Research Article Optimal and Suboptimal Resource...

Research ArticleOptimal and Suboptimal Resource Allocation inMIMO Cooperative Cognitive Radio Networks

Mehdi Ghamari Adian1 and Mahin Ghamari Adyan2

1 Electrical Engineering Department University of Zanjan Zanjan 45371-38791 Iran2 Electrical Engineering Department University of Tehran Tehran 24785-54789 Iran

Correspondence should be addressed to Mehdi Ghamari Adian mehdighamariautacir

Received 11 April 2014 Revised 19 July 2014 Accepted 21 July 2014 Published 31 August 2014

Academic Editor Kai-Kit Wong

Copyright copy 2014 M Ghamari Adian and M Ghamari Adyan This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

The core aim of this work is the maximization of the achievable data rate of the secondary user pairs (SU pairs) while ensuring theQoS of primary users (PUs) All users are assumed to be equipped with multiple antennas It is assumed that when PUs are presentthe direct communication between SU pairs introduces intolerable interference to PUs and thereby SUs transmit signal using thecooperation of one of the SUs and avoid transmission in the direct channel In brief an adaptive cooperative strategy for MIMOcognitive radio networks is proposed At the presence of PUs the issue of joint relay selection and power allocation in underlayMIMO cooperative cognitive radio networks (U-MIMO-CCRN) is addressed The optimal approach for determining the powerallocation and the cooperating SU is proposed Besides the outage probability of the proposed system is further derived Due tohigh complexity of the optimal approach a low complexity approach is further proposed and its performance is evaluated usingsimulations The simulation results reveal that the performance loss due to the low complexity approach is only about 14 whilethe complexity is greatly reduced

1 Introduction

Since the issuance of the report of Federal CommunicationsCommission (FCC) in 2002 which revealed the spectruminefficiency in the incumbent wireless communication sys-tems cognitive radio (CR) has been regarded as one potentialtechnology to activate the utilization of spectrum resourcesin the recent evolution of wireless communication systems[1] As a consequence the overlay and underlay modes canbe developed based on the definitions of spectrum holes in[1] and the operation modes in [2] to use the white and grayspectrum holes respectively

Cognitive radio (CR) MIMO communications and co-operative communications are among the most promisingsolutions to improve spectrum utilization and efficiencyDynamic and opportunistic spectrum access allows CRnodes to communicate on temporarily idle or underutilizedfrequencies MIMO systems boost spectral efficiency byhaving a multiantenna node that simultaneously transmit

multiple data streams To further enhance the performanceof cognitive radio networks a cooperative relay networkcan be incorporated Thus in the underlay CR system withan interference temperature (IT) limit the cooperative relaynetworks can also be applied to have a better capacity anderror rate performance trade-off between achievable rateand network lifetime maximum signal-to-interference-plus-noise ratio (SINR) at the destination node better channelutilization by multihop relay maximum throughput andreduced interference via beamforming and maximum SINRusing cooperative beamforming A timely issue is to embracerecent innovations of the three technologies into a singlesystem

Joint problems of relay selection and resource allocationin CR networks (CRNs) have attracted extensive researchinterests due to its more effective spectrum utilization [3ndash8] The authors in [3] consider a cooperative cognitiveradio network (CCRN) in which the relays are selectedamong the existing SUs Moreover the QoS of the relays

Hindawi Publishing CorporationJournal of OptimizationVolume 2014 Article ID 190196 13 pageshttpdxdoiorg1011552014190196

2 Journal of Optimization

should be ensured For CCRN with decode-and-forwardstrategy two relay selection schemes namely a full-channelstate information (CSI) based best relay selection (BRS)and a partial CSI based best relay selection (PBRS) wereproposed in [4] In order to obtain an optimal subcarrierpairing relay assignment and power allocation in MIMO-OFDM based CCRN the dual decomposition technique wasrecruited in [5] to maximize the sum rate subject to theinterference temperature limit of the PUs Moreover dueto high computational complexity of the optimal approacha suboptimal algorithm was further proposed in [5] Theissue of joint relay selection and power allocation in two-way CCRNwas considered in [6] A suboptimal approach forreducing the complexity of joint relay selection and powerallocation in CCRN was proposed in [7] The network cod-ing opportunities existing in cooperative communicationsthat can further increase the capacity was exploited in [8]Furthermore the reformulation and linearization techniquesto the original optimization problems with nonlinear andnonconvex objective functions were applied such that theproposed algorithms can produce high competitive solutionsin a timely manner

The issue of resource allocation in MIMO CRNs wasexplored in [9ndash14] The authors in [9] presented a lowcomplexity semidistributed algorithm for resource allocationin MIMO-OFDM based CR networks using game theoryapproach the strong duality in convex optimization and theprimal decomposition method In [11] the authors extendedthe pricing concept to MIMO-OFDM based CR networksand presented two iterative algorithms for resource allocationin such systems In order to obtain an optimal subcarrierpairing relay assignment and power allocation in MIMO-OFDM based CCRN the dual decomposition technique wasrecruited in [12] to maximize the sum rate subject to theinterference temperature limit of the PUs Moreover becauseof high computational complexity of the optimal approacha suboptimal algorithm was further proposed in [12 13]An approach for resource allocation based on beamformingwith reduced complexity in MIMO cooperative cognitiveradio networks was presented in [14] where a suboptimalapproach with reduced complexity was further proposedto jointly determine the transmit beamforming (TB) andcooperative beamforming (CB) weight vectors along withantenna subset selection in MIMO-CCRN The problemof resource allocation in a spectrum leasing scenario incooperative cognitive radio networks was addressed in [10]where the SU pair allocates the whole of its transmissionpower in a portion of transmission frame to relay the primarysignals In return the PU pairs lease their unused portion oftransmission frame to the SU pair

The optimal resource allocation in MIMO cognitiveradio networks with heterogeneous secondary users andcentralized and distributed users was investigated in [15]Theauthors in [16] modeled the problem of joint relay selectionand power allocation in MIMO-OFDM based CCRN as atwo-level cooperative game problem with two objectivesThe first objective is to assign each weak SU to one of therelays (rich SUs) through solving a problem achieved by anontransferable utility coalition graph game and the second

one is to jointly allocate available channels to the SUs suchthat no subchannel is allocated to more than one SU andsimultaneously optimize the transmit covariance matrices ofnodes based on the Nash bargaining solution which is thesecond level of the game

In this paper we consider the opportunistic spectrumaccess in MIMO cognitive radio networks (MIMO-CRN)to ensure the SUsrsquo continuous transmission and reduce itsoutage probability without interfering the PUs The desiredlink is considered as the MIMO link between two SUs SUtransmitter (SU TX) and SU receiver (SU RX) All the usersare assumed to be equipped with multiple antennas Werecruit spectrum sensing technology to detect the presenceof the PUs If the PUs are absent the SU TX communicatesthe SU RX straightly Otherwise the transmit power of SUTX has to be reduced and SU TX transmits to SU RX usingthe cooperation of one the existing SUs The cooperatingSU is determined using the best relay selection algorithmTo be more accurate when a PU transmits signal in thesystem the joint problems of opportunistic relay selectionand power allocation in the context of MIMO CRN tomaximize the end-to-end achievable data rate ofMIMOCRNneed to be considered Our focus is on the amplify-and-forward (AF) relay strategy An obvious reason is that AFhas low complexity since no decodingencoding is neededThis benefit is even more attractive in MIMO-CRN wheredecoding multiple data streams could be computationallyintensive Moreover a more important reason is that AFoutperforms decode-and-forward (DF) strategy in termsof network capacity scaling in general as the number ofrelays increases in MIMO-CRN the effective signal-to-noiseratio (SNR) under AF scales linearly as opposed to being aconstant under DF [17]

To the best of our knowledge the joint problems of relayselection and power allocation in MIMO cognitive radionetworks has not been explored yet The main contributionsof the paper are as follows

(i) The optimal structure of amplification matrix at thecooperating SU and transmit covariancematrix at thetransmitter are determined at the presence of PUs

(ii) The optimal approach for solving the problem basedon the dual method is presented and then a lowcomplexity suboptimal approach is proposed to solvethe joint problems of relay selection and powerallocation in underlay MIMO CRN

(iii) The outage performance of the desired SU link isanalyzed

The remainder of this paper is organized as followsSection 2 presents the systemmodel and general formulationof the problem In Section 3 the structure of optimal powerallocation matrices is studied Based on these structuralresults we simplify and reformulate the optimization prob-lemThe optimization algorithms including the optimal andsuboptimal approach are discussed in Section 4 In Section 5the outage probability of the desired link is analyzed Numer-ical results are provided in Section 6 and Section 7 concludesthis paper

Journal of Optimization 3

Notation Uppercase and lowercase boldface denotematricesand vectors respectivelyThe operators (sdot)119867 |sdot| Tr(sdot) and (sdot)+are Hermitian (complex conjugate) determinant trace andpseudoinverse operators respectively

2 System Model

We consider a scenario where a secondary network con-sisting of 119873SU + 2 SUs coexists with a primary networkconsisting of119873PU PU pair In this paper the communicationbetween two SUs is considered which is also referred to asthe desired SU link The SU transmitter (SU TX) transmitssignals to SU receiver (SU RX) either in the direct linkor taking advantage of the cooperation of one of the SUsdepending on the presence of the PUs When the PUs areabsent the SU TX simply communicates the SU RX directlyHowever in order not to induce intolerable interference onthe PUs transmission from SU TX to SU RX at the presenceof PUs takes place in two consecutive time-slots using thecooperation of one the existing SU

21 The Transmission Process at the Presence of PUs WhenPU pairs are present the direct communications between theSUTX and SURXmay impose intolerable interference on thePUs The cooperation of one of SUs with the desired SU linkcan provide the possibility of reducing the transmit power ofthe SUs and thereby less interference is imposed on the PUpairs This is shown in Figure 1 The SU TX is equipped withcognitive radio capabilities and senses the available spectrumbands precisely

The selected cooperating SU cannot transmit and receivein the same channel at the same time due to self-interferenceThus a transmission from SU TX to SU RX at the presenceof PUs takes two time-slots This is also depicted in Figure 1In the first time-slot the SU TX transmits signals to all theexisting SUs in the CR network In the second time-slotone of the SUs is selected to cooperate with the SU TX byamplifying its received signal and forwarding it to the SU RXAll the transmissions in the SU system need to be regulatedin order to avoid excessive interference on the PU pair Theset of candidate SUs to cooperate with the desired SU link isdenoted by S

119877 Besides the set of PU pairs is also denoted

by SPU It is further assumed that all the users includingthe SUs and the Pus are equipped with multiple antennasWithout loss of generality and for ease of exposition weassume that all the candidate SUs to cooperate with desiredlink are equipped with 119873

