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Sum-capacity of 3-user Deterministic Interference Channels with Connectivity Constraints

Sum-capacity of 3-user Deterministic Interference Channels with Connectivity Constraints. Suvarup Saha, Randy Berry Northwestern University. Linear Deterministic ICs– 2 to K users . 2-user LDIC A special case of General Deterministic IC [El Gamal, Costa ‘82].

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Sum-capacity of 3-user Deterministic Interference Channels with Connectivity Constraints

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  1. Sum-capacity of 3-user Deterministic Interference Channels with Connectivity Constraints Suvarup Saha, Randy Berry Northwestern University

  2. Linear Deterministic ICs– 2 to K users • 2-user LDIC • A special case of General Deterministic IC [El Gamal, Costa ‘82]. • Achievability can be shown either by using Han-Kobayashi strategy [Bresler, Tse ‘08] or explicit construction [Saha, Berry ‘12]. • 3 or more user LDIC – capacity is unknown in general • Alignment[Cadambe, Jafar ‘07] seems to play an important role. • Interfering links increasing exponentially in #users clutters analysis. • Gaussian HK-scheme might not be optimal. • Our approach • Consider ‘reduced link’ ICs. • Assume symmetry in parameters. • Decode sum of interfering signals aligned at the receiver. Suvarup Saha - ISIT 2012

  3. Towards Fully-connected 3-user ICs Many-to-One and One-to-Many [Bresler, Parekh, Tse ‘07] Tx-1 Rx-1 Tx-1 Rx-1 Tx-2 Rx-2 Tx-2 Rx-2 Cascade Z [Liu, Erkip ‘11] Shoe-string [Saha, Berry ‘10] Tx-3 Rx-3 Tx-3 Rx-3 Tx-1 Rx-1 Tx-1 Rx-1 Cyclic [Zhou, Yu ‘10] Tx-1 Rx-1 Tx-2 Rx-2 Tx-2 Rx-2 Tx-2 Rx-2 Crown A and B [Saha, Berry ‘12] Tx-3 Rx-3 Tx-3 Rx-3 Tx-3 Rx-3 Tx-1 Rx-1 Tx-1 Rx-1 Tx-1 Rx-1 Tx-2 Rx-2 Tx-2 Rx-2 Tx-2 Rx-2 Tx-3 Rx-3 Tx-3 Rx-3 Tx-3 Rx-3 Suvarup Saha - ISIT 2012

  4. Overview • Crown A, Crown B and Fully-connected • Consider with symmetric parameters. • #direct levels = nd , #cross (interfering) levels = nc • coupling parameter α = nc/nd . • Sum-capacity upperbounds derived using those for component Z and 2-user ICs. • Achievability shown by explicit construction. For α≥2/3, all these channels have the same sum-capacity! Suvarup Saha - ISIT 2012

  5. Sum-capacity Tx-1 Rx-1 Tx-1 Rx-1 Tx-1 Rx-1 Tx-2 Rx-2 Tx-2 Rx-2 Tx-2 Rx-2 Tx-3 Rx-3 Tx-3 Rx-3 Tx-3 Rx-3 Suvarup Saha - ISIT 2012

  6. Normalized Sum-capacity Discontinuity observed in GDoF Analysis [Vishwanath, Jafar ‘10] Suvarup Saha - ISIT 2012

  7. Upperbounds Tx-1 Rx-1 Tx-2 Rx-2 Tx-3 Rx-3 R1 +R2 < .... R2 +R3 < .... R1 +R3 < .... + 2(R1 +R2 +R3) < .... Suvarup Saha - ISIT 2012

  8. Achievability • We construct schemes in different interference regimes (values of α) that show achievability. • Key challenge – Align interference. • Schemes involve simple coding over both signal levels as well as time. [All previous constructive schemes in LDICs needed only coding over levels] • For 2/3< α<2 we may need to use 2 time instants to design codes. Related to the factor of 1/2 in the sum-capacity Suvarup Saha - ISIT 2012

  9. An Example • 3-user fully-connected LDIC with α = 3/4, nd = 4. • Sum-capacity = 3(nd –nc/2) = 15/2. • In 2-user case, sum-capacity is always an integer ; a level is used to transmit either 1 bit of information, or none. • Here, if using similar strategy, we need at least 2 time instants to show achievability! • Next, we show that 2 is enough! Suvarup Saha - ISIT 2012

  10. Coding over time Time t=1 Time t=2 a4 a4 a1 a1 a2 b2 + c2 a2 a2 a5 a5 a3 a3 + (b2 + c2) b4 b4 b1 b1 b2 a2 + c2 b2 b2 b5 b5 b3 + (a2 + c2) b3 c4 c4 c1 c1 c2 a2 + b2 c2 c2 c5 c5 c3 + (a2 + b2) c3 Each user decodes 2 more bits as well as sum of interference, yielding1 more bit from received signal at t=1 Each user decodes 2 bits at the end of t=1 Suvarup Saha - ISIT 2012

  11. Key Ideas • Align interference at each receiver. • Re-transmit interfering signal in the next time, but from a different signal level. • Decode interference to enable detection of own signal in previous time instant • No need to decode individual interference – decoding the sum is enough! Suvarup Saha - ISIT 2012

  12. Fully-connected K user LDIC • Upperbounds can similarly be derived by considering the component K(K-1)/2 two-user ICs. • Constructive strategies for 3 user cases work as well for K users! • Achievable strategy is symmetric for users. • Decode sum of (K-1) interfering signals when required. • Alignment is also preserved. Suvarup Saha - ISIT 2012

  13. Asymmetric Case? • 2-user upperbounds are not enough here. • Tightest sum-rate upperbound from 2-user cases = 21/4 = 5.5 • New upperbound derived using general deterministic model = 5. a1 a2 b1 b2 c1 Achievable! Suvarup Saha - ISIT 2012

  14. Future Work • Investigate partially-symmetric set ups – involving multiple coupling parameters • Translate understanding to Gaussian case to derive tight approximations. • Relation to recent result on approximate sum-capacity of K-user GIC [Ordentlich, Erez, Nazer ‘12] ? Suvarup Saha - ISIT 2012

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