Optimum Transmission through the Gaussian Multiple Access ...GMAC D. Calabuig, R. Gohary, H....
Transcript of Optimum Transmission through the Gaussian Multiple Access ...GMAC D. Calabuig, R. Gohary, H....
OptimumTransmissionthrough the
GMAC
D. Calabuig,R. Gohary,
H. Yanikomero.
1/22
Introduction
System model
Optimization
Algorithm
Conclusions
Optimum Transmission through theGaussian Multiple Access Channel
Daniel Calabuig1 Ramy Gohary2
Halim Yanikomeroglu2
1Institute of Telecommunications and Multimedia ApplicationsUniversidad Politécnica de Valencia
Valencia, Spain
2Department of Systems and Computer EngineeringCarleton University
Ottawa, Canada
July 8, 2013
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D. Calabuig,R. Gohary,
H. Yanikomero.
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Introduction
System model
Optimization
Algorithm
Conclusions
Outline
1 Introduction
2 System model
3 Optimum transmission parameters
4 Algorithm and application example
5 Conclusions
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Introduction
The capacity region of the GMAC is known [Wyner74]Gaussian signalingSuccessive Interference Cancelation (SIC)Time-sharing
The computation of the optimum transmissionparameters is, in general, difficult
Transmission parameters:Covariance matrices of input signalsUser decoding ordersTime-sharing weights
Solution known for linear rate objectives [Tse&Hanly98]
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Introduction
Features of linear rate objectivesComplexity reduction
The optimum user decoding order is given by the orderof the rate weightsThe optimization only has to find the optimumcovariance matrices
Convexity of the optimization problemThe linear rate objective can be expressed as a concavefunction of the covariance matrices
What about non-linear rate objectives?
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Contributions
Solution of a class of problems with non-linear objectivesObjectives convex in rates not necessarily intransmission parametersAided by variational inequalities
Complete description of the optimum parametersNecessary and sufficient condition
An algorithm that finds the optimum parametersWe can now solve problems that could not be solvedbefore
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H. Yanikomero.
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Introduction
System model
Optimization
Algorithm
Conclusions
Outline
1 Introduction
2 System model
3 Optimum transmission parameters
4 Algorithm and application example
5 Conclusions
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System model
MIMO GMACzzz
yyyHHHK
HHH1
xxxK
...
xxx1
Number of users: KAntennas of user k : Nk
Signal of user k : xxxk
yyy =K∑
k=1
HHHkxxxk + zzz
Let E[zzzzzz†] = III, QQQk = E[xxxkxxx†k ] and
Q̄QQ = QQQ1 ⊕ · · · ⊕QQQK =
QQQ1 · · · 0...
. . ....
0 · · · QQQK
Q̄QQ must satisfy certain power constraints
g`(Q̄QQ)≤ 0, ` = 1, . . . ,L
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D. Calabuig,R. Gohary,
H. Yanikomero.
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Introduction
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Problem formulation
Time-sharing implies a convex combination of (at most)K + 1 rate vectors (Carathéodory’s theorem)Achievable rate: ρρρ (ααα,Q), with the k -th entry given by
ρk (ααα,Q) =K +1∑m=1
K !∑i=1
αmi rki
(Q̄QQ
(m)), Q = {Q̄QQ(m)}K +1
m=1
We define ααα ∈ R(K +1)×K ! as the time-sharing matrixIt jointly represents time-sharing and decoding ordersThe entries must belong to the unit simplex
S ,{ααα∣∣∣K +1∑m=1
K !∑i=1
αmi = 1, αmi ≥ 0, ∀m, i}.
OptimumTransmissionthrough the
GMAC
D. Calabuig,R. Gohary,
H. Yanikomero.
9/22
Introduction
System model
Optimization
Algorithm
Conclusions
Outline
1 Introduction
2 System model
3 Optimum transmission parameters
4 Algorithm and application example
5 Conclusions
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H. Yanikomero.
10/22
Introduction
System model
Optimization
Algorithm
Conclusions
Optimization
minααα,Q
f (ρρρ (ααα,Q)) , Q = {Q̄QQ(m)}K +1m=1
subject to ααα ∈ S
Q̄QQ(m) ∈ P, m = 1, . . . ,K + 1
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H. Yanikomero.
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Problems with non-linear objectivesPreliminaries
Lemma 1 (Variational inequalities)
Let f : X → R be convex and continuously differentiableIf xxx∗ = arg min
xxx∈Xxxx†∇f (xxx∗)
then xxx∗ = arg minxxx∈X
f (xxx)
Let X be convex, and let f be continuously differentiableIf xxx∗ = arg min
xxx∈Xf (xxx)
then xxx∗ = arg minxxx∈X
xxx†∇f (xxx∗)
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H. Yanikomero.
