Microscopic Analysis for Decoupling Principle of Linear Vector Channel

Microscopic Analysis for Decoupling Principle of Linear V ector Channel Kazutaka Nakamura and T oshiyuki T anaka Departmen t of Sy stems Scie nce, G raduate Sch ool o f I nform atics, Kyoto Univ ersity 36-1, Y oshida-Honmach i, Sak yo-ku , K yoto, 6 06-85 01, Japan Email: { kna kamur, tt } @i.kyoto-u.a c.jp Abstract —This paper studies the decoupling p rinciple of a linear vec tor channel, which is an extension of CDMA and MIMO channels. W e show that the scalar -channel characterizatio n ob- tained via the decouplin g principle is va lid not only for collections of a l arge number of elements of input vector , as discu ssed in prev ious studies, but also for individual elements of inpu t vector , i.e. th e linear vector channel for in dividual elements of channel input vector is decomposed into a b ank of in dependent scalar Gaussian ch annels in the large-system limit, where dimensions of chann el input and output are both sent to infi nity while their ratio fixed. I . I N T R O D U C T I O N Recently , the replica method, developed in statistical me- chanics, has been applied to pro blems of perform ance ev alua- tion of various digital wireless commu nication systems, e spe- cially co de-division multiple-access (CDMA) and m ulti-inpu t multi-outp ut (MI MO) systems [1 ]–[4] . The replica meth od provides us with a d escription o f these c hannels, called , the decoup ling princip le ; that is, a CDMA channel, or eq uiv alently a M IMO channel, is d ecoupled , under a cer tain random ness assumption of the c hannel, into a bank of indep endent scalar Gaussian chan nels in the large-system limit, where dimen sions of chan nel input and o utput are both sent to infin ity while their ratio fixed. Existing r esults of replica an alysis, however , rely o n saddle- point evaluation of integrals, which is only valid for evaluating macr oscop ic q uantities, such as an em pirical mean of m any micr oscopic quantities, such as individual elements o f inp ut, which are many in the sense that their num ber g oes to infinity as the dimension s of the system in the large-system lim it. It is ther efore not clear as to wheth er the scalar -chann el characterizatio n of CDMA or MIMO channels ob tained via the replica analysis is still valid if we ar e interested in microsco pic quantities in the la rge-system limit. In th is p aper we show that the scalar-channel ch aracteriza- tion is still valid for m icroscopic qu antities, by perfo rming replica analysis on a linear vector channel, which is an extension of CDMA or MIMO chan nels. I I . L I N E A R V E C T O R C H A N N E L W e consid er a K -in put N -outpu t linear vector c hannel, defined as follows. Let x 0 = ( x 01 , . . . , x 0 K ) T denote the input vector of the c hannel, and y = ( y 1 , . . . , y N ) T denote the outpu