119903antennas and the PUs with 119873

119901

antennasThenumber of antennas at SUTXand SURX is also119873119904and119873

119889 respectivelyH

119904119903119894isin C119873119903times119873119904 represents the channel

matrix form SU TX to SU 119894 andH119903119889119894

isin C119873119903times119873119889 represents thechannel form SU 119894 to SU RX All the channels are modeled asRayleigh fading channels and they are invariant during onetime-slot

Remark 1 In this paper we assume that a central controlleris available so that the network channel state informationand sensing results can be reliably gathered for centralizedprocessing Notice that the centralized CRNs are valid in

Hpp

Hsp

Hp1HRP1

HSR1

HRPN119904119906

Hps

HRD1

HRDN119904119906

PU RXPU TX

SU RX

HS119877N119904119906

SUN119904119906

SU TX

SU TX PU RX (SU1 middot middot middot SUN119904119906)

PU TX PU RX SU RXSUi SU RX PU RXPU TX PU RX SU RX

Information channelInterference channel

HpN119904119906

First time-slot Second time-slot

SU1

Figure 1 Systemmodel with the two transmission hops the desiredSU link and other secondary and primary users

IEEE 80222 standard [18] where the cognitive systemsoperate on a cellular basis and the central controller canbe embedded with a base station (BS) This assumption isalso reasonable if a spectrum broker exists in CRNs formanaging spectrum leasing and access [19 20] Such central-ized approach is commonly used in a variety of CRNs (eg[19ndash24]) Compared with distributed approaches a CCRNhaving a central manager that possesses detailed informationabout the wireless network enables highly efficient networkconfiguration and better enforcement of a complex set ofpolicies [20]

22 Problem Formulation The received signal at 119894th SU canbe written as

y119903119894= H119904119903119894x119904119894+ n119903119894 forall119894 isin S

119877 (1)

where the transmit signal of SU TX intended for SU 119894 isdenoted by x

119904119894isin C119873119904times1 n

119903119894isin C119873119903times1 is a Gaussian vectorThe

entries of n119903119894are assumed to be independent and identically

distributed random variables having zero means and vari-ances 1205902

119903 that is 119899

119903119894sim CN(0 1205902

119903I119873119903

) To provide more clari-fications in order to take the effect from the PU transmissioninto consideration and similar to [25 26] the distributionof n119903119894isCN(0 (1205902

0+ sum119899isinSPU

Tr(H119901119894119899

Q119901119899H119867119901119894119899

))I119873119903

) whereH119875119894119899

represents the channel matrix from the 119899th PU to the


119894th SU Q119875119899

denotes the transmit covariance matrix of the119899th PU 1205902

0is the noise power and we assume that all the links

have the same amount of noise power Therefore 1205902119903119894= 12059020+

sum119899isinSPU

Tr(H119901119894119899

Q119901119899H119867119901119894119899

) denotes the power of noise andinterference in the link from SU TX to the 119894th SU Withoutlosing the generality and for ease of exposition we furtherassume that 1205902

119903119894= 1205902119903 for all 119894 isin S

119877 Suppose that SU 119894

is selected to cooperate with the desired SU link Then thereceived signal at the SU RX (destination) from SU 119894 is givenby

y119889= H119903119889119894

A119894y119903119894+ n119889

= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889

(2)

where A119894represents the amplification matrix used at SU 119894

similar to n119903119894 n119889isin C119873119889times1 is also a Gaussian vector and

1205902119903119894= 12059020+ sum119899isinSPU

Tr(H119901119904119899

Q119901119899H119867119901119904119899

) where H119901119904119899

denotesthe channel matrix from the 119899th PU to SU RX As a resultof cooperation of one of the SUs for example SU 119894 theachievable data rate in the desired link can be written as

119877119894=1

2log2

100381610038161003816100381610038161003816I119873119889

+H119903119889119894

A119894H119904119903119894Q119894H119867119904119903119894A119867119894H119867119903119889119894

times(1205902

119889I119873119889

+ 1205902

119903H119903119889119894

A119894A119867119894H119867119903119889119894)minus1100381610038161003816100381610038161003816

(3)

whereQ119894= 119864x

119904119894x119867119904119894 denotes the transmit covariance matrix

of SU TX intended for SU 119894 The transmit power of SUTX to each SU is restricted to 119875

119879 that is Tr(Q

119894) le 119875

119879

Furthermore the maximum transmit power of the SU 119894 ifselected as the cooperative relay is 119875

119877 that is Tr[A

119894(1205902119903I119873119903

+

H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875

119877 The coefficient 12 in (3) is due

to the fact that cooperative transmission only uses half ofresources (eg time-slots frequency bands etc) The PUsmust not be disturbed as a result of transmission by SUTX and further the cooperation of the selected SU withthe SU TX In this way the interference power constraintson the PUs are provided by Tr(H

119904119901119899Q119894H119867119904119901119899

) le 1198751198681

andTrH119894119901119899

[A119894(1205902119903I119873119903

+ H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682 for all

119899 isin SPU where the SU 119894 is selected to cooperate with the SUTX Moreover H

119894119901119899and H

119904119901119899represent the channel from

SU 119894 and SU TX to 119899th PU RX respectively Evidently themaximum tolerable interference at the PUs is 119875

1198681+ 1198751198682 One

of the aims of this work is to optimally select the cooperatingSU and also calculate the optimum power allocation (findingthe optimalQ

119894and A

119894) in the proposed system which can be

formulated as

(Qlowast119894Alowast119894) = arg max

QA119894

119877119894(Q119894A119894)

119894 = arg max119894

119877119894(Qlowast119894Alowast119894)

st Tr (Q119894) le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875119877

Tr (H119904119901119899

Q119894H119867119904119901119899

) le 1198751198681

Tr H119894119901119899

[A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682

A119894ge 0 Q

119894ge 0

(4)

for all 119899 isin SPU where Qlowast119894 and Alowast119894are the optimum transmit

covariance and amplification matrices For convenience wedefine two constraint sets according to the following

Φ119894≜ Q119894| Tr (Q

119894) ⩽ 119875119879

Tr (H119904119901119899

Q119894H119867119904119901119899

) le 1198751198681 Q119894ge 0

Ψ119894≜ A119894| Tr [A

119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875119877 A119894ge 0

Tr H119894119901119899

[A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682

(5)

for all 119899 isin SPU It is easy to verify that (4) can be decomposedinto three parts as follows

max119894isinS119877

(maxQ119894isinΦ119894

(maxA119894isinΨ119894

119877119894(Q119894A119894))) (6)

Hence solving (4) reduces to iteratively solve a subprob-lem with respect to A

119894 for all 119894 isin S

119877and 119899 isin SPU (with Q

119894

fixed) then another subproblem with respect to Q119894(with A

119894

fixed forall119894 isin S119877and 119899 isin SPU) and finally a main problem

with respect to 119894 Directly tackling problem (4) is intractablein general However we will exploit the inherent specialstructure to significantly reduce the problem complexity andconvert it to an equivalent problem with scalar parametersIn what follows we will first study the optimal structuralproperties of A

119894andQ

119894 Then we will reformulate (4)

3 Optimal Power Allocation

In the first subsection the structure of the optimal amplifica-tion matrix in 119894th SU for a given Q

119894is investigated Then the

optimal structure of Q119894is studied in the second subsection

Finally based on these optimal structures the problem in (4)is reformulated in third subsection

31 The Structure of the Optimal Amplification Matrix Fornow we assume that Q

119894is given Let the eigenvalue decom-

position ofH119904119903119894H119867119904119903119894

andH119867119903119889119894

H119903119889119894

be

H119904119903119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

H119867119903119889119894

H119903119889119894

= V119903119889119894Σ119903119889119894

V119867119903119889119894

(7)

where U119904119903119894

and V119903119889119894

are unitary matrices Σ119904119903119894

= diag1205721 1205722

120572119873119903

with 120572119897ge 0 and Σ

119903119889119894= diag120573

1 1205732 120573

119873119903

with120573119897ge 0


Proposition 2 The optimal amplification matrix of SU i A119894

has the following structure

A119894 opt = V

119903119889119894ΛA119894

U119867119904119903119894 (8)

where U119904119903119894

is obtained by eigenvalue decomposition of H119904119903119894

H119867119904119903119894

and H119904119903119894

= H119904119903119894Q119894

minus12 that is H119904119903119894H119867119904119903119894 that is H

119904119903119894

H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

Proof Please refer to the Appendix

Let the singular value decomposition (SVD) of H119904119903119894

andH119903119889119894

be

H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

H119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894

(9)

which satisfies (7) Then exploiting (8) (9) and (3) theachievable data rates of the desired link can be written as

119877119894(Q119894A119894opt)

=1

2log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Σ119904119903119894(1205902

119889I119873119889

+ 1205902

119903Λ2

119903119889119894Λ2

A119894

)minus1100381610038161003816100381610038161003816

(10)

According to (10) the achievable data rate in the desiredSU link only depends on Σ

119904119903119894but not on U

119904119903119894 Then it can be

concluded that for anymatrix Q119894which satisfiesH

119904119903119894Q119894H119867119904119903119894

=

U119904119903119894Σ119904119903119894U119867119904119903119894 the optimal data rate is the same as when the

transmit covariancematrix in the desired link is any arbitrarymatrix Q

119894 Therefore (8) can be written as

A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (11)

32 The Structure of the Optimal Transmit CovarianceMatrixIn this subsection the optimal structure of the transmitcovariance matrix of the desired link is determined

Proposition 3 The structure of optimal transmit covariancematrix of SU TX is as follows

Q119894= V119904119903119894ΛQ119894

V119867119904119903119894 (12)

where ΛQ119894

is a diagonal matrix and must be determined suchthat the achievable data rate in the desired link is maximized

Proof Suppose that Σ1199041199031198941

is 119903 times 119903 and then

H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

= [U1199041199031198941

U1199041199031198942

] [Σ119904119903119894

0] [U1199041199031198941 U1199041199031198942

]119867

(13)

where Q119894is any PSD (positive semidefinite) matrix which

satisfies H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894 Hence the singular

value decomposition of matrix H119904119903119894

with rank 119903 can beexpressed as

H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

= [U1199041199031198941 U1199041199031198942] [

Λ1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

(14)

where Λ1199041199031198941

is 119903 times 119903 It can be shown that U1199041199031198941

isorthogonal to U

1199041199031198942 Moreover U

1199041199031198942is orthogonal to U

1199041199031198941

The pseudoinverse of H119904119903119894

is denoted by H+119904119903119894 Then from

H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894 we have

H+119904119903119894H119904119903119894Q119894H119867119904119903119894H+119867119904119903119894

= [V1199041199031198941 V1199041199031198942] [

Λminus11199041199031198941

0] [U1199041199031198941 U1199041199031198942]119867

times [U1199041199031198941

U1199041199031198942

] [Σ1199041199031198941

0] [U1199041199031198941 U1199041199031198942

]119867

times [U1199041199031198941 U1199041199031198942] [

Λminus1

1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

= [V1199041199031198941 V1199041199031198942]

times [Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus11199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(15)

