Wireless Network Information Flow

Transcription

1 Š#/,% 0/,94%#(.)15% Wireless Network Information Flow Suhas iggavi School of Computer and Communication Sciences, Laboratory for Information and Communication Systems (LICOS), EPFL URL: September 9th 2008

2 Motivation Wireless characteristics State of knowledge Network communication challenges Information flow over shared networks Unicast, multicast, multiple unicast (multicommodity flow). Significant progress for graphs (routing, network coding etc). Less understood for flows over wireless networks. Network data compression: Motivation sensor networks. Some successes in side-information coding: Slepian-Wolf, Wyner Ziv etc. Many unresolved questions: istributed source coding, multiple description coding. Question: How can we make progress to fundamentally characterize flow of information over networks?

3 Motivation Wireless characteristics State of knowledge Key distinctions between wired and wireless channels Broadcast: Transmit signal potentially received by multiple receivers.

4 Motivation Wireless characteristics State of knowledge Key distinctions between wired and wireless channels Broadcast: Transmit signal potentially received by multiple receivers. Multiple access: Transmitted signals mix at the receivers.

5 Motivation Wireless characteristics State of knowledge Key distinctions between wired and wireless channels Broadcast: Transmit signal potentially received by multiple receivers. Multiple access: Transmitted signals mix at the receivers. High dynamic range: Large range in relative signal strengths.

6 Motivation Wireless characteristics State of knowledge Key distinctions between wired and wireless channels Broadcast: Transmit signal potentially received by multiple receivers. Multiple access: Transmitted signals mix at the receivers. High dynamic range: Large range in relative signal strengths. Implications: Complex signal interactions at different signal levels.

7 Motivation Wireless characteristics State of knowledge Key distinctions between wired and wireless channels Broadcast: Transmit signal potentially received by multiple receivers. Multiple access: Transmitted signals mix at the receivers. High dynamic range: Large range in relative signal strengths. Implications: Complex signal interactions at different signal levels. Interacting signals from nodes contain information (not to be treated as noise).

8 Motivation Wireless characteristics State of knowledge Key distinctions between wired and wireless channels Broadcast: Transmit signal potentially received by multiple receivers. Multiple access: Transmitted signals mix at the receivers. High dynamic range: Large range in relative signal strengths. Implications: Complex signal interactions at different signal levels. Interacting signals from nodes contain information (not to be treated as noise). Question: Can we develop cooperative mechanisms to utilize signal interaction?

9 Gaussian point-to-point channel Motivation Wireless characteristics State of knowledge Message Tx x[t] Channel h z[t] Rx ecoded message NOISY POINT TO POINT CHANNEL (SHANNON, 1948) y[t] = hx[t] + z[t] Linear channel model and additive noise model. Transmit power constraint: E[ x 2 ] P Gaussian channel due to noise z[t] N(0, 1). Capacity: C = 1 2 log(1 + h 2 P).

10 Motivation Wireless characteristics State of knowledge Gaussian multiple access channel Message 1 Tx 1 h 1 x [t] 1 z[t] ecoded messages 1,2 Message 2 Tx 2 h 2 x [t] 2 y[t] Rx MULTIPLE ACCESS CHANNEL: Ahlswede Liao (1971) y[t] = h 1 x 1 [t] + h 2 x 2 [t] + z[t] Linear signal interaction model. Transmit power constraint: E[ x i 2 ] P i, i = 1, 2 Gaussian channel due to noise z[t] N(0, 1). R log(1 + h 1 2 P 1 ) R log(1 + h 2 2 P 2 ) R 1 + R log(1 + h 1 2 P 1 + h 2 2 P 2 )

11 Gaussian broadcast channel Motivation Wireless characteristics State of knowledge Messages 1,2 Tx x[t] h 1 z [t] 1 y [t] 1 ecoded message 1 Rx 1 h 2 z [t] 2 y [t] 2 ecoded message 2 Rx 2 BROACAST CHANNEL: Cover (1972) y 1 [t] = h 1 x[t] + z 1 [t], y 2 [t] = h 2 x[t] + z 2 [t] Common transmit signal x[t], different channels. Transmit power constraint: E[ x 2 ] P. R log(1 + h 1 2 θp 1 + h 1 2 (1 θ)p ) R 2 1 Rate region region for h 1 h 2 2 log(1 + h 2 2 (1 θ)p)

12 Motivation Wireless characteristics State of knowledge Signal interaction: Gaussian wireless networks Source 1 Receiver 1 Source 2 Receiver 2 y j [t] = i H ij x i [t] + z j [t] Broadcast because transmission x i is heard by all receivers. Multiple access because transmitted signals from all nodes mix linearly at the receiver j. ynamic range depends on relative strengths of H ij.

