Cooperative Relaying & Power Allocation Strategies in Sensor Networks

Cooperative Relaying & Power Allocation Strategies in Sensor Networks Jingqiao Zhang Oct. 27, 2005

Outline • Introduction • Relay Channel • Strategies: Decode-and-Forward (DF), Amplify-and-Forward (AF), Compress-and-Forward (CF) • Power Allocation Strategies in DF & AF • Two-hop Relay Networks • Minimum Energy Accumulative Routing • Relay based routing scheme in sensor networks

Outline • Introduction • Relay Channel • Strategies: Decode-and-Forward (DF), Amplify-and-Forward (AF), Compress-and-Forward (CF) • Power Allocation Strategies in DF & AF • Two-hop Relay Networks • Minimum Energy Accumulative Routing • Relay based routing scheme in multi-hop networks

Y1 X1 X Y Y2 BC MAC X2 Information Theory • Point to point communication • Network Information Theory • Multiple Access Channel (MAC), Broadcasting Channel (BC), Relay Channel Relay Y1:X1 X Y R1 R1 The capacity is known for the special physically degraded relay channel -- p(y,y1|x,x1) = p(y1|x,x1)p(y|y1,x1) R2 R2 Binary Symmetric Broadcast Channel

Relay S D Relay Strategies • The transmission is divided into to two periods • 1st period: the source node broadcasts its signal. -- BC • 2nd period: relay nodes transmits the source’s information. -- MAC • Regenerative Decode-and-Forward (DF) • Receive  Decode  Re-encode  Transmit • Non-regenerative Amplify-and-Forward (AF) • Receive  Amplify  Transmit • Compress-and-Forward (CF) • Receive  Quantize/Compress  Transmit

Outline • Introduction • Relay Channel • Strategies: DF, AF, CF • Power Allocation Strategies in DF & AF • Two-hop Relay Networks • Minimum Energy Accumulative Routing • Relay based routing scheme in multi-hop networks

Y1 X1 X1 = √b1 Y1 (AF) Y1 = √1 X0 + Z1 √1 √1 √2 √2 √0 √m √m √M √M YM XM Channel Model 1 • A Shared Channel • Y = m (√m Xm) + Z 2 node M+1 node 0 X0 Y M−1 M • Orthogonal channels • Y = BX + Z • B = diag(√0, √1, …, √M) • X = [X0, X1, …, XM] • Z: a vector of AWGN ~ N(0,1)

Objective • Notations • Power allocation: p=[p0, p1, …, pM]; p0 is the source power, and pm is the power at relay node m. • r: the expected communication rate from the source to the destination • The objective • is to find the optimal power allocationp that supports a reliable communication at rate r between the source and the destination. • subject to Channel capacity ¸r

An Overview of Conclusions According to the two recent papers of Maric, et. al

DF: Constraints Relay s d • Relay node: • A(p0): the subset of relay nodes that decode the transmission from s • For i2 A(p0), we have W log(1+ip0 /W) ¸ r, • Destination node: • 2W IDF(p/W)¸ r • IDF(p/W): the maximum mutual information IDF(p/W) between the channel input X and output Y

log(1+x) ¼ x DF: Problem Formulation • Optimization Problem • Subject to

Subject to DF: Problem Simplification • Theorem: The wideband DF relay problem admits an optimal solution in which at most one relay node transmits. • Subject to

The optimal solution is to choose from the set U={k| k > 0 and k > 0}, and find the one minimizing the total power DF: Optimal Power Allocation • Relay node k is used (pk > 0) only if k > 0 and k > 0.

AF • Receive  Amplify  Transmit • Ym = √1 X0 + Zm, and Xm = √bm Ym • The communication rate r is no greater than the maximum mutual information 2WIAF(p/W) between the channel input X and output Y • Unlike DF, the capacity is not a increasing function of the bandwidth. To see this, • No benefit from the relay transmission

Lagrangian: AF: Problem Formulation • Define P = p/W (power per dimension) • Two sub-problems • Decide the best set of relay node {1, 2, …, K} • For a given the set {1, 2, …, K}, find the optimal power and bandwidth • Subject to

AF: Optimal Power Allocation • Theorem 3: The AF relay problem has an optimum solution in which the optimum bandwidth W*, rate r and the total transmit power p* have a linear relationship • Subproblem-1 is not solved completely. Actually, the author solves the problem for a given source power P0: • where  is the constant such that

The Effect of the Source Power • The best choice of relay nodes significantly depends on the transmit power at the source.

Outline • Introduction • Relay Channel • Strategies: DF, AF, CF • Power Allocation Strategies in DF & AF • Two-hop Relay Networks • Minimum Energy Accumulative Routing • Relay based routing scheme in multi-hop networks

v2 v3 v1 v4 Minimum Energy Accumulative Routing • Traditional multi-hop Model (TM) • Point-to-point communication in each hop • Accumulative Routing (AR) • Store the partially overheard packet (leakage) • The leakage contributes to the final reception of the packet. • For example, several copies of the same packet can be decoded by maximum ratio combining (MRC).

