Half-Duplex Gaussian Relay Networks with Interference Processing Relays

Half-Duplex Gaussian Relay Networks with Interference Processing Relays Bama Muthuramalingam Srikrishna Bhashyam Andrew Thangaraj Department of Electrical Engineering Indian Institute of Technology Madras Dec 2010 Srikrishna Bhashyam (IIT Madras) Dec 2010 1 / 26

Wireless Relay Networks Single shared resource wireless channel Wireless systems: Time-variations, Interference Relaying important in spectrally efficient wireless networks Srikrishna Bhashyam (IIT Madras) Dec 2010 2 / 26

Relaying: What is known/unknown? Single source-destination pair Gaussian relay networks Capacity unknown for arbitrary topology Cut-set upper bound Achievable rates for specific protocols and topologies Appproximate capacity Srikrishna Bhashyam (IIT Madras) Dec 2010 3 / 26

Wireless Relaying: Assumptions and Results Duplex SNR Cooperation Topology Full Large MIMO Arbitrary, Directed Half All Limited Restricted No MIMO Arbitrary Srikrishna Bhashyam (IIT Madras) Dec 2010 4 / 26

Wireless Relaying: Assumptions and Results Duplex SNR Cooperation Topology Full Large MIMO Arbitrary, Directed Half All Limited Restricted No MIMO Arbitrary [Avestimehr et al] Both, Large SNR, MIMO, Arbitrary directed Constant gap to capacity [Sreeram et al] Both, Large SNR, MIMO, Arbitrary Diversity-multiplexing trade-off [Chang et al] Half duplex, All SNR, Limited, Restricted Rates close to capacity [Bagheri et al] Half duplex, All SNR, No MIMO, Restricted Constant gap to capacity Srikrishna Bhashyam (IIT Madras) Dec 2010 4 / 26

Our Assumptions Half-duplex All SNR No MIMO/Limited cooperation Restricted, arbitrary Decode-and-Forward Srikrishna Bhashyam (IIT Madras) Dec 2010 5 / 26

Our Work Multistage relaying Heuristic scheduling of states + Coding for interference networks Flow optimization Interference processing vs. Interference avoidance Strong and weak interference conditions Srikrishna Bhashyam (IIT Madras) Dec 2010 6 / 26

States of a Half-Duplex Network Each node: Transmit, Receive, or Idle Each state is an interference network Srikrishna Bhashyam (IIT Madras) Dec 2010 7 / 26

Choice of States: Observations Interference Avoidance Only one node can transmit at any time Interference Processing Source should be in transmit mode Destination should be in receive mode Relays should be in both transmit and receive modes Required for information flow Atleast two node-disjoint paths required for source to be transmitting in all chosen states Srikrishna Bhashyam (IIT Madras) Dec 2010 8 / 26

Two-Path Two-State Schedule Shortest (three-hop) paths connecting S and D Path P1: S 2 4 D Path P2: S 3 5 D Path P3: S 2 5 D Path P4: S 3 4 D. Only two pairs of node-disjoint paths: (P1, P2) and (P3, P4). States from (P1, P2): State S1: Nodes S, 3, 4 transmit, Nodes 2, 5, D receive State S2: Nodes S, 2, 5 transmit, Nodes 3, 4, D receive Srikrishna Bhashyam (IIT Madras) Dec 2010 9 / 26

Coding for each state M N interference network [Carleial1978] Possible message from each transmitter to each subset of receivers M(2 N 1) possible rates M user Interference channel M possible messages (M rates) Achievable rate regions based on Superposition Successive interference cancellation Dirty paper coding Srikrishna Bhashyam (IIT Madras) Dec 2010 10 / 26

Common Broadcast (CB) Rate limited by weakest link Receivers employ SIC/MAC decoding Srikrishna Bhashyam (IIT Madras) Dec 2010 11 / 26

Superposition Coding (SC) Transmitters send superposed codewords Constraints involve power allocation parameters (non-linear) Larger rate region than CB Srikrishna Bhashyam (IIT Madras) Dec 2010 12 / 26

Dirty paper coding (DPC) at the source Source: origin for all messages; knows m 3 and m 4 Source does DPC to eliminate interference at receiver 2 Can be combined with CB or SC at other transmitters Srikrishna Bhashyam (IIT Madras) Dec 2010 13 / 26

