' $ 6.962 Gr aduate Seminar in Communic ation Linear Mul tiuser Receivers: Effective Interference, Effective Band width and User Cap a city 6.962 Gradua te Seminar in Communica tion No vember 9, 2000 Presenter: C. Emre K oksal & %
' $ 6.962 Gr aduate Seminar in Communic ation Outline 1. In tro duction 2. Linear m ultiuser receiv ers 3. P erformance under random spreading sequences 4. User capacit y under p o w er con trol 5. Multiple classes and e�ectiv e bandwidths 6. Summary and �nal remarks & %
' $ 6.962 Gr aduate Seminar in Communic ation In tro duction Motiv ation � High demand for all kinds of applications o v er wireless { V arious qualit y of service (bit rate, probabilit y of error) requiremen ts { Can the system accomo date another user with a QoS constrain t? � Ho w to tak e adv an tage of the additional degrees of freedom pro vided b y spread-sp ectrum tec hniques. � A t the ph ysical la y er, signal to in terference ratio (SIR) is the k ey parameter. � Previous w ork: Not m uc h insigh t on ho w a user a�ects the system except in the w orst case. & %
' $ 6.962 Gr aduate Seminar in Communic ation System Mo del W e consider a sym b ol-sync hronous m ulti-access spread-sp ectrum system X 1 s 1 X 2 s 2 . K . demodulators . X K s K Receiv ed v ector, Y : K X Y = X s + W i i i =1 � 2 � User i : X is the transmitted signal ( E [ X ] = 0 ; E X = P ; X 's 2 < i i i are iid) N 2 s is the random spreading sequence and W N (0 ; � I ) 2 < � i & % Demo dulators : Mak e a go o d estimate (soft) on the transmitted sym b ols.
' $ 6.962 Gr aduate Seminar in Communic ation Mo del - Con tin ued � W e are in terested in the SIR $ rates (bits p er sym b ol). e.g., Gaussian input distribution ) 1 log (1 + SIR ) i 2 � Successiv e cancellation is another p ossibilit y ! Linear receiv ers Receiv er 1: K X ^ T T T T X = c Y = X c s + X c s + c W i i 1 1 1 1 1 1 1 i =2 T � 2 � E ( X c s ) 1 1 1 � SIR = = 1 1 K T � T � P ) � 2 E ( X 2 ( c c + c s ) i i 1 1 i =2 1 T 2 P ( c s ) 1 1 1 = K T T P ( c c ) � 2 + P ( c s ) 2 i i 1 1 i =2 1 & %
' $ 6.962 Gr aduate Seminar in Communic ation Linear Multiuser Receiv ers Matc hed �lter � { The �lter c = s : i i T T 2 s Y P ( s s ) 1 1 ^ 1 1 X ( Y ) = ; SIR = mf ; 1 1 T K s s T P T 2 2 ( s s ) � + P ( s s ) 1 i i 1 1 1 1 i =2 ^ { X is the pro jection of Y on s . mf ;i i Decorrelator � { In the matrix form, Y can b e written as Y = S X + W T where X = [ X X ] and S = [ s s ] � � � � � � K K 1 1 { matc hed �lter outputs R form su�cien t statistic for X : & T T % R = S S X + S W
' $ 6.962 Gr aduate Seminar in Communic ation T � 1 { Decorrelating �lter is ( S S ) in addition to matc hed �lter: T � 1 T � 1 T U = ( S S ) R = X + ( S S ) S W { Decorrelating receiv er for user i is tak es the pro jection of Y on to ? (span f ( s ) g ) (Do es not exploit correlation b et w een the j j 6 = i P terms of the in terference v ector sub optimal), SIR = 1 ) 1 P ii Minim um mean square error (MMSE) receiv er � { The total in terference for user 1 is: K X Z = X s + W i i i =2 { The co v ariance matrix of Z is: T 2 K = S D S + � I Z 1 1 1 where S is the N ( K 1) matrix of signature sequences of � � 1 T & % in terferers and D is the co v ariance matrix of [ X X ] . � � � 2 K
' $ 6.962 Gr aduate Seminar in Communic ation T { Eigen v alue decomp osition K = Q � Q . K > 0 and the ! Z Z 1 � whitening �lter for the in terference is � Q : 2 1 1 1 � � � � Q Y = X � Q s + � Q Z 2 2 2 1 1 { No w that in terference is white, apply matc hed �lter to get scalar 1 1 � � su�cien t statistic for X . Pro ject � Q Y along � Q s : 2 2 1 1 T � 1 T � 1 T � 1 R = s K Y = ( s K s ) X + s K Z 1 1 1 Z 1 Z 1 Z { Finally , the MMSE estimate is the linear least squares estimate (LLSE) of X giv en the observ ation R : 1 co v ( X ; R ) P R 1 1 X ( Y ) = R = mmse v ar( R ) 1 + P R 1 T � 1 P s K Y 1 1 Z = � 1 T 1 + P s K Y 1 Z 1 { The signal to in terference ratio for user 1 is: T � 1 2 ( s K s ) P 1 1 1 T � 1 Z S I R = = P s K s 1 1 1 1 Z � 1 T s K s & % 1 Z 1