13 Motivation Wireless characteristics State of knowledge Signal interaction: Gaussian wireless networks Source 1 Receiver 1 Source 2 Receiver 2 y j [t] = i H ij x i [t] + z j [t] Broadcast because transmission x i is heard by all receivers. Multiple access because transmitted signals from all nodes mix linearly at the receiver j. ynamic range depends on relative strengths of H ij. Question: Can we characterize capacity of such networks?

14 Motivation Wireless characteristics State of knowledge Gaussian network capacity: unresolved Relay Source Sink RELAY CHANNEL: Cover, El Gamal (1979)

15 Motivation Wireless characteristics State of knowledge Gaussian network capacity: unresolved Relay Message 1 ecode message 1 Tx 1 Rx 1 Source Sink Message 2 Tx 2 ecode message 2 Rx 2 RELAY CHANNEL: Cover, El Gamal (1979) INTERFERENCE CHANNEL: Han Kobayashi (1981)

16 Motivation Wireless characteristics State of knowledge Gaussian network capacity: unresolved Relay Message 1 ecode message 1 Tx 1 Rx 1 Source Sink Message 2 Tx 2 ecode message 2 Rx 2 RELAY CHANNEL: Cover, El Gamal (1979) INTERFERENCE CHANNEL: Han Kobayashi (1981) Question: Thirty years have gone by... How can we make progress from here?

17 Simplify the model Focus on signal interaction not noise Philosophy eterministic model through examples Observation: Success of network coding was through examination of flow on wireline networks, a special deterministic channel.

18 Simplify the model Focus on signal interaction not noise Philosophy eterministic model through examples Observation: Success of network coding was through examination of flow on wireline networks, a special deterministic channel. Idea: Many wireless systems are interference rather than noise limited. Use deterministic channel model to focus on signal interaction and not noise.

19 Simplify the model Focus on signal interaction not noise Philosophy eterministic model through examples Observation: Success of network coding was through examination of flow on wireline networks, a special deterministic channel. Idea: Hope: Many wireless systems are interference rather than noise limited. Use deterministic channel model to focus on signal interaction and not noise. eterministic models more tractable. Use insight to obtain approximate characterizations for noisy (Gaussian) networks.

20 Simplify the model Focus on signal interaction not noise Philosophy eterministic model through examples Observation: Success of network coding was through examination of flow on wireline networks, a special deterministic channel. Idea: Hope: Many wireless systems are interference rather than noise limited. Use deterministic channel model to focus on signal interaction and not noise. eterministic models more tractable. Use insight to obtain approximate characterizations for noisy (Gaussian) networks. Question: Can we develop relevant models and analyze networks with deterministic signal interactions to get the insights?

21 Approximate characterizations Philosophy eterministic model through examples Philosophy: Gain insight into central difficulties of problem by identifying underlying deterministic structures. Goal: Use the insight of underlying problem to get (provable) approximate characterization for noisy problem. Underlying problem should be characterized exactly to give insight into solution structure for general case. Universal approximation: Approximation should depend only on the problem structure and not on parameters (like channel gains, SNRs etc.). Question: Can we identify the appropriate underlying problems and use them to get provable (universal) approximations.

22 Overall agenda Philosophy eterministic model through examples Introduce deterministic channel model. Motivate the utility of deterministic model with examples. evelop achievable rates for general deterministic relay networks Characterizations for linear finite field deterministic models.

23 Overall agenda Philosophy eterministic model through examples Introduce deterministic channel model. Motivate the utility of deterministic model with examples. evelop achievable rates for general deterministic relay networks Characterizations for linear finite field deterministic models. Connection to wireless networks: Use insights on achievability of deterministic networks to obtain approximate characterization of noisy relay networks.

24 Example 1: Point-to-point link Philosophy eterministic model through examples Gaussian eterministic y = 2 α/2 x + z A B Capacity is log(1 + 2 α ) α log 2 assuming unit variance noise. Receiver observes α most significant bits of transmitted signal. Number of levels received shows scale of channel strength. Scale important when signals interact in broadcast and multiple access.