Coding for the Two-Stage Relay Example DPC-SC State S1: Nodes S (1), 3, 4 transmit, Nodes 2, 5, D (6) receive Node S: Transmit to Node 2 using DPC Node 3: Transmit to Node 5 Node 4: Transmit to Nodes 5 and D using SC Srikrishna Bhashyam (IIT Madras) Dec 2010 14 / 26

Flow Optimization Joint optimization problem Scheduling constraints maximize Rate subject to State k is ON for λ k units of time Total transmission time is one unit Rate region constraints appropriate rate region depending on the coding scheme Flow constraints Total flow in a link (i, j) = k flow in link (i, j) in state k Outgoing flow from Node i - Incoming flow to Node i = Rate, if i = S, -Rate, if i = D, 0, otherwise. Srikrishna Bhashyam (IIT Madras) Dec 2010 15 / 26

Two-Stage Relay Flow optimization: DPC-SC z 1 2 z 1 4 (1 )z 1 + βz 2 1 S z 2 βz 2 z 1 6 D (1 β)z 2 + z 1 3 z 2 5 Link in State S 1 Link in State S 2 Srikrishna Bhashyam (IIT Madras) Dec 2010 16 / 26

Two-Stage Relay Flow optimization max R = z 1 + z 2, 0 λ 1,λ 2,,β 1 subject to rate constraints Flow in each link less than average rate z 1 λ 1 R S2, z 1 λ 2 R 24, z 2 λ 2 R S3, z 2 λ 1 R 35, (1 )z 1 + βz 2 λ 1 R 4D, (1 β)z 2 + z 2 λ 2 R 5D, z 1 λ 1 R 45, βz 2 λ 2 R 54, Scheduling constraint: 0 λ 1 + λ 2 1 Rates chosen according to rate region of interference network (R S2, R 35, R 45, R 4D ) R 1, (R S3, R 24, R 54, R 5D ) R 2. Srikrishna Bhashyam (IIT Madras) Dec 2010 17 / 26

Cut-Set Upper Bound S D Ω Ω c Full Duplex Network 1 R Half Duplex Network 2 R sup λ k min Ω min Ω I (X Ω ; Y Ωc X Ωc ). M k=1 λ k I (X(k) Ω ; Y (k) Ωc Ωc X(k) ). 1 T. Cover et al, Elements of information theory, John Wiley 2 M. Khojestepour,et al,,bounds on achievable rates for general multiterminal n/ws with practical constraints, IPSN, 2003 Srikrishna Bhashyam (IIT Madras) Dec 2010 18 / 26

Numerical Results Parameters: R 1 R 3 2 β 4 Tx power, P = 3 units Noise variance, σ 2 = 1 Variable channel gains S 1 δ 3 β 5 δ 6 D R 2 R 4 Comparison: Interference avoidance throughput - Lower bound Upper bound Achievable rates of the proposed relaying schemes Srikrishna Bhashyam (IIT Madras) Dec 2010 19 / 26

Two Stage Relay Network - vary β = δ, = 1, = 1 Achievable Rate 1.8 1.6 1.4 1.2 1 0.8 0.6 Bound CB SC DPC CB DPC SC IA Lower bound (DPC CB) = 1.99 db Lower bound (CB) = 5.01 db Upper bounds Cut-Set bound = C(2 2 P) = 1.40 Source rate C( 2 P) = 1 0.4 0.2 0 10 8 6 4 2 0 2 4 6 8 10 δ = β (db) Srikrishna Bhashyam (IIT Madras) Dec 2010 20 / 26

Two Stage Relay Network - vary β = δ, = 1, = 1 Achievable Rate 1.8 1.6 1.4 1.2 1 0.8 0.6 Bound CB SC DPC CB DPC SC IA Lower bound (DPC CB) = 1.99 db Lower bound (CB) = 5.01 db Upper bounds Cut-Set bound = C(2 2 P) = 1.40 Source rate C( 2 P) = 1 Large β Strong interference S R 1 R 3 D, S R 2 R 4 D 0.4 0.2 0 10 8 6 4 2 0 2 4 6 8 10 δ = β (db) Srikrishna Bhashyam (IIT Madras) Dec 2010 20 / 26

