Information flow over wireless networks: a deterministic approach

Transcription

1 Information flow over wireless networks: a deterministic approach alman Avestimehr In collaboration with uhas iggavi (EPFL) and avid Tse (UC Berkeley)

2 Overview Point-to-point channel Information theory provides an abstraction C Wireless network oes information theory give us a similar picture? Not yet.

3 Basic model for wireless channels Key features of wireless channel Broadcast Interference High dynamic range of channel variations Basic PHY layer model: additive-gaussian channel model x 2 x 1 h 2 h 1 h 3 y = i h i x i + z x 3

4 What is known? Point to point: C = 1 2 log( 1+ NR) (hannon 1948) Multiple access (Ahlswede, Liao 70 s) Broadcast (Cover, Bergmans 70 s)

5 tate of the art Unfortunately, we don t know the capacity of most other Gaussian networks Relay Relay relays Channel Many relaying strategies are developed, but Can t analyze how they perform in multi relay networks on t even know how suboptimal these schemes are Current approach tuck in some toy problems Not scalable to larger networks How can we make progress?

6 Our approach evelop a simpler model We propose a deterministic channel model e-emphasize the background noise Focus on the interaction between users signals Advantages: Far more analytically tractable calable to multi node networks Helps to visualize information flow and get insights Approximates the Gaussian model

7 Methodology AWGN Finite field Gaussian network eterministic model eterministic network Approximate analysis Perturbation Exact analysis

8 In this talk Apply our methodology to Gaussian relay networks The model to study cooperative communication strategies for next generation of wireless systems (wimax, UMB, ) communication protocols for wireless Ad-hoc networks Characterize its capacity within a constant number of bits evelop a simple, near optimal cooperative relaying strategy relays

9 Outline Introduce the deterministic channel model point-to-point multiple access broadcast Apply it to the relay network determine the capacity of deterministic relay network Going back to Gaussian relay network approximate its capacity determine a near optimal cooperative relaying strategy Other interesting applications of deterministic model

10 Point-to-point Gaussian eterministic x NR y y = NR x + C = 1 2 z log( 1+ NR) Least significant bits are truncated at noise level If n = 1 2 log 2 NR we have C ( n) = det n n 1 2 lognr n α NR on the db scale + captures channel strength

11 Multiple access Gaussian eterministic x 1 Tx 1 NR 1 Tx1 n 1 = log 2 NR 1 = 5 Rx y x 2 Tx 2 NR 2 y = NR 1 x 1 + NR 2 x 2 + z Tx2 + + Rx n 2 = log 2 NR 2 = 2 mod 2 addition Can visualize where signal interaction happens Captures channel strength variations in wireless medium Not captured in other simple models packet collision model

12 Multiple Access (cont.) Gaussian eterministic Tx 1 NR 1 Tx1 n 1 = log 2 NR 1 = 5 Rx Tx 2 NR 2 Tx2 + + Rx n 2 = log 2 NR 2 = 2 mod 2 addition R 2 log(1+nr 2 ) n 2 To within 1 bit captures interference n 1 R 1 log(1+nr 1 )

13 Broadcast Gaussian eterministic Tx NR 1 Rx 1 n 1 = log 2 NR 1 = 5 b b Rx1 n 1 NR 2 Rx 2 Tx log(1+nr 2 ) R 2 n 2 = log 2 NR 2 = 2 b Rx2 n 2 n 2 To within 1 bit R 1 captures broadcast n 1 log(1+nr 1 )

14 Apply the deterministic model to relay networks AWGN Finite field Gaussian relay eterministic model eterministic relay Approximate analysis Perturbation Exact analysis

15 eterministic relay network Link from noide i to to noide j is described by an integer n ij (channel strength) i A 1 n ij j B 1 A 2 B 2 relays

16 Algebraic representation A1 b1 b2 b3 b4 b5 B1 A2 c1 c2 c3 c4 c5 B2 Received ignal: y B1 = 0 0 y j (t) b1 = b2 b3 q=max(n ij ), : shift matrix (q q) : shift matrix of size 5 All operations are in F i N c1 j c2 q n ij = 5 3 x i (t) A1 5 2 x A 2

17 General linear finite-field model Channel from i to j is described by an arbitrary q channel matrix G ij operating on F 2 Received signal: y j ( t) = M i= 1 G ij x i ( t) (mod 2) eterministic model: G ij = q n ij Wireline network also a special case

