The Capacity Region of the Cognitive Z-interference Channel with a
Noiseless Non-cognitive Link
Nan Liu, Ivana Marić, Andrea Goldsmith and Shlomo Shamai (Shitz)
IT channel models suitable for
networks with cognitive users still
need to be proposed.
Capacity of Z-interference channel
is still unknown.
W1
source 1
NEW INSIGHTS
source 2
W2
W1
dest1
dest2
W2
In some scenarios,
interference can be
minimized by exploiting the
structure of interference and
cognition at the nodes.
Cognition should be used by
the encoder to precode
against part of the
interference caused to its
receiver.
ACHIEVEMENT DESCRIPTION
MAIN ACHIEVEMENT:
1) The capacity region of the discrete
cognitive Z- interference channel with a
noiseless non-cognitive link
2) An inner and outer bound for the
cognitive Z-interference channel
3) Solution to the generalized Gel’fandPinsker (GP) problem in which a
transmitter-receiver pair communicates in
the presence of interference non causally
known to the encoder. Our solution
determines the optimum structure of
interference.
HOW IT WORKS:
Non-cognitive encoder uses superposition
coding to enable partial decoding of
interference. The cognitive encoder
precodes against the rest of interference
using GP encoding.
ASSUMPTIONS AND LIMITATIONS:
The considered channel model:
W1 W2
cognitive
encoder
W2
non-cognitive
encoder
W
dest1 1
W2
dest2
IMPACT
Capacity of networks with
cognitive users are unknown.
Consequently, optimal ways how to
operate such networks are not
understood, nor it is clear how
cognitive nodes should exploit the
obtained information.
Motivation
1) Optimal scheme for
some channels
2) Superposition coding
and Gel’fand-Pinsker
coding may be
required in order to
minimize interference,
in some channels.
This is in contrast to
the Gaussian channel.
•
NEXT-PHASE GOALS
STATUS QUO
Summary
Introduction
•In multiuser networks:
•A key issue is how to handle and exploit interference
created by simultaneous transmissions
• Not well understood:
•Capacity of the interference channel an open problem
•Capacity of the Z-interference channel an open problem
• We aim to exploit cognition to maximize network
performance
• What is the side information at a cognitive
node?
• What is the best encoding scheme given this
side information?
• A model for cognition in IT:
• Perfect side information at the cognitive pair
• Very optimistic
• Cognitive and non-cognitive pair communication
modeled as:
3) For the GP
problem, the optimal
interference has a
superposition
structure
• Evaluate a numerical
example
• Apply proposed encoding
scheme to larger networks
and to different cognitive
node models
Previous Work
• [Marić, Yates, Kramer], [Devroye, Mitran, Tarokh]
2006
• [Wu, Vishwanath, Arapostathis], [Jovičić,
Viswanath], [Sridharan, Vishwanath] 2007
• [Marić, Goldsmith, Shamai, Kramer], [Jiang, Xin],
[Cao, Chen] 2008
• Capacity results known in special cases of ‘strong’
and ‘weak’ interference
•Cognitive radio networks:
•Multiuser networks in which some users cognitive, i.e., can
sense the environment and hence obtain side information
about transmissions in neighborhood
•How to exploit cognition in optimal ways?
Encoding scheme was proposed that exploits cognition and is optimal in certain scenarios
Achievability
Channel Model
Theorem: Achievable rate pairs (R1,R2) are given by a union
of rate regions given by
Converse
Theorem: Achievable rate pairs (R1,R2) belong to a union of
rate regions given by
R1  I (U ; Y1 | V )  I (U ; Y2 | V )
R1  I (U ; Y1 | V )  I (U ; X 2 | V )
R2  I ( X 2 ; Y2 | V )  min I (V ; Y1 ), I (V ; Y2 )
R2  I ( X 2 ; Y2 | V )  min I (V ; Y1 ), I (V ; Y2 )
where the union is over all probability distributions
p( v,u,x2 )p( x1| u,x2 )
• Two messages: Wt  1,..., M t  Rates: Rt  log 2 M t / N
• Encoding: X n  f W ,W  • Decoding: Wˆ  g Y n 
1
1
2
1
1 1
1
Wˆ 2  g 2 Y2n 
X 2n  f 2 W2 
• Alphabet for encoder t:
t
•Encoder 1 is cognitive in the sense that it knows the message of other user
where the union is over all probability distributions
p( v,u,x2 )p( x1| u,x2 )
Encoding scheme:
• In general, the achievable rates and the converse result
Encoder 2: - rate-splits its message into two messages
do no meet
- encodes using superposition coding with
• Markovity U→(V,X2)→Y2 implies:
n
n
an inner codebook v and an outer codebook x2
Encoder 1: - for each vn it performs binning, i.e., Gel’fand-Pinsker
I(U;X2 | V) ≥ I(U;Y2 | V)
encoding in order to precode against interference x2n given vn
Capacity Result
Theorem: For the cognitive ZIC with a noiseless noncognitive link i.e., p(y2|x2) is a deterministic one-to-one
function, the capacity region is given by the union of rate
regions:
R1  I (U ; Y1 | V )  I (U ; X 2 | V )
R2  I ( X 2 ; Y2 | V )  min I (V ; Y1 ), I (V ; Y2 )
where the union is over all probability distributions
p( v,u,x2 )p( x1| u,x2 )
•For Y2=X2 the two regions are the same. This leads to the following:
Implications
Connection to the Gel’fand-Pinsker Problem
•In the cognitive ZIC, when X2=Y2, X2 can be viewed as a state i.i.d.
distributed on a set of size of 2nR2
•We can design not only the codebook of the cognitive encoder, but also
the structure of the state
C ( R2 ) 
max
p ( v ,u , x2 ) p ( x1 |u , x2 )
I (U ; Y1 | V )  I (U ; X 2 | V )
H ( X 2 | V )  min I (V ; Y1 ), I (V ; X 2 )  R2
•Communication in the presence of interference (state) non-causally
known to the encoder
•Thus, for the given rate R2 of the interferer, the optimal interference
has the superposition structure
•When R2  log t reduces to the GP rate achieved in the channel with an
iid state and uniformly distributed on X
• In the considered scenario, interference can
be minimized by exploiting the structure of
interference and cognition at the nodes
• The corresponding encoding scheme requires
1) Superposition coding
2) Gel’fand-Pinsker coding
• This is in contrast to the Gaussian case
• For the GP problem, the optimum
interference has superposition structure
• We considered single-user cognitive models
that capture delay in related work [Marić,
Liu and Goldsmith 2008]
Summary and Future Work
• An achievable rate region and an outer
bound for the cognitive ZIC were derived
• The capacity result for the ZIC with a
noiseless non-cognitive link was obtained
• The Generalized Gel’fand-Pinsker problem
was solved
Future work:
•Evaluate a numerical example
• Apply proposed encoding scheme to larger
networks and to different cognitive node
models