Capacity of Multiantenna Gaussian Broadcast Channel

Transcription

1 Capacity of Multiantenna Gaussian Broadcast Channel Pramod Visanath Joint Work ith David Tse, UC Berkeley Oct 8, 2002

2 Multiple Antenna Broadcast Channel N (0, I) m 1, m 2 x y 1 ˆm 1 2 ˆm 2 y = x +

3 Overvie Nondegraded Broadcast channel Capacity region unknon Knon Result: Sum Capacity for 2 users (Caire,Shamai 00) Our Result: Sum Capacity for general number of users and antennas Simple proof and interpretation The Capacity region

4 Multiple Antenna Broadcast Channel z N (0, K) m 1, m 2 x y 1 ˆm 1 2 ˆm 2 y = x + z K ii = 1 Only marginal channels matter Noise correlation arbitrary

5 Sato Upper Bound z N (0, K) m 1, m 2 x y ˆm 1 ˆm 2 K ii = 1 Receivers cooperate

6 Upper Bound z N (0, K) Sato Broadcast channel Cooperating Receivers

7 Achievable Rates: Costa Precoding m 1 u 1 x ˆm 1 ˆm 2 m 2 u 2 Users data modulated onto spatial signatures u 1, u 2

8 Stage 1: Costa Precoding m 1 u 1 x 1 ˆm 1 m 2 u 2 oding for user 1 treating signal from user 2 as knon interference at transmitter

9 Stage 2 m 1 u 1 x m 2 u 2 ˆm 2 2 ode user 2 treating signal for user 1 as noise

10 Costa Strategy and Upper Bound z N (0, K) Sato Want to find K such that sum rate ith Costa precoding is same as capacity of cooperating receivers channel

11 DL-UL Duality A representation of sum rate achievable by Costa strategy Present a form of duality in multiantenna channels a change of variable and a conservation la Applications of this observation

12 Applications Unifies duality observations under various guises Reciprocity - Telatar (99) Virtual Uplink channel - R-Farrokhi (97), Visotsky (99) Duality beteen MAC and BC - Jindal et al (01) Extension of results on uplink to donlink Performance of linear receivers - Tse and anly (99) Achievable rate region for multiantenna broadcast channel Marton Region for Gaussian inputs

13 DL-UL Duality N (0, I) 1 x y ˆm 1, ˆm 2 2 Costa precoding over all u 1, u 2 achieves same region as uplink Jindal and Goldsmith ( 00)

14 Summary Broadcast Sato Cooperating Receivers z N (0, K) DL-UL Duality Multiple Access

15 Reciprocity Broadcast Cooperating Receivers z N (0, K) Sato DL-UL Duality N (0, I) Reciprocity x 1, x 2 independent Multiple Access E [ x t Kx ] P, K ii = 1 Cooperating Transmitters

16 Summary Broadcast Sato Cooperating Receivers z N (0, Q) DL-UL Duality Reciprocity x 1, x 2 independent Multiple Access [ E x Qx ] P Cooperating Transmitters Multiple Access Cooperating Transmitters

17 MAC and Cooperating Transmitters Channel Multiple Access Cooperating Transmitters Channel max Σx I (x ; y) max Σx I (x ; y) E [ x t x ] P E [ x t Kx ] P x 1, x 2 independent K ii 1

18 MAC and Cooperating Transmitters Channel Multiple Access Cooperating Transmitters Channel max Σx I (x ; y) max Σx I (x ; y) tr [Σ x ] P tr [Σ x K] P (Σ x ) ii 0, (Σ x ) ij = 0, i j K ii 1

19 Convex Duality MAC Problem: max Σ x [I (x ; y)] such that tr [Σ x ] P, (Σ x ) ij = 0, (Σ x ) ii 0

20 Convex Duality MAC Problem: max Σ x [I (x ; y)] such that tr [Σ x ] P, (Σ x ) ij = 0, (Σ x ) ii 0 Convex Dual: min max λ 0,λ ii 0,λ ij Σ x I (x ; y) λ (tr [Σ x ] P ) i,j λ ij (Σ x ) ij

