66183

8112019 66183

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Robust Power Allocation Algorithms for Wireless Relay Networks

Citation Quek Tony QS Moe Z Win and Marco Chiani ldquoRobust PowerAllocation Algorithms for Wireless Relay NetworksrdquoCommunications IEEE Transactions on 587 (2010) 1931-1938Copyright copy 2010 IEEE

As Published httpdxdoiorg101109tcomm201007080277

Publisher Institute of Electrical and Electronics Engineers IEEECommunications Society

Version Final published version

Accessed Thu Jul 03 022740 EDT 2014

Citable Link httphdlhandlenet1721166183

Terms of Use Article is made available in accordance with the publishers policyand may be subject to US copyright law Please refer to thepublishers site for terms of use

Detailed Terms

The MIT Faculty has made this article openly available Please share how this access benefits you Your story matters

8112019 66183


8112019 66183


1932 IEEE TRANSACTIONS ON COMMUNICATIONS VOL 58 NO 7 JULY 2010

feasibility problems which can be cast as second-order cone

programs (SOCPs)[13] We also show that the noncoherent AF

RPA problem in the presence of perfect global CSI can beapproximately decomposed into 2 quasiconvex optimization

subproblems Each subproblem can be solved ef ficiently by

the bisection method via a sequence of convex feasibility

problems in the form of SOCP We then develop the robust

optimization framework for RPA problems in the case of

uncertain global CSI We show that the robust counterparts of our convex feasibility problems with ellipsoidal uncertainty

sets can be formulated as semi-definite programs (SDPs)

Our results reveal that ignoring uncertainties associated with

global CSI in RPA algorithms often leads to poor performance

highlighting the importance of robust algorithm designs inwireless networks

The paper is organized as follows In Section II the problem

formulation is described In Section III we formulate the

coherent and noncoherent AF RPA problems as quasiconvex

optimization problems Next in Section IV we formulate therobust counterparts of our RPA problems when the global CSI

is subject to uncertainty Numerical results are presented inSection V and conclusions are given in the last section

Notations Throughout the paper we shall use the following

notation Boldface upper-case letters denote matrices boldface

lower-case letters denote column vectors and plain lower-

case letters denote scalars The superscripts () ()lowast and

()991264 denote the transpose complex conjugate and transposeconjugate respectively We denote a standard basis vector with

a one at the 1038389th element as 1103925110392511039251038389 907317 times 907317 identity matrix as 1103925

and ( )th element of as []907317 The notations tr() ∣ ∣ and

∥ ∥ denote the trace operator absolute value and the standard

Euclidean norm respectively We denote the nonnegative and

positive orthants in Euclidean vector space of dimension asℝ

+ and ℝ ++ respectively We denote ર 0 and ≻ 0 as

being positive semi-definite and positive definite respectively

II PROBLEM FORMULATION

We consider a wireless relay network consisting of r +2 nodes each with single-antenna a designated source-

destination node pair together with r relay nodes locatedrandomly and independently in a fixed area We consider a

scenario in which there is no direct link between the source

and destination nodes and all nodes are operating in a common

frequency bandTransmission occurs over two time slots In the first time

slot the relay nodes receive the signal transmitted by the

source node After processing the received signals the relay

nodes transmit the processed data to the destination nodeduring the second time slot while the source node remains

silent We assume perfect synchronization at the destination

node3 The received signals at the relay and destination nodes

can then be written as

R = ℎℎℎBS + R First slot (1)

D = ℎℎℎ FR + D Second slot (2)

3Exactly how to achieve this synchronization or the effect of smallsynchronization errors on performance is beyond the scope of this paper

where S is the transmitted signal from the source node

to the relay nodes R is the r times 1 transmitted signal

vector from the relay nodes to the destination node R

is the r times 1 received signal vector at the relay nodes

D is the received signal at the destination node R sim983772 r(000 ΣΣΣR) is the r times 1 noise vector at the relay nodes

and D sim 983772 (0 2D) is the noise at the destination node4

Note that the different noise variances at the relay nodes

are reflected in ΣΣΣR ≜ diag(2R9830841 2R9830842 983086 983086 983086 2R983084 r) MoreoverR and D are independent Furthermore they are mutually

uncorrelated with S and R With perfect global CSI at

the destination node ℎℎℎB and ℎℎℎF are r times 1 known channel

vectors from source to relay and from relay to destination

respectively where ℎℎℎB = [ℎB9830841 ℎB9830842 983086 983086 983086 ℎB983084 r ] isin ℂ r andℎℎℎF = [ℎF9830841 ℎF9830842 983086 983086 983086 ℎF983084 r] isin ℂ r For convenience we

shall refer to ℎℎℎB as the backward channel and ℎℎℎF as theforward channel

At the source node we impose an individual source power

constraint S such that 983163∣S∣2983165 le S Similarly at the

relay nodes we impose both individual relay power constraint

and aggregate relay power constraint R such that thetransmission power allocated to the 1038389th relay node 1038389 ≜

[R]10383899830841038389 le for 1038389 isin r and tr (R) le R where

R ≜ 983163R991264R∣ ℎℎℎB983165 and r = 9831631 2 983086 983086 983086 r983165

For AF relaying the relay nodes simply transmit scaled

versions of their received signals while satisfying power

constraints In this case R in (2) is given by

R = R (3)

where denotes the r times r diagonal matrix representing

relay gains and thus5

R = (

SℎℎℎBℎℎℎ991264B + ΣΣΣR

)991264983086 (4)

The diagonal structure of ensures that each relay node

only requires the knowledge about its own received signal

When each relay node has access to its locally-bidirectional

CSI it can perform distributed beamforming6 As such this

is referred to as coherent AF relaying and the 1038389th diagonal

element of is given by[4]

(1038389)coh =

radic 1038389 1038389

ℎlowastB9830841038389∣ℎB9830841038389∣

ℎlowastF9830841038389∣ℎF9830841038389∣ (5)

where 1038389 = 1983087( S∣ℎB9830841038389∣2 + 2R9830841038389) On the other hand whenforward CSI is absent at each relay node the relay node

simply forwards a scaled version of its received signal without

any phase alignment This is referred to as noncoherent AFrelaying and the 1038389th diagonal element of is given by[7] [8]