It can be verified that U1198671199041199031198941

U1199041199031198941

is a unitary matrixbecause U119867

119904119903119894U119904119903119894

is unitary Recall that if A and B are twopositive semidefinite119872times119872matrices with eigenvalues 120582

119894(A)

and 120582119894(B) arranged in the descending order respectively

then119872

sum119894=1

120582119894 (A) 120582119872+1minus119894 (B) le Tr (AB) le

119872

sum119894=1

120582119894 (A) 120582119894 (B) (16)

Then using the second inequality in (16) and knowingthat H119867

119904119903119894H+119867119904119903119894

H+119904119903119894H119904119903119894

is a project matrix with eigenvaluesbeing only 1 and 0 we have

Tr (Q119894) ge Tr (H+

119904119903119894H119904119903119894Q119894H119867119904119903119894H119867+119904119903119894)

= Tr (Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus1

1199041199031198941)

(17)

Also using the first equality in (16) we can conclude that

Tr (Q119894) ge Tr (Σ

1199041199031198941Λminus2

1199041199031198941) (18)

Therefore the structure of the optimal transmit covariancematrix in the desired link is given by

Qopt119894 = [V1199041199031198941 V1199041199031198942] [

Σ1199041199031198941

Λminus21199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(19)


which satisfies

H119904119903119894Qopt119894H

119867

119904119903119894= U119904119903119894[Σ1199041199031198941

0]U119867

119904119903119894(20)

and the proposition is proved

33 Problem Reformulation In the previous subsection weproved that the structure of the optimal amplification matrixin SU 119894 and transmit covariance matrix in the SU TX can beexpressed as

A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894

Q119894= V119904119903119894ΛQ119894

V119867119904119903119894

(21)

whereH119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

andH119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894

Recallthat the received signal in SU RX due to the cooperation ofSU 119894 is given by

y119889= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889 (22)

Using (21) y119889in (22) can be rewritten as

y119889= U119903119889119894Λ119903119889119894ΛA119894

Λ119904119903119894V119867119904119903119894x119904119894+ U119903119889119894Λ119903119889119894ΛA119894

U119867119904119903119894n119903119894+ n119889

(23)

Suppose that y119889= U119867119903119889119894

y119889 x119904119894= V119867119904119903119894x119904119894 n119903119894= U119867119904119903119894n119903119894 and

n119889= U119867119903119889119894

n119889 Then

y119889= Λ119903119889119894ΛA119894

Λ119904119903119894x119904119894+ Λ119903119889119894ΛA119894

n119903119894+ n119889 (24)

Clearly the relay channel between the SU TX and SU RXhas been decomposed into a set of parallel SISO subchannelsTherefore the achievable data rates in the desired link as aresult of the cooperation of SU 119894 can be expressed as

119877119894= log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Λ2

119904119903119894ΛQ119894

(1205902

119903Λ2

119903119889119894Λ2

A119894

+ 1205902

119889I119873119889

)minus1100381610038161003816100381610038161003816

(25)

Suppose that the eigenvalue decomposition ofH119867119904119901119899

H119904119901119899

andH119867119894119901119899

H119894119901119899

is

H119867119904119901119899

H119904119901119899

= U119904119901119899

Λ119904119901119899

U119867119904119901119899

H119867119894119901119899

H119894119901119899

= U119894119901119899

Λ119894119901119899

U119867119894119901119899

(26)

for all 119899 isin SPU We further assume that

Λ2

A119894

= diag 1198861119894 119886

119873119903119894

Λ2

119904119903119894= diag 119887

1119894 119887

119873119903119894

Λ2

119903119889119894= diag 119888

1119894 119888

119873119903119894

Λ119904119901119899

= diag 1198891119899 119889

119873119904119899 forall119899 isin SPU

Λ119894119901119899

= diag 1198901119894119899

119890119873119903119894119899 forall119899 isin SPU

ΛQ119894

= diag 1199021119894 119902

119873119904119894

(27)

Then using (27) (25) can be rewritten as

119877119894=

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

) (28)

Moreover the transmit power constraint of the SU TXand SU 119894 will become

Tr (Q119894) le 119875119879997904rArr

119873119904

sum119896=1

119902119896119894le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]

= Tr (1205902119903Λ2

A119894

+ Λ2

119904119903119894ΛQ119894

Λ2

A119894

) le 119875119877

997904rArr

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

(29)

The interference constraint on PUs due to transmissionof SU TX can be written as

Tr (H119904119901119899

Q119894H119867119904119901119899

) = Tr (H119867119904119901119899

H119904119901119899

Q119894)

= Tr (U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

V119867119904119903119894)

= Tr (V119867119904119903119894U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

)

(30)

for all 119899 isin SPU Let M119894119899 = V119867119904119903119894U119904119901119899

Then (30) can beexpressed as

Tr (M119894119899Λ119904119901119899

M119867119894ΛQ119894

) =

119873119904

sum119896=1

(

119873119904

sum119897=1

100381610038161003816100381611989811989411989611989711989910038161003816100381610038162119889119897)119902119896119894 (31)

where119898119894119896119897119899

denotes the element of 119896th row and 119897th columnof matrix M

119894119899 Let 119891

119896119894119899= sum119873119904

119897=1|119898119894119896119897119899

|2119889119897119899 Therefore the

interference constraint on PUs due to transmitting by SUTXis expressed by

Tr (H119904119901119899

Q119894H119867119904119901119899

) =

119873119904

sum119896=1

119891119896119894119899

119902119896119894le 1198751198681 (32)

The interference constraint on PUs due to the coopera-tion of SU 119894 with SU TX is written by

Tr H119894119901119899

A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894H119867119894119901119899

= Tr (V119867119903119889119894

U119894119901119899

Λ119894119901119899

U119867119894119901119899

V119903119889119894ΛA119894

times (1205902

119903I119873119903

+ Λ119904119903119894ΛQ119894

Λ119904119903119894)U119904119903119894ΛA119894

) le 1198751198682

(33)

for all 119894 isin S119877and 119899 isin SPU Let S119894119899 = V119867

119903119889119894U119894119901119899

The elementof 119896th row and 119897th column of S

119894119899is denoted by 119904

119894119896119897119899 Hence

it can be shown that (33) can be rewritten as119873119903

sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894(

119873119903

sum119897=1

100381610038161003816100381611990411989411989611989711989910038161003816100381610038162119890119897119894119899) le 119875

1198682 (34)


Let 119892119896119894119899

= sum119873119903

119897=1|119904119894119896119897119899

|2119890119897119894119899

Thus the interferenceconstraint on PUs due to the cooperation of the selected SUin relaying the signals of the SU TX is stated as

119873119903

sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894119892119896119894119899

le 1198751198682 (35)

Note that without loss of generality it was assumed in(35) that 119873

119904ge 119873119903 If this assumption is not applicable the

only necessary modification in (35) is to change the upperlimit of the inner sum to min(119873

119903 119873119904) instead of119873

119903 Let a

119894=

[1198861119894 119886

119873119903119894] and q

119894= [1199021119894 119902

119873119904119894] Finally the problem

(4) can be expressed according to the following

qlowast119894 alowast119894= arg max

q119894a119894

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

)

119894 = arg max119894

119873119903

sum119896=1

log2(1 +

119886lowast

119896119894119887119896119894119888119896119894119902lowast

119896119894

1205902119903119886lowast119896119894119888119896119894+ 1205902119889

)

subject to119873119904

sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

119873119904

sum119896=1

119891119896119894119899

119902119896119894le 1198751198681 forall119899 isin SPU

119873119903

sum119896=1

119892119896119894119899

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 1198751198682 forall119899 isin SPU

(36)

Let ℎ119896119894= 119886119896119894(1205902119903+ 119887119896119894119902119896119894) By some simple derivations

the problem in (36) is equivalent to

qlowast119894 hlowast119894= arg max

q119894h119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

ℎ119896119894le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899

ℎ119896119894le 1198751198682 forall119899 isin SPU

(37)

4 Optimization Algorithm

In this section we develop approaches for joint relay selec-tion and power allocation in cooperative cognitive radionetworks At first we provide an optimal approach and thendevelop a low complexity suboptimal approach

41 Optimal Approach Using the Lagrange multipliersmethod [27] the Lagrange function for (37) is given by

L (h119894 q119894 1205821 1205822 1205823119899 1205824119899)

= minus

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

+ 1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) + 120582

2(

119873119903

sum119896=1

ℎ119896119894minus 119875119877)

+

119873PU

sum119899=1

1205823119899(

119873119903

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681)

+

119873PU

sum119899=1

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682)

(38)

where 1205821 1205822 1205823119899 and 120582

4119899are the Lagrange multipliers and

forall119899 isin SPU According to the KKT conditions we have

1205821ge 0 120582

2ge 0 120582

3119899ge 0 120582

4119899ge 0

119902119897119894ge 0 ℎ

119896119894ge 0 119897 = 1 119873

119904 119896 = 1 119873

119903

1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) = 0

1205822(

119873119904

sum119896=1

ℎ119896119894minus 119875119877) = 0

1205823119899(

119873119904

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681) = 0

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682) = 0

120597L

120597119902119896119894

= 0 119896 = 1 119873119904

120597L

120597ℎ119896119894

= 0 119896 = 1 119873119903

(39)


for all 119899 isin SPU and 119894 isin S119877 It can be shown that ℎ

119896119894and 119902119896119894

can be obtained using the following equations

119902119896119894=

1205902119903

2119887119896119894

[[

[

radic1198882119896119894

1205904119889

ℎ2119896119894minus

4119887119896119894119888119896119894

(1205821+ 119891119896119894119899

1205823) 12059021199031205902119889ln 2

ℎ119896119894

minus(2 +119888119896119894

1205902119889

ℎ119896119894)]]

]

+

ℎ119896119894=

1205902

119889

2119888119896119894

[[

[

radic1198872119896119894

1205904119903

1199022119896minus

4119887119896119894119888119896119894

(1205822+ 119892119896119894119899

1205824) 12059021199031205902119889ln 2

119902119896

minus(2 +119887119896119894

1205902119903

119902119896119894)]]

]

+

(40)

where [sdot]+ = max(sdot 0) Using dual-domain and subgradientmethods [28] we can further obtain 120582

1 1205822 1205823119899 and 120582

4119899

through iteration

120582(119898+1)

1= [120582(119898)