t vector giv en a linear transform H x 0 of the inp uts, where H is an N × K channel matrix. Assuming the channel to be m emory less, the input-o utput character istic of the linear vector chann el is repr esented as P 0 ( y | H x 0 ) = N Y µ =1 ρ 0 y µ      h T µ x 0 √ N ! , (1) where h T µ / √ N den otes µ th row o f H . W e define a tr ue prior as P 0 ( x ) . Inf erence of the input vector x 0 , given the ou tput vector y and th e channe l matrix H , can be solved by a detection sch eme b ased on Bayesian infer ence. The detector assumes a chann el mo del to be P ( y | H x ) = Q N µ =1 ρ ( y µ | h T µ x / √ N ) , and a pr ior d istribution to b e P ( x ) . W e also assume perfect ch annel state in formatio n at the detector . Th ese a ssumptions yield the p osterior d istribution P ( x | y , H ) = P ( y | H x ) P ( x ) R P ( y | H x ) P ( x ) d x . (2) The posterio r me an estimato r (PME) ¯ x = R x P ( x | y , H ) d x is th e op timal in ference schem e to minimize th e mea n squ ared error, if the assume d mo del is matc hed to the true m odel. In this paper, we study joint distributions o f L ( ≪ K ) elements of input vector and their estimates based on the posterior distribution ( 2), given a ch annel matrix H . W ithou t loss of generality we consider the first L elements of inp ut vector , x L 0 = ( x 01 , . . . , x 0 L ) T , and their estimates x L . The joint d istribution to be stud ied is thus P ( x L 0 , x L | H ) = Z P ( x L | y , H ) P 0 ( y | H x 0 ) P 0 ( x 0 ) d x \ L 0 . (3) W e assume the channel matrix H to be rand om and ev aluate expectation of P ( x L 0 , x L | H ) over H in th e large- system limit where K , N → ∞ while β = K/ N is kept finite: P ( x L 0 , x L ) = lim K, N →∞ E H  P ( x L 0 , x L | H )  . (4) E u [ · · · ] denotes expectation over the rand om variable u . Note that if the scalar-chann el ch aracterization is der i ved fo r the joint distribution (4) using th e replica method, it is easy to show the scalar-channel characterizatio n is still valid for arbitrary microscop ic quantities dep end o n x L 0 and x L . T o simplify the analysis, we assume the following: • Rand om ch annel ma trix: T he elements { h µk } ar e in- depend ent and iden tically d istributed (i.i.d. ) with mean zero, u nit variance, odd-o rder moments being zero and (2 m ) th-or der moments being finite. • Th e first L elements of input vector x L 0 and the remaining elements x \ L 0 = ( x 0( L +1) , . . . , x 0 K ) T are ind epende nt, so that th e prior d istribution of x is factorized as P 0 ( x 0 ) = P L 0 ( x L 0 ) P \ L 0 ( x \ L 0 ) . (5) The factorized f orm P ( x ) = P L ( x L ) P \ L ( x \ L ) is also used as th e postulated p rior distribution. • Th e conditional d istributions ρ 0 ( y | u ) and ρ ( y | u ) are one and two tim es differentiable with respect to u , respec- ti vely . I I I . M A I N R E S U LT Our m ain r