Gaussian Interference Channel Capacity to Within One Bit

David TseWireless 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, 802.11n)

• 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 WiFi 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

• Characterized by 4 parameters:

– Signal-to-noise ratios SNR1, SNR2 at Rx 1 and 2.

– Interference-to-noise ratios INR2->1, INR1->2 at Rx 1 and 2.

message m1

message m2

want m1

want m2

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 INR1->2 > SNR1 and INR2->1 > SNR2, 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

• INR1->2 > SNR1, INR2->1> SNR2

• 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 SNR1=SNR2, INR1->2=INR2->1

~R1 ¸ R1 ¡ 1~R2 ¸ R2 ¡ 1

( ~R1; ~R2)(R1;R2)

Proposed Scheme for INR < SNR

W1

U1

W2

Set private power so that it is received at the level of the noise at the other receiver (INRp = 0 dB).

U2

common

private

common

private

decode

W1

W1

W2

W2

U1

U2

then

decode

then

Why set INRp = 0 dB?

• 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.

R = min

(12

log(1+INR+SNR)+12

logµ

2+SNRINR

¶¡ 1; log

µ1+ INR +

SNRINR

¶¡ 1

)

Csym · min

(12

log(1+ SNR)+12

logµ

1+SNR

1+ INR

¶; log

µ1+ INR +

SNR1+ INR

¶ )

Interference-Limited Regime

• At low SNR, links are noise-limited and interference plays little role.

• At high SNR and high INR, links are interference-limited 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) .

Cawgn = log(1+ SNR) ¼logSNR

R = log(1+SNR

1+ INR) ¼logSNR ¡ logINR

R =12

log(1+ 2SNR) ¼12

log(SNR)

Performance plot

logINR

logSNR ¡12

logINR

12

logINR

logSNR ¡ logINR

Upper Bound: Z-Channel

• Equivalently, x1 given to Rx 2 as side information.

How Good is this Bound?

23

What’s going on?

Scheme has 2 distinct regimes of operation:

Z-channel bound is tight. Z-channel bound is not tight.

logINRlogSNR

>23

logINRlogSNR

<23

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

23

Generalization

• Han-Kobayashi scheme with private power set at INRp = 0dB 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:

Q

dQ

1

Q(dB)

noise-limited regime

self-interference-limited regime

C(P ) = log2(1+ P )

=Z P

0C0(Q)dQ

=Z P

0

dQ1+ Q

log2 e

log2 e1+Q

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= 20dB INR = 10dB

Q(dB)

INRp

=0dB

private common

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.