1+ 120583(119898)

(

119873119904

sum119896=1

119902(119898)

119896119894minus 119875119879)]

+

120582(119898+1)

2= [120582(119898)

2+ 120583(119898)

(

119873119903

sum119896=1

ℎ(119898)

119896119894minus 119875119877)]

+

120582(119898+1)

3119899= [120582(119898)

3119899+ 120583(119898)

(

119873119903

sum119896=1

119891119896119894119899

119902(119898)

119896119894minus 1198751198681)]

+

120582(119898+1)

4119899= [120582(119898)

4119899+ 120583(119898)

(

119873119903

sum119896=1

119892119896119894119899

ℎ(119898)

119896119894minus 1198751198682)]

+

(41)

for all 119899 isin SPU where 119898 is the iteration index and 120583(119898) is asequence of scalar step sizes Once 120582

1 1205822 1205823119899 and 120582

4119899are

obtained we can get the optimal power allocation matricesQ119894andA

119894and the corresponding achievable data rate119877

119894when

the 119894th SU acts as the relay for the SUTX Repeating the aboveprocedures at all SUs we then find the onewith themaximumachievable data rate

42 Low Complexity Approach The optimal approach per-forms joint opportunistic relay selection andpower allocationand results in the maximum data rate However the optimalapproach is with very high complexity Here we aim todevelop an alternate low complexity suboptimal approach forproblem (37) At first we assume that the available sourcepower is distributed uniformly over the spatial modes thatis 119902uni119894

= 119875119879119873119904 Similar assumption applies for ℎ

119896119894rsquos (119896 =

1 119873119903) that is ℎuni

119894= 119875119877119873119903 Also assume that the

interference introduced to the PU by each spatial mode ofSU TX is equal and hence the maximum allowable powerthat can be allocated to the 119896th mode is 119902max

119896119894= 1198751198681119873119904119891max119896119894

where 119891max119896119894

= max119899isinSPU

119891119896119894119899

Therefore the allocated powerto the 119896th mode in the SU TX intended for SU 119894 is 119902lowast

119896119894=

min119902uni119894 119902max119896119894

for 119896 = 1 119873119904and forall119894 isin S

119877 Similarly we

assume that the interference introduced to the PU by eachspatial mode of SU 119894 is equal Therefore it can be concludedthat ℎmax

119896119894= 1198751198682119873119903119892max119896119894

where 119892max119896119894

= max119899isinSPU

119892119896119894119899

Therefore the power allocation in the SU 119894 is given by ℎlowast

119896119894=

minℎuni119894 ℎmax119896119894


119877 Afterwards

the SU 119894 is selected as the cooperative relay such that thefollowing is maximized

119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903

(42)

After determining the cooperative SU we calculate the opti-mal transmit covariancematrixQ

119894 and amplificationmatrix

A119894 using the approach provided in the optimal approach

subsection As we can see from the simulation results thisapproach is almost as good as the optimal approachHoweverit is with much lower complexity

A comparison between the computational complexity ofthe optimal and suboptimal methods is presented here Forthe optimal solution derived in the previous section 2(119873PU +1) dual variables are updated in every iteration Using thesevalues 1198732

119903function evaluations are performed to find the

power allocation Therefore the optimal solution derived inthe previous section has a complexity ofO(1198732

119903+2(119873PU+1)119879)

where 119879 is the number of iterations required to convergewhich is usually high [28] In the suboptimal scheme everyspatial mode in the source side requires notmore than (119873SU+119873SU119873119903) function evaluations Therefore the complexity ofthe proposed algorithm is O(119873SU119873119903)

5 Outage Analysis

In order to analyze the outage behaviour of the proposedsystem we consider the scenario where the PU transmittersPU TX

1 PU TX

119873PU randomly communicate with their

respective receivers PU RX1 PU RX

119873PU The interval

between two transmissions of PUs and the duration of onePU transmission are assumed being random and obeyingexponential distributionwith two parameters 120579 and 120591 respec-tively According to queuing theory the probability of theabsence of the PUs 119875(119860) and the probability of the presenceof the PUs 119875(119860) can be expressed respectively as 119875(119860) =(sum119873PU119899=0

(119873PU(119873PU minus 119899))(120579120591)119899)minus1

= 120572 and 119875(119860) = 1 minus 120572In order to facilitate the analysis of outage we modify

the system model as explained below First of all we assumethat the transmit signal at the SU TX is white and therebyQ = 120588I

119873119904

where I119873119904

represents the 119873119904times 119873119904identity matrix

and 120588119873119904is the transmit power of the SU TX Moreover

the cooperation strategy of the selected SU is assumed tobe decode-and-forward (DF) strategy This strategy switch isintended for some reasons which among them is to obtaina lower bound for the outage capacity of the desired MIMOlinkMeanwhile this assumption facilitates the analysis of theoutage probability analysis as will be shown below


In the first time-slot the spectrum sensing is used todetect whether the PUs are absent When the PUs are absentSU TX transmits data to SU RX directly When the PUs arepresent the transmit power of SUTX 120588119873

119904 should be limited

However if 120588119873119904is too low the data from SUTX cannot reach

SU RX Thus we use cooperative relaying to transmit signalfrom SUTX to SURX through the best relay which is selectedout of available SUs as described in the previous section Inthe sequel we derive the approximate outage probabilities ofthe desired SU link when the PUs are present and when noPUs transmit signals or in other words the PUs are absent

51 Absence of PUs We firstly assume that no PU link istransmitting signal Hence the SUTX communicates directlywith the SU RX and the received signal in the SU RX can bewritten as

y119889= H119904119889x119904+ n119889 (43)

Based on the assumptions expressed at the beginning ofthis section the achievable data rates of the desired link usingthe direct channel are given by

119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889

isin C119873119889times119873119904 represents the direct channel in thedesired link It is obvious that the achievable data rates inthe desired link119877119863 are a random variable which depends onthe random nature of H

119904119889 In a full-rank system (44) can be

simplified by using the singular value decomposition (SVD)as

119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898

119898 = 1 119873119889are the nonnegative eigenvalues

of the channel covariance matrix H119904119889H119867119904119889 The joint pdf of

120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)

times (prod119898lt119899

(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904

is a normalizing factor To ensure QoS for thedesired link it needs to support a minimum rate When theinstantaneous achievable data rate is less than the minimumrate 119877min an outage event occurs In quasistatic fading sincethe fading coefficients are constant over the whole frame wecannot average them with an ergodic measure In such anevent Shannon capacity does not exist in the ergodic sense[30] The probability of such an event is normally referred to

as outage probability As described in [31] the distribution ofthe random achievable data rate can be viewed as Gaussianwhen the number of transmit andor receive antennas goesto infinity It is also a very good approximation for even small119873119889and 119873

119904 for example 119873

119904= 119873119889= 2 [31] As such for a

sufficiently large 119873119889and 119873

119904 the achievable data rate of the

desired link is approximated as [31]

119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)

Then we proceed by considering the distribution of theachievable data rate in the desired link as Gaussian with thepdf given in (47) Consequently it can be shown that theoutage probability of the desired link in the absence of thePUs can be written as

119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)

where 119876(sdot) denotes the 119876-function We emphasize that (48)provides only an approximation of the outage probability inthe desired link in the absence of the PUs

52 Presence of PUs As described in the previous sectionwhen PUs transmit signals the direct communication in thedesired link must be avoided and the cooperation of the bestSU is employed instead The received signal in the SU RXusing the cooperation of 119894th SU can be expressed as

y119889= H119903119889119894

x119904119894+ n119889 (49)

Thus the achievable data rates of the desired link is givenby

119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)

The coefficient 12 is due to the cooperative transmissionand the transmission in two consecutive time-slots Similar tothe previous subsection it can be concluded that for the caseof present PUs the achievable data rates in the desired link119877119862119894 can be expressed as

119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898


of the channel covariance matrix H119903119889119894

H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903

is a normalizing factor Once again and similarto the previous discussions the achievable data rate of thedesired link is approximated as [31]

119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)

Note that the coefficient 14 in the variance of the pdf in(53) is due to the multiplication of 12 in (51) Therefore theoutage probability of the desired link in the presence of thePUs can be written as

119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)

53 The Outage Probability In this subsection the outageprobability of the system is obtained However in the casethat the DF cooperation strategy is employed and the PUsare present another possible case in the system is when noSU can decode the signal from SU TX This may be due todetrimental effects of fading and path loss in the link from theSUTX to SUs In this case the SUTX indispensably transmitsdata to SU TX directly with limited power 1205881015840119873

119904in order not

to disturb the PUs Assume that Δ119906is a nonempty subset of

the119873SU secondary users who can decode the data of SU TXthat is Δ

119906sube S119877 and assume that Δ

119906is the complementary

set of Δ119906 Suppose that 120601 is a null set Then the probability

of no existing SU to decode the data of SU TX 119875120601out can bewritten as

119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)

and 119875119877119898out where 119898 isin S119877denotes the outage probability in

the link from SU TX to the SUs in the first time-slot Similarto previous subsections a good approximate for 119875119877119898out can beobtained as

119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)

In the following theoremwe derive the outage probabilityof the desired SU link

Theorem 4 The outage probability of the desired SU link is

119875out = (1 minus 120572)(119875120601

out +2119873SUminus1

sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)

where 119875Δ 119906out is the outage probability of the desired link in thepresence of PUs and when the one SUs in the subset Δ

119906is

cooperating with desired link

Proof Consider the case that the PUs are present Then theprobability of event Δ = Δ

119906 that is there exist some SUs

which can decode the signal from SU TX can be written as

119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)

The outage probability of the desired link in the presenceof PUs and when the one SUs in the subset Δ

119906is cooperating

with desired link is given by

119875Δ119906

out = prod119894isinΔ119906

119875119862119894

out (59)

Then the outage probability of the desired SU link in thepresence of the PU signals can be written as

119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)

Finally it can be concluded that the outage probability ofthe desired link is given by

119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)

where 119875119863out is the outage probability of the desired link whenthe PUs are absent and is given in (48) and the proof iscomplete in this way

6 Simulation Results

In this section the performance of the proposed MIMOcooperative cognitive radio system is evaluated using sim-ulations More specifically we evaluate the performance ofthe proposed low complexity approach and compare it withthe optimal approach For better comprehending the meritof the proposed low complexity approach (LCA) we willalso compare the proposed approach with the approachesusing random cooperative SU selection with optimal power


0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)

Maximum transmit power of SU TX (PT) (W)10minus3 10minus2 10minus1 100

OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)

Figure 2 Achievable data rate in the desired link versus themaximum transmit power of SU TX (119875