esult is th e following claim. Claim 1: In the large-system limit and under the assump - tion of rep lica sym metry (see Sect. I V), the joint d istribution P ( x L 0 , x L ) d efined in (4) is asymptotically e quiv alen t to the joint d istribution P ( x L 0 , x L ) = Z Q L k =1 ρ G ( z k | x k ) ˜ P L ( x L ) R Q L k =1 ρ G ( z k | x k ) ˜ P L ( x L ) d x L × L Y k =1 ρ G 0 ( z k | x 0 k ) P 0 ( x L 0 ) d z L , (6) where ρ G 0 ( z | x ) and ρ G ( z | x ) rep resent inpu t-outpu t character- istics of the scalar Ga ussian chann els ρ G 0 ( z | x ) = r E 2 2 π F exp  − E 2 ( z − x ) 2 2 F  , (7) ρ G ( z | x ) = r E 2 π exp  − E ( z − x ) 2 2  , (8) respectively , an d wh ere z L = ( z 1 , . . . , z L ) T . ˜ P L ( x L ) is a “modulate d” version of the assumed prior, defined as ˜ P L ( x L ) = exp h G − F + E 2     x L     2 i P L ( x L ) R exp h G − F + E 2     x L     2 i P L ( x L ) d x L , ( 9) where     x     2 = x T x . The param eters { G, E , F } are d etermined by solving the following equation s for { G, E , F , r, m, q } , G = Z ¯ ρ 0 y      s β m 2 q t ! ¯ ρ ′′  y | √ β q t  ¯ ρ  y | √ β q t  D t dy , (10) E = Z ¯ ρ ′ 0 y      s β m 2 q t ! ¯ ρ ′  y | √ β q t  ¯ ρ  y | √ β q t  D t dy , (11) F = Z ¯ ρ 0 y      s β m 2 q t ! " ¯ ρ ′  y | √ β q t  ¯ ρ  y | √ β q t  # 2 D t dy , (12) r = lim K →∞ 1 K D D     h x i     2 E E , (13) m = lim K →∞ 1 K D D x T 0 h x i E E , (14) q = lim K →∞ 1 K D D     h x i     2 E E , (15) where R ( · · · ) D u = R ∞ −∞ ( · · · ) exp( − u 2 / 2) du/ √ 2 π . The distributions ¯ ρ 0 and ¯ ρ are defined as ¯ ρ 0 y      s β m 2 q t ! = Z ρ 0 y      s β m 2 q t + s β  r 0 − m 2 q  u ! D u, (16) ¯ ρ  y    p β q t  = Z ρ  y    p β q t + p β ( r − q ) u  D u, (17) respectively , where f ′ ( y | u ) = ∂ ∂ u f ( y | u ) , and wher e r 0 = lim K →∞ 1 K Z     x 0     2 P 0 ( x 0 ) d x 0 . (18) The bra ckets h h· · ·i i and h· · · i den ote the averages with resp ect to the jo int d istribution of x 0 and z = ( z 1 , . . . , z K ) T ,   · · ·   = Z Z ( · · · ) K Y k =1 ρ G 0 ( z k | x 0 k ) P 0 ( x 0 ) d z d x 0 , (19) and the p osterior distribution of x given z , h· · · i = R ( · · · ) Q K k =1 ρ G ( z k | x k ) ˜ P ( x ) R Q K k =1 ρ G ( z k | x k ) ˜ P ( x ) d x , (20) respectively , where ˜ P ( x ) = exp h G − F + E 2     x     2 i P ( x ) R exp h G − F + E 2     x     2 i P ( x ) d x . (21) If more th an one solu tion exists for (10)–(15), the co rrect solution is th e on e tha t m inimizes th e fu nction F defin ed as F = 1 β Z Z ¯ ρ 0 y      s β m 2 q t ! log ¯ ρ  y    p β q t  D t dy + 1 2 Gr − E m + 1 2 F q + F 2 E + 1 2 E r 0 − 1 2 log E 2 π + lim K →∞ 1 K Z Z K Y k =1 ρ G 0 ( z k | x 0 k ) P 0 ( x 0 ) × ( log Z K Y k =1 ρ G ( z k | x k ) ˜ P ( x ) d x ) d x 0 d z . (22) Detailed de riv ation o f the claim is giv e n in Section IV. The claim implies that the scalar-channel ch aracterization is valid for