Junshan Zhang Dept. of Electrical Engineering Arizona State University MSRI 2006, Berkeley CA

1. Throughput Scaling in Wideband Sensory Relay Networks: Cooperative Relaying, Power Allocation and Scaling Laws Junshan Zhang Dept. of Electrical Engineering Arizona State University MSRI 2006, Berkeley CA Joint work with Bo Wang and Lizhong Zheng

2. Wireless Ad-Hoc/Sensor Networks • Potential applications: • Battlefield wireless networks, • Monitoring chemical/biological warfare agents, • Homeland security. • Basic network models: • (1) Many-to-one networks; • (2) Multi-hop wireless networks; • (3) Sensory relay networks. • Two key features of sensor networks: node cooperation and data correlation

3. Large Scale Wireless Relay Networks • One source node, one destination node and n relay nodes • Two-hop transmissions: Source to relays in first hop and relays to destination in second hop

4. Related Work (on large-scale networks) • [Gupta-Kumar 00] investigated throughput-scaling in many-to-many multi-hop networks. • [Gastpar-Vitterli 02] considered relay traffic pattern and studied coherent relaying: perfect channel information available at each relay node  throughput scales as log(n) ; non-coherentrelaying throughput scales as O(1) • [Grossglauser-Tse01][Bolsckei04] [Dousse-Franceschetti-Thiran 04] [Dana-Hassibi 04] [Oyman-Paulraj 05]…

5. Related Work (on finite-node relay networks) • [Kramer-Gastpar-Gupta 05] provided comprehensive studies on Cooperative strategies and capacity for multi-hop relay networks. • [Wang-Zhang-Host Madsen 05] studied ergodic capacity for 3-node relay channel and provided capacity-achieving conditions (not necessarily degraded) • Independent codebooks at source and relay • Channel uncertainty (randomness) at transmitters make the two codebooks independent • Many many more ….

6. Outline • Model for wideband sensory relay networks; • Cooperative relaying by using AF with network training; • Narrowband relay networks in the low SNR regime; • Power-constrained wideband relay networks; • Conclusions and ongoing work Technical details can be found in our preprint: • B. Wang, J. Zhang & L. Zheng, “Achievable Rates and Scaling Laws of Power-constrained Sensory Relay Networks,”

7. Our Relay Network Model • Large bandwidth W (wideband regime) • Each node has an average power constraint P • All source-relay and relay-destination links are under Rayleigh fading; there is no a priori information on channel conditions • Relay nodes amplify-and-forward (AF) to relay data.

8. Motivation • Under what conditions can the throughput gap between coherent relay networks and non-coherent relay networks be closed? • Study scaling behavior of achievable rates for AF with network training, in asymptotic regime of number of relays and bandwidth. • Characterize scaling laws of sensory relay networks

9. Two relaying strategies: AF vs. DF Amplify-and-forward with network training AF Relaying with Network Training

10. More energy for training  more precise estimation but less energy for data rate Question: how much energy for training? Optimal energy allocation for training  maximize overall SNR at destination e.g., narrowband model: Energy tradeoff: training vs. data transmission

11. Joint Asymptotic Regime • Key parameters: bandwidth W; number of relays n • Coherence interval spans L-symbol duration • Approach: decompose power-constrained wideband relay networks to a set of narrowband relay networks ; • Joint asymptotic regime (a natural choice) • Wideband: L andWscale withn • Narrowband: L andρscale withn • L scales between 0 and ∞: from non-coherent to coherent

12. Joint Asymptotic Regime (cont’) • Exponents:

13. Narrowband Relay Networksin the Low SNR Regime

14. AF Relaying at Node i • Estimate (MMSE) channel conditions for backward and forward channels prior to data transmission • Amplify and forward received signals using network training • Data transmission: source -> relays • Phase-alignment and power amplification at relays • Data transmission: relays -> destination Phase alignment Amplification factor

