1. Joint Physical Layer Coding and Network Coding for Bi-Directional Relaying Makesh Wilson, Krishna Narayanan, Henry Pfister and Alex Sprintson Department of Electrical and Computer Engineering Texas A&M University, College Station, TX TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AA A A A

2. Network Coding  Network coding is the idea of mixing packets at nodes Nov 5, 2008 Wireless Communications Lab, TAMU 2

3. Characteristics of Wireless Systems  Superposition of signals – signals add at the PHY layer  Broadcast nature – node can broadcast to nodes naturally Nov 5, 2008 Wireless Communications Lab, TAMU 3

4.  Half-duplex and no direct path between Nodes A and B  Wireless Gaussian links with signal superposition  Same Tx power constraint P and receiver noise variance  Metric: Exchange rate per channel use Bi-Directional Relaying Problem Node A Node B Relay Node V xAxA xBxB xVxV xVxV y B = x V + n B y A = x V + n A

5. A Naive Scheme  Rate A - B = (1/8) log(1 + snr)  Rate B - A = (1/8) log(1 + snr)  R ex = (1/4) log(1 + snr) xAxA xBxB xBxB xAxA A - Relay B - Relay Relay - B Relay - A Total Transmission Time

6. Network Coding Solution Ref : Katti et al, “ XORs in the Air: Practical Wireless Network Coding ”, ACM SIGCOMM 2006 xVxV xVxV  Rate A - B = (1/6) log(1 + SNR)  Rate B - A = (1/6) log(1 + SNR)  R ex = (1/3) log(1 + SNR) A - Relay B - Relay Relay - A, B Total Transmission Time xBxB xAxA

7. Recent related work – Scale and Forward  Forward link y = x A + x B + n  Reverse link: scale y and broadcast  Can achieve Katti et al, “ Embracing Wireless Interference: Analog Network Coding ” ACM SIGCOMM 2007 xAxA xBxB kyky kyky y = x A + x B + n

8.  Forward link – MAC Phase y = x A + x B + n  Reverse link – Broadcast phase  Same Power P and noise variance  Exchange rate per channel use = (R AB + R BA ) Two Phase Schemes xAxA xBxB xVxV xVxV y = x A + x B + n y B = x V + n B y A = x V + n A

9. Main Results in this Talk – Two Phase Schemes  An upper bound on the exchange “ capacity ” is  Coding Schemes Lattice coding with lattice decoding Lattice coding with minimum angle decoding MAC channel decoding  Essentially optimal at high and low SNRs  Extends to other Network coding problems, asymmetric SNRs 9

10.  Forward link – Code for MAC channel (R A, R B )  Reverse link – Code for the broadcast channel   Do we have to decode (x A, x B ) at the relay ? Coding for the MAC Channel xAxA xBxB xVxV xVxV y = x A + x B + n Decode (x A, x B ) y B = x V + n B y A = x V + n A

11. Motivation – BSC (p) Example MAC Phase  x A, x B, x V 2 { 1,0 } n and channel performs Binary sum  Relay Node V receives y = x A © x B © e, e 2 { 1,0} n  Relay Node V transmits x V  y A = x V © e A, y B = x V © e B, e A,e B 2 { 1,0 } n, BSC(p) xAxA xBxB xVxV y = x A © x B © e

12. BSC(p) channel – Upper bound BSC(p) channel  Cut-set to bound  C = 1-H(p)

13. Coding in the MAC Phase  Coding Scheme: A, B use same linear code at rate R = 1-H(p)  Relay Node V receives y = x A © x B © e  Relay Node V decodes x V = x A © x B BSC channel with binary addition xAxA xBxB xVxV y = x A © x B © e

14. Reverse Link – BSC case  Relay broadcasts x V  Nodes A and B decode x V from x V © e ’  Nodes obtain x B and x A by XOR at rate R xAxA xBxB xVxV xVxV xBxB xAxA  In BSC R = 1 – H(p) is the best achievable rate (R ex ) y = x A © x B © e