25 Philosophy eterministic model through examples Example 2: Multiple access channel ETERMINISTIC MOEL Rx GAUSSIAN MOEL B 1 R 2 y = 2 α1/2 x α2/2 x 2 + z. n 2 log(1 + SNR 2 ) B 2 Mod 2 addition log(1 + SNR 1 ) n 1 R 1

26 Philosophy eterministic model through examples Example 3: Scalar broadcast channel Tx GAUSSIAN MOEL ETERMINISTIC MOEL A 1 S y 1 = 2 α1/2 x + z 1, y 2 = 2 α2/2 x + z 2 R 2 A 2 n 2 log(1 + SNR 2 ) Approximation of 1 bit log(1 + SNR 1 ) n 1 R 1

27 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R S

28 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R R S S ETERMINISTIC MOEL

29 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R R S S Source cut = 3 bits ETERMINISTIC MOEL

30 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R R S S ETERMINISTIC MOEL estination cut = 3 bits

31 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R R S S ETERMINISTIC MOEL

32 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R R S S ETERMINISTIC MOEL

33 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R R S S ETERMINISTIC MOEL

34 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R R S S ETERMINISTIC MOEL Cut-set bound achievable. ecode and forward is optimal.

35 Philosophy eterministic model through examples Relay channel: deterministic approximation GAUSSIAN RELAY CHANNEL R S R S gap h R 2 / h S h SR 2 / h S Result: Gap from cut-set less than 1 bit, on average much less. ETERMINISTIC MOEL Cut-set bound achievable. ecode and forward is optimal.

36 iamond network Philosophy eterministic model through examples Gaussian A h SA1 h A1 S h SA2 h A2 B

37 iamond network Philosophy eterministic model through examples eterministic Cut value = 3 Gaussian A A S h SA1 h A1 S h SA2 h A2 B B

38 iamond network Philosophy eterministic model through examples eterministic Gaussian A Cut value = 3 A h SA1 h A1 S S h SA2 h A2 B B

39 iamond network Philosophy eterministic model through examples Gaussian eterministic Cut value = 6 A A h SA1 h A1 S S h SA2 h A2 B B

40 iamond network Philosophy eterministic model through examples Gaussian A eterministic A Cut value = 3 h SA1 h A1 S S h SA2 h A2 B B

41 iamond network Philosophy eterministic model through examples eterministic Gaussian A A h SA1 h A1 S S h SA2 h A2 B Result: Gap from cut-set less 1 bit. Cut-set bound achievable. Partial decode-forward is optimal. B

42 Two-layer network Philosophy eterministic model through examples Gaussian eterministic A 1 h A1 B 1 B 1 S h S,A1 h S,A2 h A1 B 2 h A2 B 1 h B1 hb2 A 2 h A2 B B 2 2

43 Two-layer network Philosophy eterministic model through examples Gaussian eterministic A 1 h A1 B 1 B 1 h S,A1 h B1 A1 B1 h A1 B 2 S h S,A2 h A2 B 1 hb2 S A 2 h A2 B B 2 2 A2 B2 Cut-set bound achievable. Linear map and forward is optimal.

44 Two-layer network Philosophy eterministic model through examples Gaussian eterministic A 1 h A1 B 1 B 1 h S,A1 h B1 A1 B1 h A1 B 2 S h S,A2 h A2 B 1 hb2 S A 2 h A2 B B 2 2 Result: Gap from cut-set less than constant number of bits. A2 Cut-set bound achievable. Linear map and forward is optimal. B2

45 Questions Philosophy eterministic model through examples Is the cut-set bound achievable for the deterministic model in arbitrary networks?

46 Questions Philosophy eterministic model through examples Is the cut-set bound achievable for the deterministic model in arbitrary networks? What is the structure of the optimal strategy?

47 Questions Philosophy eterministic model through examples Is the cut-set bound achievable for the deterministic model in arbitrary networks? What is the structure of the optimal strategy? Can we use insight from deterministic analysis to get approximately optimal strategy for Gaussian networks?

