Sum capacity of 3 user deterministic interference channels with connectivity constraints
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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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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].

    • 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


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


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


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


Normalized Sum-capacity

Discontinuity observed in GDoF Analysis [Vishwanath, Jafar ‘10]

Suvarup Saha - ISIT 2012


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


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


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


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


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


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


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


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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