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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. Wireless Ad-Hoc/Sensor Networks.
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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
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
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
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]…
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 ….
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,”
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.
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
Two relaying strategies: AF vs. DF Amplify-and-forward with network training AF Relaying with Network Training
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
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
Joint Asymptotic Regime (cont’) • Exponents:
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
Equivalent End-to-end Model • Destination collects signals from relays:
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]
Achievable Rates of AF using Network Training • Equivalent SNR • Achievable rate using AF with network training
Upper Bound on Capacity of Narrowband Relay Networks • Cut-set theorem: broadcast cut (BC) provides upper bound • Scaling order of upper bound
Scaling Behavior of Achievable Rate R • Case 1: • Case 2:
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
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?
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)
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
k-th Sub-band (narrowband) : Equal Power Allocation at Relays • For k-th sub-band (narrowband) • Equivalent SNR for k-th sub-channel
Scaling Behavior of Achievable Rates: Equal Power Allocation at Relays • If • If and • If
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
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
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
Upper Bound on Achievable Rate (cont.) • Cauchy-Schwarz’s Inequality and convex analysis gives upper bound on SNR • Upper bound on achievable rate
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.
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
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
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
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
Event Sensing • Event-sensing: Each sensor detects events
Random Field Monitoring • 2-D random field sensing