mutual informatio n b etw een x and y give n S . W e discuss maxi- mization of C with resp ect to c har a cteristics of t he random matrix S . 3. Analysis 3.1. Evaluation of aver a ge mutual information via r eplic a metho d The av erage m utual info rmation (5) is decomp osed into t w o terms, C = lim K →∞ 1 K [ E y ,S { log p ( y | S ) } − E y , x ,S { log p ( y | x , S ) } ] (6) = F − 1 2 β  1 + log( 2 π σ 2 )  , (7) with F ≡ − lim K →∞ 1 K E y ,S { log p ( y | S ) } , (8) where E y ,S denotes exp ectation with respect to y and S . Direct calculation of the righ t-ha nd side of (8) is in general computationally in tractable. In order to ev aluate (8), w e in v oke the replica metho d. Substituting the iden tit y lim n → 0 ∂ ∂ n ( p ( y | S )) n = log p ( y | S ) (9) to the right-hand side of (8), w e obta in F = − lim K →∞ 1 K lim n → 0 ∂ ∂ n log E y ,S { ( p ( y | S )) n } . (10) 5 W e assume that the limit with resp ect to K and the limit a nd the differen ti- ation with resp ect to n a re in terc hangeable without altering the final result, obtaining F = − lim n → 0 ∂ ∂ n lim K →∞ 1 K log E y ,S { ( p ( y | S )) n } . (11) The limit K → ∞ allo ws us t o apply the saddle-p o in t metho d to ev aluate a relev an t quan tity . W e apply replica t r ick in order to ev a luate (1 1), in whic h w e first ev aluate E y ,S { ( p ( y | S )) n } assuming that n is a non-negativ e in teger, and then p erform the limit and the differen tiation with resp ect to n , assuming that n is real. Ev aluation of (11) basically go es in a similar ma nner as [8]. Detailed analysis is describ ed in the app endix. Here, we only sho w the result. The a v erage m utual information in the large-system limit is given b y C = − 1 2 θ E − 1 2 G  − E σ 2  − 1 2 log 2 π θ − 1 2 − Z p ( u ; θ ) log p ( u ; θ ) d u, (12) where {E , θ } are parameters whose v alues are to b e determined later, and where p ( u ; θ ) is a probabilit y densit y function of output u of a scalar A WGN c hannel with 1 /θ the noise v ariance, whe n the c hannel input x is generated from the distribution p ( x ). The function G ( t ) is defined as G ( t ) = Z t 0 R ( z ) d z (13) where R ( z ) is the R-transform [12] of the limiting eigen v alue distribution ρ ( λ ) of the correlatio n matrix R , whic h is defined o n the basis of the Hilb ert transform 2 of ρ ( λ ), C ( γ ) = Z ρ ( λ ) γ − λ d λ, γ < λ min , (14) as C  R ( z ) + 1 z  = z . (15) 2 It s ho uld b e noted that the so-called Ca uch y transform is defined by the sa me formula as the Hilb ert transform (14), but with γ in the upp er half o f co mplex plane. 6 The parameters {E , θ } are to b e determined fr om the followin g saddle-p oint equations: E = E { ( x − h x i ) 2 ; θ } (16) θ = 1 σ 2 R  − E σ 2  (17) where h x i denotes p osterior mean estimate of the c hannel input x of the scalar A W GN c hannel in tro duced a b ov e, defined as h x i = Z x q θ 2 π e − θ ( u − x ) 2 / 2 p ( x ) d x Z q θ 2 π e − θ ( u − x ) 2 / 2 p ( x ) d x , (18) and where E in (16) denotes expectation with respect to the c hannel input x and output u of the scalar A W GN channel. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 P S f r a g r e p la c e m e n t s λ ρ MP ρ W B E C [b it / u s e r / H z ] S N R [d B ] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1 2 3 4 5 6 P S f r a g r e p la c e m e n t s λ ρ M P ρ WBE C [b it / u s e r / H z ] S N R [d B ] Figure 1: Eigen v alue distributions ρ MP (left fig ure) and ρ WBE (right figur e) with β = 1 . 