Overview of Gaussian MIMO (Vector) BC

Transcription

1 Overview of Gaussia MIMO (Vector) BC Gwamo Ku Adaptive Sigal Processig ad Iformatio Theory Research Group Nov. 30, 2012

2 Outlie / Capacity Regio of Gaussia MIMO BC System Structure Kow Capacity Regios - Aliged ad Icosistetly Degraded MIMO BC Superpositio - Aliged MIMO BC without Commo Message Writig o Dirty Paper - Degraded Message Sets (A Commo & Oe Private Message) Duality of Gaussia MIMO BC & MAC Gaussia MIMO MAC Gaussia MIMO BC & MAC

3 Gaussia MIMO (Vector) BC 3/11 System Structure M 0, M 1, M 2 M 0 : A Commo Message M 1 : A Private Message to Rx. 1 M 2 : A Private Message to Rx. 2 Z 1 t : # Tx. At. G 1 X dim G Ecoder 1 dim G dim x 2 1 = t 1 G 2 Z 2 dim z 1 = r 1 Y 1 = r t = r t Y 2 Decoder 1 r : # Rx. At. dim y 1 = r 1 dim y 2 = r 1 r : # Rx. At. Decoder 2 dim z 2 = r 1 M 01, M 1 M 02, M 2 1 i=1 Power Costrait x T m 0, m 1, m 2, i x(m 0, m 1, m 2, i ) P m 0, m 1, m 2 1: 2 R 0 1: 2 R 1 [1: 2 R 2] chael Y 1 = G 1 X + Z 1 Y 2 = G 2 X + Z 2 Z 1 N(0, I r ) Z 2 N(0, I r )

4 Capacity Regio of Gaussia MIMO BC 3/11 Special Cases Kow Capacity Regio Aliged ad Icosistetly Degraded MIMO BC t = r, diagoal G 1, G 2 (G T 1 G 1 ad G T 2 G 2 have the same set of Eigevalue) : A Product of Gaussia BC Superpositio Codig Aliged MIMO BC (M 0 = 0) Oly Private Messages without a Commo Message Vector Writig o Dirty Paper Degraded a Private Message ad a Commo Message Either M 0 = 0 or M 0 = 0 Degraded Message Set Superpositio Codig

5 Case 1 : Gaussia Product BC Parallel Gaussia BCs Y 1k = X k + Z 1k Y 2k = X k + Z 2k k [1: r] Z jk N(0, N jk ) j = 1,2 M. I. Not Degraded, but Icosistetly Degraded BC Z 1 l N(0, N 1 ) Z 2 l N(0, N 2 N 1 ) X 1 l + Y 1 l + Y 2 l N 1k N 2k k [1: l] r Z 2,l+1 N(0, N 2 ) r Z 1,l+1 N(0, N 1 N 2 ) r X l+1 + r Y 2,l+1 + r Y 1,l+1 N 2k > N 1k k [l + 1: r]

6 Case 1 : Gaussia Product BC Capacity Regio R 0 + R 1 C β kp l k=1 N 1k r + C( k=l+1 α k β k P 1 α k β k P + N 1k ) R 0 + R 2 l k=1 C α k β k P 1 α k β k P + N 2k r + C β kp k=l+1 N 2k R 0 + R 1 + R 2 C β kp R 0 + R 1 + R 2 l k=1 l k=1 [C N 1k r + [C k=l+1 α k β k P + C β kp ] 1 α k β k P + N 2k N 1k α k β k P + C( 1 α k β k P )] 1 α k β k P + N 1k N 2k r + C( β kp ) N 2k k=l+1 For some α k, β k [0,1], k [1: r], with r k=1 β k = 1

7 Case 1 : Gaussia Product BC Rate Regio R 0 + R 1 I X 1 ; Y 11 + I(U 2 ; Y 12 ) R 0 + R 2 I X 2 ; Y 22 + I(U 1 ; Y 21 ) R 0 + R 1 + R 2 I X 1 ; Y 11 + I U 2 ; Y 12 + I X 2 ; Y 22 U 2 ) R 0 + R 1 + R 2 I X 2 ; Y 22 + I U 1 ; Y 21 + I X 1 ; Y 11 U 1 ) For some pmf p u 1, x 1 p( u 2, x 2 ) Achievability & Coverse Proof Superpositio Codig (Degraded Gaussia BC)