(1038389)noncoh =

radic 1038389 1038389983086 (6)

4 983772 (983084 10383892) denotes a complex circularly symmetric Gaussian distributionwith mean and variance 10383892 Similarly 983772 (983084ΣΣΣ) denotes a complex 1103925 -variate Gaussian distribution with a mean vector and a covariance matrixΣΣΣ

5Note that in (4) the source employs the maximum allowable power 907317 S

in order to maximize the SNR at the destination node6Here locally-bidirectional CSI refers to the knowledge of only ℎB9830841038389 andℎF9830841038389 at the th relay node

8112019 66183


QUEK et al ROBUST POWER ALLOCATION ALGORITHMS FOR WIRELESS RELAY NETWORKS 1933

It follows from (1)-(3) that the received signal at the destina-

tion node can be written as

D = ℎℎℎ FℎℎℎBS + ℎℎℎ

FR + D ≜983772D

(7)

where 983772D represents the effective noise at the destination node

and the instantaneous SNR at the destination node conditioned

on ℎℎℎB

and ℎℎℎF

is then given by

SNR( ) ≜

∣ℎℎℎ

FℎℎℎBS∣2 ∣ℎℎℎB ℎℎℎF

983163∣983215D∣2 ∣ℎℎℎF983165

= Sℎℎℎ

FℎℎℎBℎℎℎ991264B991264ℎℎℎlowastF

ℎℎℎ FΣΣΣR991264ℎℎℎlowastF + 2

D

(8)

where = [ 1 2 983086 983086 983086 r ] Our goal is to maximize system

performance by optimally allocating transmission power of

the relay nodes We adopt the SNR at the destination nodeas the performance metric and formulate the RPA problem as

follows

max SNR( )st tr (R) le R

0 le [R]10383899830841038389 le forall1038389 isin r983086

(9)

Note that the optimal solution to the problem in (9) maximizes

the capacity of the AF relay network under perfect global

CSI since this capacity given by 12 log(1 + SNR) i s a

monotonically increasing function of SNR

III OPTIMAL RELAY POWER ALLOCATION

A Coherent AF Relaying

First we transform (9) for the coherent AF RPA problem

into a quasiconvex optimization problem as given in thefollowing proposition

Proposition 1 The coherent AF relay power allocation

problem can be transformed into a quasiconvex optimization

problem as

1038389 coh max

coh( ) ≜ S2D

( )2

∥ ∥2+1

st isin 1103925 (10)

and the feasible set 1103925 is given by

1103925 =983163

isin ℝ r+

sum1038389isin r 21038389 le 1 0 le 1038389 le radic

p forall1038389 isin r983165

where 1038389 ≜991770

1038389 R

is the optimization variable and p ≜

983087 R In addition = [1 2 983086 983086 983086 r ] isin ℝ r+ and =

diag(1 2 983086 983086 983086 r) isin ℝ rtimes r+ are de fined for notational

convenience where

1038389 =radic

1038389 R ∣ℎB9830841038389∣∣ℎF9830841038389∣ (11)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (12)

Proof See Appendix A

Remark 1 Note that 1038389 denotes the fractional power allo-

cated to the 1038389th relay node and p denotes the ratio betweenthe individual relay power constraint and the aggregate relay

power constraint where 0 lt p le 1 It is well-known that

we can solve 1038389 coh ef ficiently through a sequence of convex

feasibility problems using the bisection method[14] In our

case we can always let min corresponding to the uniform RPAand we only need to choose max appropriately We formalize

these results in the following algorithm

Algorithm 1 The program 1038389 coh in Proposition 1 can be

solved numerically using the bisection method

0 Initialize min

= coh

( min

) max

= coh

( max

) where

coh( min) and coh( max) de fine a range of relevant

values of coh( ) and set tolerance isinℝ++

1 Solve the convex feasibility program 1038389 (SOCP)coh () in (13)

by fi xing = (max + min)9830872

2 If 1103925 coh() = empty then set max = else set min =

3 Stop if the gap (max minus min) is less than the tolerance

Go to Step 1 otherwise

4 Output opt obtained from solving 1038389 (SOCP)coh () in Step

1

where the convex feasibility program can be written in SOCP

form as

1038389 (SOCP)coh () find

st isin 1103925 coh() (13)

with the set 1103925 coh() given by

coh() =

⎧⎨⎩1038389 1038389 1038389 isin ℝ1103925 r

+

⎡⎢⎣

110392511039251103925907317 1038389 1038389 1038389 radic

S2D983080

19073179073179073171038389 1038389 1038389

983081⎤⎥⎦ ર 0983084

983131 11038389 1038389 1038389

983133 ર 0983084

⎡⎣

p+1

2983080 1038389 1038389 1038389 907317 1038389pminus1

2

983081⎤⎦ ર 0983084 forall isin 1038389 r

⎫⎬⎭ 983086

(14)

Proof See Appendix B

B Noncoherent AF Relaying

Similar to the formulation of coherent AF RPA problem in

(10) we can expressed the noncoherent AF RPA problem as

1038389 noncoh max

S2D

∣ ∣2

∥ ∥2+1

st isin 1103925 (15)

where 1103925

and are given in Proposition 1 The difference is

in = [1 2 983086 983086 983086 r ] isin ℂ r where

1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389983086

As a result we cannot directly apply Algorithm 1 to solve

1038389 noncoh in (15) Instead we introduce the following lemma

which enables us to decompose 1038389 noncoh into 2 quasiconvexoptimization subproblems each of which can then be solved

ef ficiently via the algorithm presented in Algorithm 1

Lemma 1 (Linear Approximation of Modulus[15]) The

modulus of a complex number isin ℂ can be linearly

approximated with the polyhedral norm given by

( ) = maxisinℒ

1048699real983163 983165 cos

983080

983081+ image1038389 983163 983165 sin

983080

9830811048701

8112019 66183



where ℒ = 9831631 2 983086 983086 983086 2983165 real983163 983165 and image1038389983163 983165 denote the

real and imaginary parts of and the polyhedral norm ( )is bounded by

( ) le ∣ ∣ le ( )sec(

2

)983086

and is a positive integer such that ge 2

Proposition 2 The noncoherent AF relay power allocation

problem can be approximately decomposed into 2 quasi-

convex optimization subproblems The master problem can be

written as

maxisinℒ

noncoh( opt ) (16)

where

noncoh( )

