The Dirty MIMO Multiple-Access Channel

The Dirty MIMO Multiple-Access Channel Anatoly Khina, Caltech Joint work with: Yuval Kochman, Hebrew University Uri Erez, Tel-Aviv University ISIT 2016 Barcelona, Catalonia, Spain July 12, 2016

Motivation: MIMO Dirty-Paper Channel

Motivation: MIMO Dirty Multiple-Access Channel

Multiple-Access Channel (MAC) Scenarios

Model Capacity region Single-Input Single-Output (SISO) MAC w 1 Encoder 1 x 1 z y Decoder ŵ 1,ŵ 2 w 2 Encoder 2 x 2 y = x 1 +x 2 +z x i Scalar input of Encoder i of power P i y Scalar output z White Gaussian noise N(0,1)

SISO Multiple-Access Channel Model Capacity region Capacity region [Ahleswede 71][Liao 72] R 1 1 2 log(1+p 1) R 2 1 2 log(1+p 2) R 1 +R 2 1 2 log(1+p 1 +P 2 ) Capacity region at high SNRs R 1 1 2 log(p 1) R 2 1 2 log(p 2) R 1 +R 2 1 2 log(p 1 +P 2 ) Sum-capacity strictly greater than individual capacities

The Dirty MAC Model Capacity region

SISO Dirty Multiple-Access Channel Model Capacity region s 1 z w 1 x 1 Encoder 1 y ŵ 1,ŵ 2 Decoder w 2 Encoder 2 x 2 s 2 y = x 1 +x 2 +s 1 +s 2 +z x i Scalar input of Encoder i of power P i s i Arbitrary side-information sequence known to Encoder i y Scalar output z White Gaussian noise N(0,1)

Model Capacity region SISO Dirty Multiple-Access Channel Random binning schemes are bad

SISO Dirty Multiple-Access Channel Model Capacity region Random binning schemes are bad Costa-like auxiliaries U i = X i +αs i achieve zero rate!

SISO Dirty Multiple-Access Channel Model Capacity region Random binning schemes are bad Costa-like auxiliaries U i = X i +αs i achieve zero rate! Structure can help!

SISO Dirty Multiple-Access Channel Model Capacity region Random binning schemes are bad Costa-like auxiliaries U i = X i +αs i achieve zero rate! Structure can help! Achievable rate region using lattices [Philosof et al. IT 11] R 1 +R 2 1 ( ) 1 2 log 2 +min{p 1,P 2 } Outer region [Philosof et al. IT 11] R 1 +R 2 1 2 log(1+min{p 1,P 2 }) DMAC capacity region at high SNR R 1 +R 2 1 2 log(min{p 1,P 2 }) MAC sum-capacity at high SNR R 1 +R 2 1 2 log(p 1 +P 2 )

Model Capacity region What about Multiple-Input Multiple-Output (MIMO)?

MIMO MAC Model Capacity region w 1 Encoder 1 x 1 H 1 z y Decoder ŵ 1,ŵ 2 w 2 Encoder 2 x 2 H 2 y = H 1 x 1 +H 2 x 2 + z x i N 1 input vector of Encoder i of power P i H i N N unit-determinant channel matrix of Encoder i y N 1 output vector z N 1 white Gaussian noise N(0,I N )

MIMO MAC Model Capacity region Capacity region [Yu PhD 02] tr{cx i } P i Cx i 0 (R 1,R 2 ) : R 1 1 I+H 2 log T 1 Cx 1 H T 1 R 2 1 2 log I+H T 2 Cx 2 H T 2 R 1 +R 2 1 2 log I+H T 1 Cx 1 H T 1 +H T 2 Cx 2 H T 2

MIMO MAC Model Capacity region Capacity is achieved using random coding Practical schemes based on matrix decompositions can be used SVD, QR decomposition, GMD,... Reduces MIMO coding task to coding over parallel SISO links Use good SISO codes to approach capacity

Dirty MIMO MAC Model Outer bound Direct SVD LQ JET Achievable s 1 z w 1 x 1 Encoder 1 H 1 y ŵ 1,ŵ 2 Decoder w 2 Encoder 2 H x 2 2 s 2 y = H 1 x 1 +H 2 x 2 + s 1 + s 2 + z x i N 1 input vector of Encoder i of power P i H i N N unit-determinant channel matrix of Encoder i s i Arbitrary side-information sequence known to Encoder i y output of length N z white Gaussian noise N(0,I N )