the joint distribution P ( x L 0 , x L ) , this is, the joint distribution P ( x L 0 , x L ) d efined in (4) can be asymptotically identified as the join t distribution of x L 0 and x L where the elements of x L 0 are indep endently transm itted over th e scalar Gaussian chann el ρ G 0 ( z | x ) and where th e detector p ostulates the ch annel model ρ G ( z | x ) and the m odulated version of the assumed p rior ˜ P ( x L ) (Fig. 1). This result is a finer version of the d ecoup ling p rinciple, which is first stated by Tse and Hanly [5], an d n amed by Guo and V erd ´ u [2]. Linear Vector Channel Detector PME . . . . . . . . . . . . . . . . . . postulates P S f r a g r e p l a c e m e n t s x 0 ∼ P 0 ( x 0 ) x 01 x 02 x 0 L x 0 K x 1 x 2 x L x K ˆ x 1 ˆ x 2 ˆ x L ˆ x K P 0 ( y | H x 0 ) y y P ( x | y ,H ) P ( x ) P ( y | H x ) (a) Scalar Gaussian Channels Detector PME . . . . . . . . . . . . . . . . . . . . . postulates P S f r a g r e p l a c e m e n t s x 0 ∼ P 0 ( x 0 ) x 01 x 02 x 0 L x 0 K x 1 x 2 x L x K ˆ x 1 ˆ x 2 ˆ x L ˆ x K P 0 ( y | H x 0 ) y y P ( x | y , H ) P ( x ) P ( y | H x ) x L 0 ∼ P L 0 ( x L 0 ) ρ G 0 ( z 1 | x 01 ) ρ G 0 ( z 2 | x 02 ) ρ G 0 ( z L | x 0 L ) ρ G ( z 1 | x 1 ) ρ G ( z 2 | x 2 ) ρ G ( z L | x L ) z 1 z 2 z L z 1 z 2 z L P ( x | z ) ˜ P L ( x L ) (b) Fig. 1. The linea r vector channel and the correspon ding detector (a). The bank of scalar Gaussian chann els and their corresponding detector (b). I V . D E R I V A T I O N O F T H E C L A I M A. Rep lica meth od W e ev aluate P ( x L 0 , x L ) d efined in (4) v ia replica method. Introd ucing a r eal numbe r n , ( 4) can b e rewritten as P ( x L 0 , x L ) = lim K, N →∞ lim n → 0 E H " Z  Z P ( y | H x ) P ( x ) d x \ L  ×  Z P ( y | H x ) P ( x ) d x  n − 1 P 0 ( y | H x 0 ) P 0 ( x 0 ) d x \ L 0 # . (23) According to the standa rd p rescription of replica metho d, we first evaluate Z n ( x L 0 , x L 1 ) = lim K, N →∞ E H " n Y a =0 ( Z P a ( y | H x a ) P a ( x a ) ) × d x \ L 0 d x \ L 1 n Y a =2 d x a # (24) for a positiv e integer n , where P a ( y | H x a ) = P ( y | H x a ) and P a ( x ) = P ( x ) for a = 1 , . . . , n , and then the result is co ntinuated to real n in ord er to take the limit n → 0 to obtain lim n → 0 Z n ( x L 0 , x L 1 )    x L 1 = x L = P ( x L 0 , x L ) . (25) Although there is no rigo rous justification for the re plica method, w e assume validity of th e replica meth od a nd r elated technique s throug hout this paper . B. A verage over chan nel matrix T o ev aluate (24), we first take th e a vera ge over th e channel matrix H . Using the assump tions o f ran dom channe l matrix and memory less channels, on e h as Z n ( x L 0 , x L 1 ) = lim K, N →∞ Z · · · Z ( E h " Z n Y a =0 ρ a y      h T x a √ N ! dy #) N × n Y a =0 P a ( x a ) d x \ L 0 d x \ L 1 n Y a =2 d x a , (26) where ρ a ( y | u ) = ρ ( y | u ) for a = 1 , . . . , n . W e let A = ( E h " Z n Y a =0 ρ a y      h T x a √ N ! dy #) N (27) and introdu ce auxiliary rando m variables v = ( v 0 , . . . , v n ) T , v a = h T x a / √ K . T he a verage over h in (27) can b e rewritten in terms of an integral over the condition al distribution o f v giv en { x a ; a = 0 . . . , n } , den oted by V ( v |{ x a } ) , as A = ( Z V ( v |{ x a } ) Z n Y a =0 ρ a  y | p β v a  dy d v ) N . (28) T o obtain an explicit expr ession for V ( v |{ x a } ) , we ev alu ate the ch aracteristic f unction of V ( v |{ x a } ) , as ˆ V ( ˆ v |{ x a } ) = Z e i ˆ v T v V ( v |{ x a } ) d v = e xp  − 1 2 ˆ v T Q ˆ v  ×    1 − 3 − κ 24 K n X a,b,c,d =0 W abcd ˆ v a ˆ v b ˆ v c ˆ v d + O  K − 2     , (29) where ˆ v = ( ˆ v 0 , . . . , ˆ v n ) T , where κ is fo urth-o rder momen t of h µk , and where ( n + 1) × ( n + 1) symmetric matrix Q an d fourth -order symmetric ten sor W are defined a s Q ab = 1 K K X k =1 x ak x bk (0 ≤ a ≤ b ≤ n ) , (30) W abcd = 1 K K X k =1 x ak x bk x ck x dk (0 ≤ a ≤ b ≤ c ≤ d ≤ n ) . (31) Note that in the ab ove we have to ev aluate ˆ V ( ˆ v |{ x a } ) up to O ( K − 1 ) terms. The in verse Fourier tran sform yields V ( v |{ x a } ) = V G ( v ) − 1 K V ∆ ( v ) + O ( K − 2 ) , (32) where V G ( v ) = h (2 π ) n +1 det ( Q ) i − 1 2 exp  − 1 2 v T Q − 1 v  , (33) V ∆ ( v ) = 3 − κ 24 n X a,b,c,d =0 W abcd ∂ 4 ∂ v a ∂ v b ∂ v c ∂ v d V G ( v ) . (34) Collecting these expressions, we have A = exp h N G 0 ( Q ) − G 1 ( Q, W ) + O  K − 1  i , (35) where G 0 ( Q ) = log Z V G ( v ) Z n Y a =0 ρ a  y | p β v a  dy d v , (36) G 1 ( Q, W ) = R V ∆ ( v ) R Q n a =0 ρ a  y | √ β v a  dy d v β R V G ( v ) R Q n a =0 ρ a  y | √ β v a  dy d v . (37) C. In te g ral over Q and W Since the q uantity A de pends on { x a } only th rough Q an d W , o ne can rewrite (26) in terms o f an integral over Q and W , as Z n ( x L 0 , x L 1 ) = lim K, N →∞ Z Z exp h N G 0 ( Q ) − G 1 ( Q, W ) + O  K − 1  i × µ K ( Q, W ; x L 0 , x L 1 ) dQ dW, (38) where µ K ( Q, W ; x L 0 , x L 1 ) = Z · · · Z Y 0 ≤ a ≤ b ≤ n δ Q ab − 1 K K X k =1 x ak x bk ! × Y 0 ≤ a ≤ b ≤ c ≤ d ≤ n δ W abcd − 1 K K X k =1 x ak x bk x ck x dk ! × n Y a =0 P a ( x a ) d x \ L 0 d x \ L 1 n Y a =2 d x a , (39) and dQ = Q 0 ≤ a ≤ b ≤ n dQ ab , dW = Q 0 ≤ a ≤ b ≤ c ≤ d ≤ n dW abcd . W e ev aluate (39) in the large- system limit by following the deriv atio n i n [6], [7]. W e in troduce parameters ˆ Q = { ˆ Q ab ; 0 ≤ a ≤ b ≤ n } and ˆ W = { ˆ W abcd ; 0 ≤ a ≤ b ≤ c ≤ d ≤ n } , which are conjugates to Q and W , respectively , an d d efine some f unctions o f them for later u se: Λ( ˆ Q, ˆ W ) = 1 K log Z · · · Z K Y k =1 exp " X 0 ≤ a ≤ b ≤ n ˆ Q ab x ak x bk + X 0 ≤ a ≤ b ≤ c ≤ d ≤ n ˆ W abcd x ak x bk x ck x dk # n Y a =0  P a ( x a ) d x a  , (40) λ x ( ˆ Q, ˆ W ; x L 0 , x L 1 ) = log Z · · · Z L Y k =1 exp " X 0 ≤ a ≤ b ≤ n ˆ Q ab x ak x bk + X 0 ≤ a ≤ b ≤ c ≤ d ≤ n ˆ W