≜ S

2D

852059real

852091

852093cos(983087) + image1038389

852091

852093sin(983087)

8520612∥ ∥2 + 1

and opt is the optimal solution of the following subproblem

1038389 noncoh() max

1103925 noncoh(

)

st real852091

852093

cos(983087)+image1038389

852091

852093sin(983087) ge 0

isin 1103925 983086(17)

The feasible set 1103925 is given by

1103925 =983163

isin ℝ r+

sum1038389isin r

21038389 le 1 0 le 1038389 le radic p forall1038389 isin r

983165

where 1038389 ≜991770

1038389 R


983087 R In addition = [1 2 983086 983086 983086 r ] isin ℂ r and

= diag(1 2 983086 983086 983086 r)

isinℝ rtimes r+ are de fined as

1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389 (18)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (19)

Proof Similar to the proof of Proposition 1 Remark 2 Note that 1103925 and in Proposition 2 are exactly

the same as that in Proposition 1 The difference is in only Unlike 1038389 coh we now need to solve 2 quasiconvex

optimization subproblems due to the approximation of ∣ ∣using Lemma 1

Algorithm 2 Each of the 2 subproblems 1038389 noncoh() in

Proposition 2 can be solved ef ficiently by the bisection method

via a sequence of convex feasibility problems in the form of

SOCP The 2 solutions 983163 opt 9831652

=1 then forms a candidate set

for the optimal optthat maximizes our master problem

Proof The proof follows straightforwardly from Algo-

rithm 1

IV ROBUST RELAY POWER ALLOCATION


Using the above methodology we formulate the robust

counterpart of our AF RPA problem in Proposition 1 withuncertainties in and as follows

max coh( )st isin 1103925 forall ( ) isin 907317 (20)

where the feasible set 1103925 is given in Proposition 1 and 907317 is

an uncertainty set that contains all possible realizations of

and To solve for the above optimization problem weincorporate the uncertainties associated with and into the

convex feasibility program in (13) of Algorithm 1 Since ( )only appears in the first constraint of (14) we simply need to

focus on this constraint and build its robust counterpart as

follows

ge1057306

2D

S(1 + ∥ ∥2) forall( ) isin 907317 983086 (21)

In the following we adopt a conservative approach which

assumes that 907317 affecting (21) is sidewise ie the uncer-

tainty affecting the right-hand side in (21) is independentof that affecting the left-hand side Specifically we have

907317 = 907317 R times 907317 L Without such an assumption it is known

that a computationally tractable robust counterpart for (21)

does not exist which makes the conservative approach rather

attractive[2] Our results are summarized in the next theorem

Theorem 1 The robust coherent AF relay power allocation problem in (20) can be solved numerically via Algorithm 1 ex-

cept that the convex feasibility program is now conservatively

replaced by its robust counterpart given as follows

1103925 (robust)coh () find 1038389 1038389 1038389

st 1038389 1038389 1038389 isin coh(983084907317907317907317983084110392511039251103925)983084 forall907317907317907317 isin 907317 R983084 110392511039251103925 isin 907317 L(22)

with the sidewise independent ellipsoidal uncertainty sets 907317 Rand 907317 L given by

907317 R =

⎧⎨⎩

= 0 +

sumisin 907317 ∥∥ le 1

⎫⎬⎭

(23)

907317 L =

⎧⎨⎩ = 0 +

sumisin

∥∥ le 2

⎫⎬⎭ (24)

where = 9831631 2 983086 983086 983086 983165 = 9831631 2 983086 983086 983086 983165 and and are the dimensions of and respectively Then

the approximate robust convex feasibility program 1038389 (robust)coh ()

can be written in SDP form as

find ( )st ( ) isin coh()

(25)

such that ( ) isin ℝ r+ times ℝ+ times ℝ+ and the feasible

set coh() is shown at the top of this page where 983768983768983768 =

[1 2 983086 983086 983086 907317 ] and 983768983768983768 =852059 1 2 983086 983086 983086

852061

Proof The proof follows similar steps as in the proof of

Theorem 6 in [1]


In the next theorem we formulate the robust counterparts

of the 2 subproblems in Algorithm 2 with uncertainties

associated with and

Theorem 2 The robust noncoherent AF relay power alloca-

tion problem can be approximately decomposed into 2 sub-

problems Under sidewise independent ellipsoidal uncertainty

8112019 66183


8112019 66183



0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100

Uniform RPA ( r = 10)

Optimal RPA ( r = 10)



S9830872D (dB)

O u t a g e P r o b a b i l i t y

Fig 1 Outage probability as a function of 907317 S98308710383892D for the coherent AF

relay network with p = 09830861

0 2 4 6 8 10 1210

minus2

10minus1

100





S9830872D (dB)


Fig 2 Outage probability as a function of 907317 S98308710383892D for the noncoherent AF


result in the same performance This can be explained by the

fact that it is optimal for each relay node to transmit at the

maximum transmission power when p = 983087 R = 09830861When r = 20 we first observe that lower outage proba-

bilities can be achieved for both power allocations comparedto the case with r = 10 due to the presence of diversity

gains in coherent AF relay network In addition significant

performance improvements with optimal RPA compared to

uniform RPA can be observed since optimal RPA can exploit

the channel variation more effectively for larger r to enhance

the effective SNR at the destination node

Similar to Fig 1 we show the outage probability as a

function of S9830872D for the noncoherent AF relay network

with p = 09830861 in Fig 2 Under uniform RPA we observe

that the increase in the number of relay nodes does not yield

any performance gain This behavior of noncoherent AF relay

network is consistent with the results of [17] and can beattributed to the lack of locally-bidirectional CSIs at the relay

nodes making coherent combining at the destination node im-

0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100

= 0 = 0 98308601 = 0 9830861 = 0 98308625

S9830872D (dB)