Dirty MIMO MAC Model Outer bound Direct SVD LQ JET Achievable Outer bound In SISO case: Sum-capacity { Individual capacities 1 R 1 +R 2 min 2 log(1+p 1), 1 } 2 log(1+p 2) Straightforward adaptation { for the MIMO case: 1 R 1 +R 2 min max Cx 1 :tr(cx 1 ) P 1 2 log I+H1 Cx 1 H T 1, 1 max Cx 1 :tr(cx 1 ) P 1 2 log I+H2 Cx 2 H T 2 High SNR: White inputs become optimal R 1 +R 2 1 { } 2 logmin P 1 N H 1H T 1, P 2 N H 2H T 2 = N ( ) 2 log min{p1,p 2 } N }

Dirty MIMO MAC Model Outer bound Direct SVD LQ JET Achievable How to approach the outer bound at high SNRs? Cannot use random coding/binning Achieves zero rate even in SISO case... Need to incorporate structure

Model Outer bound Direct SVD LQ JET Achievable Singular Value Decomposition (SVD) Based Scheme? H 1 = Q 1 D 1 V T 1 H 2 = Q 2 D 2 V T 2 Q i,v i Orthogonal matrices D i Diagonal matrices Problem 1: Common left operation at decoder Decoder matrix Q i depends on channel matrix H i But Q is shared by both users: Q T y = Q T H 1 V 1 x 1 +Q H 2 V 2 x 2 +Qs 1 +Qs 2 +Qz? =D 1? =D 2 Problem 2: Rate/SNR bottleneck problem Even if H i are diagonal Rate limited by minimal diagonal value

Bottleneck Problem: Example P 1 = 60, H 1 = ( ) 2 0 0 1/2 Dirty MIMO point-to-point setting R 1 1 2 log (30 1/2 2) 1.45 R 2 1 2 log (30 2 2) 3.45 R tot = R 1 +R 2 4.90 Model Outer bound Direct SVD LQ JET Achievable P 2 = 60, H 2 = ( ) 1/2 0 0 2 Dirty MIMO point-to-point setting R 1 1 2 log (30 2 2) 3.45 R 2 1 2 log (30 1/2 2) 1.45 R tot = R 1 +R 2 4.90 Dirty MIMO MAC setting R 1 1 { (min 2 log 30 2 2,30 1/2 2}) 1.45 R 2 1 { (min 2 log 30 1/2 2,30 2 2}) 1.45 R tot = R 1 +R 2 2 1.45 = 2.90 < 4.90

Bottleneck Problem: Example P 1 = 60, H 1 = ( ) 2 0 0 1/2 Dirty MIMO point-to-point setting R 1 1 2 log (30 1/2 2) 1.45 R 2 1 2 log (30 2 2) 3.45 R tot = R 1 +R 2 4.90 Model Outer bound Direct SVD LQ JET Achievable P 2 = 60, H 2 = ( ) 1/2 0 0 2 Dirty MIMO point-to-point setting R 1 1 2 log (30 2 2) 3.45 R 2 1 2 log (30 1/2 2) 1.45 R tot = R 1 +R 2 4.90 Dirty MIMO MAC setting R 1 1 { (min 2 log 3.529 2 2,56.471 1/2 2}) 1.91 R 2 1 { (min 2 log 56.471 1/2 2,3.529 2 2}) 1.91 R tot = R 1 +R 2 2 1.91 = 3.82 < 4.90

Model Outer bound Direct SVD LQ JET Achievable Resolving Problem 1 (Common Left Operation) X Joint diagonalization is not possible in general... LQ decomposition (QR decomposition transposed) H 1 = T 1 V T 1 H 2 = T 2 V T 2 T i Triangular matrices V i Orthogonal matrices Off-diagonal values treated as part of the side information Always possible No left matrix is needed Problem 1 solved! X But... diag(t 1 ) diag(t 2 ) Bottleneck problem! Can we make the diagonals equal?

Model Outer bound Direct SVD LQ JET Achievable Resolving Problem 1 (Common Left Operation) X Joint diagonalization is not possible in general... LQ decomposition (QR decomposition transposed) H 1 = I N T 1 V T 1 H 2 = I N T 2 V T 2 T i Triangular matrices V i Orthogonal matrices Off-diagonal values treated as part of the side information Always possible No left matrix is needed Problem 1 solved! X But... diag(t 1 ) diag(t 2 ) Bottleneck problem! Can we make the diagonals equal?

