Information Theory of Wireless Networks: A Deterministic Approach

1. Information Theory of Wireless Networks:A Deterministic Approach David Tse Wireless Foundations U.C. Berkeley CISS 2008 March 21, 2008 Joint work with Salman Avestimehr, Guy Bresler, Suhas Diggavi, Abhay Parekh. TexPoint fonts used in EMF: AAAAAAAAAAAAAA

2. The Holy Grail • Shannon’s information theory provides the basis for all modern-day communication systems. • His original theory was point-to-point. • After 60 years we are still very far away from generalizing the theory to networks. • We propose an approach to make progress in the context of wireless networks.

3. Modeling the Wireless Medium • broadcast • superposition • high dynamic range in channel strengths between different nodes • Basic model: additive Gaussian channel:

4. Gaussian Network Capacity: What We Know Tx Rx point-to-point (Shannon 48) Tx 1 Rx1 Rx Tx Tx 2 Rx 2 multiple-access (Alshwede, Liao 70’s) broadcast (Cover, Bergmans 70’s)

5. What We Don’t Know Unfortunately we don’t know the capacity of most other Gaussian networks. Tx 1 Rx 1 Tx 2 Rx 2 Interference (Best known achievable region: Han & Kobayashi 81) Relay S D relay (Best known achievable region: El Gamal & Cover 79)

6. 30 Years Have Gone by….. We are still stuck. How to make progress?

7. It’s the model. • Shannon focused on noise in point-to-point communication. • But many wireless networks are interference rather than noise-limited. • We propose a deterministic channel model emphasizing interaction between users’ signals rather than on background noise. • Far more analytically tractable and can be used to determine approximate Gaussian capacity

8. Agenda Warmup: • point-to-point channel • multiple access channel • broadcast channel The meat: • relay networks (Avestimehr, Diggavi & T. 07) • interference channels (Bresler &T. 08, Bresler,Parekh & T. 08)

9. Gaussian Transmit a real number If we have Example 1: Point-to-Point Link Deterministic n /SNR on the dB scale Least significant bits are truncated at noise level.

10. Gaussian Example 2: Multiple Access Deterministic user 2 mod 2 addition user 1 sends cloud centers, user 2 sends clouds. user 1

11. Comparing Multiple Access Capacity Regions Gaussian Deterministic user 2 mod 2 addition user 1 accurate to within 1 bit per user

12. To within 1 bit Example 3: Broadcast Gaussian Deterministic user 1 user 2

13. Agenda Warmup: • point-to-point channel • multiple access channel • broadcast channel The meat: • relay networks • interference channels

14. History • The (single) relay channel was first proposed by Van der Meulen in 1971. • Cover and El Gamal (1979) provided a whole array of achievable strategies. • Recent generalization of these techniques to more than 1 relay. • Do not know how far they are from optimal • General upper bound: cutset bound

15. The Relay Channel Deterministic Gaussian R hSR hRD hSD D nSR S nRD x Theorem (Avestimehr et al 07) Gap from cutset bound is at most 1 bit. x nSD gap Cutset bound is achievable. Decode-Forward is optimal On average it is much less than 1-bit Decode-Forward is near optimal

16. Generalization to Relay Networks • Can the cutset bound be achievable in the deterministic model? • Can one always achieve to within a contant gap of the cutset bound in the Gaussian case?

17. General Relay Networks Main Theorem: Cutset bound is achievable for deterministic networks. (Avestimehr, Diggavi & T. 07)

18. Main Theorem Theorem generalizes to arbitrary linear MIMO channels on finite fields. In the case of wireline graph, rank is the number of links crossing the cut. Our theorem is a generalization of Ford-Fulkerson max-flow min-cut theorem.

19. Connections to Network Coding • Achievability: random linear coding at relays • Proof style: similar to Ahlswede et al 2000 for wireline networks. • Technical innovation: dealing with “inter-symbol interference” between signals arriving along paths of different lengths.

20. Back to Gaussian Relay Networks Approximation Theorem: There is a scheme that achieves within a constant gap to the cutset bound, independent of the SNR’s of the links. (Avestimehr, Diggavi and T. 2008)

21. Agenda Warmup: • point-to-point channel • multiple access channel • broadcast channel The meat: • relay networks • interference channels

22. Interference • So far we have looked at single source, single destination networks. • All the signals received is useful. • With multiple sources and multiple destinations, interference is the central phenomenon. • Simplest interference network is the two-user interference channel.

23. Two-User Gaussian Interference Channel • Capacity region unknown • Best known achievable region: Han & Kobayashi 81. message m1 want m1 message m2 want m2

24. Gaussian to Deterministic Interference Channel GaussianDeterministic In symmetric case, channel described by two parameters: SNR, INR Capacity can be computed using a result by El Gamal and Costa 82.

25. Symmetric Deterministic Capacity 1 1/2

26. Back to Gaussian • Theorem: Constant gap between capacity regions of the two-user deterministic and Gaussian interference channels. (Bresler & T. 08) • A deeper view of earlier 1-bit gap result on two-user Gaussian interference channel (Etkin,T. & Wang 06). • Bounds further sharpened to get exact results in the low-interference regime ( < 1/3) (Shang et al 07,Annaprueddy&Veeravalli08,Motahari&Khandani07)

27. Extension:Many-to-One Interference Channel Gaussian Deterministic Deterministic capacity can be computed exactly . Gaussian capacity to within constant gap, using structured codes and interference alignment. (Bresler, Parekh & T. 07)

28. Example • Interference from users 1 and 2 is aligned at the MSB at user 0’s receiver in the deterministic channel. • How can we mimic it for the Gaussian channel ? Rx0 Tx0 Rx1 Tx1 Rx2 Tx2

29. Suppose users 1 and 2 use a random Gaussian codebook: Gaussian Han-Kobayashi Not Optimal Lattice codes can achieve constant gap Rx0 Tx0 Random Code Rx1 Tx1 Sum of Two Random Codebooks Lattice Code for Users 1 and 2 Rx2 Tx2 Interference from users 1 and 2 fills the space: no room for user 0. User 0 Code

30. Interference Channels: Recap • In two-user case, we showed that an existing strategy can achieve within 1 bit to optimality. • In many-to-one case, we showed that a new strategy can do much better. • General K-user interference channel still open.

31. Evolution of Ideas • deterministic network capacity in 1980’s: • broadcast channels (Marton 78, Pinsker 79) • 2-user interference channel (El Gamal & Costa 82) • single-relay channel (El Gamal & Aref 82) • relay networks with broadcast but no interference (Aref 79) • inspired by network coding in early 2000’s: • finite-field model with erasures (Gupta et al 06) but connection to Gaussian networks missing. • 2-user Gaussian interference channel capacity to within 1 bit (Etkin, T & Wang 06) • Linear deterministic model (Avestimehr, Diggavi & T 07) and applied to relay networks.

32. Parting Words • Main message: If something can’t be computed exactly, approximate. • Similar evolution has happened in other fields: • fluid and heavy-traffic approximation in queueing networks • approximation algorithms in CS theory • Approximation should be good in engineering-relevant regimes.