119879)

allocation matrices (transmit covariance matrix and amplifi-cation matrix) referred to as RS-OPA (random SU-optimalpower allocation) and nonoptimal power allocation that isthe amplification matrix of the randomly selected SU andthe transmit covariance matrix are obtained as describedin Section 42 respectively which is referred to as RS-EPA (random SU-equal power allocation) The simulationassumptions are as follows otherwise stated

(i) All users are assumed to be equipped with the samenumber of antennas denoted by119873

(ii) We set interference limits 1198751198681

= 1198751198682

= 001Wotherwise stated

(iii) There exist 5 PU pairs in the system otherwise stated(iv) The elements of the channel matrices follow a

Rayleigh distribution and are independent of eachother

(v) TheSUs are uniformly located between the SUTXandSU RX

(vi) The path-loss exponent is 4 and the standard devia-tion of shadowing is 6 dB

(vii) The number of existing SUs in the system119873SU is 20otherwise stated

(viii) The level of noise is assumed identical in the systemand 1205902

119903= 1205902119889= 10minus6WHz

The achievable data rate in the desired SU link versus themaximum transmit power of SU TX for different numberof antennas and various scenarios is shown in Figure 2The maximum transmit power of each SU 119894 for all 119894 isin

S119877 is 119875

119877= 02W Using the low complexity approach

(LCA) 50 achievable data rate gain over the RS-OPA isobtained when 119873 = 2 Moreover LCA leads to only 14

0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)

10minus3 10minus2 10minus1 100

OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)

Maximum transmit power of cooperating SU (PR) (W)

Figure 3 Achievable data rate in the desired link versus themaximum transmit power of cooperating SU (119875

119877)

data rate degradation compared with OA with much lowercomplexity When 119875

119879is small the achievable data rate in

the desired SU link increases rapidly with 119875119879 However for

large amounts of 119875119879 due to restrictions by the interference

limits the achievable data rate is not sensitive to the 119875119879 As

another observation it can also be seen that the random SUand optimal power allocation scheme (RS-OPA) achieves asignificant gain in the achievable data rate over the randomSU and nonoptimal (equal) power allocation scheme (RS-EPA) especially when 119875

119879is small

The achievable data rate of the desired SU link versusthe maximum transmit power of the cooperating SU (119875

119877) is

depicted in Figure 3Themaximum transmit power of the SUTX is fixed at 119875

119879= 04W and the number of existing SUs in

the secondary network 119873SU is 20 It can be concluded thatthe achievable data rate in the system is mainly determinedby the transmit power of the cooperating SU when themaximum transmit power of the SU TX is a constant

As shown in Figure 4 the achievable data rate in thedesired link grows with the number of the existing SUs in theCRnetworkHowever this growth saturates from a particularnumber of SUs which shows that the increasing number ofexisting SUs will not necessarily result in the similar increasein the data rate of the desired linkMoreover deploying largernumber of antennas in users that is larger 119873 compensatesfor the less maximum transmit power of SU TX and thecooperating relay It must also be noted that the achievabledata rate in the system is increased with the number ofexisting SUs due to multiuser diversity

As a final note all the simulation results are indicating theundeniable effect of the deploying multiple antennas at theSUs and cooperation of other SUs on the performance of thesecondary spectrum access in the cognitive radio networksThe results provided in this paper suggest that it is inevitableto take advantage ofMIMO systems and cooperation of other


2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)

Number of existing SUs (NSU)

OA N = 3 PT = PR = 07 WLCA N = 3 PT = PR = 07 WOA N = 2 PT = 2 PR = 07 WLCA N = 2 PT = 2 PR = 07 WOA N = 2 PT = PR = 07 WLCA N = 2 PT = PR = 07 W

Figure 4 Achievable data rate in the desired link versus the numberof existing SUs

SUs for the aim of opportunistic and dynamic spectrumaccess and to achieve larger data rates without inducingintolerable interference on the PUs

7 Conclusions

In this work an adaptive transmission strategy for underlayMIMO cooperative cognitive radio networks was proposedIt is assumed that when the PUs are present the directtransmission by the SUs introduces intolerable interferenceon PUs As a remedy and tomaintain the performance qualityof the SUs the cooperation of one of the existing SUs wasproposed not only to reduce the imposed interference onPUs but also to maximize the data rates in the desired SUlink Afterwards the optimal solution of the joint problemsof power allocation (both in the SU TX and the cooperatingSU) and relay selection was presented However due to highcomplexity of the optimal approach a suboptimal approachwith less complexity was further proposed Meanwhile theexpected degradation in the system performance due tosuboptimal approach was proved to be negligible usingsimulations Finally an outage probability analysis was pro-vided to examine the performance of the proposed MIMOcooperative cognitive radio network

Appendix

Proof of Proposition 2 Our convention is that all eigenvaluesare arranged in descending order It was shown in [32] that ifthe SU TX works in spatial multiplexing mode that is the

SU TX transmits independent data streams from differentantennas the amplification matrix of SU 119894 can be written as

A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894

is a diagonal matrix Therefore A119894can be con-

sidered as a matched filter along the singular vectors ofthe channel matrices In order to use the results of [32]for the case of nonwhite transmit data of the SU TX andequivalently the transmit covariance matrix which is anyarbitrary matrix Q

119894 we define the equivalent channel matrix

H119904119903119894

= H119904119903119894Qminus12119894

Hence by adopting the same method asin [32] for any given pair of A

119894and Q

119894 there always exists

another pair A119894opt and Q

119894that achieves better or equal data

rate in the desired link In this case for the case of knownQ119894

(A1) must be modified as

A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894

is obtained by eigenvalue decomposition ofH119904119903119894H119867119904119903119894 that is H

119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005

[2] Q Zhao and B M Sadler ldquoA survey of dynamic spectrumaccessrdquo IEEE Signal Processing Magazine vol 24 no 3 pp 79ndash89 2007

[3] Y Yu W Wang C Wang F Yan and Y Zhang ldquoJointrelay selection and power allocation with QoS support forcognitive radio networksrdquo in Proceedings of the IEEE WirelessCommunications and Networking Conference (WCNC rsquo13) pp4516ndash4521 Shanghai China April 2013

[4] K J Kim T Q Duong and X Tran ldquoPerformance analysisof cognitive spectrum-sharing single-carrier systems with relayselectionrdquo IEEE Transactions on Signal Processing vol 60 no12 pp 6435ndash6449 2012

[5] M G Adian and H Aghaeinia ldquoOptimal resource allo-cation for opportunistic spectrum access in multiple-inputmultiple-output-orthogonal frequency division multiplexingbased cooperative cognitive radio networksrdquo IET Signal Process-ing vol 7 no 7 pp 549ndash557 2013

[6] P Ubaidulla and S Aıssa ldquoOptimal relay selection and powerallocation for cognitive two-way relaying networksrdquo IEEEWireless Communications Letters vol 1 no 3 pp 225ndash228 2012

[7] L Li X Zhou H Xu G Y Li D Wang and A SoongldquoSimplified relay selection and power allocation in cooperativecognitive radio systemsrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 33ndash36 2011

[8] P Li S Guo W Zhuang and B Ye ldquoOn efficient resource allo-cation for cognitive and cooperative communicationsrdquo IEEEJournal on Selected Areas in Communications vol 32 no 2 pp264ndash273 2014


[9] M G Adian H Aghaeinia and Y Norouzi ldquopectrum sharingand power allocation in multi-input-multi-output multi-bandunderlay cognitive radio networksrdquo IET Communications vol7 no 11 pp 1140ndash1150 2013

[10] M G Adian andH Aghaeinia ldquoAn auction-based approach forspectrum leasing in cooperative cognitive radio networks whento lease and how much to be leasedrdquoWireless Networks vol 20pp 411ndash422 2014

[11] M G Adian and H Aghaeinia ldquoSpectrum sharing and powerallocation in multiple-in multiple-out cognitive radio networksvia pricingrdquo IET Communications vol 6 no 16 pp 2621ndash26292012

[12] M G Adian andH Aghaeinia ldquoResource allocation inMIMO-OFDM based cooperative cognitive radio networksrdquo IEEETransactions on Communications vol 62 no 7 pp 2224ndash22352014

[13] M G Adian and H Aghaeinia ldquoLow complexity resourceallocation in MIMO-OFDM-based cooperative cognitive radionetworksrdquo Transactions on Emerging Telecommunications Tech-nology 2014

[14] MGAdian ldquoBeamformingwith reduced complexity inMIMOcooperative cognitive radio networksrdquo Journal of Optimizationvol 2014 Article ID 325217 10 pages 2014

[15] M G Adian and H Aghaeinia ldquoOptimal resource allocationin heterogeneous MIMO cognitive radio networksrdquo WirelessPersonal Communication vol 76 no 1 pp 23ndash39 2014

[16] M G Adian and H Aghaeinia ldquoA two-level cooperative game-based approach for joint relay selection and distributed resourceallocation in MIMO-OFDM-based cooperative cognitive radionetworksrdquo Transactions on Emerging Telecommunications Tech-nology 2014

[17] J Liu N B Shroff andH D Sherali ldquoOptimal power allocationin multi-relay MIMO cooperative networks theory and algo-rithmsrdquo IEEE Journal on Selected Areas in Communications vol30 no 2 pp 331ndash340 2012

[18] C Stevenson G Chouinard Z Lei W Hu S ShellhammerandW Caldwell ldquoIEEE 80222 the first cognitive radio wirelessregional area network standardrdquo IEEE Communications Maga-zine vol 47 no 1 pp 130ndash138 2009

[19] J Jia J Zhang and Q Zhang ldquoCooperative relay for cognitiveradio networksrdquo in Proceedings of the 28th Conference onComputer Communications (INFOCOM rsquo09) pp 2304ndash2312IEEE Rio de Janeiro Brazil April 2009

[20] V Brik E Rozner S Banerjee and P Bahl ldquoDSAP a protocolfor coordinated spectrum accessrdquo in Proceedings of the 1st IEEEInternational Symposium on New Frontiers in Dynamic Spec-trum Access Networks (DySPAN rsquo05) pp 611ndash614 November2005

[21] G Zhao C Yang G Y Li D Li and A C K Soong ldquoPowerand channel allocation for cooperative relay in cognitive radionetworksrdquo IEEE Journal on Selected Topics in Signal Processingvol 5 no 1 pp 151ndash159 2011

[22] G Bansal M J Hossain and V K Bhargava ldquoOptimal and sub-optimal power allocation schemes for OFDM-based cognitiveradio systemsrdquo IEEE Transactions onWireless Communicationsvol 7 no 11 pp 4710ndash4718 2008

[23] Y Ma D I Kim and Z Wu ldquoOptimization of OFDMA-based cellular cognitive radio networksrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2265ndash2276 2010

