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

Wireless Communications Lab, TAMU 2

Network Coding

Network coding is the idea of mixing packets at nodes

Nov 5, 2008

Wireless Communications Lab, TAMU 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

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 ANode B

Relay Node V

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xA

xB

xV xV

yB = xV + nByA = xV + nA

y = xA + xB + n

A Naive Scheme

Rate A - B = (1/8) log(1 + snr) Rate B - A = (1/8) log(1 + snr)

Rex = (1/4) log(1 + snr)

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xA xB

xB

xA

A - Relay

B - Relay

Relay - B

Relay - A

Total Transmission Time

Network Coding Solution

Ref : Katti et al, “XORs in the Air: Practical Wireless Network Coding”, ACM SIGCOMM 2006

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xV

xV

Rate A - B = (1/6) log(1 + SNR) Rate B - A = (1/6) log(1 + SNR)

Rex = (1/3) log(1 + SNR)

A - Relay

B - Relay

Relay - A, B

Total Transmission Time

xBxA

Recent related work – Scale and Forward

Forward link y = xA + xB + n

Reverse link: scale y and broadcast

Can achieve

Katti et al, “Embracing Wireless Interference: Analog Network Coding” ACM SIGCOMM 2007

xA

xB

ky ky

y = xA + xB + n

Forward link – MAC Phase y = xA + xB + n

Reverse link – Broadcast phase

Same Power P and noise variance

Exchange rate per channel use = (RAB + RBA)

Two Phase Schemes

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xA

xB

xV xV

y = xA + xB + n

yB = xV + nByA = xV + nA

¾2

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

Forward link – Code for MAC channel (RA , RB )

Reverse link – Code for the broadcast channel

Do we have to decode (xA , xB ) at the relay ?

Coding for the MAC Channel

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xA

xB

xV xV

y = xA + xB + nDecode (xA ,

xB )

yB = xV + nByA = xV + nA

Motivation – BSC (p) Example MAC Phase

xA , xB , xV 2 {1,0}n and channel performs Binary sum

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Relay Node V receives y = xA © xB © e , e 2 {1,0}n

Relay Node V transmits xV

yA = xV © eA , yB = xV © eB , eA ,eB 2 {1,0}n , BSC(p)

xA

xB

xV

y = xA © xB © e

BSC(p) channel – Upper bound

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BSC(p) channel

Cut-set to bound

C = 1-H(p)C = 1-H(p)

Coding in the MAC Phase

Coding Scheme: A, B use same linear code at rate R = 1-H(p)

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Relay Node V receives y = xA © xB © e

Relay Node V decodes xV = xA © xB

BSC channel with binary addition

xA

xB

xV

y = xA © xB © e

Reverse Link – BSC case

Relay broadcasts xV

Nodes A and B decode xV from xV © e’

Nodes obtain xB and xA by XOR at rate R

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xAxB

xV

xV

xB xA

In BSC R = 1 – H(p) is the best achievable rate (Rex)

y = xA © xB © e

Main point

Linearity was important in the uplink

Structured codes outperform random codes

16

1-D Example – with uniform noise

Upper Bound on Rex is 1 bit

Can we get this 1 bit?

Rx Noise Distribution

0-1 +1

Peak Tx Power Constraint

0-1 +1

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xAxB

xV

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!

+2-2 0 +1-1

-1 +1

(y’ mod 4) -1

Main point

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

Gaussian Channel – Upper bound

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Using cut-set to bound by capacity of each link

Upper bound on Rex is (1/2) log(1 + snr)

C = (1/2) log(1 + snr)C = (1/2) log(1 + snr)

Structured Codes - Lattices

Lattice ¤ is a sub-group of Rn under vector addition

Q¤(x) – closest lattice point

x mod ¤ = x – Q¤(x)

¸1

¸2

¸1 + ¸2

xQ¤(x)

0

(fundamental) voronoi region

Nested lattices

Coarse lattice within a fine lattice

V, V1 – vol. of Voronoi regions

V1V

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 = tA

xB = tB

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XAXB

y = xA + xB + n

Decode to (xA + xB) mod

¤

Reverse link

t at relay is function of tA , tB t = (tA + tB) mod ¤

t is transmitted back and decoded at nodes A and B

Notice that t satisfies the power constraint

tA = (t-tB) mod ¤ and tB = (t-tA) mod ¤

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Encoding with Dither

xA = (tA + uA) mod ¤

xB = (tB + uB) mod ¤

y = xA + xB + n

We want to decode (tA + tB) mod ¤ from y

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®® (1)®®

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xA

tA + uA

Decoding – lattice decoding with MMSE

(® y + uA + uB ) mod ¤

Equals (tA + tB – (1 - ®) (xA + xB) + ® n) mod ¤

Define t = (tA + tB) mod ¤

Define Neq = – (1 - ®) (xA + xB) + ® n

Form (t + Neq) mod ¤

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Achievable rate

Theorem: Using Nested lattices a rate of (1/2) log (0.5 + snr) is achievable

The second moment of Neq is (2P ¾2)/(2P + ¾2) 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

More Proof details

Follows Erez and Zamir results for Modulo Lattice Additive Noise (MLAN) Channel

(t mod ¤ + Neq) mod ¤

Neq can be approximated by Gaussian

t mod ¤ is uniformly distributed

Poltyrev exponents can be calculated

Low/medium SNR regime

Does Gaussian MAC to achieve CAB + CBA = (1/4) log(1 + 2 snr)

Note relay can decode both xA and xB

Does Slepian Wolf Coding in reverse link

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0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

SNR

Exchange C

apacity

Lattice codingscheme

Upper Bound

Amplify andForward

Achievable rates

Rex

MAC

LATTICE CODING

SNR

SNR

30

Where does the suboptimality come from?

