Two memoryless channels

1
Superposition Encoding Is Optimal for a Class of
Z-interference Channels
Nan Liu and Andrea Goldsmith
Single User Capacity Intuition
Han/Kobayashi Achievable Region
Introduction
Superposition Encoding Is Optimal for a Class of
Z-interference Channels
ACHIEVEMENT DESCRIPTION
Superposition encoding and partial decoding is
proved to be the best way to handle interference
under certain scenarios Results can only be
proved for interference channels that satisfy
certain conditions
MAIN RESULT A single-letter capacity region
for a class Z-interference channels are found,
and for this class, superposition encoding and
partial decoding is shown to be optimal. Examples
of interference channels that belong to this
class are provided. HOW IT WORKS
Superposition encoding schemes are enough to
achieve all possible output space sizes at the
receivers, and thus there is no benefit to
considering other encoding schemes. ASSUMPTIONS
AND LIMITATIONS The result applies only to
interference channels that satisfy the properties
that, 1) only one transmitter-receiver pair faces
interference, 2) for the transmitter-receiver
pair that suffers from interference, a) the
conditional output entropy is invariant with
respect to the input and b) the maximum output
entropy is achieved by a single input
distribution irrespective of the interference
distribution.

• Investigate if the insight and tools are
  applicable to
• interference channels rather than Z-interference
  channels
• Gaussian interference channels rather than
  discrete channels

END-OF-PHASE GOAL
STATUS QUO
The multi-user information theoretic tool of
images of sets via two channels can be applied
to interference channels, and as a result, it
provides upper bounds for the capacity region of
some interference channels.

• Largest achievable region Han/Kobayashi in 81
• Is it enough to consider only superposition
  encoding?
• Our work
• Apply converse technique by Korner/Marton in 77
• Superposition Encoding enough to achieve
  capacity

COMMUNITY CHALLENGE
Utilizing the tool of images of sets via two
channels to derive upper bounds in a wider
variety of wireless communication problems

• In wireless communications, interference is
  unavoidable
• The interference channel (IC)
• Capacity region open in most cases
• Z-interference channel
• Capacity region open in most cases
• Our work contribute to capacity for
  Z-interference channels

NEW INSIGHTS

• Number of codewords
• output space of codebook / output space per
  codeword

The result expand the set of interference
channels with known capacity regions and indicate
that superposition encoding is enough to achieve
capacity for certain interference channels.
Application to Z-interference Channels
Condition 1 on Z-interference channels
Images of Sets via Two Channels
Intuition for Z-interference Channels

• Two memoryless channels
• Achievability superposition encoding
• Converse
• Enough to consider superposition encoding
• BC with degraded message set, Gelfand-Pinsker
  problem

• 0000 10011
  or
• Invariant to sequence
• Condition 1 For any
  , when
• evaluated with the distribution
  is
• independent of for any .
• Gaussian channel satisfies condition 1

• Channel not memoryless
• In general, three spaces depend on codebook 1

• Tradeoff between two output space sizes for
  codebook 1

Conclusions and Future Work
Previous Work
Condition 2 on Z-interference Channels
Main Results and Example

• Benzel 79
• Special case of our example
• Degradedness treating interference as noise
  optimal
• We remove the degradedness constraint
• El Gamal/Costa 82
• Figure 3 is a special case of our example
• Deterministic Superposition encoding is optimal
• We remove the deterministic constraint
• Telatar/Tse 07
• Our example is a special case of their channel
• They show achievability and converse within
  finite gap
• We show in our example achievability is tight

• For Z-interference channels that satisfy
  conditions 1 and 2
• Korner/Marton technique applicable
• Enough to consider superposition encoding
• Han/Kobayashi achievable region is the capacity
  region
• Example

• Conclusions
• Adapted the converse of Korner/Marton to
  interference channels
• Found the capacity region for a class of
  Z-interference channels
• Expand the set of interference channels with
  known capacity
• Showed the optimality of superposition encoding
• Future work
• Derive upper bounds for a wider class of
  Z-interference channels
• Extend the result to two-sided interference
  channels
• Extend the result to Gaussian interference
  channels
• Apply the converse of Korner/Marton to a wider
  variety of channels

• Largest output space by codebook 2 alone
• Condition 2 Define as
  , there
• exists a such that ,
  when evaluated with
• the distribution
  is equal to
• for any .
• Mod-sum channel satisfies condition 2
• Gaussian channel does not satisfy