Fig 3 Effect of uncertain global CSI on the outage probability of thecoherent AF relay network using non-robust algorithm for p = 09830861 and r = 20

possible However we can see that performance improves with

optimal RPA compared to uniform RPA and this improvement

increases with r Comparing Figs 1 and 2 even with optimal

RPA the noncoherent AF relay network performs much worse

than the coherent AF case since optimal RPA is unable to fully

reap the performance gain promised by coherent AF case due

to the lack of distributed beamforming gainFigure 3 shows the effect of uncertainties associated with

the global CSI on the outage probability of coherent AF

networks using non-robust RPAs when r = 20 and p = 09830861

By non-robust algorithms we refer to optimization algorithms

in Section III that optimize RPAs based only on 0 and 0

instead of the true global CSI and where = 0 + 1

and = 0 + 19 Clearly we see that ignoring CSI

uncertainties in our designs can lead to drastic performance

degradation when the uncertainty size becomes large In

this figure we can see that when is less than 098308601 we may

ignore CSI uncertainties since the performance degradation

is negligible However performance deteriorates rapidly as increases

Figure 4 shows the outage probabilities of coherent AF relay

networks as a function of the size of the uncertainty set using

robust RPAs when r = 20 and p = 09830861 For comparison we

also plot the performance of uniform and non-robust RPAs in

these plots We observe that non-robust RPAs still offer someperformance improvements over uniform RPAs as long as is not large When is large the effectiveness of non-robust

RPA algorithms is significantly reduced On the other handwe see that robust RPAs provide significant performance gain

over non-robust RPAs over a wide range of showing the

effectiveness of our robust algorithms in the presence of global

CSI uncertainty

VI CONCLUSION

In this paper we developed RPA algorithms for coherentand noncoherent AF relay networks We showed that these

9These results are generated based on the worst case scenario where = and = minus

8112019 66183



01 015 02 025 03 035 04 045 050

01

02

03

04

05

06

07

08

09

1

Uniform RPA

Nonrobust RPA

Robust RPA


Fig 4 Outage probability as a function of size of uncertainty set forcoherent AF relay network with 907317 S9830871038389

2D = 3 dB p = 09830861 and r = 20

RPA problems in the presence of perfect global CSI can beformulated as quasiconvex optimization problems Thus the

RPA problems can be solved ef ficiently using the bisectionmethod through a sequence of convex feasibility problems

which can be cast as SOCPs We developed the robust

optimization framework for RPA problems when global CSI

is subject to uncertainties We showed that the robust coun-

terparts of our convex feasibility problems with ellipsoidal

uncertainty sets can be formulated as SDPs Conventionally

uncertainties associated with the global CSI are ignored and

the optimization problem is solved as if the given global

CSI is perfect However our results revealed that such a

naive approach often leads to poor performance highlighting

the importance of addressing CSI uncertainties by designing

robust algorithms in realistic wireless networks

APPENDIX A

PROOF OF PROPOSITION 1

First to show that 1038389 coh is a quasiconvex optimization

problem we simply need to show that the objective function coh( ) is quasiconcave and the constraint set in (10) is

convex The constraint set in (10) is simply the intersectionof a hypercube with an SOC Since the intersection of convex

sets is convex the constraint set in (10) is again convex For

any isin ℝ+ the upper-level set of coh( ) that belongs to 1103925 is given by

( coh ) =

1048699 isin 1103925 S

2D

( )2

∥ ∥2 + 1 ge

1048701

=

⎧⎨⎩ isin 1103925

⎡⎢⎣

991770 S2D983080

1

983081⎤⎥⎦ ર1038389 0

⎫⎬⎭ 983086 (29)

It is clear that ( coh ) is a convex set since it can be

represented as an SOC Since the upper-level set ( coh )is convex for every isin ℝ+ coh( ) is thus quasiconcave

Note that a concave function is also quasiconcave We now

show that coh( ) is not concave by contradiction Supposethat coh( ) is concave We consider and such that

= 11103925110392511039251 and = 11103925110392511039251 for 0 le 1 le radic p 21 le 1

and 0 lt lt 1 Clearly and isin 1103925 For any isin [0 1]we have

coh( + (1 minus ) ) = S9830872

D2121

+ 1 21 [1+1(1minus)]2

≜ ( 1) (30)

where ( 1) is clearly convex in 1 Due to convexity of ( 1)

the following inequality must hold

( (1)1 + (1 minus )

(2)1 ) le (

(1)1 ) + (1 minus )(

(2)1 )983086 (31)

Now by letting (1)1 = 1983087[ + (1 minus )] and

(2)1 = 1983087[ +

(1 minus )] we can rewrite (31) as

coh( + (1 minus ) ) le coh( ) + (1 minus ) coh( )983086

(32)

Thus we have showed that there exists isin 1103925 and isin[0 1] such that (32) holds By contradiction coh( ) is not

concave on 1103925

APPENDIX BPROOF OF ALGORITHM 1

We first show that for each given the convex feasibility

program is an SOCP For each the first constraint in (14)

follows immediately from (29) which is an SOC constraint

Clearly the aggregate relay power constraint in (10) can be

cast as an SOC constraint Lastly the individual relay power

constraints can be cast as SOC constraints as follows

1103925110392511039251038389 le radic p hArr

983080 1103925110392511039251038389pminus1

2

983081

le p + 1

2 983086 (33)

In summary 1038389

(SOCP)

coh is an SOCP since

1103925 coh() is equivalent

to the intersection of ( r + 2) SOC constraints and the

objective function is linear

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[1] T Q S Quek H Shin and M Z Win ldquoRobust wireless relay networksslow power allocation with guaranteed QoS IEEE J Sel Topics Signal