15. Main point Linearity was important in the uplink Structured codes outperform random codes

16. 1-D Example – with uniform noise  Upper Bound on R ex is 1 bit  Can we get this 1 bit? 16 Rx Noise Distribution 0 +1 Peak Tx Power Constraint 0 +1 xAxA xBxB xVxV

17. 1-D example with uniform noise  Node A and B transmit +1 or -1  Relay receives 2, 0 or -2 with noise and decodes y ’  Map using modulo and transmit  We can indeed achieve an Exchange rate of 1bit! 17 +2 -20 +1 +1 (y ’ mod 4) -1

18. Main point Modulo operation is important to satisfy the power constraint at the relay

19. Gaussian Channel – Upper bound  Using cut-set to bound by capacity of each link  Upper bound on R ex is (1/2) log(1 + snr) C = (1/2) log(1 + snr)

20. Structured Codes - Lattices   Lattice ¤ is a sub-group of R n under vector addition  Q ¤ (x) – closest lattice point  x mod ¤ = x – Q ¤ (x) ¸1¸1 ¸2¸2 ¸ 1 + ¸ 2 x Q ¤ (x) 0  (fundamental) voronoi region

21. Nested lattices  Coarse lattice within a fine lattice  V, V 1 – vol. of Voronoi regions  V1V1 V There exist good nested lattices such that coarse lattice is a good quantizer and fine lattice is good channel code

22. Structured coding - MAC Phase  x A = t A x B = t B XAXA XBXB  y = x A + x B + n  Decode to (x A + x B ) mod ¤

23. Reverse link  t at relay is function of t A, t B t = (t A + t B ) mod ¤  t is transmitted back and decoded at nodes A and B  Notice that t satisfies the power constraint  t A = (t-t B ) mod ¤ and t B = (t-t A ) mod ¤

24. Encoding with Dither  x A = (t A + u A ) mod ¤  x B = (t B + u B ) mod ¤  y = x A + x B + n  We want to decode (t A + t B ) mod ¤ from y xAxA t A + u A

25. Decoding – lattice decoding with MMSE  (® y + u A + u B ) mod ¤  Equals (t A + t B – (1 - ®) (x A + x B ) + ® n) mod ¤  Define t = (t A + t B ) mod ¤  Define N eq = – (1 - ®) (x A + x B ) + ® n  Form (t + N eq ) mod ¤

26. Achievable rate  Theorem: Using Nested lattices a rate of (1/2) log (0.5 + snr) is achievable  The second moment of N eq is (2P ¾ 2 )/(2P + ¾ 2 )  The second moment of t is P  Hence rate is (1/2) log (0.5 + snr) – so many fine lattice points in Coarse lattice  At high SNR approx equal to (1/2) log (1 + snr), the upper bound

27. More Proof details  Follows Erez and Zamir results for Modulo Lattice Additive Noise (MLAN) Channel  (t mod ¤ + N eq ) mod ¤  N eq can be approximated by Gaussian  t mod ¤ is uniformly distributed  Poltyrev exponents can be calculated

28. Low/medium SNR regime  Does Gaussian MAC to achieve C AB + C BA = (1/4) log(1 + 2 snr)  Note relay can decode both x A and x B  Does Slepian Wolf Coding in reverse link

29. Achievable rates R ex MAC LATTICE CODING SNR

30. Where does the suboptimality come from ?  Replace lattice decoding with minimum angle decoding  Still gives the same rate !  No dither here 30 P 2P

31. Where does the sub-optimality come from?  Not all lattice points at radius of 2P are code words at low rates!  Prior distribution of (x A + x B ) is not uniform ! 31 P 2P

32. Does it Generalize?  Exchanging information between nodes in Multi-hop  Holds for general networks like the Butterfly network  Asymmetric channel gains