v2 v3 p2 = H / g2,3 p3 = H / g3,4 p1 = H / g1,2 v1 v4 TM: Point-to-Point Transmission • TM: Point-to-point transmission (without relay): pi¸ H/gi,j • Total tx power: ipi = H/gi,i+1 gi,j/ di,j pjr= pigi,j ¸ H pi vj vi

p1 = H / g1,2 TR: Relay-based Routing • Relay transmission: the previous transmission of the same packet can be overheard and stored by the latter node. • E.g., before vivi+1, the leakage from v1, v2, …, vi-1 is: li+1 = vj2{v1, …, vi-1}pj gj,i+1 • Total tx power: ipi =  (H – li+1)/gi,i+1 p2 = (H – p1 g1,3) / g2,3 v2 v3 p3 = (H – p1 g1,4– p2 g2,4) / g3,4 v1 v4

r r H 0.5 H H H overheard s d s d An Example • Assume the power attenuation exponent  = 2. • In simulation, 30% energy saving is achieved Total energy = 2H Total energy = 1.5H

Problem Formulation • To find a feasible transmission scheduleS = [(v1, p1), …, (vw, pw))] • S is an ordered list of node-power pair • Feasible: • v1 = s, vw = d • relay: decode – transmit • A general AR routing: j=1i-1 pj gj,i ¸ H, 8 i > 1 • A k-Relay case: j=i-ki-1 pj gj,i ¸ H, 8 i > 1 • Minimum Energy Accumulative Routing: MEAR (V, s, d) • For a source-destination pair (s, d), • to look for a feasible transmission schedule S = [(v1, p1), …, (vw, pw))], • such that the total tx energy E(S) ,i=1wpi is minimized

Complexity of the Problem • Theorem 1: The MEAR(V, s, t) problem is NP-complete for a general graph with arbitrary link gains and a cap on the transmission energy a node can spend on one packet. • D-MEAR (V,s,t) problem: Given (V,s,t), is there a feasible tx schedule S for (s,t) such that E(S) · P?

Shortest Path Heuristic (SPH) Algorithm • Shortest path algorithm • point-to-point communication in each hop • pi =H/gi,i+1 as the weight of edge (i,i+1) • Theorem 2: In a general graph model, E(OPT) ∈ o( E(SPH) ) in the worst case • SPHbe the solution from the shortest path heuristic, and • OPTbe the optimal solution. • Proof: consider a special case where H =1, gi,j=(|i - j|+)-1 • SPH: from node v1 to vn directly. E(SPH) = n+ • OPT:

p1 = H / g1,2 (1,2) , p2 = (H – p1 g1,3) / g2,3 (2,3) , (3,4) , p3 = (H – p1 g1,4– p2 g2,4) / g3,4 (u,v): cost (or distance) from node u to node v The Structure of Optimal Schedules • Theorem 3:A MEAR(V, s, t) problem always has an optimal schedule that is a wavepath. • Wavepath: A schedule S is a wavepath iff • vivj (8ij ) ---- no loop in the route • j=1i-1 pj gj,i = H ---- wavepath property

(u, v) (u, v) e(u) u s v e(v) RPAR Algorithm • Initialization • RPAR generate a tree T rooted as s • Calculate the cost (distance) from s to any other node v: e(v) = H / gs,v • At each iteration, • Find a new node u that is closest to the source node. Add it to the tree T. • Update the cost (distance) from the rest node v to the root s.

Comparison (OPT, RPAR, SPH) • A number of nodes are randomly distributed in a 1000m £ 1000m region. Attenuation exponent  = 2 • RPAR is very close to the optimal solution. E(RPAR) / E(OPT) < 1.1 SPH RPAR

Effects of Node Density

Implementation Issues - 1 • Identify each received packet before MRC • Reliable header & partially overhead payload • Adopt a strong modulation/coding of the packet header • Store all the partially received packets • the same source address and SN. • Combine various copies and decode the packet • Maximum Ratio Combining v2 v3 v4 v1

Implementation Issues - 2 • Wireless channel: RTS/CTS handshake • IEEE 802.11-like MAC protocol • RTS/CTS collision avoidance mechanism should be modified • to prevent interference at overhearing nodes • E.g., let RTS/CTS cover the all routing area -- only suitable for a low-load network

Conclusions • Communication performance can be improved by the cooperation of intermediate relay nodes • In a two-hop network, • DF: At most one relay node at the “best” position assists the communication • AF: There exist an optimal bandwidth. Optimal power allocation is like water-filling (orthogonal channels) or CMR (shared channels) • In a multi-hop network, • Energy saving can be achieved by a relay-based routing algorithm • A heuristic algorithm achieves the sub-optimal solution in polynomial time.

Future Work • What if both DF and AF modes are allowed? What will be the optimal power allocation? • Is there any analytical result on the RPAR, since seemingly we can transform its operation into a Trellis structure? • Is it possible to find the near-optimal set of participating nodes distributedly? • If yes, the distributed power allocation can be easily obtianed according to the wavepath property.

References • G. Kramer, M. Gastpar, and P. Gupta, Cooperative Strategies and Capacity Theorems for Relay Networks, IEEE Transactions on Information Theory, vol. 51, no. 9, pp. 3037 – 3063, Sept. 2005 • I. Maric and D. Yates, Bandwidth and Power Allocation for Coopertive Strategies in Gaussian Relay Network, In Proc. of Asilomar, Nov. 2004. Monterey, CA. (invited) • I. Maric and D. Yates, Forwarding Strategies for Gaussian Parallel-Relay Networks, Intl. Symposium on Information Theory, pp. 269, June 2004 • J. Chen, L. Jia, X. Liu, G. Noubir, and R. Sundaram, Minimum Energy Accumulative Routing in Wireless Networks, Infocom 2005, vol. 3, pp. 1875 – 1886,March 2005. • T. M. Cover and J.A. Thomas, Elements of Information Theory, Wiley, 1991

Thank you!

Cooperative Relaying & Power Allocation Strategies in Sensor Networks