Process vol 1 no 4 pp 700-713 Dec 2007[2] A Ben-Tal and A Nemirovski ldquoRobust convex optimization Math

Oper Res vol 21 no 4 pp 769-805 Nov 1998[3] L E Ghaoui F Oustry and H Lebret ldquoRobust solutions to uncertain

semidefinite programs SIAM J Optim vol 9 no 1 pp 33-52 1998[4] M Gastpar and M Vetterli ldquoOn the capacity of large Gaussian relay

networks IEEE Trans Inf Theory vol 51 no 3 pp 765-779 Mar2005

[5] I Maric and R D Yates ldquoBandwidth and power allocation for coop-erative strategies in Gaussian relay networks in Proc Asilomar ConfSignals Syst Comput vol 2 Pacific Grove CA Nov 2004 pp 1907-1911

[6] A Hoslashst-Madsen and J Zhang ldquoCapacity bounds and power allocationfor wireless relay channels IEEE Trans Inf Theory vol 51 no 6pp 2020-2040 June 2005

[7] I Hammerstrom and A Wittneben ldquoOn the optimal power allocation fornonregenerative OFDM relay links in Proc IEEE Int Conf Communvol 10 Istanbul Turkey June 2006 pp 4463-4468

[8] M O Hasna and M-S Alouini ldquoOptimal power allocation for relayedtransmissions over Rayleigh-fading channels IEEE Trans Wireless

Commun vol 3 no 6 pp 1999-2004 Nov 2004[9] T Marzetta and B Hochwald ldquoFast transfer of channel state information

in wireless systems IEEE Trans Signal Process vol 54 no 4 pp1268-1278 Apr 2006

[10] W M Gifford M Z Win and M Chiani ldquoDiversity with practicalchannel estimation IEEE Trans Wireless Commun vol 4 no 4 pp1935-1947 July 2005

8112019 66183



[11] T Q S Quek H Shin M Z Win and M Chiani ldquoOptimal power allo-cation for amplify-and-forward relay networks via conic programmingin Proc IEEE Int Conf Commun Glasgow Scotland June 2007 pp5058-5063

[12] mdashmdash ldquoRobust power allocation for amplify-and-forward relay net-works in Proc IEEE Int Conf Commun Glasgow Scotland June2007 pp 957-962

[13] M S Lobo L Vandenberghe S Boyd and H Lebret ldquoApplicationsof second-order cone programming Linear Algebra its Appl vol 284pp 193-228 Nov 1998

[14] S Boyd and L Vandenberghe Convex Optimization Cambridge UK

Cambridge University Press 2004[15] K Glashoff and K Roleff ldquoA new method for chebyshev approximation

of complex-valued functions Mathematics Computation vol 36 no153 pp 233-239 Jan 1981

[16] J Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox for optimizationover symmetric cones Optim Meth Softw vol 11-12 pp 625-653Aug 1999

[17] H Boumllcskei R Nabar O Oyman and A Paulraj ldquoCapacity scalinglaws in MIMO relay networks IEEE Trans Wireless Commun vol 5no 6 pp 1433-1444 June 2006

8112019 66183


8112019 66183

































































R ≜ 983163R991264R∣ ℎℎℎB983165 and r = 9831631 2 983086 983086 983086 r983165




R = R (3)



R = (


)991264983086 (4)







(1038389)coh =

radic 1038389 1038389

ℎlowastB9830841038389∣ℎB9830841038389∣

ℎlowastF9830841038389∣ℎF9830841038389∣ (5)




(1038389)noncoh =

radic 1038389 1038389983086 (6)




8112019 66183






FR + D ≜983772D

(7)



on ℎℎℎB

and ℎℎℎF

is then given by

SNR( ) ≜

∣ℎℎℎ


983163∣983215D∣2 ∣ℎℎℎF983165

= Sℎℎℎ



D

(8)




follows



(9)











problem as

1038389 coh max

coh( ) ≜ S2D

( )2

∥ ∥2+1

st isin 1103925 (10)


1103925 =983163

isin ℝ r+



where 1038389 ≜991770

1038389 R




convenience where

1038389 =radic

1038389 R ∣ℎB9830841038389∣∣ℎF9830841038389∣ (11)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (12)











0 Initialize min

= coh

( min

) max

= coh

( max

) where









1


form as


st isin 1103925 coh() (13)


coh() =

⎧⎨⎩1038389 1038389 1038389 isin ℝ1103925 r

+

⎡⎢⎣

110392511039251103925907317 1038389 1038389 1038389 radic

S2D983080

19073179073179073171038389 1038389 1038389

983081⎤⎥⎦ ર 0983084

983131 11038389 1038389 1038389

983133 ર 0983084

⎡⎣

p+1

2983080 1038389 1038389 1038389 907317 1038389pminus1

2

983081⎤⎦ ર 0983084 forall isin 1038389 r

⎫⎬⎭ 983086

(14)





1038389 noncoh max

S2D

∣ ∣2

∥ ∥2+1

st isin 1103925 (15)

where 1103925


in = [1 2 983086 983086 983086 r ] isin ℂ r where

1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389983086








( ) = maxisinℒ

1048699real983163 983165 cos

983080

983081+ image1038389 983163 983165 sin

983080

9830811048701

8112019 66183





( ) le ∣ ∣ le ( )sec(

2

)983086





written as

maxisinℒ

noncoh( opt ) (16)

where

noncoh( )

≜ S

2D

852059real

852091

852093cos(983087) + image1038389

852091

852093sin(983087)

8520612∥ ∥2 + 1



1103925 noncoh(

)

st real852091

852093

cos(983087)+image1038389

852091

852093sin(983087) ge 0

isin 1103925 983086(17)


1103925 =983163

isin ℝ r+

sum1038389isin r


983165

where 1038389 ≜991770

1038389 R



= diag(1 2 983086 983086 983086 r)


