Information flow over wireless networks: a deterministic approach alman Avestimehr In collaboration with uhas iggavi (EPFL) and avid Tse (UC Berkeley)
Overview Point-to-point channel Information theory provides an abstraction C Wireless network oes information theory give us a similar picture? Not yet.
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
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)
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?
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
Methodology AWGN Finite field Gaussian network eterministic model eterministic network Approximate analysis Perturbation Exact analysis
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
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
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
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
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 )
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 )
Apply the deterministic model to relay networks AWGN Finite field Gaussian relay eterministic model eterministic relay Approximate analysis Perturbation Exact analysis
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
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 2 0 0 0 i N c1 j c2 q n ij = 5 3 x i (t) A1 5 2 x A 2
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
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 )
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 )
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
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
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
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
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
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
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
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
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
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)
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
The End