5. Both distributions have probability w eig ht 1 − 1 /β at λ = 0. The a v erag e m utual information (12) dep ends on the limiting eigenv alue distribution ρ ( λ ) of R as w ell as the prior distribution p ( x k ). When we assume that s µk are i.i.d. random v a riables whose mean and v aria nce are zero and one, resp ective ly , our result is r educed to that obtained b y Guo and V erd ´ u [4 ]. In this cas e, the limiting eigen v alue distribution ρ ( λ ) is giv en b y the so-called Mar˘ cenk o -P astur law [13], ρ MP ( λ ) =  1 − 1 β  + δ ( λ ) + p ( λ − a ) + ( b − λ ) + 2 π β λ , (19) 7 where ( x ) + = max(0 , x ), and a = (1 − √ β ) 2 , b = (1 + √ β ) 2 (see figure 1), whose R-tra nsform is giv en by R MP ( z ) = 1 1 − β z . (20) Substituting (20) to (13), o ne can confirm the ab ov e-men tioned fact. Our analysis a lso includes the case of the system with WBE spreading sequences. Since the c haracteristic of WBE spreading sequences is expressed as (4 ), the correlation ma t r ix o f WBE spreading sequences has t r ivial zero eigen v alue and the eigenv alue λ = β , with multiplic ities ( K − L ) and L , resp ectiv ely (see figure 1), and therefore ρ WBE ( λ ) =  1 − 1 β  δ ( λ ) + 1 β δ ( λ − β ) , (21) whose R-tra nsform is giv en by R WBE ( z ) = 2 1 − β z + p ( β z − 1) 2 + 4 z . (22) One can ev aluat e, via R WBE ( z ), the mutual information when orthog o nally- in v aria n t random WBE spreading sequences a r e emplo y ed. In fig ur e 2, w e sho w a comparison of the m utual infor mation when the ab o v e tw o spreading sequence s are employ ed, a nd when probability distribution of { x k } is giv en by p ( x k ) = ( δ ( x k − 1) + δ ( x k + 1)) / 2, k = 1 , · · · , K . One can confirm that WBE spreading seque nces a chiev e higher m utual information than i.i.d. random spreading sequences do. 3.2. Optimizin g sp r e ading s e quenc es Choices of spreading se quences affect t he a verage m utual information C through the limiting eigen v alue distribution ρ ( λ ) of the correlation matr ix R . Then, w e regard the a v erage m utual information C as a functional of ρ ( λ ), and seek the eigen v a lue distribution ρ ∗ ( λ ) whic h maximizes the av erag e m utual information C . Hereafter, we consider the case of β > 1 since optimal spreading sequences in the case of β ≤ 1 ar e obviously orthogonal spreading sequence s. In optimizing C with resp ect to ρ ( λ ), the fo llowing t w o constrain ts should b e imp osed on ρ ( λ ): First, since β > 1 , the K × K matrix R ha s trivial zero eigen v alues with m ultiplicity ( K − L ). Second, since w e ha ve norma lized 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 -20 -15 -10 -5 0 5 10 15 20 Random WBE P S f r a g r e p la c e m e n t s λ ρ M P ρ W B E C [bit/ user/Hz] SNR [dB] Figure 2: The av er age mutual infor mation C when ρ WBE (WBE) and ρ MP (Random) are sp ecified, in ca se of β = 1 . 