[24] X Kang H Garg Y Liang and R Zhang ldquoOptimal powerallocation for OFDM-based cognitive radio with new primary

transmission protection criteriardquo IEEE Transactions onWirelessCommunications vol 9 no 6 pp 2066ndash2075 2010

[25] L Zhang Y-C Liang and Y Xin ldquoJoint beamforming andpower allocation for multiple access channels in cognitive radionetworksrdquo IEEE Journal on Selected Areas in Communicationsvol 26 no 1 pp 38ndash51 2008

[26] L Zhang Y Xin Y Liang and H V Poor ldquoCognitive multipleaccess channels optimal power allocation forweighted sum ratemaximizationrdquo IEEE Transactions on Communications vol 57no 9 pp 2754ndash2762 2009

[27] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004

[28] W Yu and R Lui ldquoDual methods for nonconvex spectrumoptimization of multicarrier systemsrdquo IEEE Transactions onCommunications vol 54 no 7 pp 1310ndash1322 2006

[29] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash598 1999

[30] E Biglieri G Caire and G Taricco ldquoLimiting performance ofblock-fading channels with multiple antennasrdquo IEEE Transac-tions on Information Theory vol 47 no 4 pp 1273ndash1289 2001

[31] B M Hochwald T L Marzetta and V Tarokh ldquoMultiple-antenna channel hardening and its implications for rate feed-back and schedulingrdquo IEEETransactions on InformationTheoryvol 50 no 9 pp 1893ndash1909 2004

[32] X Tang and Y Hua ldquoOptimal design of non-regenerativeMIMO wireless relaysrdquo IEEE Transactions on Wireless Commu-nications vol 6 no 4 pp 1398ndash1407 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


should be ensured For CCRN with decode-and-forwardstrategy two relay selection schemes namely a full-channelstate information (CSI) based best relay selection (BRS)and a partial CSI based best relay selection (PBRS) wereproposed in [4] In order to obtain an optimal subcarrierpairing relay assignment and power allocation in MIMO-OFDM based CCRN the dual decomposition technique wasrecruited in [5] to maximize the sum rate subject to theinterference temperature limit of the PUs Moreover dueto high computational complexity of the optimal approacha suboptimal algorithm was further proposed in [5] Theissue of joint relay selection and power allocation in two-way CCRNwas considered in [6] A suboptimal approach forreducing the complexity of joint relay selection and powerallocation in CCRN was proposed in [7] The network cod-ing opportunities existing in cooperative communicationsthat can further increase the capacity was exploited in [8]Furthermore the reformulation and linearization techniquesto the original optimization problems with nonlinear andnonconvex objective functions were applied such that theproposed algorithms can produce high competitive solutionsin a timely manner

The issue of resource allocation in MIMO CRNs wasexplored in [9ndash14] The authors in [9] presented a lowcomplexity semidistributed algorithm for resource allocationin MIMO-OFDM based CR networks using game theoryapproach the strong duality in convex optimization and theprimal decomposition method In [11] the authors extendedthe pricing concept to MIMO-OFDM based CR networksand presented two iterative algorithms for resource allocationin such systems In order to obtain an optimal subcarrierpairing relay assignment and power allocation in MIMO-OFDM based CCRN the dual decomposition technique wasrecruited in [12] to maximize the sum rate subject to theinterference temperature limit of the PUs Moreover becauseof high computational complexity of the optimal approacha suboptimal algorithm was further proposed in [12 13]An approach for resource allocation based on beamformingwith reduced complexity in MIMO cooperative cognitiveradio networks was presented in [14] where a suboptimalapproach with reduced complexity was further proposedto jointly determine the transmit beamforming (TB) andcooperative beamforming (CB) weight vectors along withantenna subset selection in MIMO-CCRN The problemof resource allocation in a spectrum leasing scenario incooperative cognitive radio networks was addressed in [10]where the SU pair allocates the whole of its transmissionpower in a portion of transmission frame to relay the primarysignals In return the PU pairs lease their unused portion oftransmission frame to the SU pair

The optimal resource allocation in MIMO cognitiveradio networks with heterogeneous secondary users andcentralized and distributed users was investigated in [15]Theauthors in [16] modeled the problem of joint relay selectionand power allocation in MIMO-OFDM based CCRN as atwo-level cooperative game problem with two objectivesThe first objective is to assign each weak SU to one of therelays (rich SUs) through solving a problem achieved by anontransferable utility coalition graph game and the second

one is to jointly allocate available channels to the SUs suchthat no subchannel is allocated to more than one SU andsimultaneously optimize the transmit covariance matrices ofnodes based on the Nash bargaining solution which is thesecond level of the game

In this paper we consider the opportunistic spectrumaccess in MIMO cognitive radio networks (MIMO-CRN)to ensure the SUsrsquo continuous transmission and reduce itsoutage probability without interfering the PUs The desiredlink is considered as the MIMO link between two SUs SUtransmitter (SU TX) and SU receiver (SU RX) All the usersare assumed to be equipped with multiple antennas Werecruit spectrum sensing technology to detect the presenceof the PUs If the PUs are absent the SU TX communicatesthe SU RX straightly Otherwise the transmit power of SUTX has to be reduced and SU TX transmits to SU RX usingthe cooperation of one the existing SUs The cooperatingSU is determined using the best relay selection algorithmTo be more accurate when a PU transmits signal in thesystem the joint problems of opportunistic relay selectionand power allocation in the context of MIMO CRN tomaximize the end-to-end achievable data rate ofMIMOCRNneed to be considered Our focus is on the amplify-and-forward (AF) relay strategy An obvious reason is that AFhas low complexity since no decodingencoding is neededThis benefit is even more attractive in MIMO-CRN wheredecoding multiple data streams could be computationallyintensive Moreover a more important reason is that AFoutperforms decode-and-forward (DF) strategy in termsof network capacity scaling in general as the number ofrelays increases in MIMO-CRN the effective signal-to-noiseratio (SNR) under AF scales linearly as opposed to being aconstant under DF [17]

To the best of our knowledge the joint problems of relayselection and power allocation in MIMO cognitive radionetworks has not been explored yet The main contributionsof the paper are as follows

(i) The optimal structure of amplification matrix at thecooperating SU and transmit covariancematrix at thetransmitter are determined at the presence of PUs

(ii) The optimal approach for solving the problem basedon the dual method is presented and then a lowcomplexity suboptimal approach is proposed to solvethe joint problems of relay selection and powerallocation in underlay MIMO CRN

(iii) The outage performance of the desired SU link isanalyzed

The remainder of this paper is organized as followsSection 2 presents the systemmodel and general formulationof the problem In Section 3 the structure of optimal powerallocation matrices is studied Based on these structuralresults we simplify and reformulate the optimization prob-lemThe optimization algorithms including the optimal andsuboptimal approach are discussed in Section 4 In Section 5the outage probability of the desired link is analyzed Numer-ical results are provided in Section 6 and Section 7 concludesthis paper



2 System Model







119901







Hpp

Hsp

Hp1HRP1

HSR1

HRPN119904119906

Hps

HRD1

HRDN119904119906

PU RXPU TX

SU RX

HS119877N119904119906

SUN119904119906

SU TX




HpN119904119906


SU1





119877 (1)


119904119894isin C119873119904times1 n




119903 that is 119899

119903119894sim CN(0 1205902

119903I119873119903


0+ sum119899isinSPU

Tr(H119901119894119899

Q119901119899H119867119901119894119899

))I119873119903

) whereH119875119894119899



119894th SU Q119875119899




sum119899isinSPU

Tr(H119901119894119899

Q119901119899H119867119901119894119899


119903119894= 1205902119903 for all 119894 isin S



y119889= H119903119889119894

A119894y119903119894+ n119889

= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889

(2)



1205902119903119894= 12059020+ sum119899isinSPU

Tr(H119901119904119899

Q119901119899H119867119901119904119899

) where H119901119904119899


119877119894=1

2log2

100381610038161003816100381610038161003816I119873119889

+H119903119889119894

A119894H119904119903119894Q119894H119867119904119903119894A119867119894H119867119903119889119894

times(1205902

119889I119873119889

+ 1205902

119903H119903119889119894

A119894A119867119894H119867119903119889119894)minus1100381610038161003816100381610038161003816

(3)

whereQ119894= 119864x



119879 that is Tr(Q

119894) le 119875

119879


119877 that is Tr[A

119894(1205902119903I119873119903

+

H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875



119904119901119899Q119894H119867119904119901119899

) le 1198751198681

andTrH119894119901119899

[A119894(1205902119903I119873119903

+ H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682 for all


119894119901119899and H



1198681+ 1198751198682 One


119894and A


formulated as


QA119894

119877119894(Q119894A119894)

119894 = arg max119894

119877119894(Qlowast119894Alowast119894)

st Tr (Q119894) le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875119877

Tr (H119904119901119899

Q119894H119867119904119901119899

) le 1198751198681

Tr H119894119901119899

[A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682

A119894ge 0 Q

119894ge 0

(4)



Φ119894≜ Q119894| Tr (Q

119894) ⩽ 119875119879

Tr (H119904119901119899

Q119894H119867119904119901119899

) le 1198751198681 Q119894ge 0

Ψ119894≜ A119894| Tr [A

119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875119877 A119894ge 0

Tr H119894119901119899

[A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682

(5)





119877119894(Q119894A119894))) (6)


119894 for all 119894 isin S


119894


119894



119894andQ









position ofH119904119903119894H119867119904119903119894

andH119867119903119889119894

H119903119889119894

be

H119904119903119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

H119867119903119889119894

H119903119889119894

= V119903119889119894Σ119903119889119894

V119867119903119889119894

(7)

where U119904119903119894

and V119903119889119894


= diag1205721 1205722

120572119873119903

with 120572119897ge 0 and Σ

119903119889119894= diag120573

1 1205732 120573

119873119903

with120573119897ge 0




A119894 opt = V

119903119889119894ΛA119894

U119867119904119903119894 (8)

where U119904119903119894


H119867119904119903119894

and H119904119903119894

= H119904119903119894Q119894


119904119903119894

H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



andH119903119889119894

be

H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

H119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894

(9)


119877119894(Q119894A119894opt)

=1

2log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Σ119904119903119894(1205902

119889I119873119889

+ 1205902

119903Λ2

119903119889119894Λ2

A119894

)minus1100381610038161003816100381610038161003816

(10)


119904119903119894but not on U

119904119903119894 Then it can be


119904119903119894Q119894H119867119904119903119894

=




A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (11)



Q119894= V119904119903119894ΛQ119894

V119867119904119903119894 (12)

where ΛQ119894




H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

= [U1199041199031198941

U1199041199031198942

] [Σ119904119903119894

0] [U1199041199031198941 U1199041199031198942

]119867

(13)


satisfies H119904119903119894Q119894H119867119904119903119894




H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

= [U1199041199031198941 U1199041199031198942] [

Λ1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

(14)

where Λ1199041199031198941


isorthogonal to U

1199041199031198942 Moreover U


1199041199031198941



H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894 we have

H+119904119903119894H119904119903119894Q119894H119867119904119903119894H+119867119904119903119894

= [V1199041199031198941 V1199041199031198942] [

Λminus11199041199031198941

0] [U1199041199031198941 U1199041199031198942]119867

times [U1199041199031198941

U1199041199031198942

] [Σ1199041199031198941

0] [U1199041199031198941 U1199041199031198942

]119867

times [U1199041199031198941 U1199041199031198942] [

Λminus1

1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

= [V1199041199031198941 V1199041199031198942]

times [Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus11199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(15)


U1199041199031198941


119904119903119894U119904119903119894


119894(A)


then119872

sum119894=1


119872

sum119894=1

120582119894 (A) 120582119894 (B) (16)


119904119903119894H+119867119904119903119894

H+119904119903119894H119904119903119894


Tr (Q119894) ge Tr (H+

119904119903119894H119904119903119894Q119894H119867119904119903119894H119867+119904119903119894)

= Tr (Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus1

1199041199031198941)

(17)


Tr (Q119894) ge Tr (Σ

1199041199031198941Λminus2

1199041199031198941) (18)


Qopt119894 = [V1199041199031198941 V1199041199031198942] [

Σ1199041199031198941

Λminus21199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(19)


which satisfies

H119904119903119894Qopt119894H

119867

119904119903119894= U119904119903119894[Σ1199041199031198941

0]U119867

119904119903119894(20)



A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894

Q119894= V119904119903119894ΛQ119894

V119867119904119903119894

(21)

whereH119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

andH119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894


y119889= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889 (22)


y119889= U119903119889119894Λ119903119889119894ΛA119894

Λ119904119903119894V119867119904119903119894x119904119894+ U119903119889119894Λ119903119889119894ΛA119894

U119867119904119903119894n119903119894+ n119889

(23)

Suppose that y119889= U119867119903119889119894

y119889 x119904119894= V119867119904119903119894x119904119894 n119903119894= U119867119904119903119894n119903119894 and

n119889= U119867119903119889119894

n119889 Then

y119889= Λ119903119889119894ΛA119894

Λ119904119903119894x119904119894+ Λ119903119889119894ΛA119894

n119903119894+ n119889 (24)


119877119894= log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Λ2

119904119903119894ΛQ119894

(1205902

119903Λ2

119903119889119894Λ2

A119894

+ 1205902

119889I119873119889

)minus1100381610038161003816100381610038161003816

(25)


H119904119901119899

andH119867119894119901119899

H119894119901119899

is

H119867119904119901119899

H119904119901119899

= U119904119901119899

Λ119904119901119899

U119867119904119901119899

H119867119894119901119899

H119894119901119899

= U119894119901119899

Λ119894119901119899

U119867119894119901119899

(26)


Λ2

A119894

= diag 1198861119894 119886

119873119903119894

Λ2

119904119903119894= diag 119887

1119894 119887

119873119903119894

Λ2

119903119889119894= diag 119888

1119894 119888

119873119903119894

Λ119904119901119899

= diag 1198891119899 119889

119873119904119899 forall119899 isin SPU

Λ119894119901119899

= diag 1198901119894119899

119890119873119903119894119899 forall119899 isin SPU

ΛQ119894

= diag 1199021119894 119902

119873119904119894

(27)


119877119894=

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

) (28)


Tr (Q119894) le 119875119879997904rArr

119873119904

sum119896=1

119902119896119894le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]

= Tr (1205902119903Λ2

A119894

+ Λ2

119904119903119894ΛQ119894

Λ2

A119894

) le 119875119877

997904rArr

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

(29)


Tr (H119904119901119899

Q119894H119867119904119901119899

) = Tr (H119867119904119901119899

H119904119901119899

Q119894)

= Tr (U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

V119867119904119903119894)

= Tr (V119867119904119903119894U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

)

(30)



Tr (M119894119899Λ119904119901119899

M119867119894ΛQ119894

) =

119873119904

sum119896=1

(

119873119904

sum119897=1

100381610038161003816100381611989811989411989611989711989910038161003816100381610038162119889119897)119902119896119894 (31)

where119898119894119896119897119899


119894119899 Let 119891

119896119894119899= sum119873119904

119897=1|119898119894119896119897119899

|2119889119897119899 Therefore the


Tr (H119904119901119899

Q119894H119867119904119901119899

) =

119873119904

sum119896=1

119891119896119894119899

119902119896119894le 1198751198681 (32)


Tr H119894119901119899

A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894H119867119894119901119899

= Tr (V119867119903119889119894

U119894119901119899

Λ119894119901119899

U119867119894119901119899

V119903119889119894ΛA119894

times (1205902

119903I119873119903

+ Λ119904119903119894ΛQ119894

Λ119904119903119894)U119904119903119894ΛA119894

) le 1198751198682

(33)


119903119889119894U119894119901119899


119894119899is denoted by 119904

119894119896119897119899 Hence


sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894(

119873119903

sum119897=1

100381610038161003816100381611990411989411989611989711989910038161003816100381610038162119890119897119894119899) le 119875

1198682 (34)


Let 119892119896119894119899

= sum119873119903

119897=1|119904119894119896119897119899

|2119890119897119894119899


119873119903

sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894119892119896119894119899

le 1198751198682 (35)




119903 119873119904) instead of119873

119903 Let a

119894=

[1198861119894 119886

119873119903119894] and q

119894= [1199021119894 119902




q119894a119894

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

)

119894 = arg max119894

119873119903

sum119896=1

log2(1 +

119886lowast

119896119894119887119896119894119888119896119894119902lowast

119896119894

1205902119903119886lowast119896119894119888119896119894+ 1205902119889

)


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899

119886119896119894(1205902


(36)




q119894h119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

ℎ119896119894le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899


(37)




L (h119894 q119894 1205821 1205822 1205823119899 1205824119899)

= minus

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

+ 1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) + 120582

2(

119873119903

sum119896=1

ℎ119896119894minus 119875119877)

+

119873PU

sum119899=1

1205823119899(

119873119903

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681)

+

119873PU

sum119899=1

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682)

(38)

where 1205821 1205822 1205823119899 and 120582



1205821ge 0 120582

2ge 0 120582

3119899ge 0 120582

4119899ge 0

119902119897119894ge 0 ℎ

119896119894ge 0 119897 = 1 119873

119904 119896 = 1 119873

119903

1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) = 0

1205822(

119873119904

sum119896=1

ℎ119896119894minus 119875119877) = 0

1205823119899(

119873119904

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681) = 0

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682) = 0

120597L

120597119902119896119894

= 0 119896 = 1 119873119904

120597L

120597ℎ119896119894

= 0 119896 = 1 119873119903

(39)



119896119894and 119902119896119894


119902119896119894=

1205902119903

2119887119896119894

[[

[

radic1198882119896119894

1205904119889

ℎ2119896119894minus

4119887119896119894119888119896119894

(1205821+ 119891119896119894119899

1205823) 12059021199031205902119889ln 2

ℎ119896119894

minus(2 +119888119896119894

1205902119889

ℎ119896119894)]]

]

+

ℎ119896119894=

1205902

119889

2119888119896119894

[[

[

radic1198872119896119894

1205904119903

1199022119896minus

4119887119896119894119888119896119894

(1205822+ 119892119896119894119899

1205824) 12059021199031205902119889ln 2

119902119896

minus(2 +119887119896119894

1205902119903

119902119896119894)]]

]

+

(40)


1 1205822 1205823119899 and 120582

4119899

through iteration

120582(119898+1)

1= [120582(119898)

1+ 120583(119898)

(

119873119904

sum119896=1

119902(119898)

119896119894minus 119875119879)]

+

120582(119898+1)

2= [120582(119898)

2+ 120583(119898)

(

119873119903

sum119896=1

ℎ(119898)

119896119894minus 119875119877)]

+

120582(119898+1)

3119899= [120582(119898)

3119899+ 120583(119898)

(

119873119903

sum119896=1

119891119896119894119899

119902(119898)

119896119894minus 1198751198681)]

+

120582(119898+1)

4119899= [120582(119898)

4119899+ 120583(119898)

(

119873119903

sum119896=1

119892119896119894119899

ℎ(119898)

119896119894minus 1198751198682)]

+

(41)


1 1205822 1205823119899 and 120582

4119899are



119894when




119896119894rsquos (119896 =

1 119873119903) that is ℎuni



119896119894= 1198751198681119873119904119891max119896119894

where 119891max119896119894

= max119899isinSPU

119891119896119894119899


119896119894=

min119902uni119894 119902max119896119894


119877 Similarly we


119896119894= 1198751198682119873119903119892max119896119894

where 119892max119896119894

= max119899isinSPU

119892119896119894119899


119896119894=

minℎuni119894 ℎmax119896119894


119877 Afterwards


119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903

(42)








119903+2(119873PU+1)119879)


5 Outage Analysis


1 PU TX





(119873PU(119873PU minus 119899))(120579120591)119899)minus1



119873119904

where I119873119904










y119889= H119904119889x119904+ n119889 (43)


119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889




119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898



120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)


(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904




119904= 119873119889= 2 [31] As such for a




119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)


119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)



y119889= H119903119889119894

x119904119894+ n119889 (49)


119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)


119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2 System Model







119901







Hpp

Hsp

Hp1HRP1

HSR1

HRPN119904119906

Hps

HRD1

HRDN119904119906

PU RXPU TX

SU RX

HS119877N119904119906

SUN119904119906

SU TX




HpN119904119906


SU1





119877 (1)


119904119894isin C119873119904times1 n




119903 that is 119899

119903119894sim CN(0 1205902

119903I119873119903


0+ sum119899isinSPU

Tr(H119901119894119899

Q119901119899H119867119901119894119899

))I119873119903

) whereH119875119894119899



119894th SU Q119875119899




sum119899isinSPU

Tr(H119901119894119899

Q119901119899H119867119901119894119899


119903119894= 1205902119903 for all 119894 isin S



y119889= H119903119889119894

A119894y119903119894+ n119889

= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889

(2)



1205902119903119894= 12059020+ sum119899isinSPU

Tr(H119901119904119899

Q119901119899H119867119901119904119899

) where H119901119904119899


119877119894=1

2log2

100381610038161003816100381610038161003816I119873119889

+H119903119889119894

A119894H119904119903119894Q119894H119867119904119903119894A119867119894H119867119903119889119894

times(1205902

119889I119873119889

+ 1205902

119903H119903119889119894

A119894A119867119894H119867119903119889119894)minus1100381610038161003816100381610038161003816

