The Capacity Region of the Gaussian MIMO Broadcast Channel

Transcription

1 0-0 The Capacity Region of the Gaussian MIMO Broadcast Channel Hanan Weingarten, Yossef Steinberg and Shlomo Shamai (Shitz)

2 Outline Problem statement Background and preliminaries Capacity region of the degraded Gaussian vector broadcast channel Capacity region of the non-degraded Gaussian vector broadcast channel (without use of the DSM bound)

3 Problem Statement 2 y k = H k x + n k, k =,...,m x : Transmitted vector (t ). y k : Received vector by k th user (r k ). H k : Gain matrix between channel input and user k (r k t - fixed and known to both transmitter and receivers). n k N(0, N k ): Additive noise vector (r k ). Power constraint, P. C MIMO (P, N...m, H...m ) =?

4 Known Results - Degraded BC placemen Y 3 X P Y X P Y2 Y Y 2 In general, C depends only on P Y X and P Y2 X Stochastically degraded: If there exist Y and Y2 such that X Y Y2 form a Markov chain and such that P Y X = P Y X and P Y 2 X = P Y2 X C Deg = where U is an auxiliary r.v. P U,X,Y,Y 2 = P U,X P Y2,Y X { } R I(X; Y U), (R, R 2 ) R 2 I(U; Y 2 )

5 Known Results for Degraded BC Capacity Region - Scalar Gaussian BC 4 y = x + n, y 2 = x + n 2 n N(0, N ), n 2 N(0, N 2 ) (N 2 N ) C sclr = 0 α R ( (R, R 2 ) 2 log + αp ), N R 2 ( ) 2 log ( α)p + N 2 + αp Superposition coding and successive decoding

6 Known Results for Degraded BC Capacity Region - Scalar Gaussian BC 5 Scalar Gaussian BC is always stochastically degraded, yet the converse is not trivial Bergmans, IEEE IT, 974: Converse requires Entropy Power Inequality (EPI) Bergman s proof Can not be directly extended to vector Gaussian BCs

7 Downside 6 MIMO BC is not degraded in general. Capacity of a degraded Gaussian vector BC - Unknown.

8 Recent Results 7 Sum Capacity: Caire and Shamai, IEEE IT, 2003 Yu and Cioffi,ISIT 2002 Viswanath and Tse, IEEE IT, 2003 Vishwanath, Jindal and Goldsmith, IEEE IT, 2003 Degraded Same Marginal (DSM) Bound: Vishwanath, Kramer, Shamai, Jafar and Goldsmith, DIMACS 2003 Tse and Viswanath, DIMACS 2003

9 The Broadcast channel and DSM channel with π j = j j 8 y DSM n y DSM 2 H y n 2 H 2 y 2 x y DSM m n m H m y m

10 DSM Bound Results 9 Conclusion [Vishwanath et al 2003, Tse et al 2003]: If Gaussian coding is optimal for the degraded MIMO BC, then C BC (P, H...m ) = R DPC (P, H...m ). Conjecture [Vishwanath et al 2003, Tse et al 2003]: Gaussian inputs are optimal for the DSM channel (and hence also for the general).

11 Subclasses of the Gaussian MIMO BC - AMBC and ADBC 0 Aligned BC - AMBC: t = r = = r m and H = = H m = I y k = x + n k, k =,...,m Aligned and Degraded BC - ADBC: if N N 2 N m, then y = x + n, y k = y k + ñ k, Ñ k = E[ñ k ñ T k ] = N k N k k = 2, 3,...,m The capacity region of a general Gaussian MIMO BC can be obtained by a limit process on the capacity region of an AMBC as some of the eigenvalues of some of the noise covariances go to infinity.

12 Total Power Constraint Vs. Covariance Matrix Const (P S) Proposition. where C(P, N...m ) = S 0 tr{s} P C(S, N...m ) C(P, N...m ) - capacity region of an AMBC under a total power constraint, tr { E[x x T ] } P. C(S, N...m ) - capacity region of an AMBC under a covariance matrix constraint, E[x x T ] S. Note: Capacity results can be extended to per-antenna power constraints.

13 Capacity Region of the ADBC (Degraded, N N 2 N m ) 2 Gaussian superposition coding and successive decoding: R k Rk G (B...m, N...m ) = 2 log k i= B i + N k k, k =,...,m i= B i + N k B,...,B m : User power allocation. m matrices of size (t t) Gaussian rate region: R G (S, N...m ) = B i 0 i s.t. m i= B i S ( ) R G (B...m, N...m ),...,Rm(B G...m, N...m ) Theorem. Let C(S, N...m ) denote the capacity region of the ADBC, then C(S, N...m ) = R G (S, N...m ).

14 Proof Technique 3 Entropy Power Inequality (EPI). Use Bergmans (IEEE IT, 974) style proof on enhanced channel instead of the original channel. Note: Without an enhanced channel, extending Bergmans proof to the vector case is not trivial.