8 Case 1 : Gaussia Product BC Achievability Proof (X 1, p y 11 x 1 p y 21 y 11, Y 11 Y 21 ) X 1 p(y 11 x 1 ) Y 11 p(y 21 y 11 ) Y 21 X 2 p(y 22 x 2 ) Y 22 p(y 12 y 22 ) Y 12 (X 2, p y 22 x 2 p y 12 y 22, Y 12 Y 22 ) Rate Splittig Divide M j, j = 1,2 ito two idep. Messages : M j0 at rate R j0, M jj at rate R jj

9 Case 1 : Gaussia Product BC Codebook Geeratio Fix a pmf p u 1, x 1 p(u 2, x 2 ). Radomly ad idep. Geerate 2 (R 0+R 10 +R 20 ) sequece pairs u 1, u 2 m 0, m 10, m 20 accordig to i=1 p U1 u 1i p U2 (u 2i ) m 0, m 10, m 20 1: 2 R 0 1: 2 R 10 [1: 2 R 20] For m 0, m 10, m 20, radomly ad coditioally idep. Geerate 2 R jj sequeces Ecodig x j (m 0, m 10, m 20, m jj ) accordig to i=1 p Xj U j (x ji u ji (m 0, m 10, m 20 ) m jj [1: 2 R jj], j = 1,2 To sed the message triple m 0, m 1, m 2 = (m 0, m 10, m 11, m 20, m 22 ) Trasmit (x 1 m 0, m 10, m 20, m 11, x 2 m 0, m 10, m 20, m 22 )

10 Case 1 : Gaussia Product BC Decodig ad aalysis of the probability of error Decoder 1 : fid uique triple (m 01, m 10, m 11 ) such that ((u 1, u 2 )(m 01, m 10, m 11 ),x 1 m 01, m 10, m 20, m 11 ), y 1, y 2 T ε () For some m 10. Probability error for decoder 1 R 0 + R 1 + R 20 < I U 1, U 2, X 1 ; Y 11, Y 12 δ(ε) = I X 1 ; Y 11 + I U 2 ; Y 12 δ(ε) R 11 < I X 1 ; Y 11 U 1 Probability error for decoder 2 δ(ε) R 0 + R 10 + R 2 < I(X 2 ; Y 22 ) + I(U 1 ; Y 21 ) δ(ε) R 22 < I X 2 ; Y 22 U 2 δ(ε)

11 Case 2 : Private Messages Capacity Regio C = R WDP = co(r 1 R 2 ) R 1 : DPC with No-causal State X 2 R 1 < 1 2 log G 1K 1 G 1 T + I r R 2 < 1 2 log G 2K 2 G T 2 + G 2 K 1 G T 2 + I r G 2 K 1 G T 2 + I r R 2 : DPC with No-causal State X 1 R 1 < 1 2 log G 1K 1 G 1 T + G 1 K 2 G 1 T + I r G 1 K 2 G 1 T + I r R 2 < 1 2 log G 2K 2 G 2 T + I r

12 Vector Writig o Dirty Paper (1) Vector Writig o Dirty Paper S S N(0, K S ) Secod oise chael (AWGN) Z N(0, I r ) M Y W Ecoder Decoder Average power costrait P X C = Y = GX + S + Z max tr K X P 1 2 log G K XG T + I r

13 Vector Writig o Dirty Paper (2) Proof of Capacity C = 1 max tr K X P 2 log G K XG T + I r C = sup [ I U; Y I U; S ] p u s,x u s :E X T X P Let U = X + AS, where X N(0, K X ) is idepedet of S A = K X G T G K X G T + I r 1 I U; Y I U; S = h U S h(u Y) = h X + AS S h(x + AS Y) = h(x) h(x GX + Z) = I(X; GX + Z) = 1 2 log I r + G K X G T h X + AS Y = h(x + AS AY Y) = h(x + A(S Y) Y) = h(x + A(GX + Z) Y) = h(x + A(GX + Z)) = h(x + A(GX + Z) GX + Z) = h(x GX + Z)

