The Capacity of Interference Channels with Partial Transmitter Cooperation

1. The Capacity of Interference Channels with Partial Transmitter Cooperation The theme that I am going to talk about today considers cooperation among the nodes in wireless networks to improve the network performance. This topic has been in a focus for a several years now and has been analyzed from many aspects to understand how to cooperate. We use network information theory to determine the fundamental limits to the performance and propose cooperative schemes that can achieve or come close to these limits. I am going to talk about a specific problem that analyzes a communication situation in which two senders wish to communicate with two receivers, that is. the interference channel. At the same time, we want to allow cooperation between transmitters and investigate gains from transmitter cooperation, i.e. channel models that allow transmitters to partially cooperate. So, we introduce the limited transmitter cooperation in the interference channel. For two different scenarios in the interference channel with limited cooperation, we will show conditions under which, we determine the capacity region. This is joint work with Prof. Roy Yates in Winlab and Gerhard Kramer, Bell Labs.The theme that I am going to talk about today considers cooperation among the nodes in wireless networks to improve the network performance. This topic has been in a focus for a several years now and has been analyzed from many aspects to understand how to cooperate. We use network information theory to determine the fundamental limits to the performance and propose cooperative schemes that can achieve or come close to these limits. I am going to talk about a specific problem that analyzes a communication situation in which two senders wish to communicate with two receivers, that is. the interference channel. At the same time, we want to allow cooperation between transmitters and investigate gains from transmitter cooperation, i.e. channel models that allow transmitters to partially cooperate. So, we introduce the limited transmitter cooperation in the interference channel. For two different scenarios in the interference channel with limited cooperation, we will show conditions under which, we determine the capacity region. This is joint work with Prof. Roy Yates in Winlab and Gerhard Kramer, Bell Labs.

2. 2 Interference Channel More specifically, the interference channel is given by two sets-input alphabets X1 and X2, two output alphabets Y1 and Y2 and conditional probability distributions More specifically, the interference channel is given by two sets-input alphabets X1 and X2, two output alphabets Y1 and Y2 and conditional probability distributions

3. 3 Strong Interference Capacity region known if there is strong interference: Strong interference channel: the mutual information across is larger. In the Gaussian channel, the channel gain on the interference path is stronger (at least as strong as) the channel gain on the desired path.Strong interference channel: the mutual information across is larger. In the Gaussian channel, the channel gain on the interference path is stronger (at least as strong as) the channel gain on the desired path.

4. 4 Compound MAC

5. 5 Cooperation in Interference Channel More specifically, the interference channel is given by two sets-input alphabets X1 and X2, two output alphabets Y1 and Y2 and conditional probability distributions More specifically, the interference channel is given by two sets-input alphabets X1 and X2, two output alphabets Y1 and Y2 and conditional probability distributions

6. 6 Full cooperation: MIMO broadcast channel DPC optimal [Weingarten, Steinberg & Shamai, 04], [Caire & Shamai, 01], [Viswanath, Jindal & Goldsmith, 02] Several cooperation strategies proposed  [Host-Madsen, Jindal, Mitra& Goldsmith, 03, Ng & Goldsmith, 04] [Jindal, Mitra& Goldsmith, Ng& Goldsmith]: Gain from DPC when sources close together When apart, relaying outperforms DPC Assumptions: Dedicated orthogonal cooperation channels Total power constraint Transmitter Cooperation for Gaussian Channels How should nodes cooperate? Transmitter cooperation in Gaussian channel with two senders and two receivers have been considered by several authors. Different coding cooperative strategies were proposed. Host-Madsen and Ng and Goldsmith demonstrated the different performance depending on the relative network geometry. The real difference is depending whether sources clustered together or not. Under full cooperation: this channel is a MIMO broadcast and DPC optimal. How should nodes cooperate? Transmitter cooperation in Gaussian channel with two senders and two receivers have been considered by several authors. Different coding cooperative strategies were proposed. Host-Madsen and Ng and Goldsmith demonstrated the different performance depending on the relative network geometry. The real difference is depending whether sources clustered together or not. Under full cooperation: this channel is a MIMO broadcast and DPC optimal.

7. 7 Transmitter Cooperation In Discrete Memoryless Channel Simple model for cooperation in DMC was proposed by Willems for the MAC channel. Cooperation using links with finite capacities We want to model cooperation between sources for discrete channel. Here is a simple model for cooperation that was used/proposed by Willems:Simple model for cooperation in DMC was proposed by Willems for the MAC channel. Cooperation using links with finite capacities We want to model cooperation between sources for discrete channel. Here is a simple model for cooperation that was used/proposed by Willems:

8. 8 Cooperation through Conference h is a set of communicating functionsh is a set of communicating functions

9. 9

10. 10 Transmitter Cooperation

11. 11 Compound MAC with Common Information

12. 12 The Compound MAC with Common Information Encoding Decoding The error probability (R0, R1, R2) achievable if, for any , there is an (M0, M1, M2, N, Pe) code such that The capacity region is the closure of the set of all achievable (R0, R1, R2) Easy to determine given the result by [Slepian & Wolf, 1973] After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.

13. 13 The MAC with Common Information After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.

14. 14 The MAC with Common Information After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.

15. 15 The Compound MAC with Common Information After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.

16. 16 union over p(u)p(x1|u)p(x2|u)p(y1,y2|x1,x2) For each p : ( R0,R1,R2 ) is an intersection of rate regions RMACt achieved in two MACs with common information: The Compound MAC with Common Information After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.

