Gaussian Interference Channel Capacity to Within One Bit

Gaussian Interference Channel Capacity to Within One Bit David Tse Wireless Foundations U. C. Berkeley MIT LIDS May 4, 2007 Joint work with Raul Etkin (HP) and Hua Wang (UIUC)

State-of-the-Art in Wireless • Two key features of wireless channels: – fading – interference • Fading: – information theoretic basis established in 90’s – leads to new points of view (opportunistic and MIMO communication). – implementation in modern standards (eg. CDMA 2000 EV-DO, HSDPA, 802. 11 n) • What about interference?

Interference • Interference management is an central problem in wireless system design. • Within same system (eg. adjacent cells in a cellular system) or across different systems (eg. multiple Wi. Fi networks) • Two basic approaches: – orthogonalize into different bands – full sharing of spectrum but treating interference as noise • What does information theory have to say about the optimal thing to do?

Two-User Gaussian Interference Channel message m 1 want m 1 message m 2 want m 2 • Characterized by 4 parameters: – Signal-to-noise ratios SNR 1, SNR 2 at Rx 1 and 2. – Interference-to-noise ratios INR 2 ->1, INR 1 ->2 at Rx 1 and 2.

Related Results • If receivers can cooperate, this is a multiple access channel. Capacity is known. (Ahlswede 71, Liao 72) • If transmitters can cooperate , this is a MIMO broadcast channel. Capacity recently found. (Weingarten et al 04) • When there is no cooperation of all, it’s the interference channel. Open problem for 30 years.

State-of-the-Art • If INR 1 ->2 > SNR 1 and INR 2 ->1 > SNR 2, then capacity region Cint is known (strong interference, Han. Kobayashi 1981, Sato 81) • Capacity is unknown for any other parameter ranges. • Best known achievable region is due to Han. Kobayashi (1981). • Hard to compute explicitly. • Unclear if it is optimal or even how far from capacity. • Some outer bounds exist but unclear how tight (Sato 78, Costa 85, Kramer 04).

Review: Strong Interference Capacity • INR 1 ->2 > SNR 1, INR 2 ->1> SNR 2 • Key idea: in any achievable scheme, each user must be able to decode the other user’s message. • Information sent from each transmitter must be common information, decodable by all. • The interference channel capacity region is the intersection of the two MAC regions, one at each receiver.

Our Contribution • We show that a very simple Han-Kobayashi type scheme can achieve within 1 bit/s/Hz of capacity for all values of channel parameters: For any in Cint, this scheme can achieve with: • Proving this result requires new outer bounds. • In this talk we focus mainly on the symmetric capacity Csym and symmetric channels SNR 1=SNR 2, INR 1 ->2=INR 2 ->1

Proposed Scheme for INR < SNR decode common private then Set private power so that it is received at the level of the noise at the other receiver (INRp = 0 d. B).

Why set INR p = 0 d. B? • This is a sweet spot where the damage to the other link is small but can get a high rate in own link since SNR > INR.

Main Results (SNR > INR) • The scheme achieves a symmetric rate per user: • The symmetric capacity is upper bounded by: • The gap is at most one bit for all values of SNR and INR.

Interference-Limited Regime • At low SNR, links are noise-limited and interference plays little role. • At high SNR and high INR, links are interferencelimited and interference plays a central role. • In this regime, capacity is unbounded and yet our scheme is always within 1 bit of capacity. • Interference-limited behavior is captured.

Baselines • Point-to-point capacity: • Achievable rate by orthogonalizing: • Achievable rate by treating interference as noise: • Similar high SNR approximation to the capacity can be made using our bounds (error < 1 bit).

Performance plot

Upper Bound: Z-Channel • Equivalently, x 1 given to Rx 2 as side information.

How Good is this Bound?

What’s going on? Scheme has 2 distinct regimes of operation: Z-channel bound is tight. Z-channel bound is not tight.

New Upper Bound • Genie only allows to give away the common information of user i to receiver i. • Results in a new interference channel. • Capacity of this channel can be explicitly computed!

New Upper Bound + Z-Channel Bound is Tight

Generalization • Han-Kobayashi scheme with private power set at INRp = 0 d. B is also within 1 bit to capacity for the entire region. • Also works for asymmetric channel. • Is there an intuitive explanation why this scheme is universally good?

Review: Rate-Splitting Capacity of AWGN channel: d. Q Q 1 noise-limited regime self-interference-limited regime

Rate-Splitting View Yields Natural Private-Common Split • Think of the transmitted signal from user 1 as a superposition of many layers. • Plot the marginal rate functions for both the direct link and cross link in terms of received power Q in direct link. SNR= 20 d. B INR = 10 d. B private common INRp =0 d. B Q(d. B)

Conclusion • We present a simple scheme that achieves within one bit of the interference channel capacity. • All existing upper bounds require one receiver to decode both messages and can be arbitrarily loose. • We derived a new upper bound that requires neither user to decode each other’s message. • Results have interesting parallels with El Gamal and Costa (1982) for deterministic interference channels.