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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

- 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

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

- 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

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

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

Suvarup Saha - ISIT 2012

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

- 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

- 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

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

- 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

- 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

- 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

- 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