15 Entropy Power Inequality (EPI) 4 Shannon, Bell Sys. Tech. J., 948 Blachman, IEEE IT, 965 Let Z and Y be two independent random vectors, then e 2 n H(Y +Z) e 2 n H(Y ) + e 2 n H(Z) with equality iff Y N(K) and Z N(α K). Bergmans extension to EPI: where P Y,Z W = P Y W P Z. H(Y + Z W) n 2 log ( e 2 n H(Y W) + e 2 n H(Z))

16 Bergman s converse proof - Scalar case 5 Assume (R, R 2 ) / C sclr (R > 0 and R 2 > 0) be achievable, then R ( ) αp + 2 log N, N R 2 > ( ) P + 2 log N2 N 2 + αp for some 0 α. By Fano s Inequality: n I(W ; Y W 2 ) R 2 log n I(W 2; Y 2 ) R 2 > 2 log ( ) αp + N N ( P + N2 N 2 + αp ) () (2) Thus (from ()) n H(Y W 2 ) 2 log (2πe(αP + N )) (3)

17 Bergman s converse proof - Scalar case 6 Note: Y 2 = Y + Z where Z has IID elements N(0, N 2 N ) By EPI: H(Y + Z W 2 ) n 2 log ( e 2 n H(Y W 2 ) + e 2 n H(Z)) (by (3)) n H(Y 2 W 2 ) 2 log (2πe(αP + N 2)) (4) Thus, by Fano s inequality (by (2)): n H(Y 2) > log (2πe(P + N 2 )) In contradiction with upper bound on differential entropy!

18 Bergman s converse proof - Degraded vector channel (pitfalls) y = x + n y 2 = x + n 2 where n N(0, N ), n 2 N(0, N 2 ), (N 2 N 0) In Bergmans development: Scalar channel Vector Channel P S αp B( S) 2 log( ) 2 log det( ) 7 By EPI: n H(Y 2 W 2 ) t ( 2 log e 2 n H(Y W 2 ) {}}{ 2πe(B + N ) t + e 2 n H(Z) {}}{ 2πe(N 2 N ) t )

19 Bergman s converse proof - Degraded vector channel (pitfalls) 8 However, due to Minkowski inequality, t ( ) 2 log 2πe(B + N ) t + 2πe(N2 N ) t t ( ) 2 log 2πe(B + N 2 ) t Minkowski inequality: Let K 0, K 2 0 with equality iff K K 2. K + K 2 t K t + K2 t i.e. proof will hold if (B + N ) (N 2 N )

20 Preliminaries - Enhanced Channel 9 Definition (Enhanced Channel). An AMBC, N,...,N m, is an enhanced version of another AMBC, N,...,N m, if N k N k k =,...,m Clearly, R G (S, N...m ) R G (S, N...m) and C(S, N...m ) C(S, N...m)

21 ADBC - Enhanced Channel Existence Theorem Theorem 2. Consider an ADBC with noise covariances N,...,N m. If the power allocation B O,...,Bm O achieves a boundary point of R G (S, N...m ) under a covariance constraint, S 0, then there exists an enhanced ADBC with noise covariances N k such that k. (Enhanced Channel) 20 0 N k N k and N k N k+ 2. (Proportionality) for some α k 0. α k k Bj O + N k = N k+ N k j= 3. (Rate preservation and Optimality preservation) R G k (B O...m, N...m ) = R G k (B O...m, N...m)

22 Example Gaussian rate region of a two user 2 2 ADBC R G (S, N...m), tangential to R G (S, N...m ) at R = 0. R R G (S, N...m), departs from R G (S, N...m ) at R = R G (S, N...m ) R

23 Application of the Enhanced Channel Theorem 22 Assume (R, R 2 ) / R G (S, N...m ), then by Fano Inequality + Theorem 2 (Rate Preservation) + Information processing inequality: n I(W ; Ȳ W 2 ) n I(W ; Ȳ W 2 ) 2 log B O + N N = B O 2 log + N (5) N n I(W 2; Ȳ 2) n I(W 2; Ȳ 2 ) > 2 log S + N 2 B O + N 2 = 2 log S + N 2 B O + N 2 (6)

24 Capacity of the AMBC 23 DPC precoding in a given order π: R k R DPC π (k) (π, B...m, N...m ), k =,...,m where Rk DPC (π, B...m, N...m ) = 2 log k i= B π(i) + N π(k) k i= B π(i) + N π(k) where π: A given ordering of the users (a permutation of the set {,...,m}) B,...,B m : User power allocation. A set of positive semi-definite matrices (t t)

25 Capacity of the AMBC 24 DPC rate region: R DPC (S, N...m ) = co π Π B,...,B m s.t. B i 0, i m i= B i S ( R DPC π () (π, B...m, N...m ),..., R DPC π (m) (π, B...m, N...m ) ) Theorem 3. Let C(S, N...m ) denote the capacity region of the AMBC, then C(S, N...m ) = R DPC (S, N...m ).

26 AMBC - Enhanced Channel Existence Theorem 25 Theorem 4. Let { (R,...,R m ) R m m i= γ ir i = b } (γ i > 0, b > 0) be a supporting hyperplane of R DPC (S, N...m ). Then, there exists an enhanced ADBC, (Ñ,...,Ñ m) such that R G (S, N...m) is supported by the same hyperplane. 2.5 DPC rate region of a two user 4 4 AMBC 2 Supporting Hyperplane.5 R G (S, N...m) R2 R DPC (S, N...m ) R

27 Converse for AMBC 26 3 DPC rate region of a two user 4 4 AMBC R2.5 R DPC (S, N...m ) External point R

28 Converse for AMBC 27 3 DPC rate region of a two user 4 4 AMBC 2.5 Supporting Hyperplane 2 R2.5 R DPC (S, N...m ) External point R

29 Converse for AMBC 28 3 DPC rate region of a two user 4 4 AMBC 2.5 Supporting Hyperplane 2 R G (S, N...m) R2.5 R DPC (S, N...m ) External point R

30 Converse for AMBC 29 3 DPC rate region of a two user 4 4 AMBC 2.5 Supporting Hyperplane 2 R G (S, N...m) = C(S, N...m) C(S, N...m ) R2.5 R DPC (S, N...m ) External point R