Joint Physical Layer Coding and Network Coding for

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

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

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

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

A Naive Scheme x. B x. A x. B A - Relay o o Rate A - B = (1/8) log(1 + snr) Rate B - A = (1/8) log(1 + snr) o Rex = (1/4) log(1 + snr) B - Relay - B Relay - A Total Transmission Time

Network Coding Solution x. V x. B x. A A - Relay B - Relay o o Rate A - B = (1/6) log(1 + SNR) Rate B - A = (1/6) log(1 + SNR) o Rex = (1/3) log(1 + SNR) Relay - A, B Total Transmission Time Ref : Katti et al, “XORs in the Air: Practical Wireless Network Coding”, ACM SIGCOMM 2006

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

Two Phase Schemes x. V y. A = x V + n. A y. B = x V + n. B y = x A + x. B + n x. A x. B o Forward link – MAC Phase y = x. A + x. B + n o Reverse link – Broadcast phase o Same Power P and noise variance o Exchange rate per channel use = (RAB + RBA)

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

Coding for the MAC Channel x. V y. A = x V + n. A y. B = x V + n. B x. A y = x A + x. B + n Decode (x. A , x. B ) x. B o Forward link – Code for MAC channel (RA , RB ) o Reverse link – Code for the broadcast channel o o Do we have to decode (x. A , x. B ) at the relay ?

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

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

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

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

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

1 -D Example – with uniform noise x. V x. B x. A Rx Noise Distribution Peak Tx Power Constraint -1 0 +1 -1 +1 0 o Upper Bound on Rex is 1 bit o Can we get this 1 bit? 16

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

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

Gaussian Channel – Upper bound C = (1/2) log(1 + snr) o Using cut-set to bound by capacity of each link o Upper bound on Rex is (1/2) log(1 + snr)

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

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

Structured coding - MAC Phase XA o x. A = t A x. B = t B o y = x. A + x. B + n o Decode to (x. A + x. B) mod ¤ XB

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

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

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

Achievable rate o Theorem: Using Nested lattices a rate of (1/2) log (0. 5 + snr) is achievable o The second moment of Neq is (2 P ¾ 2)/(2 P + ¾ 2) o The second moment of t is P o Hence rate is (1/2) log (0. 5 + snr) – so many fine lattice points in Coarse lattice o At high SNR approx equal to (1/2) log (1 + snr) , the upper bound

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

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

Achievable rates LATTICE CODING Rex MAC SNR

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

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

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

Multiple hops o Multi-hop with each node can communicate with only two immediate neighbors o Each node can not broadcast and listen in the same transmission slot o Again (1/2) log (0. 5 + snr) can be achieved

Example 3 relays a 2)+mod a + (4 a(2+a 2 a +2 a+2 a +32+ 4 b(b 1(2 a 1 b 1+mod a 21+ 2 b 1 +a )¤mod (a 2(2+aa 11++b mod b 2 + 1 ¤ 3 ¤ 1 (a 1+ b 1 ) mod ¤ 1) mod 2 b 1 +b 2) mod ¤ +b 21 b +12 b+2)bmod ¤ ¤ +b 2 + b¤ 3) mod ¤ 3 Node A a 4132 a 43 b 1 b 2 6 25 4 1 3 a 4 At each stage one packet is unknown. Hence we can always decode a 1 a 2 Node B b 4132 b 43 b 4

Fading with asymmetric channel gain Fading links o Transmission occurs in L coherence time intervals o Pai , Pbi , Pri are powers at nodes A, B and V in ith coherence time o Channel is symmetric and ha, hb (L£ 1)vectors known to all nodes o Total sum power constraint on the nodes 35

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

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

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

Analysis for L = 1 o Here · 2 Pa = Pb 39

Comparison of the Bounds for arbitrary L o 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

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

Conclusion o Structured codes are advantageous in wireless networking o Results for high and low snr show are nearly optimal o Extension to multihop channels and asymmetric gains o Many challenges remain n n 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

Structured coding - MAC Phase Node A Node B x. V x. A x. B t. A t t. B t. A mod ¤ t = (t. A + t. B )mod ¤ t. B mod ¤

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

Conjecture ML decoding Lattices o Can’t do better than (1/2)log ( 0. 5 + SNR) with ML o Minkowski-Hlawka for existence of lattices o Blichfeldts Principle for Concentration of codewords ideas o Assuming uniform distribution yeilds good concentration

Concentration at 2 P o Sum x. A + x. B will be in 2 P x 1 Use Blichfeldt for lattice equivalent x 2

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

Connection to Lattices o Using Blichfeldts theorem we can establish concentration for lattices also o Minkowski-Hlawka theorem can be used to perform ML Decoding o Again it appears we can get only (1/2) log (0. 5 + SNR)

References o