Replace lattice decoding with minimum angle decoding

Still gives the same rate ! No dither here

P2P

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 (xA + xB) is not uniform !

P2P

Does it Generalize?

Exchanging information between nodes in Multi-hop

Holds for general networks like the Butterfly network

Asymmetric channel gains

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

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Example 3 relays

At each stage one packet is unknown. Hence we can always decode

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Node A

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Node Ba1

a2

a3

a4

b1

b2

b3

b4

12

a1 mod ¤ b1 mod ¤

a2

a3

a4

b2

b3

b4

(a1 + b1) mod ¤

3a3

a4

b3

b4

(a2 + a1 +b1) mod ¤ (b2 + b1 +a1) mod ¤

4b1a1

(2 a1 + a2 + 2 b1 + b2) mod ¤

5a4 b4

(2 a1 + a2 + a3 + 2 b1 +b2) mod ¤

(2 a1 + a2 + 2 b1 +b2 + b3) mod ¤

(4 a1 + 2 a2 + a3 + 4 b1 + 2 b2 + b3) mod ¤

6b2

a2

35

Fading with asymmetric channel gain

Transmission occurs in L coherence time intervals

Pai , Pbi , Pri are powers at nodes A, B and V in ith coherence time

Channel is symmetric and ha, hb (L£1)vectors known to all nodes

Total sum power constraint on the nodes

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Fading links

ha 2 CL hb 2 CL

36

Upper Bound

Upper bound using cut-set arguments

C(x):= log(1 + x)

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Fading links

maximizeLX

i=1

minfC(jhai j2Pai );C(jhbi j2Pr i )g+ minfC(jhbi j2Pbi );C(jhai j2Pr i )g

subject toLX

i=1

Pai +LX

i=1

Pbi +LX

i=1

Pr i · 2LP

Pai ;Pbi ;Pr i ¸ 0; i 2 1;2:::L

37

Analysis for L = 1

Hence

Or

Pa = P Pb = P· 2

1+ · 2

Pb = PPa = P1

1+ · 2

For ·2 small

For ·2 large 0 0.5 1 1.5 2 2.5 3 3.5 4

20

40

60

80

100

2

Pa

0 0.5 1 1.5 2 2.5 3 3.5 40

50

100

2

Pb

0 0.5 1 1.5 2 2.5 3 3.5 460

70

80

90

100

2

Pr

· 2 := jhaj2=jhbj2

jhaj2Pa ¼jhbj2Pb

· 2Pa ¼Pb

38

Achievable scheme using channel inversion

D(snr):= Rate using Lattice/MAC based scheme for given snr

maximize minfD(jhai j2Pai );D(jhbi j2Pr i )g+ minfD(jhbi j2Pbi );D(jhai j2Pr i )g

subject toLX

i=1

Pai +LX

i=1

Pbi +LX

i=1

Pr i · 2LP;

jhai j2Pai = jhbi j2Pbi ;

Pr i ¸ Pai ;

Pr i ¸ Pbi ;

Pai ;Pbi ;Pr i ¸ 0; i 2 1;2:::L

39

Analysis for L = 1

Here ·2 Pa = Pb

· 2 := jhaj2=jhbj2

0 0.5 1 1.5 2 2.5 3 3.5 420

40

60

80

100

2

Pa

0 0.5 1 1.5 2 2.5 3 3.5 40

50

100

2

Pb

0 0.5 1 1.5 2 2.5 3 3.5 460

70

80

90

100

2

Pr

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!

0 100 200 300 400 500 6003

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Average Channel Power

Avera

ge E

xchange r

ate

in B

its

Upper Bound

Channel inversion scheme with Lattice coding

Does it Generalize?

Exchanging information between nodes in Multi-hop

Holds for general networks like the Butterfly network

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

Structured coding - MAC Phase

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Node A Node B

xA xB

tA mod ¤tB mod ¤

t = (tA + tB )mod ¤

tA

tB

t

xVxV

Bi-AWGN Channel ML decoding

Equivalent channel from b1 © b2 to y

Linear codes so b1 © b2 is a codeword

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Node A

Relay Node V

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Node B

x1 2 {-1,1}nx2 2 {-1,1}n

x1 + x2 2 {-2,0,2}n

b1 2 {1,0}n b2 2 {1,0}n

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

Concentration at 2 P

Sum xA + xB will be in 2 P

x1 x2

Use Blichfeldt for lattice equivalent

Projecting to inner sphere

Project Codewords to inner sphere

Use Minkowski – Hlawka to show existence of Lattice

Calculate probability of error

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)

References