1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389 (18)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (19)











rithm 1











follows

ge1057306

2D

S(1 + ∥ ∥2) forall( ) isin 907317 983086 (21)











1103925 (robust)coh () find 1038389 1038389 1038389



907317 R =

⎧⎨⎩

= 0 +


⎫⎬⎭

(23)

907317 L =

⎧⎨⎩ = 0 +

sumisin

∥∥ le 2

⎫⎬⎭ (24)





(25)



[1 2 983086 983086 983086 907317 ] and 983768983768983768 =852059 1 2 983086 983086 983086

852061


Theorem 6 in [1]




associated with and




8112019 66183


8112019 66183



0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100





S9830872D (dB)




0 2 4 6 8 10 1210

minus2

10minus1

100





S9830872D (dB)




















0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100

= 0 = 0 98308601 = 0 9830861 = 0 98308625

S9830872D (dB)





























CSI uncertainty

VI CONCLUSION



8112019 66183



01 015 02 025 03 035 04 045 050

01

02

03

04

05

06

07

08

09

1

Uniform RPA

Nonrobust RPA

Robust RPA



2D = 3 dB p = 09830861 and r = 20














APPENDIX A







( coh ) =

1048699 isin 1103925 S

2D

( )2

∥ ∥2 + 1 ge

1048701

=

⎧⎨⎩ isin 1103925

⎡⎢⎣

991770 S2D983080

1

983081⎤⎥⎦ ર1038389 0

⎫⎬⎭ 983086 (29)







coh( + (1 minus ) ) = S9830872

D2121

+ 1 21 [1+1(1minus)]2

≜ ( 1) (30)



( (1)1 + (1 minus )

(2)1 ) le (

(1)1 ) + (1 minus )(

(2)1 )983086 (31)


(2)1 = 1983087[ +



(32)


concave on 1103925








1103925110392511039251038389 le radic p hArr

983080 1103925110392511039251038389pminus1

2

983081

le p + 1

2 983086 (33)

In summary 1038389

(SOCP)





REFERENCES













8112019 66183











8112019 66183

































































R ≜ 983163R991264R∣ ℎℎℎB983165 and r = 9831631 2 983086 983086 983086 r983165




R = R (3)



R = (


)991264983086 (4)







(1038389)coh =

radic 1038389 1038389

ℎlowastB9830841038389∣ℎB9830841038389∣

ℎlowastF9830841038389∣ℎF9830841038389∣ (5)




(1038389)noncoh =

radic 1038389 1038389983086 (6)




8112019 66183






FR + D ≜983772D

(7)



on ℎℎℎB

and ℎℎℎF

is then given by

SNR( ) ≜

∣ℎℎℎ


983163∣983215D∣2 ∣ℎℎℎF983165

= Sℎℎℎ



D

(8)




follows



(9)











problem as

1038389 coh max

coh( ) ≜ S2D

( )2

∥ ∥2+1

st isin 1103925 (10)


1103925 =983163

isin ℝ r+



where 1038389 ≜991770

1038389 R




convenience where

1038389 =radic

1038389 R ∣ℎB9830841038389∣∣ℎF9830841038389∣ (11)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (12)











0 Initialize min

= coh

( min

) max

= coh

( max

) where









1


form as


st isin 1103925 coh() (13)


coh() =

⎧⎨⎩1038389 1038389 1038389 isin ℝ1103925 r

+

⎡⎢⎣

110392511039251103925907317 1038389 1038389 1038389 radic

S2D983080

19073179073179073171038389 1038389 1038389

983081⎤⎥⎦ ર 0983084

983131 11038389 1038389 1038389

983133 ર 0983084

⎡⎣

p+1

2983080 1038389 1038389 1038389 907317 1038389pminus1

2

983081⎤⎦ ર 0983084 forall isin 1038389 r

⎫⎬⎭ 983086

(14)





1038389 noncoh max

S2D

∣ ∣2

∥ ∥2+1

st isin 1103925 (15)

where 1103925


in = [1 2 983086 983086 983086 r ] isin ℂ r where

1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389983086








( ) = maxisinℒ

1048699real983163 983165 cos

983080

983081+ image1038389 983163 983165 sin

983080

9830811048701

8112019 66183





( ) le ∣ ∣ le ( )sec(

2

)983086





written as

maxisinℒ

noncoh( opt ) (16)

where

noncoh( )

≜ S

2D

852059real

852091

852093cos(983087) + image1038389

852091

852093sin(983087)

8520612∥ ∥2 + 1



1103925 noncoh(

)

st real852091

852093

cos(983087)+image1038389

852091

852093sin(983087) ge 0

isin 1103925 983086(17)


1103925 =983163

isin ℝ r+

sum1038389isin r


983165

where 1038389 ≜991770

1038389 R



= diag(1 2 983086 983086 983086 r)


1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389 (18)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (19)











rithm 1











follows

ge1057306

2D

S(1 + ∥ ∥2) forall( ) isin 907317 983086 (21)











1103925 (robust)coh () find 1038389 1038389 1038389



907317 R =

⎧⎨⎩

= 0 +


⎫⎬⎭

(23)

907317 L =

⎧⎨⎩ = 0 +

sumisin

∥∥ le 2

⎫⎬⎭ (24)





(25)



[1 2 983086 983086 983086 907317 ] and 983768983768983768 =852059 1 2 983086 983086 983086

852061


Theorem 6 in [1]




associated with and




8112019 66183


8112019 66183



0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100





S9830872D (dB)




0 2 4 6 8 10 1210

minus2

10minus1

100





S9830872D (dB)




















0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100

= 0 = 0 98308601 = 0 9830861 = 0 98308625

S9830872D (dB)





