15. Equivalent End-to-end Model • Destination collects signals from relays:

16. Equivalent End-to-end Model (cont’) • : estimate error, signal-dependent, non-Gaussian • : “amplified” noise from relays, non-Gaussian • : signal-dependent, non-Gaussian • : ambient noise at destination, Gaussian  achievable rate under uncertainty [Medard 00]

17. Achievable Rates of AF using Network Training • Equivalent SNR • Achievable rate using AF with network training

18. Upper Bound on Capacity of Narrowband Relay Networks • Cut-set theorem: broadcast cut (BC) provides upper bound • Scaling order of upper bound

19. Scaling Behavior of Achievable Rate R • Case 1: • Case 2:

20. Scaling Law of Narrowband Relay Networks in Low SNR Regime • Theorem: As , if there exist , such that , then the capacity of relay networks scales as: • Intuition for scaling law achieving condition: normalized energy per fading block, , is bounded below

21. Power Constrained Wideband Sensory Relay Networks • Total achievable rate is sum of achievable rates across sub-bands • Key question: what is good power allocation policies across subbandsat relay nodes?

22. Upper Bound on Capacity of Wideband Relay Networks: • Cut-set theorem: broadcast cut provides upper bound • Scaling order of upper bound (limited by node diversity n and bandwidth W)

23. Achievable Rates of AF Using Network Training • Power allocation policy across subbands. Consider two policies at relays: • Uniformly distribute power among sub-bands • Optimally distribute power across fading blocks and among sub-bands • Each subband points to a narrowband relay network in low SNR regime

24. k-th Sub-band (narrowband) : Equal Power Allocation at Relays • For k-th sub-band (narrowband) • Equivalent SNR for k-th sub-channel

25. Scaling Behavior of Achievable Rates: Equal Power Allocation at Relays • If • If and • If

26. Equivalent Wideband Network Model: Optimal Power Allocation at Relays • Allow each relay allocate power in time and freq. domains. • For k-th sub-channel • Equivalent SNR for k-th sub-channel

27. Optimal Power Allocation at Relays • Finding achievable rate using optimal power allocation at relays boils down to solving • Challenges • Non-convex optimization • As bandwidth grows, complexity increases exponentially

28. Throughput Scaling by using Optimal Power Allocation • Our approach: • Find an upper bound on achievable rate using optimal power allocation • Find a lower bound on achievable rate • Apply a “sandwich” argument

29. Upper Bound on Achievable Rate (cont.) • Cauchy-Schwarz’s Inequality and convex analysis gives upper bound on SNR • Upper bound on achievable rate

30. Scaling of Achievable Rate Using Optimal Power Allocation • Lower bound on achievable rate using optimal power allocation: achievable rate using equal power allocation serves as a lower bound • Somewhat surprising: scaling order of achievable rate using optimal power allocation is the same as that using equal power allocation  Equal power allocation at relays is asymptotically order-optimal to achieve scaling laws • Intuition: regardless of power allocation, power amplification factor is same for desired signal and noise.

31. Scaling Law of Wideband Relay Networks • Theorem: As , if there exist , 1 such that and , then capacity of wideband relay networks scales as • Intuition: Conditions to achieve scaling law • 1st condition: normalized energy per block is bounded below • 2nd condition: W is sub-linear in n

32. Conditions to achieve scaling law: Engineering intuition virtually noise free • Aggregated noise from relays is , and ambient noise at destination is . • When W is sub-linear in n: relay network can be viewed as a SIMO system • The cut-set upper bound is obtained by treating the system as SIMO

33. Discussion • Aggregated noise from relays is , and ambient noise at destination is . • When  “SIMO” • Open question: Scaling behavior when W is super-linear in n ? • Amplify-forward vs. Decode-forward

34. Ongoing work • In previous studies, only source node has data • Ongoing work: all nodes have sensed data • Applications: event-sensing and random field monitoring in large-scale sensory relay networks • Goal: maximize mutual info. between sensors and received signal at sink

35. Event Sensing • Event-sensing: Each sensor detects events

36. Random Field Monitoring • 2-D random field sensing

37. Thank You!