21 Convex Duality MAC Problem: max Σ x [I (x ; y)] such that tr [Σ x ] P, (Σ x ) ij = 0, (Σ x ) ii 0 Convex Dual: min max λ 0,λ ii,λ ij Σ x I (x ; y) λ (tr [Σ x ] P ) i,j λ ij (Σ x ) ij In Matrix Form: K ii = 1 λ ii /λ, K ij = λ ij /λ min K,K ii 1,λ 0 max Σ x [I (x ; y) λ (tr [KΣ x ] P )]

22 Convex Duality MAC Problem: max Σ x [I (x ; y)] such that tr [Σ x ] P, (Σ x ) ij = 0 Convex Dual: min max λ 0,λ ii 0,λ ij Σ x I (x ; y) λ (tr [Σ x ] P ) i,j λ ij (Σ x ) ij In Matrix Form: K ii = 1 λ ii /λ, K ij = λ ij /λ min max [I (x ; y) λ (tr [KΣ x ] P )] K,K ii 1,λ 0, Σ x Finally: K ii 1, tr [KΣ x ] P min K max Σ x [I (x ; y)]

23 Convex Duality: Positive Semidefinite Constraints Convex Dual of MAC Problem: K ii 1, tr [KΣ x ] P min K max Σ x [I (x ; y)] Cooperating Transmitters Channel: K ii 1, tr [KΣ x ] P min max [I (x ; y)] K 0 Σ x 0

24 Convex Duality: Positive Semidefinite Constraints Convex Dual of MAC Problem: K ii 1, tr [KΣ x ] P min K max Σ x [I (x ; y)] p.s.d. constraints Cooperating Transmitters Channel: K ii 1, tr [KΣ x ] P min max [I (x ; y)] K 0 Σ x 0

25 Main Result Broadcast Sato Cooperating Receivers z N (0, Q) DL-UL Duality Reciprocity x 1, x 2 independent Multiple Access Convex Duality [ E x Qx ] P Cooperating Transmitters Multiple Access Cooperating Transmitters

26 Capacity Region Focus on ar 1 + R 2 R 1 = rate of user 1 R 2 = rate of user 2 a < 1: user 1 has less eight

27 Cooperating Upper Bound Doesnt Work z N (0, K) m 1, m 2 x y ˆm 1 ˆm 2 No separation of rates R 1, R 2 ence no control over ar 1 + R 2

28 Degraded Receivers z N (0, K) m 1, m 2 x y ˆm 1 ˆm 2 User 1 is privy to signal of user 2 No a degraded Gaussian broadcast channel

29 Degraded Upper Bound z N (0, K) K ii = 1

30 Costa Coding Achievability Degraded Receivers z N (0, K) Degraded Upper Bound DL-UL Duality Multiple Access

31 What is the 4th System? Degraded z N (0, K) Degraded Upper Bound DL-UL Duality? Multiple Access

32 Degraded Transmitters MAC Degraded Receivers z N (0, K) Degraded Bound DL-UL Duality E [ x xt 1 Kx 1] P Multiple Access Degraded Transmitters MAC

33 Degraded Transmitters Bound Degraded Receivers z N (0, K) Degraded Bound DL-UL Duality E [ x xt 1 Kx 1] P Multiple Access Degraded Transmitters MAC

34 Convex Duality Choose cost function K such that User 1 (stronger one) does not use user 2 s input. for input that maximizes ar 1 + R 2

35 Convex Duality Degraded Receivers z N (0, K) Degraded Bound DL-UL Duality Convex Duality Optimal K E [ x xt 1 Kx 1] P Multiple Access Degraded Transmitters MAC

36 Reciprocity: Almost There Degraded Receivers z N (0, K) Degraded Bound DL-UL Duality for Gaussian Degraded Inputs Reciprocity Convex Duality Optimal K E [ x xt 1 Kx 1] P Multiple Access Degraded Transmitters MAC

37 Inequalities Not in the Correct Direction Degraded Receivers z N (0, K) Degraded Bound DL-UL Duality With general degraded inputs Convex Duality Optimal K E [ x xt 1 Kx 1] P Multiple Access Degraded Transmitters MAC

38 The Final Step Degraded Receivers z N (0, K) Degraded Bound DL-UL Duality Gaussian Inputs Suffice Kramer 07 Oct 2002 Convex Duality Optimal K E [ x xt 1 Kx 1] P Multiple Access Degraded Transmitters MAC