Two Stage Relay Network - vary β = δ, = 1, = 1 Achievable Rate 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Bound CB SC DPC CB DPC SC IA Lower bound (DPC CB) = 1.99 db 0 10 8 6 4 2 0 2 4 6 8 10 δ = β (db) Lower bound (CB) = 5.01 db Upper bounds Cut-Set bound = C(2 2 P) = 1.40 Source rate C( 2 P) = 1 Large β Strong interference S R 1 R 3 D, S R 2 R 4 D Small β S R 1 R 4 D, S R 2 R 3 D Srikrishna Bhashyam (IIT Madras) Dec 2010 20 / 26

Two Stage Relay Network - vary β = δ, = 1, = 1 Achievable Rate 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Bound CB SC DPC CB DPC SC IA Lower bound (DPC CB) = 1.99 db 0 10 8 6 4 2 0 2 4 6 8 10 δ = β (db) Lower bound (CB) = 5.01 db Upper bounds Cut-Set bound = C(2 2 P) = 1.40 Source rate C( 2 P) = 1 Large β Strong interference S R 1 R 3 D, S R 2 R 4 D Small β S R 1 R 4 D, S R 2 R 3 D β = 0 db DPC is better SC and CB are same Srikrishna Bhashyam (IIT Madras) Dec 2010 20 / 26

Two Stage Relay Network - vary β = δ, = 1, = 1 Achievable Rate 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Bound CB SC DPC CB DPC SC IA Lower bound (DPC CB) = 1.99 db 0 10 8 6 4 2 0 2 4 6 8 10 δ = β (db) Lower bound (CB) = 5.01 db Upper bounds Cut-Set bound = C(2 2 P) = 1.40 Source rate C( 2 P) = 1 Large β Strong interference S R 1 R 3 D, S R 2 R 4 D Small β S R 1 R 4 D, S R 2 R 3 D β = 0 db DPC is better SC and CB are same HD relaying meets the bound for some channel gains Srikrishna Bhashyam (IIT Madras) Dec 2010 20 / 26

Strong Interference h 46, h 56 2-user MAC: 2C( 2 P) C(2h 2 min P) where h min = min ( h 45, h 35 ) Srikrishna Bhashyam (IIT Madras) Dec 2010 21 / 26

Weak Interference h 46, h 56 2-user MAC: C( 2 P) C ( ) h 2 35 P 1+h45 2 P Srikrishna Bhashyam (IIT Madras) Dec 2010 22 / 26

Strong and Weak Interference Bounds 1.1 1 0.9 Weak interference bound = 3.63 db Strong interference bound = 2.68 db (A) Achievable Rate 0.8 0.7 0.6 0.5 0.4 0.3 Weak interference condition not satisfied Strong interference bound = 0.88 db (B) Bound MDF Protocol 0.2 10 8 6 4 2 0 2 4 6 8 10 (db) (A): h 23 = h 25 = h 45 = h 34 =, h S2 = h S3 = h 5D = h 4D = 1, h 24 = h 35 = 1.25 (B): h 23 = h 25 = h 45 = h 34 = h 5D =, h S2 = h S3 = h 24 = h 35 = h 4D = 0.58 Srikrishna Bhashyam (IIT Madras) Dec 2010 23 / 26

Grid network Source 2 4 10 7 6 9 12 3 1 11 5 Sink 8 β β β β β β β β β Three states S D 6 8 4 9 β β D S 5 7 4 9 β β S D 5 7 6 8 β Srikrishna Bhashyam (IIT Madras) Dec 2010 24 / 26

Performance in Grid Network, β = 1, = 1, vary 1.5 Achievable Rate 1 0.5 Bound CB SC DPC CB DPC SC IA 0 10 8 6 4 2 0 2 4 6 8 10 (db) Srikrishna Bhashyam (IIT Madras) Dec 2010 25 / 26

Summary States of a relay network as interference networks Scheduling of states using path heuristic Interference processing receivers at the relays No cooperative decoding between relays Strong and weak interference conditions on channel gains Srikrishna Bhashyam (IIT Madras) Dec 2010 26 / 26