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

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.

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

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

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

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

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

General Methodology Deterministic networkGaussian network Deterministic model Exact analysisApproximate analysis Perturbation AWGN Linear Finite field

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)

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.

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

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

High SNR Convergence

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

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

History The (single) relay channel was first proposed by Van der Meulen in 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

The Relay Channel Gaussian Deterministic S R D h SR h RD h SD Decode-Forward is near optimal Decode-Forward is optimal On average it is much less than 1-bit x x n SR n RD n SD gap Cutset bound is achievable. Theorem (Avestimehr et al 07) Gap from cutset bound is at most 1 bit.

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?

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

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.

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.

Achievability (Special Case): Equal Path Lengths Lengths of all paths from source to destination are the same. Source maps each message into a random codeword. Each relay randomly bins its received signal and transmits the bin index. Same achievability strategy as Ashlwede et al 00. Key simpification of equal path lengths: messages don’t mix. Can generalize to arbitrary networks via time-expansion.

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)

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

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.

Two-User Gaussian Interference Channel Capacity region unknown Best known achievable region: Han & Kobayashi 81. message m 1 message m 2 want m 1 want m 2

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

Symmetric Deterministic Capacity 1 1/2

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)

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)

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 ? Tx 0 Tx 1 Tx 2 Rx 0 Rx 1 Rx 2

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

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.

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.

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.