CSI uncertainty

VI CONCLUSION



8112019 66183



01 015 02 025 03 035 04 045 050

01

02

03

04

05

06

07

08

09

1

Uniform RPA

Nonrobust RPA

Robust RPA



2D = 3 dB p = 09830861 and r = 20














APPENDIX A







( coh ) =

1048699 isin 1103925 S

2D

( )2

∥ ∥2 + 1 ge

1048701

=

⎧⎨⎩ isin 1103925

⎡⎢⎣

991770 S2D983080

1

983081⎤⎥⎦ ર1038389 0

⎫⎬⎭ 983086 (29)







coh( + (1 minus ) ) = S9830872

D2121

+ 1 21 [1+1(1minus)]2

≜ ( 1) (30)



( (1)1 + (1 minus )

(2)1 ) le (

(1)1 ) + (1 minus )(

(2)1 )983086 (31)


(2)1 = 1983087[ +



(32)


concave on 1103925








1103925110392511039251038389 le radic p hArr

983080 1103925110392511039251038389pminus1

2

983081

le p + 1

2 983086 (33)

In summary 1038389

(SOCP)





REFERENCES













8112019 66183











8112019 66183






FR + D ≜983772D

(7)



on ℎℎℎB

and ℎℎℎF

is then given by

SNR( ) ≜

∣ℎℎℎ


983163∣983215D∣2 ∣ℎℎℎF983165

= Sℎℎℎ



D

(8)




follows



(9)











problem as

1038389 coh max

coh( ) ≜ S2D

( )2

∥ ∥2+1

st isin 1103925 (10)


1103925 =983163

isin ℝ r+



where 1038389 ≜991770

1038389 R




convenience where

1038389 =radic

1038389 R ∣ℎB9830841038389∣∣ℎF9830841038389∣ (11)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (12)











0 Initialize min

= coh

( min

) max

= coh

( max

) where









1


form as


st isin 1103925 coh() (13)


coh() =

⎧⎨⎩1038389 1038389 1038389 isin ℝ1103925 r

+

⎡⎢⎣

110392511039251103925907317 1038389 1038389 1038389 radic

S2D983080

19073179073179073171038389 1038389 1038389

983081⎤⎥⎦ ર 0983084

983131 11038389 1038389 1038389

983133 ર 0983084

⎡⎣

p+1

2983080 1038389 1038389 1038389 907317 1038389pminus1

2

983081⎤⎦ ર 0983084 forall isin 1038389 r

⎫⎬⎭ 983086

(14)





1038389 noncoh max

S2D

∣ ∣2

∥ ∥2+1

st isin 1103925 (15)

where 1103925


in = [1 2 983086 983086 983086 r ] isin ℂ r where

1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389983086








( ) = maxisinℒ

1048699real983163 983165 cos

983080

983081+ image1038389 983163 983165 sin

983080

9830811048701

8112019 66183





( ) le ∣ ∣ le ( )sec(

2

)983086





written as

maxisinℒ

noncoh( opt ) (16)

where

noncoh( )

≜ S

2D

852059real

852091

852093cos(983087) + image1038389

852091

852093sin(983087)

8520612∥ ∥2 + 1



1103925 noncoh(

)

st real852091

852093

cos(983087)+image1038389

852091

852093sin(983087) ge 0

isin 1103925 983086(17)


1103925 =983163

isin ℝ r+

sum1038389isin r


983165

where 1038389 ≜991770

1038389 R



= diag(1 2 983086 983086 983086 r)


1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389 (18)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (19)











rithm 1











follows

ge1057306

2D

S(1 + ∥ ∥2) forall( ) isin 907317 983086 (21)











1103925 (robust)coh () find 1038389 1038389 1038389



907317 R =

⎧⎨⎩

= 0 +


⎫⎬⎭

(23)

907317 L =

⎧⎨⎩ = 0 +

sumisin

∥∥ le 2

⎫⎬⎭ (24)





(25)



[1 2 983086 983086 983086 907317 ] and 983768983768983768 =852059 1 2 983086 983086 983086

852061


Theorem 6 in [1]




associated with and




8112019 66183


8112019 66183



0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100





S9830872D (dB)




0 2 4 6 8 10 1210

minus2

10minus1

100





S9830872D (dB)




















0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100

= 0 = 0 98308601 = 0 9830861 = 0 98308625

S9830872D (dB)





























CSI uncertainty

VI CONCLUSION



8112019 66183



01 015 02 025 03 035 04 045 050

01

02

03

04

05

06

07

08

09

1

Uniform RPA

Nonrobust RPA

Robust RPA



2D = 3 dB p = 09830861 and r = 20














APPENDIX A







( coh ) =

1048699 isin 1103925 S

2D

( )2

∥ ∥2 + 1 ge

1048701

=

⎧⎨⎩ isin 1103925

⎡⎢⎣

991770 S2D983080

1

983081⎤⎥⎦ ર1038389 0

⎫⎬⎭ 983086 (29)







coh( + (1 minus ) ) = S9830872

D2121

+ 1 21 [1+1(1minus)]2

≜ ( 1) (30)



( (1)1 + (1 minus )

(2)1 ) le (

(1)1 ) + (1 minus )(

(2)1 )983086 (31)


(2)1 = 1983087[ +



(32)


concave on 1103925








1103925110392511039251038389 le radic p hArr

983080 1103925110392511039251038389pminus1

2

983081

le p + 1

2 983086 (33)

In summary 1038389

(SOCP)





REFERENCES













8112019 66183











8112019 66183





( ) le ∣ ∣ le ( )sec(

2

)983086





written as

maxisinℒ

noncoh( opt ) (16)

where

noncoh( )

≜ S

2D

852059real

852091

852093cos(983087) + image1038389

852091

852093sin(983087)

8520612∥ ∥2 + 1



1103925 noncoh(

)

st real852091

852093

cos(983087)+image1038389

852091

852093sin(983087) ge 0

isin 1103925 983086(17)


1103925 =983163

isin ℝ r+

sum1038389isin r


983165

where 1038389 ≜991770

1038389 R



= diag(1 2 983086 983086 983086 r)


1038389 =radic

1038389 R ℎB9830841038389ℎF9830841038389 (18)

1038389 =

radic 1038389 R ∣ℎF9830841038389∣R9830841038389

D983086 (19)











rithm 1











follows

ge1057306

2D

S(1 + ∥ ∥2) forall( ) isin 907317 983086 (21)