48 Algebraic representation Philosophy eterministic model through examples S A1 b 1 y 1 2 b 2 y 2 b y 3 3 b y 4 4 b 5 y 5 S = 6 4 A2 B1 B c 1 c 2 c 3 c 4 c 5 S is shift matrix of size q = max i,j n i,j. y B1 = y 1 y 2 y 3 y 4 y 5 = 0 0 b 1 b 2 b c 1 c 2 = S5 3 x A1 S 5 2 x A2 = S 5 3 b S 5 2 c

49 Generalizations Linear finite field model Philosophy eterministic model through examples Channel from i to j is described by channel matrix G ij operating over F 2. Received signal at node j: y j [t] = N G ij x i [t] i=1 Special case: our model given in examples G ij = S q α ij General deterministic network: y[t] = G(x 1 [t],..., x N [t]) Observation: Wireline networks are a special case.

50 Information-theoretic cut-set Cut-set upper bound Results: deterministic model Results: Gaussian model A1 B1 S A2 B2 Cut: Separates S from Ω Λ Ω c Cut transfer matrix G Ω,Ω c: Transfer function from nodes in Ω to Ω c.

51 Cutset upper bound Cut-set upper bound Results: deterministic model Results: Gaussian model General relay network: C relay C = max min I(X Ω ; Y Ω c X Ω c) p(x 1,...,x N ) Ω

52 Cutset upper bound Cut-set upper bound Results: deterministic model Results: Gaussian model General relay network: C relay C = General deterministic relay network: C relay C = max min I(X Ω ; Y Ω c X Ω c) p(x 1,...,x N ) Ω max min H(Y Ω c X Ω c) p(x 1,...,x N ) Ω

53 Cutset upper bound Cut-set upper bound Results: deterministic model Results: Gaussian model General relay network: C relay C = General deterministic relay network: C relay C = max min I(X Ω ; Y Ω c X Ω c) p(x 1,...,x N ) Ω max min H(Y Ω c X Ω c) p(x 1,...,x N ) Ω Linear finite field network: Optimal input distribution x 1,..., x N independent and uniform C relay C = min Ω rank(g Ω,Ω c) where G Ω,Ω c is the transfer matrix X Ω Y Ω c.

54 Cut-set upper bound Results: deterministic model Results: Gaussian model Main results: eterministic relay networks Theorem (Avestimehr, iggavi and Tse, 2007) Given a general deterministic relay network (with broadcast and multiple access), we can achieve all rates R upto Multicast information flow: Qmax min H(Y Ω c X Ω c) i p(x i ) Ω Theorem (Avestimehr, iggavi and Tse, 2007) Given a general deterministic relay network (with broadcast and multiple access), we can achieve all rates R from S multicasting to all destinations up to, Qmax min min H(Y Ω c X Ω c) i V p(x i) Ω Λ

55 Application Linear deterministic models Cut-set upper bound Results: deterministic model Results: Gaussian model Corollary (Avestimehr, iggavi and Tse, 2007) Given a linear finite-field relay network (with broadcast and multiple access), the capacity C of such a relay network is given by, Multicast information flow: C = min Ω Λ rank(g Ω,Ω c). Corollary (Avestimehr, iggavi and Tse, 2007) Given a linear finite-field relay network (with broadcast and multiple access), the multicast capacity C of such a relay network is given by, C = min min Ω Λ rank(g Ω,Ω c).

56 Cut-set upper bound Results: deterministic model Results: Gaussian model Consequences: eterministic Relay Networks General deterministic networks: Cutset upper bound was C relay max p(x 1,...,x N ) min Ω H(Y Ω c X Ω c) = achievable if optimum was product distribution. Linear finite field model: Cutset upper bound was C relay min Ω rank(g Ω,Ω c) = cutset bound achievable For wireline graph model rank(g Ω,Ω c) is number of links crossing the cut. Observation: We have a generalization of Ford-Fulkerson max-flow min-cut theorem to linear finite field relay networks with broadcast and multiple access.

57 Cut-set upper bound Results: deterministic model Results: Gaussian model Main results: Gaussian relay networks Theorem (Avestimehr, iggavi and Tse, 2007) Given a Gaussian relay network, G, we can achieve all rates R up to C κ. Therefore the capacity of this network satisfies C κ C C, where C is the cut-set upper bound on the capacity of G, and κ is a constant independent of channel gains. Theorem (Multicast information flow) Given a Gaussian relay network, G, we can achieve all multicast rates R up to C mcast κ, i.e., for C mcast = min C, C mcast κ C C mcast

58 Ingredients and insights Cut-set upper bound Results: deterministic model Results: Gaussian model Main steps: Gaussian strategy Relay operation: Quantize received signal at noise-level. Relay function: Random mapping from received quantized signal to transmitted signal. Handle unequal (multiple) paths between nodes like inter-symbol interference. Consequences: With probabilistic method we demonstrate min-cut achievability for linear deterministic networks. Gaussian networks constant gap independent of SNR operating point. Engineering insight of (almost) optimal coding strategies.

