Approaching the Capacity of the Multi-Pair Bidirectional Relay Network via a Divide and Conquer Strategy

Transcription

1 Approaching the Capacity of the Multi-Pair Bidirectional elay Network via a Divide and Conquer Strategy Salman Avestimehr Cornell In collaboration with: Amin Khajehnejad (Caltech), Aydin Sezgin (UC Irvine) Babak Hassibi (Caltech) CTW 2009

2 Overview K pairs communicate via a relay What is the optimal relaying strategy? message m A1 wants m B1 A 1 B 1 message m B1 wants m A1 A 2 B 2 message m AK wants m BK A K B K message m BK wants m AK

3 State of the art Two-way relay channel has been studied extensively Decode & Broadcast: (Oechtering et. al. ) PHY network coding: (Katti et. al., Narayanan et. al., Hausl et. al., Baik et. al.) Max achievable rate: Equal channel gains: (Narayanan-Wilson-Sprintson) AB = BA = log( SN) In general:(nam-chung-lee., Gunduz-Tuncel-Nayak, Avestimehr-Sezgin-Tse) A 1- bit gap approximation B

4 Deterministic channel model (ADT 07) point to point MAC Tx1 x + mod 2 + addition Tx2 eceiver gets those bits that arrive above noise level eceiver gets the modulo sum of those bits that arrive at the same signal level

5 Single pair Optimal scheme (deterministic): eorder-forward equations uplink A b 1 b 2 b 3 n 1 b 3 c 1 n 2 c 1 c 2 B b 4 b 4 c 2 downlink c 1, c 2 c 1 A n 3 b 3 c 1 b 4 c 2 b 1 b 3 c 1 b 2 b 4 c 2 n 4 bb 3 c b c 1 B b 1, b 2,b 3,b 4 b 4 c 2 b 2 AB min(n 1,n AB 4 AB )=4, BA BA =2 min(n =1 2,n 3 )=2 Near optimal scheme (Gaussian): elay decodes the strong codeword first Quantizes the remaining message (AST 2008), or decode the lattice point (NCL 2008), then broadcast

6 Single pair Just need to forward enough new equations (random mixing suffices) In the Gaussian case CF also achieves within 1-bit of the cut-set (Gunduz et. al.) uplink A b 1 b 2 b 3 n 1 b 3 c 1 n 2 c 1 c 2 B b 4 b 4 c 2 n 3 n 4 downlink c 1, c 2 A B b 1, b 2,b 3,b 4

7 Multiple pairs andom mixing of equations does not seem to work for multiple pairs We need to manage interference and carefully create and forward the equations λ 11 a 1 +λ 12 b 1 +λ 13 a 2 +λ 14 b 2 λ 21 a 1 +λ 22 b 1 +λ 23 a 2 +λ 24 b 2 a 1 b 1 b 1 a 1, b 1, a 2, b a 1 A B 2 2 a 2 b A B 2 2 a 2 b 2

8 A simple divide and conquer strategy Orthogonalize the pairs on signal levels to create and forward equations Assume we want to achieve =( A1, B1, A2, B2 ) 1. Pick a pair with non-zero rates (say A1 0, B1 0) Assign the strongest signal level in UL that is connected to both Assign the weakest signal level in DL that is connected to both Update =( A1-1, B1-1, A2, B2 ). 2. Pick a user with non-zero rate (say A1 0) Assign the strongest signal level in UL that is connected to A 1 Assign the weakest signal level in DL that is connected to B 1 Update =( A1-1, B1, A2, B2 ). A 2 B 2 UL A 2 B 2 DL

9 Example Is =(2, 1, 2, 1) achievable? 1. Pick (A 1,B 1 ) 2. pick (A 2,B 2 ) =(1, 0, 2, 1) =(1, 0, 1, 0) 3. pick A 1 4. pick A 2 =(0, 0, 1, 0) =(0, 0, 0, 0) A 2 B 2 UL A 2 B 2 DL