Resolving the Bottleneck Problem Model Outer bound Direct SVD LQ JET Achievable LQ decomposition did not use any left orthogonal matrix Use left orthogonal matrix to make diagonals equal! Joint Equi-diagonal Triangularization (JET) [Kh.-Kochman-Erez SP 12] H 1 = H 2 H 1 = QT 1 V1 T H 2 = QT 2 V2 T Q,V 1,V 2 Orthogonal matrices T 1,T 2 Triangular matrices diag(t 1 ) = diag(t 2 )

Resolving the Bottleneck Problem: Example Model Outer bound Direct SVD LQ JET Achievable H 1 = H 2 = ( ) 2 0 = 0 1/2 Q T 1 { (}} ){{ ( }} ) {{( }} ){ 1 1 1 1.458 0 0.243 0.970 2 1 1 1.286 0.686 0.970 0.243 Q ( ) { (}} ){ ( {}} ) {{( }} ){ 1/2 0 1 1 1 1.458 0 0.243 0.970 = 0 2 2 1 1 1.286 0.686 0.970 0.243 T 2 V T 2 V T 1

Achievable Region Model Outer bound Direct SVD LQ JET Achievable Apply the JET to the channel matrices: (t 1,...,t N ) diag(t 1) = diag(t 2) Achievable rate region All (R 1,R 2 ) satisfying: R 1 +R 2 N 2 log ( min{p1,p 2 } N ) Proof R 1 +R 2 = N (r 1;j +r 2;j ) j=1 N j=1 ( { 1 min P1 2 log tj 2,P 2tj 2 N = N 2 log ( min{p1,p 2 } N ) +log }) N j=1 t j = H i =1

Achievable Region Model Outer bound Direct SVD LQ JET Achievable Apply the JET to the channel matrices: (t 1,...,t N ) diag(t 1) = diag(t 2) Achievable rate region All (R 1,R 2 ) satisfying: R 1 +R 2 N 2 log ( min{p1,p 2 } N Outer bound at high SNRs Capacity region at high SNRs R 1 +R 2 N 2 log ( min{p1,p 2 } N All (R 1,R 2 ) satisfying: R 1 +R 2 N 2 log ( min{p1,p 2 } N ) ) )

K-user Dirty MAC K-User General matrices SISO capacity at high SNRs [Philosof et al. IT 11] K R i 1 ( ) 2 log min P i i=1,...,k i=1 MIMO outer bound at high SNRs Again, sum-capacity individual capacities K R i N min 2 log N i=1 i=1,...,k P i Achievable rate region Problem: JET not possible for K > 2 matrices, in general

K-User General matrices K-user JET via Space Time Coding [Kh.-Livni-Hitron-Erez IT 15] H i = QT i V i Bunch two channel uses together: H {}} i Q T ( ) {{( }} ){{}} i {{ ( )( }} ) { Hi 0 Q 0 Ti 0 V = i 0 0 H i 0 Q 0 T i 0 V i H i have a block-diagonal structure Use general Q, V i (not block-diagonal): H i ( {}} ) { Hi 0 0 H i = ( Q ) ( Ti ) ( Vi ) Exploiting off-diagonal 0s enables JET of more users! X Edge effect: Fixed number of unbalanced parallel channels Negligible by processing together large number of channel uses X V i X

K-user Dirty MIMO MAC K-User General matrices Achievable rate region All (R 1,...,R K ) satisfying: K R i N 2 log i=1 min i=1,...,k P i N

K-user Dirty MIMO MAC K-User General matrices Achievable rate region All (R 1,...,R K ) satisfying: K R i N 2 log i=1 min i=1,...,k P i N Outer bound at high SNRs K R i N 2 log i=1 min Capacity region at high SNRs K All (R 1,R 2 ) satisfying: R i N 2 log i=1 i=1,...,k P i N min i=1,...,k P i N

General Matrices K-User General matrices Thus far, we considered N N full-rank matrices What about the more general case? Case 1: Interference and signal live in same subspace y = H 1 (x 1 + s 1 )+H 2 (x 2 + s 2 )+z The case of the two base-station motivating example Simple extension of the discussed solution Case 2: Signal limited to subspace; interference is not y = H 1 x 1 +H 2 x 2 + s 1 + s 2 + z Constructing a tight outer bound is more challenging Sum-rate bounding by the individual capacities seems loose

General Matrices K-User General matrices Example [ ] [ ] 1 0 1 0 H 1 =, P 0 1 1 = 100, H 2 =, P 0 0 2 = 50 2 C 1 2 1 ( ) 100 2 log = 2 1 2 2 log(50) C 2 1 2 log(502 ) C 1 Current outer bound: R 1 +R 2 min{c 1,C 2 } 2 1 2 log(50) Clearly not achievable... Proposed inner bound 1 Project onto common subspace 2 Apply previous result In example: Projection y 1 +y 2 achieves R 1 +R 2 1 2 log(2 50)