14 Vector Writig o Dirty Paper (3) 3/11 R 1 M 1 M 1 - Ecoder X 1 M 2 M 2 -Ecoder X 2 X R 1 < I X 1 ; G 1 X 1 + Z 1 = 1 2 log G 1K 1 G 1 T + I r G 1 G 2 Z 1 Z 2 Y 1 Y 1 = G 1 X 1 + G 1 X 2 + Z 1 Y 2 = G 2 X 2 + G 2 X 1 + Z 2 Y 2 R 2 < I X 2 ; G 2 X 1 + G 2 X 2 + Z 2 = 1 2 log G 2K 1 G 2 T + G 2 K 2 G 2 T + I r G 2 K 1 G 2 T + I r

15 Vector Writig o Dirty Paper 3/11 R 2 M 1 M 2 M 1 -Ecoder M 2 - Ecoder X 1 X 2 X G 1 G 2 Z 1 Z 2 Y 1 Y 1 = G 1 X 1 + G 1 X 2 + Z 1 Y 2 = G 2 X 2 + G 2 X 1 + Z 2 Y 2 R 1 < I X 1 ; G 1 X 1 + G 1 X 2 + Z 1 = 1 2 log G 1K 1 G 1 T + G 1 K 2 G 1 T + I r G 1 K 2 G 1 T + I r R 2 < I X 2 ; G 2 X 2 + Z 2 = 1 2 log G 2K 2 G 2 T + I r

16 Capacity Regio of Gaussia MIMO BC

17 BC-MAC Duality Z 1 N(0, I r ) X G 1 Y 1 X 1 G 1 T Z N(0, I t ) Y G 2 G 2 T Y 2 X 2 Z 2 N(0, I r ) C DP BC P; G 1, G 2 = C MAC (P 1, P 2 ; G T 1, G T 2 ) 2 i=1 tr P i P

18 MIMO Multiple Access Chael 3/11 System Structure M 1 Ecoder 1 X 1 G 1 Z N(0, I r ) Y Decoder M 2 Ecoder 2 X 2 G 2 1 i=1 Power Costrait x T j m j, i x j (m j, i ) P m j 1: 2 R j, j = 1,2 chael Y = G 1 X 1 + G 2 X 2 + Z Z N(0, I r )

19 MIMO MAC Capacity Regio R log G 1K 1 G 1 T + I r R log G 2K 2 G 2 T + I r R 1 + R log G 1K 1 G 1 T + G 2 K 2 G 2 T + I r Boudary Poit R R 1 = 1 2 log G 1K 1 G 1 T + I r R 2 = 1 2 log G 1K 1 G 1 T + G 2 K 2 G 2 T + I r 1 2 log G 1K 1 G 1 T + I r = 1 2 log G 1K 1 G 1 T + G 2 K 2 G 2 T + I r G 1 K 1 G 1 T + I r

20 Achievability Proof : DPC Capacity Regio / Usig Dual MAC R WDP = C DMAC = R(K 1, K 2 ) K 1,K 2 0:tr K 1 +tr K 2 P R 1, R 2 of C DMAC lies o the boudary of (K 1, K 2 ) max [αr 1 + αr 2 ] α 0,1, R 1,R 2 C DMAC max α 0,1,tr K 1 +tr K 2 P,K 1,K 2 0} [α 2 log G 1K 1 G 1 T + G 2 K 2 G 2 T + I r + α α 2 log G 2 K 2 G 2 T + I r ] Itroducig Dual Variables tr K 1 + tr K 2 P K 1, K 2 0 λ 0 γ 1, γ 2 0

21 Achievability Proof : DPC Capacity Regio L K 1, K 2, γ 1, γ 2, λ = α 2 log G 1K 1 G 1 T + G 2 K 2 G 2 T + I r + α α 2 log G 2 K 2 G 2 T + I r +tr γ 1 K 1 + tr γ 2 K 2 λ[tr K 1 + tr K 2 P) Applyig KKT λ G 1 S 1 G 1 T + γ 1 λ I r = 0 λ G 2 S 2 G 2 T + γ 2 λ I r = 0 λ tr K 1 + tr K 2 P = 0 tr γ 1 K 1 = tr(γ 2 K 2 ) = 0 S 1 = α 2λ G 1 T K 1 G 1 + G T 2 K 2 G 2 + I 1 r S 2 = α 2λ G 1 T K 1 G 1 + G T 2 K 2 G 2 + I 1 α α r + 2λ G T 2 K 2 G 2 + I 1 r K 1 = α 2λ G 2 T K 2 G 2 + I 1 r S 1 K 2 = α 2λ I r K 1 S 2