33. Multiple hops  Multi-hop with each node can communicate with only two immediate neighbors  Each node can not broadcast and listen in the same transmission slot  Again (1/2) log (0.5 + snr) can be achieved

34. Example 3 relays At each stage one packet is unknown. Hence we can always decode Node ANode B a1a1 a2a2 a3a3 a4a4 b1b1 b2b2 b3b3 b4b4 12 a 1 mod ¤ b 1 mod ¤ a2a2 a3a3 a4a4 b2b2 b3b3 b4b4 (a 1 + b 1 ) mod ¤ 3 a3a3 a4a4 b3b3 b4b4 (a 2 + a 1 +b 1 ) mod ¤ (b 2 + b 1 +a 1 ) mod ¤ 4 b1b1 a1a1 (2 a 1 + a 2 + 2 b 1 + b 2 ) mod ¤ 5 a4a4 b4b4 (2 a 1 + a 2 + a 3 + 2 b 1 +b 2 ) mod ¤ (2 a 1 + a 2 + 2 b 1 +b 2 + b 3 ) mod ¤ (4 a 1 + 2 a 2 + a 3 + 4 b 1 + 2 b 2 + b 3 ) mod ¤ 6 b2b2 a2a2

35. Fading with asymmetric channel gain  Transmission occurs in L coherence time intervals  P ai, P bi, P ri are powers at nodes A, B and V in i th coherence time  Channel is symmetric and h a, h b (L £ 1)vectors known to all nodes  Total sum power constraint on the nodes 35 Fading links

36. Upper Bound  Upper bound using cut-set arguments  C(x):= log(1 + x) 36 Fading links

37. Analysis for L = 1  Hence  Or 37  For · 2 small  For · 2 large

38. Achievable scheme using channel inversion  D(snr):= Rate using Lattice/MAC based scheme for given snr 38

39. Analysis for L = 1  Here · 2 P a = P b 39

40. Comparison of the Bounds for arbitrary L  Theorem: For the problem setup, for arbitrary L and ¢ = 0.5, under the high snr approximation, the channel inversion scheme with Lattices is at max a constant (0.09 bits per complex channel use), away from the upper bound! 40

41. Does it Generalize?  Exchanging information between nodes in Multi-hop  Holds for general networks like the Butterfly network

42. Conclusion  Structured codes are advantageous in wireless networking  Results for high and low snr show are nearly optimal  Extension to multihop channels and asymmetric gains  Many challenges remain Capacity is unknown – only 2-phase schemes were considered Channel has to be known Even if channel is known can we get 0.5 log(1+snr) ? Practical lattice codes to achieve these rates

43. Structured coding - MAC Phase Node A Node B xAxA xBxB t A mod ¤ t B mod ¤ t = (t A + t B )mod ¤ tAtA tBtB t xVxV xVxV

44. Bi-AWGN Channel ML decoding  Equivalent channel from b 1 © b 2 to y  Linear codes so b 1 © b 2 is a codeword Node A Relay Node V Node B x 1 2 {-1,1} n x 2 2 {-1,1} n x 1 + x 2 2 {-2,0,2} n b 1 2 { 1,0} n b 2 2 { 1,0} n

45. Conjecture ML decoding Lattices  Can ’ t do better than (1/2)log ( 0.5 + SNR) with ML  Minkowski-Hlawka for existence of lattices  Blichfeldts Principle for Concentration of codewords ideas  Assuming uniform distribution yeilds good concentration

46. Concentration at 2 P  Sum x A + x B will be in 2 P x1x1 x2x2 Use Blichfeldt for lattice equivalent

47. Projecting to inner sphere Project Codewords to inner sphere Use Minkowski – Hlawka to show existence of Lattice Calculate probability of error

48. Connection to Lattices  Using Blichfeldts theorem we can establish concentration for lattices also  Minkowski-Hlawka theorem can be used to perform ML Decoding  Again it appears we can get only (1/2) log (0.5 + SNR)

49. References 