10 Is divide & conquer strategy optimal? Does it achieve the cut-set bound? Ai min(n Ai,n Bi ) Bi min(n Bi,n Ai ) A1 + A 2 min{max(n A1,n A 2 ), max(n B1,n B 2 )} B1 + B 2 min{max(n B1,n B 2 ), max(n A1,n A 2 )} A1 + B 2 min{max(n A1,n B 2 ), max(n B1,n A 2 )} B1 + A 2 min{max(n B1,n A 2 ), max(n A1,n B 2 )} Yes! (proof by induction on A1 + B1 + A2 + B2 )

11 Main result (for deterministic networks) Theorem (AKSB 09): For deterministic multi-pair bidirectional relay networks Capacity region= cut-set bound region It is achieved by a divide-conquer equation forwarding scheme

12 Capacity achieving scheme (deterministic) Structure of the optimal solution: (assume Ai Bi ) Forwards Bi equations for each pair Forwards Ai - Bi bits from A i to B i Order of these chunks is the same as the order of the channel gains A 2 B 2 n A1 n A 2 n B1 n B 2 A 2 B 2 n B 2 n B1 n A 2 n A1 A1 - B1 bits of A 1 B1 eqns of pair 1 eceived signal levels at the relay A2 - B2 bits of A 2 B1 eqns of pair 1 B2 eqns of pair 2 Transmit signal levels at the relay B2 eqns of pair 2 A1 - B1 bits of A 1 A2 - B2 bits of A 2

13 Transition to Gaussian channel model Three challenges 1. Additive noise 2. Power leakage from the signals of lower levels to those transmitted at higher levels 3. Decoding the equations Solutions 1. Coding 2. Leakage inevitable, however: Treat all lower level interferences as noise Decrease the rates to be able to decode (loosing constant bits) 3. Use appropriate lattice code (superposition of two codewords is a valid codeword)

14 elaying scheme (Gaussian) Weak users: use a lattice code Strong users: use a superposition of a lattice code and a random code A 2 B 2 x A1 = x B1 = α (1) A1 x (1) A1 + α (2) (2) A1 x A1 α B1 x B1 x A 2 = α (1) A 2 x (1) A 2 + α (2) (2) A 2 x A 2 x B 2 = α B 2 x B 2 h A1 h A 2 h B1 h B 2 eceived signal levels at the relay A1 - B1 bits of A 1 A2 - B2 bits of A 2 B1 eqns of pair 1 B2 eqns of pair 2 h A1 2 α (1) A1 2 A1 + A 2 h A 2 2 (1) α A 2 2 A1 + B1 h B1 2 α B1 = h A1 2 α (2) A1 2 B1 + B 2 h B 2 2 α B 2 = h A 2 2 α (2) A 2 = 2 B 2

15 elaying scheme (Gaussian) elay successively decodes the received equations and bits by treating interference at lower levels as noise Uses a superposition codeword to broadcasts the decoded information A 2 B 2 A 2 B 2 h A1 h A 2 h B1 h B 2 h B 2 h B1 h A 2 h A1 A1 - B1 bits of A 1 B1 eqns of pair 1 eceived signal levels at the relay A2 - B2 bits of A 2 B1 eqns of pair 1 B2 eqns of pair 2 Transmit signal levels at the relay B2 eqns of pair 2 A1 - B1 bits of A 1 A2 - B2 bits of A 2

16 Gaussian two-pair two-way relay network Theorem (SKAB 09): If =( A1, B1, A2, B2 ) is in the cut-set region of the twopair two-way relay network, then ( A1-2, B1-2, A2-2, B2-2) is achievable. A 2 B 2

17 Summary Orthogonalizing the pairs over signal levels is optimal (under the deterministic channel model) Proposed a divide and conquer equation forwarding scheme for optimal signal level allocation Showed that a similar strategy achieves within 2 bits/user of the capacity when there are two pairs and the channel model is AWGN

18 Questions?