1103925 (robust)coh () find 1038389 1038389 1038389



907317 R =

⎧⎨⎩

= 0 +


⎫⎬⎭

(23)

907317 L =

⎧⎨⎩ = 0 +

sumisin

∥∥ le 2

⎫⎬⎭ (24)





(25)



[1 2 983086 983086 983086 907317 ] and 983768983768983768 =852059 1 2 983086 983086 983086

852061


Theorem 6 in [1]




associated with and




8112019 66183


8112019 66183



0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100





S9830872D (dB)




0 2 4 6 8 10 1210

minus2

10minus1

100





S9830872D (dB)




















0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100

= 0 = 0 98308601 = 0 9830861 = 0 98308625

S9830872D (dB)





























CSI uncertainty

VI CONCLUSION



8112019 66183



01 015 02 025 03 035 04 045 050

01

02

03

04

05

06

07

08

09

1

Uniform RPA

Nonrobust RPA

Robust RPA



2D = 3 dB p = 09830861 and r = 20














APPENDIX A







( coh ) =

1048699 isin 1103925 S

2D

( )2

∥ ∥2 + 1 ge

1048701

=

⎧⎨⎩ isin 1103925

⎡⎢⎣

991770 S2D983080

1

983081⎤⎥⎦ ર1038389 0

⎫⎬⎭ 983086 (29)







coh( + (1 minus ) ) = S9830872

D2121

+ 1 21 [1+1(1minus)]2

≜ ( 1) (30)



( (1)1 + (1 minus )

(2)1 ) le (

(1)1 ) + (1 minus )(

(2)1 )983086 (31)


(2)1 = 1983087[ +



(32)


concave on 1103925








1103925110392511039251038389 le radic p hArr

983080 1103925110392511039251038389pminus1

2

983081

le p + 1

2 983086 (33)

In summary 1038389

(SOCP)





REFERENCES













8112019 66183











8112019 66183


8112019 66183



0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100





S9830872D (dB)




0 2 4 6 8 10 1210

minus2

10minus1

100





S9830872D (dB)




















0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100

= 0 = 0 98308601 = 0 9830861 = 0 98308625

S9830872D (dB)





























CSI uncertainty

VI CONCLUSION



8112019 66183



01 015 02 025 03 035 04 045 050

01

02

03

04

05

06

07

08

09

1

Uniform RPA

Nonrobust RPA

Robust RPA



2D = 3 dB p = 09830861 and r = 20














APPENDIX A







( coh ) =

1048699 isin 1103925 S

2D

( )2

∥ ∥2 + 1 ge

1048701

=

⎧⎨⎩ isin 1103925

⎡⎢⎣

991770 S2D983080

1

983081⎤⎥⎦ ર1038389 0

⎫⎬⎭ 983086 (29)







coh( + (1 minus ) ) = S9830872

D2121

+ 1 21 [1+1(1minus)]2

≜ ( 1) (30)



( (1)1 + (1 minus )

(2)1 ) le (

(1)1 ) + (1 minus )(

(2)1 )983086 (31)


(2)1 = 1983087[ +



(32)


concave on 1103925








1103925110392511039251038389 le radic p hArr

983080 1103925110392511039251038389pminus1

2

983081

le p + 1

2 983086 (33)

In summary 1038389

(SOCP)





REFERENCES













8112019 66183











8112019 66183



0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100





S9830872D (dB)




0 2 4 6 8 10 1210

minus2

10minus1

100





S9830872D (dB)




















0 2 4 6 8 10 1210

minus3

10minus2

10minus1

100

= 0 = 0 98308601 = 0 9830861 = 0 98308625

S9830872D (dB)





























CSI uncertainty

VI CONCLUSION



8112019 66183



01 015 02 025 03 035 04 045 050

01

02

03

04

05

06

07

08

09

1

Uniform RPA

Nonrobust RPA

Robust RPA



2D = 3 dB p = 09830861 and r = 20














APPENDIX A







( coh ) =

1048699 isin 1103925 S

2D

( )2

∥ ∥2 + 1 ge

1048701

=

⎧⎨⎩ isin 1103925

⎡⎢⎣

991770 S2D983080

1

983081⎤⎥⎦ ર1038389 0

⎫⎬⎭ 983086 (29)







coh( + (1 minus ) ) = S9830872

D2121

+ 1 21 [1+1(1minus)]2

≜ ( 1) (30)



( (1)1 + (1 minus )

(2)1 ) le (

(1)1 ) + (1 minus )(

(2)1 )983086 (31)


(2)1 = 1983087[ +



(32)


concave on 1103925








1103925110392511039251038389 le radic p hArr

983080 1103925110392511039251038389pminus1

2

983081

le p + 1

2 983086 (33)

In summary 1038389

(SOCP)





REFERENCES













8112019 66183











8112019 66183



01 015 02 025 03 035 04 045 050

01

02

03

04

05

06

07

08

09

1

Uniform RPA

Nonrobust RPA

Robust RPA



2D = 3 dB p = 09830861 and r = 20














APPENDIX A







( coh ) =

1048699 isin 1103925 S

2D

( )2

∥ ∥2 + 1 ge

1048701

=

⎧⎨⎩ isin 1103925

⎡⎢⎣

991770 S2D983080

1

983081⎤⎥⎦ ર1038389 0

⎫⎬⎭ 983086 (29)







coh( + (1 minus ) ) = S9830872

D2121

+ 1 21 [1+1(1minus)]2

≜ ( 1) (30)



( (1)1 + (1 minus )

(2)1 ) le (

(1)1 ) + (1 minus )(

(2)1 )983086 (31)


(2)1 = 1983087[ +



(32)


concave on 1103925








1103925110392511039251038389 le radic p hArr

983080 1103925110392511039251038389pminus1

2

983081

le p + 1

2 983086 (33)

In summary 1038389

(SOCP)





REFERENCES













8112019 66183











8112019 66183











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66183