17. 17 Converse Error probability in MACt Error probability in CMAC  ? ? Necessary condition for : ? Rates confined to R MAC1 (p) and R MAC2 (p) for every p

18. 18 From Slepian and Wolf result, choosing the rates ( R0,R1,R2 ) in will guarantee that Pe1 and Pe2 can be made arbitrarily small  ? Pe will be arbitrarily small Achievability

19. 19 Implications We can use this result to determine the capacity region of several channel with partial transmitter cooperation: The Strong Interference Channel with Common Information The Strong Interference Channel with Unidirectional Cooperation The Compound MAC with Conferencing Encoders

20. 20 After The Conference After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.

21. 21 Theorem For the interference channel with common information satisfying for all product input distributions the capacity region C is

22. 22 Proof Achievability Follows directly from the achievability in the Compound MAC with common information Decoding constraint is relaxed Converse Using Fano�s inequality In the interference channel with no cooperation: outer bounds rely on the independence of X1 and X2 Due to cooperation: Codewords not independent Theorem conditions obtained from the converse

23. 23 Relationship to the Strong Interference Conditions ?

24. 24 Interference Channel With Unidirectional Cooperation Encoding functions Decoding functions General theme of this work is cooperation in the interference channel, in this talk I will focus on the case of the unidirectional cooperation. It is assumed that a message W1 at sender 1 is also know to the other encoder, but not vice versa.General theme of this work is cooperation in the interference channel, in this talk I will focus on the case of the unidirectional cooperation. It is assumed that a message W1 at sender 1 is also know to the other encoder, but not vice versa.

25. 25 Cognitive Radio Settings Cognitive Radio Channel [Devroye, Mitran, Tarokh, 2005] An achievable rate region Consider simple two-transmitter, two-receiver network: Assume that one transmitter is cognitive It can �overhear� transmission of the primary user It obtains partially the primary user�s message ? it can cooperate At this point, it is unclear what exactly the transmitter cooperation isAt this point, it is unclear what exactly the transmitter cooperation is

26. 26 Interference Channel with Unidirectional Cooperation  The Interference Channel with Unidirectional Cooperation [Maric, Yates & Kramer, 2005] Capacity in very strong interference The Interference Channel with Degraded Message Set [Wu, Vishwanath & Arapostathis, 2006] Capacity for weak interference and for Gaussian channel in weak interference Cognitive Radio Channel [Jovicic& Viswanath, 2006] Capacity for Gaussian channel in weak interference The Interference Channel with a Cognitive Transmitter [Maric, Goldsmith, Kramer& Shamai, 2006] New outer bounds and an achievable region

27. 27 For the interference channel with unidirectional cooperation satisfying for all joint input distributions p(x1,x2), the capacity region C is  Capacity region = capacity region of the Compound MAC channel with Common Information Theorem

28. 28 Achievability: Compound MAC with Common Information

29. 29 Converse Using Fano�s inequality Interference channel with no cooperation: outer bounds rely on the independence of X1 and X2 Due to cooperation: Codewords not independent Theorem conditions obtained from the converse

30. 30 Converse (Continued ) Lemma: If per-letter conditions are satisfied for all p(x1,x2), then

31. 31 Converse (Continued ) Recall that the achievable rates are By assumption, for all p(x1,x2) The rates union over all p(x1,x2) are thus achievable and are an outer bound

32. 32 Gaussian Channel Channel outputs:

33. 33 Capacity Achieving Scheme in Strong Interference Encoder 1: Codebook x1(w1)

34. 34 Gaussian Channel- Strong Interference Conditions

35. 35 Gaussian Channel with Unidirectional Cooperation

36. 36 Gaussian Channel With Conferencing Channel outputs:

37. 37

38. 38 Sender t power, t=1,2:  Pc-conference power

39. 39 Full Cooperation in Gaussian Channel

40. 40 Interference Channel With Conferencing Relax the constraint ? Each user decodes only one message: Strong interference conditions? After the conference

41. 41 Interference Channel With Conferencing After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.After the conference, there is a part of both W1 and W2 that is known to BOTH encoders: we refer to it as a common message, it contains the cell indexes exchanged during the conference.

42. 42 Discussion Introduced cooperation into the Interference Channel: Capacity results for scenarios of strong interference If the interference is strong : decoders can decode the interference No partial decoding at the receivers ? easier problem Ongoing work: Channel models that incorporate node cooperation Capture characteristics of cognitive radio networks Capacity results and cooperation gains for more general settings

43. 43 Ongoing Work Determining the fundamental limits of wireless networks requires understanding the optimal ways of cooperation Hierarchy of cooperation Encoding schemes Optimum resource allocation How to cope with interference

44. 44 Not needed

45. 45

46. 46 Interference Channel More specifically, the interference channel is given by two sets-input alphabets X1 and X2, two output alphabets Y1 and Y2 and conditional probability distributions More specifically, the interference channel is given by two sets-input alphabets X1 and X2, two output alphabets Y1 and Y2 and conditional probability distributions

47. 47 Interference Channel with Confidential Messages Joint work with: Ruoheng Liu, Predrag Spasojevic and Roy D. Yates Developed inner and outer bounds

48. 48 Initailly, the decoding constraint�Initailly, the decoding constraint�

49. 49 Theorem The Compound MAC capacity region C (C12,C21) is union over p(u)p(x1|u)p(x2|u)p(y1,y2|x1,x2) denoted p For each p : Intersection between rate regions of two MACs with partially cooperating encoders

50. 50 Gaussian Channel With Conferencing Union over all