5. the p ow er of spreading sequences as P L µ =1 ( s µk / √ L ) 2 = 1 , k = 1 , · · · , K , the matrix S should satisfy T r S T S = K X k =1 λ k = K , (23) where { λ k } are the eigen v alues of K × K matrix S T S . In terms o f ρ ( λ ), the constrain t (2 3) is expressed as Z λ ρ ( λ ) d λ = 1 . (24) W e rewrite ρ ( λ ) in view of these constrain ts as ρ ( λ ) =  1 − 1 β  δ ( λ ) + 1 β π ( λ ) , (25) where π ( λ ) satisfies Z π ( λ ) d λ = 1 , (26) as the normalizatio n as a probability distribution, a nd Z λ π ( λ ) d λ = β , (27) 9 whic h corresp onds to the nor malizat io n of the p o w er o f spreading sequence s ( 2 4). In or der to discuss the extrem um of C with resp ect to ρ ( λ ), w e consider first-order p erturbations of C . Since the parameters affected by the p ertur- bation of ρ ( λ ) are { G ( t ) , E , θ } , the functional deriv ativ e of C with resp ect to ρ is expressed as δ C δ ρ = δ C δ G · δ G δ ρ + ∂ C ∂ E · δ E δ ρ + ∂ C ∂ θ · δ θ δ ρ . (28) Since {E , θ } should satisfy the saddle-p oint equations (16) and (17), the deriv ativ es of C with r esp ect to the pa r a meters {E , θ } should b e zero at the saddle p oint. Therefore, one can safely ignore the effects of p erturbatio ns via E and θ . Our next observ a tion is that, if one can find a n eigen v alue distribution whic h maximizes − (1 / 2) G ( −E /σ 2 ), whic h is the only term ha ving the first- order effect in (12), it also maximizes the av erage m utual information (12). W e rewrite − G ( −E /σ 2 ) as − G  − E σ 2  = − Z −E /σ 2 0 R ( z ) d z = Z 0 −E /σ 2 R ( z ) d z . (29) Since −E /σ 2 < 0 and R ( z ) > 0, one can mak e the follow ing statemen t: If there is a distribution ρ ∗ ( λ ) whose R- t ransform R ∗ ( z ) satisfies R ∗ ( z ) ≥ R ( z ) , for ∀ z ∈ ( −E / σ 2 , 0) , (30) for any R-t r ansform R ( z ) of the distribution ρ ( λ ) whic h satisfies the con- strain ts (25)–(27), ρ ∗ ( λ ) also maximizes − G ( −E /σ 2 ). W e summarize the ab ov e ar g umen ts in the next prop osition. Prop osition 1 If one c an find an eigenvalue distribution ρ ∗ ( λ ) which m ax- imizes R-tr ansform for ∀ z ∈ ( −E /σ 2 , 0) , ρ ∗ ( λ ) als o maximizes the aver age mutual information C . It should b e noted that the existence of ρ ∗ ( λ ) is not g uaran teed at this stage. Ho w ev er, in the follo wing, w e sho w tha t t here is a distribution whic h satisfies the condition of Prop osition 1. As a next step, we conv ert the optimization pro blem in terms of R- transform into the one in terms of Hilb ert tr a nsform. Since Hilb ert transform C ( γ ) is a monoto nically decreasing function of γ , and since Hilb ert tr a nsform 10 has the relation (15) with R-transform, it follo ws that R ( z ) + 1 /z also de- creases monotonically with respect to z . This fact leads us to the following statemen t: If one can find an eigen v alue distribution ρ ∗ ( λ ) which maximizes C ( γ ) for ∀ γ < λ min , and whic h satisfies (25)–(2 7 ), ρ ∗ ( λ ) also maximizes R ( z ) for ∀ z ∈ ( z min , 0), with z min = lim γ → λ min − 0 C ( γ ) . (31) Summarizing the argumen ts so far, w e can state the next prop osition. Prop osition 2 If one c an find an eigenvalue distribution ρ ∗ ( λ ) whose Hilb ert tr ansform C ∗ ( γ ) satisfies C ∗ ( γ ) ≥ C ( γ ) , ∀ γ < λ min , (32) for a n y eigenvalue distribution ρ ( λ ) with Hilb ert tr ansform C ( γ ) , which sat- isfies ( 2 5) – (27) , ρ ∗ ( λ ) also maxi m izes the aver a ge mutual informa tion C . F ollow ing the ab ov e proposition, w e consider the maximization problem of the Hilb ert transform for γ < λ min . Substituting (25) to (14) , t he Hilb ert transform C ( γ ) is rewritten as C ( γ ) =  1 − 1 β  1 γ + 1 β Z π ( λ ) γ − λ d λ. (33) Since the first term, whic h is deriv ed from the trivial zero eigen v alues, has no ro om for optimization, w e maximize the second term under the con- strain ts (26 ) and (27). Let us consider the following in tegral: Z π ( λ )  1 γ − λ − f ( λ )  d λ, (34) where f ( λ ) is a linear function tangen tial to 1 / ( γ − λ ) at λ = β . Since the function f ( λ ) is a linear function, the exp ectation of f ( λ ) with resp ect to π ( λ ) is constan t unde r the constrain ts (26 ) and (27). Th us, the quan tities (3 4) and (3 3 ) a r e maximized b y the same eigen v alue distribution. W e here con- sider a maximization problem of the ob jectiv e f unction ( 34) for γ < λ min with the constrain t ( 2 6), but wi thout the constrain t (27) . Since we hav e only to consider λ ∈ [ λ min , λ max ], we can assume λ > γ . Since 1 / ( γ − λ ) is con v ex up w ard in λ for λ > γ , one has 1 γ − λ − f ( λ ) ≤ 0 , λ > γ , (35) 11 where the equalit y holds if a nd o nly if λ = β . Then, the ob jectiv e func- tion (34) is maximized for γ < λ min b y the probability distribution π ( λ ) = δ ( λ − β ) . (36) Since the distribution (36) inciden tally satisfies the pow er constrain t (27), the distribution (36) is also a maximizer of the ob jectiv e function (34) with b oth of the constraints (26) a nd (27). Since the t w o functions (33) a nd (34) are maximized b y the same probabilit y distribution, the distribution (36) is also the optimal solution of the maximization problem of the Hilb ert trans- form. Thus , w e obtain the maximizer of the av erage m utual informatio n (1 2). Substituting (36) t o (25), one can confirm that t he optimal eigen v alue distri- bution is the one of WBE spreading sequence s ρ WBE , whic h is giv en b y (21). W e hav e so far shown that WBE spreading sequences are also asymptot- ically optimal in CDMA s ystems with a non- Gaussian input dis tribution in the large-system lim it. This finding is an extension o f the optimalit y result of WBE spreading sequence s fo r Gaussian-input CDMA systems. 4. Conclusion W e ha ve demonstrated that the statistical-mec hanics approach is useful not only in analyzing theoretical p erformance of wireless comm unication sys- tems but also in pro viding clu es to ho w to design them, via the pro blem of optimizing spreading sequenc es in CDMA systems. W e h av e ev aluated, via replica metho d, av erage m utual info r ma t ion b et w een input and output of the system in the large-system limit, and discussed the optimization problem of the a v erage m utual informa t ion in terms of characteris tics of random spread- ing seq uences. It has been sho wn that the a ve ra ge m utual information is maximized in the larg e-system limit b y orthogona lly- inv a rian t random WBE spreading sequences ev en whe n the inputs a re non-Gaussian. Altho ug h in this pap er w e ha v e only studied a f ully-sync hrono us CDMA mo del with p er- fect p ow er con trol, o ne can consider the same problem in more general CDMA systems , suc h as the one with unequal-p o w er users, and that is deferred to our future w or k. A. Details of replica analysis In this app endix, w e explain ho w to ev aluate F giv