59 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Achievability program: eterministic networks Layered (equal path) networks S A 1 B 1 Lengths of ALL paths from source to destination are the same. Broadcast and multiple access for general deterministic functions. A 2 B 2

60 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Achievability program: eterministic networks Layered (equal path) networks S A 1 B 1 Lengths of ALL paths from source to destination are the same. Broadcast and multiple access for general deterministic functions. A 2 B 2 Illustrate analysis through equal path network.

61 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Achievability program: eterministic networks Layered (equal path) networks S A 1 B 1 Lengths of ALL paths from source to destination are the same. Broadcast and multiple access for general deterministic functions. A 2 B 2 Illustrate analysis through equal path network. Extend to unequal path networks through time-expansion.

62 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Achievability: eterministic networks Map each message into random codeword of length T symbols. Each relay randomly independently maps its received signal onto transmit codewords = transmit distributions independent across relays. x j = f j (y j ) Strategy similar to network coding for wireline graphs (Ahlswede et al 2000). For linear deterministic network, simplification in relay function: x j = F j y j where F j is randomly chosen matrix.

63 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Achievability: eterministic networks Y = B 1 G X SA S 1 X = F A A Y Y A A X A = F A Y A X S A1 A2 Y = G A2 X G A X B B S B1 B2 Y = B 1 G SA X S 2 X = F B Y A B Y B 2 X = F Y B B B 2 2 2

64 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Linear deterministic networks: relay strategy Key simplification for staged networks: In equal path networks all nodes in a stage are transmitting information about same message.

65 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Linear deterministic networks: relay strategy Key simplification for staged networks: In equal path networks all nodes in a stage are transmitting information about same message. Time Block 1 A A 1 2 W 3 W 2 W 1 S B 1 B 2

66 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Linear deterministic networks: relay strategy Key simplification for staged networks: In equal path networks all nodes in a stage are transmitting information about same message. Time Block 2 A A 1 2 W 3 W 2 W 1 S B 1 B 2

67 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Linear deterministic networks: relay strategy Key simplification for staged networks: In equal path networks all nodes in a stage are transmitting information about same message. Time Block 3 A A 1 2 W 3 W 2 W 1 S B 1 B 2 Implication: Focus on message w = W 1, which passes through layer l at block time l.

68 eterministic layered network General deterministic network Gaussian layered network Compound relay networks : Layered (equal path) deterministic networks Focus on one message w of RT bits. P {error} 2 RT P {w w } istinguishablity: Nodes that receive distinct signals under w and w can disambiguate between them = received signals under two message distinct or y j y j when j can distinguish. P w w = X Ω P Nodes in Ω can distinguish w, w and nodes in Ω c cannot {z } P where Ω is a source-destination separation cut.

69 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Linear deterministic networks: Confusability analysis Key idea: Nodes in Ω c (w, w ), will transmit same codeword/signal under both w, w. Consequence: Error event being analyzed when y Ω c = y Ωc, i.e., Can distinguish A 1 Ω Can distinguish A 2 Ω G Ω,Ω cx Ω = G Ω,Ω cx Ω G Ω,Ω c (x Ω x Ω ) = 0 G Ω,Ω c S Ω Cannot distinguish B 1 Ω c Transmits same signal under w, w Ω c Cannot distinguish B 2 Ω c For x j = F j y j with F j uniform i.i.d. random matrix over F 2, we need, (. F j yj y ) j } {{ } z j. = G Ω,Ω cz = 0 j Ω.

70 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Confusability analysis: Linear deterministic networks ) Since F j is a uniform i.i.d. matrix over F 2, z j = F j (y j y j is uniform vector over F q 2, due to y j y j for j Ω. Therefore z is a uniform vector over F q Ω 2 and we are calculating its probability of being in null space of G Ω,Ω c. This probability is: 2 T rank(g Ω,Ω c), i.e., P {Ω(w, w ) = Ω} = 2 T rank(g Ω,Ω c) Hence for linear deterministic layered networks taking union bound, P {error} 2 RT Ω 2 T rank(g Ω,Ω c) Implication: R < min Ω rank(g Ω,Ω c) is achievable.