' $ 6.962 Gr aduate Seminar in Communic ation P erformance Under Random Spreading Sequences 1 T � Spreading sequences: s = [ V � � � V ] , where V 's are iid p i i 1 iN ik N � 2 � 0 mean and v ariance 1 ) E k s k = 1. e.g., i 1 . . . user 1 -1 1 . . . user 2 -1 . . . . . . user K degrees of freedom 1 2 3 N K � W e are in terested in the case K ; N ! 1 ; = � . Assume that N asymptotically empirical distribution of the p o w ers of users 1 2 P X P F ( P (i.e., = ) con v erge to ) & % i i K
' $ 6.962 Gr aduate Seminar in Communic ation Matc hed Filter ( N ) Pr op osition 3.3: L et � b e the (r andom) SIR of the c onventional 1 ;M F matche d �lter r e c eiver for user 1. Then, with pr ob ability 1: P ( N ) � 1 � � = ! 1 ;M F 1 ;M F 2 � + � E [ P ] F Sketch of the Pr o of: By de�nition, T 2 P ( s s ) 1 1 1 � = 1 ;M F K T P T 2 2 ( s s ) � + P ( s s ) i i 1 1 1 i =2 T T Note that s s 1 w.p. 1 and expanding s s , it w as sho wn that ! i 1 1 1 h i K T 2 P v ar P ( s s ) j P ; P ; : : : = 0 ; realizations of P 's. Th us, 8 i i 1 2 i 1 i =2 " # K K K K 1 X X X T T 2 2 P ( s s ) = E P ( s s ) = P � E [ P ] ! i i i i i F 1 1 N K i =2 i =2 i =2 P P with probabilit y 1. Hence, N 1 , P( in terferer ) = P(in terferer ) ! i i P 1 � = 1 ;M F K 1 P & � 2 + P % i i =2 N
' $ 6.962 Gr aduate Seminar in Communic ation & %
' $ 6.962 Gr aduate Seminar in Communic ation MMSE Receiv er ( N ) The or em 3.1: L et � b e the (r andom) SIR of the MMSE r e c eiver 1 ;M M S E for user 1. Then, with pr ob ability 1: P ( N 1 ) � � � = ! 1 ;M M S E 1 ;M M S E � � 2 + � E [ I ( P ; P ; � )] F 1 1 ;M M S E where P P � 1 I ( P ; P ; � ) = 1 1 ;M M S E � P + P � 1 1 ;M M S E Notes on the Pr o of: W e will use the follo wing theorem due to Silv erstein and Bai ab out the limiting eigen v alue distribution of large matrices. X th ij Let A b e a n m matrix whose ( i; j ) en try is where X 's are p � n � m ij n iid with unit v ariance. Let T b e a m m diagonal matrix whose en tries � m T are real v alued random v ariables. The matrix A T A has real n � m m n � m non-negativ e eigen v alues with empirical distribution G ( � ). Note that n m G ( � ) is a random v ariable. As n; m 1 ; = � , G ( � ) approac hes to ! n n n & % a deterministic function, G ( � ).
' $ 6.962 Gr aduate Seminar in Communic ation Some Observ ations T 2 The co v ariance matrix K = S D S + � I has exactly the desired � Z 1 1 1 form. The asymptotic eigen v alue distribution, G ( � ) is not degenerate in � ) this asymptotic regime, in terference is not white. { If K w ere �nite and N 1 , in terference w ould b e white w.p. 1. ! { If N w ere �nite and K is increased, in terference will b e increasingly white. { If the in terference is white, matc hed �lter and MMSE receiv er K are iden tical. Th us, only if is constan t, the MMSE receiv er N outp erfroms the matc hed �lter. & %
' $ 6.962 Gr aduate Seminar in Communic ation Pro of - con tin ued One last complication left: Just the eigen v alue distribution ma y not � b e su�cien t for SIR c haracterization: T { K = U � U where � is diagonal and U is orthogonal for ev ery Z realization of Z. { Recall that T � 1 T � 1 � = P s K s = P ( U s ) � ( U s ) 1 :M M S E 1 1 1 1 1 1 Z th us the relativ e p osition of s wrt. eigen v ectors of K also 1 Z matters. { It is sho wn in Lemma 4.2 that as N 1 ; s is white in an y ! 1 co ordinate system and k U s k is constan t for an y realization of Z . Th us, � can b e c haracterized using only the eigen v alue � 1 ;M M S E distribution K . Z & %
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