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Problems with non-linear objectives
Theorem 1
Let www = −∇f (ρρρ(ααα∗,Q∗)) for some ααα∗, Q∗ and convex fLet users be labelled so that w1 ≤ · · · ≤ wK
Then, ααα∗ and Q∗ are optimum if and only if for eachstrictly positive α∗mi ∈ ααα∗
1 the decoding order i is ordered as w1, . . . ,wK , and2 Q̄QQ
∗(m)solves
maxQ̄QQ
K∑k=1
(wk − wk−1) log∣∣∣III +
∑j≥k
HHH jQQQjHHH†j
∣∣∣, s.t. Q̄QQ ∈ P
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Problems with non-linear objectivesRemarks on Theorem 1
maxQ̄QQ
K∑k=1
(wk − wk−1) log∣∣∣III +
∑j≥k
HHH jQQQjHHH†j
∣∣∣, s.t. Q̄QQ ∈ P
Theorem 1 gives a complete description of the optimumtransmission parameters for convex rate objectivesThe above optimization problem
is independent of the decoding order, andis convex if the power constraints are convexTheorem 1 is easily testable
Computation of the optimum pair (ααα∗,Q∗) is still difficultNeed to find all the solutions and try all combinationsNext theorem solves this problem
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Problems with convex and monotonic powerconstraints
Theorem 2
Let the power constraint functions, g`, ` = 1, . . . ,L, beconvex and monotonicThen, any achievable rate vector in the correspondingGMAC can be achieved with one collection ofcovariance matrices
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Problems with convex and monotonic powerconstraintsRemarks on Theorem 2
maxQ̄QQ
K∑k=1
(wk − wk−1) log∣∣∣III +
∑j≥k
HHH jQQQjHHH†j
∣∣∣, s.t. Q̄QQ ∈ P
Theorem 2 is true for the entire capacity regionIt is true for any objective f
We just need one collection of covariance matricesThe search of optimum parameters is simplified
All solutions of the above problem are equally optimum
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H. Yanikomero.
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Introduction
System model
Optimization
Algorithm
Conclusions
Outline
1 Introduction
2 System model
3 Optimum transmission parameters
4 Algorithm and application example
5 Conclusions
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Algorithm
We use Theorems 1 and 2 to design an algorithm thatconverges to the optimum ααα∗ and Q∗
Each algorithm iteration is divided into two steps1 We fix the time-sharing matrix and compute the
covariance matrices that satisfy condition 2 ofTheorem 1
2 We fix the covariance matrices and compute theoptimum time-sharing matrix
This time-sharing matrix necessarily satisfies condition 1of Theorem 1
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H. Yanikomero.
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Algorithm
Proposition 1
If the rate objective f is bounded below for rates insidethe capacity regionthen, the previous algorithm converges to the optimumpair (ααα∗,Q∗)
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H. Yanikomero.
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Application
We used the previous algorithm to minimize the totalcompletion time of a two-user GMACCompletion time [Liu&Erkip2011]: Time required totransmit the data stored in the buffer→ bits / bit rateLet bk be the number of bits in the buffer of user kLet P be the total available power at each transmitter
minα1,α2,QQQ1,QQQ2
b1
ρ1 (α1, α2,QQQ1,QQQ2)+
b2
ρ2 (α1, α2,QQQ1,QQQ2)
subject to α1 + α2 = 1, α1 ≥ 0, α2 ≥ 0QQQ1 � 0, tr (QQQ1) ≤ P, QQQ2 � 0, tr (QQQ2) ≤ P
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H. Yanikomero.
20/22
Introduction
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Algorithm
Conclusions
Application
2.5
3
3.54
571020
0 2 4 6 80
1
2
3
4
5
6
Rate of user 1 (bits/s/Hz)
Rateof
user2(bits/s/Hz)
π1(j) = 1, 2, j = 1, 2
π2 (j
)=
2,1,j=
1,2
Objective contour lines
One ordering zones
Time-sharing zone
Optimum rate vector
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D. Calabuig,R. Gohary,
H. Yanikomero.
21/22
Introduction
System model
Optimization
Algorithm
Conclusions
Outline
1 Introduction
2 System model
3 Optimum transmission parameters
4 Algorithm and application example
5 Conclusions
OptimumTransmissionthrough the
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D. Calabuig,R. Gohary,
H. Yanikomero.
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Introduction
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Optimization
Algorithm
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Conclusions
GMAC problems with non-linear rate objectivesMain results
Complete description of the optimum pair (ααα∗,Q∗) forconvex rate objectives
Convex in rates, not in ααα and QVariational inequalities as a linear-to-convex bridge
A simplification of the problem when the powerconstraints are convex and monotonic
An algorithm has been proposedIt converges to the optimum pair (ααα∗,Q∗)