71 Large deviations result eterministic layered network General deterministic network Gaussian layered network Compound relay networks When y j = f(x 1,...,x N ) for general deterministic functions, we need a more sophisticated error calculation. Basic large-deviations result: If a T -length sequence is generated i.i.d. according to probability law q the probability that its emperical behavior is like that of sequence generated as p is given by: P(q p) = 2 T(p q), where (p q) is the relative entropy given by: (p q) = u p(u) log p(u) q(u) For q = p(x)p(y), p = p(x, y) this probability is 2 TI(X;Y) since (p q) = u p(u) log p(u) q(u) = p(x, y) log x,y p(x, y) = I(X; Y) p(x)p(y) }{{} mutual information.

72 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Confusability analysis: eterministic networks Can distinguish A 1 Ω Can distinguish A 2 Ω S Ω Ω c Cannot distinguish Transmits same signal under B 1 Ω c w, w Cannot distinguish B 2 Ω c

73 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Confusability analysis: eterministic networks Can distinguish A 1 Ω Can distinguish A 2 Ω S Ω Ω c Cannot distinguish Transmits same signal under B 1 Ω c w, w Cannot distinguish B 2 Ω c Consequence: Error for e.g., when w codewords generated like p(x B1, y B2 )p(x A1 ), are jointly typical with y B2, occurs with probability 2 T E E = (p(x B1, y B2, x A1 ) p(x B1, y B2 )p(x A1 ) = I(X A1 ; Y B2, X B1 ) = I(X A1 ; X B1 ) +I(X A1 ; Y B2 X B1 ) }{{} 0

74 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Confusability analysis: eterministic networks Continuing this way we get P 2 T[I(X S;Y B1 )+I(X A1 ;Y B2 X B1 )+I(X A2 ;Y X B2 )] = 2 TH(Y Ω c X Ω c) because we have a layered network and using Markov relationship.

75 eterministic layered network General deterministic network Gaussian layered network Compound relay networks Confusability analysis: eterministic networks Continuing this way we get P 2 T[I(X S;Y B1 )+I(X A1 ;Y B2 X B1 )+I(X A2 ;Y X B2 )] = 2 TH(Y Ω c X Ω c) because we have a layered network and using Markov relationship. Implication: For layered networks an achievable rate is R < Qmax min H(Y Ω c X Ω c) i p(x i ) Ω

76 General deterministic networks eterministic layered network General deterministic network Gaussian layered network Compound relay networks Unequal path networks = message synchronization lost. Approach: Consider time-expanded network over K blocks. Observation: Time-expanded network is an equal path network for (super) message of KRT bits. Convert multi-letter achievable rate to single-letter using submodularity of entropy. Observation: Time-expansion was needed in network coding for cyclic networks, but in our case it is useful even in acyclic networks.

77 eterministic layered network General deterministic network Gaussian layered network Compound relay networks General networks: Time expansion S kq t 1 kq t 2 kq t 3 t k 2 kq t k 1 kq t k kq S 1 A 1 kq α 1 α 2 α 3 α4 S 2 A 2 kq S 3 A 3 S k 2 A k 2 S k 1 A k 1 S k A k h SA A h A B 1 α 5 B 2 B 3 B k 2 B k 1 B k S h AB k 2 k 1 k r 1 kq kq r 2 kq r 3 r k 2 r k 1 r k kq kq kq kq kq h SB B h B Create a virtual node for every time-block = new network is layered = previous results apply. Many cuts in time-expanded network, not a cut in original network. Only horizontal cuts matter using submodularity of entropy.

78 Gaussian coding strategy eterministic layered network General deterministic network Gaussian layered network Compound relay networks Encoding Each relay quantizes received signal to noise-level distortion. Each relay independently randomly maps quantized signal to a Gaussian transmit signal satisfying power constraints. Caution: This is not a compress-forward strategy since we are not trying to reconstruct any of the quantized outputs. ecoding estination finds all the messages w that are jointly typical with received quantized sequence. Note: The relays do not decode any part of the message.