18 Cut-set upper bound A1 B1 A2 B2 Ω c Ω C relay C = max P X 1,..., XM min I(X Ω Ω ;Y Ω c X Ω c ) For deterministic, linear finite field model C relay C = min Ω rank( G Ω Ω c )

19 Main result Theorem: Cutset bound is achievable, C relay = C = min Ω rank( G Ω Ω c ) In wireline networks, rank( G Ω Ω of the capacity of links from Ω to Ω c is just summation Our theorem is a generalization of Ford-Fulkerson max-flow min-cut theorem Also holds in the multicast scenario Generalization of network coding to achieve the multicast capacity of wireline networks (Ahlswede-Cai-Li-Yeung) c )

20 Example: one relay How to achieve the capacity? Routing! How to achieve the capacity in other networks? n R R n R n C = min ( max( n, n ),max( n, n )) R R ( n n ) +,( n n + ) = n + min ) R R

21 Multi-stage network (special case) A1 B1 m 3 m 2 m 1 m ˆ, m ˆ 1 2,... A2 B2 Lengths of all paths from to are the same Major simplification messages do not mix in the network Use a network coding strategy (similar to Ahlswede et. al. 2000): : map each message into a random codeword of length T symbol times Each relay randomly maps the received signal into a transmit codeword Min-cut is achieved

22 General networks Consider the time-expanded network with k stages, each T symbol times long It is a multi-stage network! Apply the same strategy on (super) messages Can achieve 1/k of the min-cut of time-expanded network ( ) r[1] r[2] r[3] r[4] A [1] [2] [3] [4] C k A[1] B[1] n 1 n 2 n 3 n 4 A[2] B[2] n 1 n 2 n 3 n 4 A[3] B[3] n 1 n 2 n 3 n 4 A[4] B[4] n 1 n 3 n 4 [1] t[1] n 5 [2] t[2] n 5 k=4 [3] t[3] n 5 [4] t[4] n 2 B n 5

23 General networks (cont.) Key Question: Is lim = C, min-cut of the original network? k k C k There are more cuts in the time expanded graph Yes! (proof based on submodularity of entropy function) Min-cut is achieved r[1] [1] r[2] [2] r[3] [3] r[4] [4] A A[1] B[1] n 1 n 2 n 3 n 4 A[2] B[2] n 1 n 2 n 3 n 4 A[3] B[3] n 1 n 2 n 3 n 4 A[4] B[4] n 1 n 3 n 4 [1] t[1] n 5 [2] t[2] n 5 k=4 [3] t[3] n 5 [4] t[4] n 2 B n 5

24 Back to the Gaussian relay network AWGN Finite field Gaussian relay eterministic model eterministic relay Approximate analysis Perturbation Exact analysis - Capacity characterization - Optimal communication scheme

25 Example: one relay Gaussian h R R h R eterministic h n R R n R C 1 C = C? C n 1 Gap is at most 1-bit On average it is much less than 1-bit gap h h R 2 2 h h R 2 2 ecode-forward ecode-forward is near optimal? Routing is optimal

26 Relaying scheme eterministic encodes the message over T symbol times Each relay randomly maps the received signal into a transmit codeword decodes the message deterministically optimal Gaussian encodes the message over T symbol times Each relay, Quantizes the received signal at noise level Randomly maps it into a Gaussian codeword decodes the message by finding the one that is jointly ˆ y A1 : x A1 y ˆ B1 : x B1 typical with y x y ˆ y A 2 : x A 2 ˆ y B 2 : x B 2

27 Properties of the scheme imple Quantize Map to a transmit codeword Relays don t need any channel information How does it perform? m ball A 1 m ball m ball B 1 m ball ˆ y m m m ball m ball A 2 B 2 m ball m ball m ball m ball

28 Main result Theorem: for any Gaussian relay network C κ C C C κ - is the cut-set upper bound on the capacity - is a constant that depends on size of the network, but not the channel gains or NR s of the links Uniform approximation of the capacity for all channel parameters Much stronger than degrees of freedom calculation

29 Extensions We generalize the result to the following scenarios: Multicast to multiple destinations Nodes have multiple antennas Half-duplex constraint Fading (channel variations over time)

30 ummary Complexity of Gaussian model prevents further development in understanding wireless networks evelopment of a linear deterministic model to: Help obtain intuitive engineering insights Help make progress in wireless network information theory Future interesting applications of the deterministic model Information theory Wireless communications Networking

31 The End