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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

2/22

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

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

3/22

Introduction

System model

Optimization

Algorithm

Conclusions

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]

OptimumTransmissionthrough the

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D. Calabuig,R. Gohary,

H. Yanikomero.

4/22

Introduction

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Algorithm

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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?

OptimumTransmissionthrough the

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D. Calabuig,R. Gohary,

H. Yanikomero.

5/22

Introduction

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Algorithm

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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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

6/22

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

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

7/22

Introduction

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Optimization

Algorithm

Conclusions

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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

8/22

Introduction

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Optimization

Algorithm

Conclusions

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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

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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D. Calabuig,R. Gohary,

H. Yanikomero.

11/22

Introduction

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Algorithm

Conclusions

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∗)

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

12/22

Introduction

System model

Optimization

Algorithm

Conclusions

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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

13/22

Introduction

System model

Optimization

Algorithm

Conclusions

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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

14/22

Introduction

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Optimization

Algorithm

Conclusions

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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

15/22

Introduction

System model

Optimization

Algorithm

Conclusions

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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

16/22

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.

17/22

Introduction

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Optimization

Algorithm

Conclusions

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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

18/22

Introduction

System model

Optimization

Algorithm

Conclusions

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∗)

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

19/22

Introduction

System model

Optimization

Algorithm

Conclusions

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

OptimumTransmissionthrough the

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

20/22

Introduction

System model

Optimization

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

OptimumTransmissionthrough the

GMAC

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

GMAC

D. Calabuig,R. Gohary,

H. Yanikomero.

22/22

Introduction

System model

Optimization

Algorithm

Conclusions

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∗)