79 Typicality and typical sequences Important tool in information theory eterministic layered network General deterministic network Gaussian layered network Compound relay networks Typical sequence: For i.i.d. generated sequence u 1,..., u T is typical with respect to probability measure p if, 1 T log p(u 1,...,u T ) = 1 T t log p(u t ) T H(U) Jointly typical sequences: Sequences {(u t, v t } generated i.i.d. are jointly typical if individually they are typical and 1 T log p(u 1,..., u T, v 1,..., v T ) = 1 T t log p(u t, v t ) T H(U, V) Facts: For i.i.d. generated sequences, the probability of getting atypical sequence is exponentially small. All typical sequences are asymptotically equally likely.

80 Ingredients of analysis Perturbation of deterministic case eterministic layered network General deterministic network Gaussian layered network Compound relay networks w ball w ball Can distinguish w, w ŷ A1 A 1 Ω B 1 Ω Cannot distinguish w, w w ball ŷ A2 S w Ω ball A 2 Ω c B 2 Ω c Ω c Typicality Message w multiple transmitted signals. Message jointly typical (quantized) signal, any plausible sequence that is typical with it.

81 Ingredients of analysis Perturbation of deterministic case eterministic layered network General deterministic network Gaussian layered network Compound relay networks S Ω Can distinguish w, w Cannot distinguish w, w A 1 Ω B 1 Ω A 2 Ω c B 2 Ω c Ω c w ball ŷ w ball Error events estination quantized signal typical with w and w = cannot distinguish between them. signals. istinguishablity: Nodes that are not jointly typical with both w and w. ivide network into nodes that can and cannot distinguish w, w = defines a source-destination separation cut.

82 Ingredients of analysis Perturbation of deterministic case Can distinguish w, w eterministic layered network General deterministic network Gaussian layered network Compound relay networks w ball w ball A 1 Ω B 1 Ω ŷ B1 S Ω A 2 Ω c B 2 Ω c Ω c w ball w ball Cannot distinguish w, w ŷ B2 Putting it together: For particular plausible signal under w is confusable with probability: P 2 T»I(X Ω ;ŶΩ c X Ω c ) Union bound: P X(w T ) 2 {z } 2 Tγ»I(X Ω ;ŶΩ c X Ω c )»I(X 2 T Ω ;ŶΩ c X Ω c ) γ

83 Finishing touches eterministic layered network General deterministic network Gaussian layered network Compound relay networks Components of gap: κ = β + γ + δ Lose β bits due to noise-level quantization. Lose γ bits due to transmit list. Lose δ bits for independent distribution beamforming loss. Layered general networks: Time expansion and single-letterization like in deterministic case. Implication: For cut-set bound C, C κ C C

84 Compound relay networks eterministic layered network General deterministic network Gaussian layered network Compound relay networks Compound model: Channel realizations from a set h i,j H i,j, unknown to sender. Observations: Theorem Relay strategy does not depend on the channel realization. Overall network from source to destination behaves like a compound channel. Utilize point-to-point compound channel ideas get approximate characterization for compound network. Given a compound Gaussian relay network the capacity C cn satisfies C cn κ C cn C cn, where C cn = max p({xj } j V ) inf h H min Ω Λ I(Y Ω c; X Ω X Ω c).

85 Relay networks: Open questions and extensions Š#/,% 0/,94%#(.)15% Extensions: Outage set behavior for full duplex networks. of half-duplex systems with fixed transmit fractions. Ergodic channel variations. Open questions: -M trade-off for channel dependent half-duplex systems. Tightening gap to cut-set bound. Use deterministic model directly to get Gaussian result.

86 Extensions of deterministic approach Š#/,% 0/,94%#(.)15% Interference channel: Successfully used to generate approximate characterization (Bresler and Tse, 2007), K -user interference channel: Used to demonstrate new phenomenon of interference alignment (Bresler-Tse, 2007, Jafar 2007). Relay-interference networks: Extension of multiple unicast to wireless networks (Mohajer, iggavi, Fragouli and Tse, 2008). Wireless network secrecy: Used to demonstrate secrecy over networks (iggavi, Perron and Telatar, 2008). Network data compression: Identify correct multi-terminal lossless structures to get approximation to multiple-description data compression (Tian, Mohajer and iggavi, 2007).

87 Program: Hope: Focus on underlying deterministic coding problem. Obtain exact characterization this is a central challenge. Use insight to obtain approximate characterization of noisy problem. The program will yield insight to network flow problems. Exposes the central difficulties, solution insights and new schemes? Approximations may be sufficient for engineering practice. In this talk: Obtained approximate max-flow min-cut characterization for noisy relay networks. Papers/preprints: