Gaussian Interference Channel Capacity to Within One Bit

1
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)

TexPoint fonts used in EMF AAAAAAAAAAA
2
State-of-the-Art in Wireless

• Two key features of wireless channels
• fading
• interference
• Fading
• information theoretic basis established in 90s
• 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?

3
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?

4
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-gt1, INR1-gt2 at
  Rx 1 and 2.

message m1
want m1
message m2
want m2
5
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, its the
  interference channel. Open problem for 30 years.

6
State-of-the-Art

• If INR1-gt2 gt SNR1 and INR2-gt1 gt 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).

7
Review Strong Interference Capacity

• INR1-gt2 gt SNR1, INR2-gt1gt SNR2
• Key idea in any achievable scheme, each user
  must be able to decode the other users 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.

8
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 SNR1SNR2,
  INR1-gt2INR2-gt1

9
Proposed Scheme for INR lt SNR

• Split each users signal into two streams
• Common information to be decoded by all.
• Private information to be decoded by own receiver
  and appear as noise in the other link.
• Set private power so that it is received at the
  level of the noise at the other receiver
• (INRp 0 dB).
• Each receiver decodes all the common information
  first, cancel them and then decodes its own
  private information.

10
Proposed Scheme for INR lt SNR
decode
common
then
private
decode
common
private
then
Set private power so that it is received at the
level of the noise at the other receiver (INRp
0 dB).
11
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 gt INR.

12
Main Results (SNR gt 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.

13
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.

14
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 lt 1 bit) .

15
Performance plot
16
Upper Bound Z-Channel

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

17
How Good is this Bound?

18
Whats going on?

• Scheme has 2 distinct regimes of operation

Z-channel bound is tight.
Z-channel bound is not tight.
19
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!

20
New Upper Bound Z-Channel Bound is Tight
21
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?

22
Review Rate-Splitting

• Capacity of AWGN channel

dQ
Q
1
noise-limited regime
self-interference-limited regime
23
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
private
common
INRp 0dB
Q(dB)
24
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 others message.
• Results have interesting parallels with El Gamal
  and Costa (1982) for deterministic interference
  channels.