Based on Partial CSI in MIMO Ad-Hoc Networks

Transcription

1 s Based on Partial CSI in MIMO Ad-Hoc Networks Y. Richter & I. Bergel Faculty of Engineering, Bar-Ilan University Austin, May

2 Outline s &. Optimal routing in random ad-hoc networks. antenna routing. MIMO routing. 2

3 Statistical optimal routing in Wireless Ad Hoc Network s 3

4 Statistical optimal routing in Wireless Ad Hoc Network s 4

5 Statistical optimal routing in Wireless Ad Hoc Network s 5

6 System model s Slotted ALOHA MAC (p tx ). PPP distributed nodes (λ). / antennas. Local knowledge on nodes in routing zone. j } M j {(r i, H i,j ) : i N j 6

7 Ergodic Rate Density (ERD) R(λ) = λp tx E {log 2 (1 + SIR)} s Achievable upper bound on WANETs performance. Convenient for analysis. Good bounds. High complexity, large delay. George et al, Rate Density 10 0 Novel Upper Bound ERD Lower bound N=9 N= N= λ 7

8 Asymptotic density of rate and progress (ADORP) ( ) D (f ( )) λp tx E {r } f (M) log SIRf (M) s WANETs performance is measured by Rate Progress. f ( ) is the routing function. Use opportunistic relaying. No delay constraints. Implicit mobility. 8

9 Optimal routing s In the single antenna case ( D (f ( )) = λp tx E {r f (M) log S )} f (M) J f (M) Using the Law of Total Expectation D (f ( )) = λp tx E M { E J M { r f (M) log 2 ( 1 + S f (M) J f (M) ) M }} SO: Statistical Optimal routing function ( f SO (M) = argmax r i E {log S ) M } i i N J i Depends on the distribution of J i M. 9

10 Evaluation of the SO metric Uses only local knowledge. Takes into account interference statistics. Can be evaluated using Monte Carlo Simulations. Different statistics of nodes inside/outside the routing zone. s p tx p tx 10

11 BO routing s Lower bound on ERD { ( E log S ) } i M S ) i p Z (i, M) log J 2 (1 + i E{J i r > r Z, M} BO routing function f BO (M) = argmax p Z (i, M)r i log 2 (1 + i N S i J i 1 + J i 2 + J i 3 ) J 1 J J 2 3 r Z 11

12 NSO routing Narrow knowledge s M i = {r i, h i } Narrow Statistically Optimal routing function ( f NSO (M) = argmax r i E {log S ) M i i} i N J i Without knowledge on neighbors, the distribution of J i is identical for all nodes. Interference distribution can be measured locally. Complexity is still quite high. 12

13 NBO routing s Lower bound on ERD p Z (i, M) log 2 (1 + = const log 2 (1 + γs i ) NBO routing function ) S i E{J i r min,i > r Z, M i } f NBO (M) = argmax r i log 2 (1 + γ S i ) i N where γ = α 2 ( α 2 απλp tx ) α 2 Considers network parameters. Low complexity! 13

14 Numerical Results s Previously published geographic routing schemes NiC: Nearest in a cone f Nearest (M) = argmin r i i MPR: Most progress within radius f MPR (M) = argmax i: r i,j r max r i Need to optimize the routing parameter (MPR)! 14

15 ADORP vs p tx s ADORP [bps/hz/km] antenna, α = 3 SO BO NSO NBO Nearest MPR p (ALOHA transmission probability) tx 15

16 ADORP vs p tx 0.35 antenna, α = s ADORP [bps/hz/km] SO BO NSO NBO Nearest MPR p (ALOHA transmission probability) tx 16

17 antennas WANET s Optimal MIMO routing needs to select Destination. Number of streams. Precoding vectors. Challenges: Need to take into account the processing at the receiver. The number of streams affects the distribution of the interference. Simplifying assumptions: Tx: eigenbeamforming (equal power streams). Rx: Partial ZF (of N ZF nearest transmitters). Consider only narrow knowledge. 17

18 antennas WANET s ADORP Define ( )} D o (f ( )) = λp tx E M {B f (M), M, K(M) B(i, M, K) E {r K ( ) } i log K r α i W i,k M i J i,k. k=1 W i,k : Effective channel gain at the k-th stream Partially known at the transmitter. 18

19 NSO routing s Using narrow knowledge f NSO (M), K(M) = argmax r i i,k K k=1 { ( ) E log K r α i γi,k 2 Y } i,k M i J i,k The distribution of J depends on the distribution of K(M). High complexity. Intractable. 19

20 FSO routing s Fixed number of streams per user f FSO (M, K) = argmax r i i K k=1 ( E {log K r α i γ 2 i,k Y i,k J i,k ) } M i The distribution of J i,k is identical for any i, k. 20

21 LC routing function s Taking the expectation of the numerator and the denominator f LC (M), K(M) = argmax r i i,k where K 1 K log 2 (1 + r α ) i γi,kȳ 2 σn 2 + C α,nzf k=1 Ȳ N R + 1 K T ZF N R + 1 K γ 2 i,k is the k-th singular value, and TZF is the average number of zeroed streams, and C α,nzf 2(λp txπ) α 2 (N ZF α 4 )1 α 2 E{W } α 2 21

22 ADORP vs p tx s ADORP [bps/hz/km] antenna, N ZF = 1, N R = 10, α = 3 LC FSO Nearest p (ALOHA probability) 22

23 ADORP vs p tx s ADORP [bps/hz/km] antenna, N ZF = 2, N R = 10, α = 3 LC FSO Nearest p (ALOHA probability) 23

24 ADORP vs p tx s ADORP [bps/hz/km] antenna, N ZF = 3, N R = 10, α = 3 LC FSO Nearest p (ALOHA probability) 24

25 s Presented novel routing metrics. Based on optimization of the ADORP bound. Simple to evaluate locally. Uses only local knowledge and exploits the statistics of the interference. Close to optimal, outperforms traditional schemes. 25

26 Thank you! s Questions? 26

27 ERD Lower Bound Rewrite } D(f ( )) = λpe M {G (f (M), M)) s where { ( G(i, M) = E J M r i,0 log ρs ) M } i. J i Zone and Threshold Zone 33

28 ERD Lower Bound p Z (i, M): probability that no transmitter within distance r Z from node i: { (1 p) N Z,i, if r i + r Z < r R p Z (i, M) = e λpb T,i (1 p) N Z,i, o.w. s r R r Z B T 34

29 ERD Lower Bound s Lower bound ( G(i, M) = (1 p Z (i, M))E {r i log ρ S i ( + p Z (i, M)E {r i log ρ S i ( p Z (i, M)E {r i log 2 = p Z (i, M) r i log 2 (1 + J i 1 + ρ S i J i J i ) } rmin,i r Z, M ) } rmin,i > r Z, M ) } rmin,i > r Z, M ) ρ S i E{J i r min,i > r Z, M} Eventually, E{J i r min,i > r Z, M} = J i 1 + J i 2 + J i 3 35

30 Partial Zero Forcing (PZF) s Cancels its N ZF nearest transmitters. T ZF : # of inter-streams to be canceled. Remaining degrees of freedom (DOF) L = N R T ZF (K 0 1). Set of N ZF indices of undesired transmitters N Inter ZF = {j Φ T : r j r ZF }. Set of its (K 1) intra-streams N Intra ZF,k = {1, 2,..., k 1, k + 1,..., K}. Continue 36