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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

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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

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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

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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-coherent relaying throughput scales as O(1)

[Grossglauser-Tse01][Bolsckei04] [Dousse-Franceschetti-Thiran 04] [Dana-Hassibi 04] [Oyman-Paulraj 05] …

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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 ….

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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: 1. B. Wang, J. Zhang & L. Zheng, “Achievable Rates and Scaling

Laws of Power-constrained Sensory Relay Networks,”

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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.

W(Hz)B

k- thsubband

K total subbands

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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

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

sT sLT sLT2L( sT)1+ 0

Data Data

Relays Relays

Pilot

Data

First hop Second hop

Source Destination

LTs

Pilot Pilot

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Energy tradeoff: training vs. data transmission

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:

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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 and W scale with n Narrowband: L and ρ scale with n

L scales between 0 and ∞: from non-coherent to coherent

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Joint Asymptotic Regime (cont’)

Exponents:

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Narrowband Relay Networksin the Low SNR Regime

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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

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Equivalent End-to-end Model

Destination collects signals from relays:

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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]

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Achievable Rates of AF using Network Training

Equivalent SNR

Achievable rate using AF with network training

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Upper Bound on Capacity of Narrowband Relay Networks

Cut-set theorem: broadcast cut (BC) provides upper bound

Scaling order of upper bound

RelaysSource Destination

BroadcastCut

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Scaling Behavior of Achievable Rate R Case 1:

Case 2:

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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

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Power Constrained Wideband Sensory Relay Networks

W(Hz)B

k- thsubband

K total subbands

Total achievable rate is sum of achievable rates across sub-bandsKey question: what is good power allocation policies across subbands at relay nodes?

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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)

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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

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k-th Sub-band (narrowband) : Equal Power Allocation at Relays

For k-th sub-band (narrowband)

Equivalent SNR for k-th sub-channel

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Scaling Behavior of Achievable Rates: Equal Power Allocation at Relays

If

If and

If

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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

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Finding achievable rate using optimal power allocation at relays boils down to solving

Challenges Non-convex optimization As bandwidth grows, complexity increases

exponentially

Optimal Power Allocation at Relays

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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

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Upper Bound on Achievable Rate (cont.)

Cauchy-Schwarz’s Inequality and convex analysis gives upper bound on SNR

Upper bound on achievable rate

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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.

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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

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Conditions to achieve scaling law: Engineering intuition

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

RelaysSource Destination

Virtual ReceiveAntenna Array

virtually noise free

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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

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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

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Event Sensing

Event-sensing: Each sensor detects events

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Random Field Monitoring

2-D random field sensing

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Thank You!