en by (1 1). F irst, w e calculate the expectation E y ,S { p ( y | S ) n } assuming that n is a non-negativ e 12 in teger. In tro ducing replicated random ve ctors x a = [ x a 1 , · · · , x aK ] T ∈ R K , a = 0 , · · · , n , whic h are dra wn from the same probabilit y distribution as x , w e rewrite E y ,S { p ( y | S ) n } as E S ( Z Z n Y a =0 p ( y | x a , S ) p ( x a )d x a d y ) . (37) P erforming t he in tegral with resp ect to y , we obtain E y ,S { ( p ( y | S )) n } = E x a ,S  exp  K 2 T r RV − L 2 log( n + 1) − nL 2 log(2 π σ 2 )   . (3 8) where K × K matrix V is given b y V = 1 ( n + 1) K σ 2 n X a =0 x a ! n X a =0 x a ! T − 1 K σ 2 n X a =0 x a x T a . (39) The exp ectation with resp ect to S can b e p erformed via the so-called Itzyks on- Zub er in tegral [14, 12] (see a lso [8 , 1 5]), since R = S T S is assumed orthogo- nally in v arian t and rank of V is a t most ( n + 1), as lim K →∞ 1 K log E S  exp  K 2 T r RV  = 1 2 T r G ( V ) , (40) where G ( x ) is the function defined in (13). Th us, we obtain the following equation, ignoring v anishing terms in the larg e- system limit, E y ,S { ( p ( y | S )) n } = E { x a }  exp  K 2 T r G ( V ) − L 2 log( n + 1) − nL 2 log(2 π σ 2 )  . (4 1) W e next tak e exp ectation of (41) with resp ect to { x a } . Since eigenv alues of the matrix V are functions of { x a } o nly through their inner pro ducts x a · x b , a, b = 0 , · · · , n , we rewrite t he exp ectation with resp ect t o { x a } in to the one with respect to the ( n + 1) × ( n + 1) matr ix Q = ( Q ab ) , Q ab = 1 K K X k =1 x ak x bk , (42) 13 as Z exp[ K G ( Q )] µ K ( Q )d Q, (43) where K G ( Q ) is the exp onen t of (41), G ( Q ) =  1 2 T r G ( V )  ( Q ) − 1 2 β log( n + 1) − n 2 β log(2 π σ 2 ) , (44) and where µ K is the following measure µ K ( Q ) = E { x a } ( n Y 0 ≤ a ≤ b δ K X k =1 x ak x bk − K Q ab !) . (45) Utilizing the saddle-p oin t metho d [3 , 4], w e ev aluate (43) in the limit K → ∞ as lim K →∞ 1 K log E { ( p ( y | S )) n } = sup Q {G ( Q ) − I ( Q ) } , (46) where I ( Q ) is the rat e function of the empirical means (42), defined via a Legendre transform a s I ( Q ) = sup ˜ Q " X 0 ≤ a ≤ b Q ab ˜ Q ab − log M ( ˜ Q ) # , (47) where ˜ Q = ( ˜ Q ab ) is an ( n + 1) × ( n + 1) symmetric matrix. The cum ulan t generating function log M ( ˜ Q ) o f { x a } is defined as log M ( ˜ Q ) = lo g E { x a } ( exp " X 0 ≤ a ≤ b ˜ Q ab x a x b #) . (48) In order to pro ceed further, we assume the so-called r eplic a symmetry : W e assume that the extrem um of (46) is in v arian t under exc hanges of the replica indexes. Under the assumption of replica symmetry , we in tro duce new parameters, Q aa = p, Q ab = q , a 6 = b. (49) Using these para meters, t he eigen v alues of V are expresse d as λ 1 = − p − q σ 2 , (50) 14 λ 2 = 0 , (51) whose m ultiplicities are n and ( K − n ), respectiv ely . Since G ( x ) is an analytic function and G (0) = 0, we can express G ( Q ) as G ( Q ) = n 2 G  − p − q σ 2  − 1 2 β log( n + 1) − n 2 β log(2 π σ 2 ) . (52) W e also apply the assumption of replica symmetry to ˜ Q as ˜ Q aa = c, ˜ Q ab = θ , a 6 = b, (5 3) and rewrite the rate function I ( Q ) as I ( Q ) = sup ˜ Q  ( n + 1) cp + n ( n + 1) 2 θ q − log E { x a }  exp " c n X a =0 x a + θ X 0 ≤ a

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