abcd x ak x bk x ck x dk # n Y a =0 P L a ( x L a ) n Y a =2 d x L a , (41) λ ( ˆ Q, ˆ W ) = log Z · · · Z L Y k =1 exp " X 0 ≤ a ≤ b ≤ n ˆ Q ab x ak x bk + X 0 ≤ a ≤ b ≤ c ≤ d ≤ n ˆ W abcd x ak x bk x ck x dk # n Y a =0  P L a ( x L a ) d x L a  . (42) W e fu rther assume that Λ( ˆ Q, ˆ W ) h as a limit as K → ∞ . Using the functio ns (40)–(42), the Fourier transform of (3 9) is g iv en by ˆ µ K ( ˆ Q, ˆ W ; x L 0 , x L 1 ) = exp " K Λ i ˆ Q K , i ˆ W K ! + λ x i ˆ Q K , i ˆ W K ; x L 0 , x L 1 ! − λ i ˆ Q K , i ˆ W K ! # , (43) and its inverse Fourier tran sform yield s µ K ( Q, W ; x L 0 , x L 1 ) =  K 2 π  { ( n +2 2 ) + ( n +4 4 ) } × Z Z exp h K n − iQ · ˆ Q − iW · ˆ W + Λ( i ˆ Q, i ˆ W ) oi × ex p h λ x ( i ˆ Q, i ˆ W ; x L 0 , x L 1 ) − λ ( i ˆ Q, i ˆ W ) i d ˆ Q d ˆ W , (44) where Q · ˆ Q and W · ˆ W are abbreviations of P 0 ≤ a ≤ b ≤ n Q ab ˆ Q ab and P 0 ≤ a ≤ b ≤ c ≤ d ≤ n W abcd ˆ W abcd , respectively . T o evaluate the integral over ˆ Q and ˆ W in (4 4), let ˆ Q ∗ = { ˆ Q ∗ ab ; 0 ≤ a ≤ b ≤ n } an d ˆ W ∗ = { ˆ W ∗ abcd ; 0 ≤ a ≤ b ≤ c ≤ d ≤ n } denote the solution of the equation s Q ab = ∂ Λ( ˆ Q, ˆ W ) ∂ ˆ Q ab , W abcd = ∂ Λ( ˆ Q, ˆ W ) ∂ ˆ W abcd . (45) Applying thr ee operatio ns to ( 44); a ch ange of variables i ˆ Q ab → i ˆ Q ab √ K + ˆ Q ∗ ab , i ˆ W abcd → i ˆ W abcd √ K + ˆ W ∗ abcd , (46) T aylor expansion of Λ , λ x and λ , and a change of integration paths to real axes, on e can find that the in tegral in ( 44) lead s to a Gau ssian integration. Then, one obtains µ K ( Q, W ; x L 0 , x L 1 ) =  K 2 π  1 2 { ( n +2 2 ) + ( n +4 4 ) } det  H (Λ | ˆ Q ∗ , ˆ W ∗ )  − 1 2 × ex p h K n − Q · ˆ Q ∗ − W · ˆ W ∗ + Λ ( ˆ Q ∗ , ˆ W ∗ ) o + λ x ( ˆ Q ∗ , ˆ W ∗ ; x L 0 , x L 1 ) − λ ( ˆ Q ∗ , ˆ W ∗ ) + O  K − 1 2  i , (47) where H ( f | u ∗ ) repr esents a Hessian matrix of the f unction f ( u ) at u = u ∗ . Use of Gaussian integration requires the Hessian matr ix H (Λ | ˆ Q ∗ , ˆ W ∗ ) b eing positive defin ite. No te that a similar evaluation is still possible when H (Λ | ˆ Q ∗ , ˆ W ∗ ) is n on-negative definite [ 8]. D. Sad dle-po int evaluation W e ev a luate the integral over Q and W in (38) via the saddle-po int method [9]. W e obtain Z n ( x L 0 , x L 1 ) = lim K, N →∞ D exp h K n F n ( Q ∗ , W ∗ ) − G 1 ( Q ∗ , W ∗ ) + λ x ( ˆ Q ∗ , ˆ W ∗ ; x L 0 , x L 1 ) − λ ( ˆ Q ∗ , ˆ W ∗ ) + O  K − 1  i , (48) where the f unction F n ( Q, W ) is defin ed as F n ( Q, W ) = 1 n  1 β G 0 ( Q ) − Q · ˆ Q ∗ − W · ˆ W ∗ + Λ ( ˆ Q ∗ , ˆ W ∗ )  (49) Note that ˆ Q ∗ and ˆ W ∗ depend on Q a nd W via (45). T he saddle points Q ∗ = { Q ∗ ab ; 0 ≤ a ≤ b ≤ n } and W ∗ = { W ∗ abcd ; 0 ≤ a ≤ b ≤ c ≤ d ≤ n } are d etermined as the solution o f ∂ F n ( Q, W ) ∂ Q ab = 0 , ∂ F n ( Q, W ) ∂ W abcd = 0 . (50) If more than one solution exists for (50), the corre ct solution is the on e that maximize s (49). The normalization factor D is giv en by D = h det  H (Λ | ˆ Q ∗ , ˆ W ∗ )  det  H ( − n F n | Q ∗ , W ∗ ) i − 1 2 . (51) Application o f the saddle-po int metho d here requir es that the Hessian matrix H ( − n F n | Q ∗ , W ∗ ) is positi ve definite. Since our final r esult will b e a function of x L 0 and x L 1 , we can ign ore terms in (48) which ar e indepen dent o f the se variables, obtaining Z n ( x L 0 , x L 1 ) ∝ exp h λ x ( ˆ Q ∗ , 0; x L 0 , x L 1 ) i . (52) Note that o ne obtain s ˆ W abcd = 0 by solving (50), an d that th e overall factor , which we hav e just ign ored, can be deter mined via norm alization. I t turns out, from ˆ W abcd = 0 , (45), and (50), th at Q ∗ and ˆ Q ∗ do n ot depen d on W ∗ . E. Rep lica symmetric an satz T o p roceed fur ther, we assume r eplica sym metry (RS) [1 0], under w hich w e let Q ∗ 00 = r 0 , Q ∗ aa = r, Q ∗ 0 a = m, Q ∗ ab = q , ˆ Q ∗ 00 = 1 2 G 0 , ˆ Q ∗ aa = 1 2 G, ˆ Q ∗ 0 a = E , ˆ Q ∗ ab = F , (5 3) for positi ve integers a < b . Then, F ≡ lim n → 0 F n ( Q, W ) is reduced to (22), and the saddle- point equations (45) and (50) become (10 )–(15), (18) a nd G 0 = 0 (For d etailed deriv atio n, see [10]). Notice that the conditio n f or the Hessian matrix H ( − n F n | Q ∗ , W ∗ ) b eing positiv e definite yie lds the de Almeida-T houless (A T) cond ition for local stability of RS solutions [11]. Inserting th e RS assum ption (5 3) into ( 52), o ne o btains Z n ( x L 0 , x L 1 ) ∝ Z " L Y k =1 ρ G ( z k | x 1 k ) e G − F + E 2     x L 1     2 P L 1 ( x L 1 ) # × " Z L Y k =1 ρ G ( z k | x k ) e G − F + E 2     x L     2 P L ( x L ) d x L # n − 1 × L Y k =1 n ρ G 0 ( z k | x 0 k ) e 1 2 ( nE z 2 k + G 0 x 2 0 k ) o P L 0 ( x L 0 ) d z L . (54) T aking the limit n → 0 , one finally arrives at ( 6). V . C O N C L U S I O N In this paper, we have con sidered the d ecouplin g princip le of the linear vector channe l. W e h ave shown that the scalar- channel characte rization obtained via decou pling principle is valid for the joint distributions of L ( ≪ K ) elements of input vector an d their e stimates based on the posterior probability , in the large-system limit. This implies that the scalar-chann el characterizatio n is valid not on ly for macrosco pic qu antities, but also for micr oscopic qu antities on the lin ear vector cha n- nel. A C K N O W L E D G M E N T The authors would lik e to acknowledge sup port from the Gr ant-in-Aid fo r Scientific Research on Pr iority Area s (No. 1807 9010 ), the Ministry of Education, Culture, Sports, Science an d T echno logy , Japan. R E F E R E N C E S [1] T . T anaka, “ A statistica l-mechan ics approach to large- system analysis of CDMA multiuser detec tors, ” IEEE Tr ans. Inf. Theory , vol. 48, no. 11, pp. 2888–2910, Nov . 2002. [2] D. Guo and S. V erd ´ u, “Randomly spread CDMA: Asym ptotic s via statisti cal physics, ” IEE E T rans. Inf. 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