(3)

whereQ119894= 119864x



119879 that is Tr(Q

119894) le 119875

119879


119877 that is Tr[A

119894(1205902119903I119873119903

+

H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875



119904119901119899Q119894H119867119904119901119899

) le 1198751198681

andTrH119894119901119899

[A119894(1205902119903I119873119903

+ H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682 for all


119894119901119899and H



1198681+ 1198751198682 One


119894and A


formulated as


QA119894

119877119894(Q119894A119894)

119894 = arg max119894

119877119894(Qlowast119894Alowast119894)

st Tr (Q119894) le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875119877

Tr (H119904119901119899

Q119894H119867119904119901119899

) le 1198751198681

Tr H119894119901119899

[A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682

A119894ge 0 Q

119894ge 0

(4)



Φ119894≜ Q119894| Tr (Q

119894) ⩽ 119875119879

Tr (H119904119901119899

Q119894H119867119904119901119899

) le 1198751198681 Q119894ge 0

Ψ119894≜ A119894| Tr [A

119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875119877 A119894ge 0

Tr H119894119901119899

[A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682

(5)





119877119894(Q119894A119894))) (6)


119894 for all 119894 isin S


119894


119894



119894andQ









position ofH119904119903119894H119867119904119903119894

andH119867119903119889119894

H119903119889119894

be

H119904119903119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

H119867119903119889119894

H119903119889119894

= V119903119889119894Σ119903119889119894

V119867119903119889119894

(7)

where U119904119903119894

and V119903119889119894


= diag1205721 1205722

120572119873119903

with 120572119897ge 0 and Σ

119903119889119894= diag120573

1 1205732 120573

119873119903

with120573119897ge 0




A119894 opt = V

119903119889119894ΛA119894

U119867119904119903119894 (8)

where U119904119903119894


H119867119904119903119894

and H119904119903119894

= H119904119903119894Q119894


119904119903119894

H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



andH119903119889119894

be

H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

H119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894

(9)


119877119894(Q119894A119894opt)

=1

2log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Σ119904119903119894(1205902

119889I119873119889

+ 1205902

119903Λ2

119903119889119894Λ2

A119894

)minus1100381610038161003816100381610038161003816

(10)


119904119903119894but not on U

119904119903119894 Then it can be


119904119903119894Q119894H119867119904119903119894

=




A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (11)



Q119894= V119904119903119894ΛQ119894

V119867119904119903119894 (12)

where ΛQ119894




H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

= [U1199041199031198941

U1199041199031198942

] [Σ119904119903119894

0] [U1199041199031198941 U1199041199031198942

]119867

(13)


satisfies H119904119903119894Q119894H119867119904119903119894




H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

= [U1199041199031198941 U1199041199031198942] [

Λ1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

(14)

where Λ1199041199031198941


isorthogonal to U

1199041199031198942 Moreover U


1199041199031198941



H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894 we have

H+119904119903119894H119904119903119894Q119894H119867119904119903119894H+119867119904119903119894

= [V1199041199031198941 V1199041199031198942] [

Λminus11199041199031198941

0] [U1199041199031198941 U1199041199031198942]119867

times [U1199041199031198941

U1199041199031198942

] [Σ1199041199031198941

0] [U1199041199031198941 U1199041199031198942

]119867

times [U1199041199031198941 U1199041199031198942] [

Λminus1

1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

= [V1199041199031198941 V1199041199031198942]

times [Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus11199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(15)


U1199041199031198941


119904119903119894U119904119903119894


119894(A)


then119872

sum119894=1


119872

sum119894=1

120582119894 (A) 120582119894 (B) (16)


119904119903119894H+119867119904119903119894

H+119904119903119894H119904119903119894


Tr (Q119894) ge Tr (H+

119904119903119894H119904119903119894Q119894H119867119904119903119894H119867+119904119903119894)

= Tr (Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus1

1199041199031198941)

(17)


Tr (Q119894) ge Tr (Σ

1199041199031198941Λminus2

1199041199031198941) (18)


Qopt119894 = [V1199041199031198941 V1199041199031198942] [

Σ1199041199031198941

Λminus21199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(19)


which satisfies

H119904119903119894Qopt119894H

119867

119904119903119894= U119904119903119894[Σ1199041199031198941

0]U119867

119904119903119894(20)



A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894

Q119894= V119904119903119894ΛQ119894

V119867119904119903119894

(21)

whereH119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

andH119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894


y119889= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889 (22)


y119889= U119903119889119894Λ119903119889119894ΛA119894

Λ119904119903119894V119867119904119903119894x119904119894+ U119903119889119894Λ119903119889119894ΛA119894

U119867119904119903119894n119903119894+ n119889

(23)

Suppose that y119889= U119867119903119889119894

y119889 x119904119894= V119867119904119903119894x119904119894 n119903119894= U119867119904119903119894n119903119894 and

n119889= U119867119903119889119894

n119889 Then

y119889= Λ119903119889119894ΛA119894

Λ119904119903119894x119904119894+ Λ119903119889119894ΛA119894

n119903119894+ n119889 (24)


119877119894= log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Λ2

119904119903119894ΛQ119894

(1205902

119903Λ2

119903119889119894Λ2

A119894

+ 1205902

119889I119873119889

)minus1100381610038161003816100381610038161003816

(25)


H119904119901119899

andH119867119894119901119899

H119894119901119899

is

H119867119904119901119899

H119904119901119899

= U119904119901119899

Λ119904119901119899

U119867119904119901119899

H119867119894119901119899

H119894119901119899

= U119894119901119899

Λ119894119901119899

U119867119894119901119899

(26)


Λ2

A119894

= diag 1198861119894 119886

119873119903119894

Λ2

119904119903119894= diag 119887

1119894 119887

119873119903119894

Λ2

119903119889119894= diag 119888

1119894 119888

119873119903119894

Λ119904119901119899

= diag 1198891119899 119889

119873119904119899 forall119899 isin SPU

Λ119894119901119899

= diag 1198901119894119899

119890119873119903119894119899 forall119899 isin SPU

ΛQ119894

= diag 1199021119894 119902

119873119904119894

(27)


119877119894=

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

) (28)


Tr (Q119894) le 119875119879997904rArr

119873119904

sum119896=1

119902119896119894le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]

= Tr (1205902119903Λ2

A119894

+ Λ2

119904119903119894ΛQ119894

Λ2

A119894

) le 119875119877

997904rArr

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

(29)


Tr (H119904119901119899

Q119894H119867119904119901119899

) = Tr (H119867119904119901119899

H119904119901119899

Q119894)

= Tr (U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

V119867119904119903119894)

= Tr (V119867119904119903119894U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

)

(30)



Tr (M119894119899Λ119904119901119899

M119867119894ΛQ119894

) =

119873119904

sum119896=1

(

119873119904

sum119897=1

100381610038161003816100381611989811989411989611989711989910038161003816100381610038162119889119897)119902119896119894 (31)

where119898119894119896119897119899


119894119899 Let 119891

119896119894119899= sum119873119904

119897=1|119898119894119896119897119899

|2119889119897119899 Therefore the


Tr (H119904119901119899

Q119894H119867119904119901119899

) =

119873119904

sum119896=1

119891119896119894119899

119902119896119894le 1198751198681 (32)


Tr H119894119901119899

A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894H119867119894119901119899

= Tr (V119867119903119889119894

U119894119901119899

Λ119894119901119899

U119867119894119901119899

V119903119889119894ΛA119894

times (1205902

119903I119873119903

+ Λ119904119903119894ΛQ119894

Λ119904119903119894)U119904119903119894ΛA119894

) le 1198751198682

(33)


119903119889119894U119894119901119899


119894119899is denoted by 119904

119894119896119897119899 Hence


sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894(

119873119903

sum119897=1

100381610038161003816100381611990411989411989611989711989910038161003816100381610038162119890119897119894119899) le 119875

1198682 (34)


Let 119892119896119894119899

= sum119873119903

119897=1|119904119894119896119897119899

|2119890119897119894119899


119873119903

sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894119892119896119894119899

le 1198751198682 (35)




119903 119873119904) instead of119873

119903 Let a

119894=

[1198861119894 119886

119873119903119894] and q

119894= [1199021119894 119902




q119894a119894

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

)

119894 = arg max119894

119873119903

sum119896=1

log2(1 +

119886lowast

119896119894119887119896119894119888119896119894119902lowast

119896119894

1205902119903119886lowast119896119894119888119896119894+ 1205902119889

)


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899

119886119896119894(1205902


(36)




q119894h119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

ℎ119896119894le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899


(37)




L (h119894 q119894 1205821 1205822 1205823119899 1205824119899)

= minus

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

+ 1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) + 120582

2(

119873119903

sum119896=1

ℎ119896119894minus 119875119877)

+

119873PU

sum119899=1

1205823119899(

119873119903

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681)

+

119873PU

sum119899=1

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682)

(38)

where 1205821 1205822 1205823119899 and 120582



1205821ge 0 120582

2ge 0 120582

3119899ge 0 120582

4119899ge 0

119902119897119894ge 0 ℎ

119896119894ge 0 119897 = 1 119873

119904 119896 = 1 119873

119903

1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) = 0

1205822(

119873119904

sum119896=1

ℎ119896119894minus 119875119877) = 0

1205823119899(

119873119904

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681) = 0

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682) = 0

120597L

120597119902119896119894

= 0 119896 = 1 119873119904

120597L

120597ℎ119896119894

= 0 119896 = 1 119873119903

(39)



119896119894and 119902119896119894


119902119896119894=

1205902119903

2119887119896119894

[[

[

radic1198882119896119894

1205904119889

ℎ2119896119894minus

4119887119896119894119888119896119894

(1205821+ 119891119896119894119899

1205823) 12059021199031205902119889ln 2

ℎ119896119894

minus(2 +119888119896119894

1205902119889

ℎ119896119894)]]

]

+

ℎ119896119894=

1205902

119889

2119888119896119894

[[

[

radic1198872119896119894

1205904119903

1199022119896minus

4119887119896119894119888119896119894

(1205822+ 119892119896119894119899

1205824) 12059021199031205902119889ln 2

119902119896

minus(2 +119887119896119894

1205902119903

119902119896119894)]]

]

+

(40)


1 1205822 1205823119899 and 120582

4119899

through iteration

120582(119898+1)

1= [120582(119898)

1+ 120583(119898)

(

119873119904

sum119896=1

119902(119898)

119896119894minus 119875119879)]

+

120582(119898+1)

2= [120582(119898)

2+ 120583(119898)

(

119873119903

sum119896=1

ℎ(119898)

119896119894minus 119875119877)]

+

120582(119898+1)

3119899= [120582(119898)

3119899+ 120583(119898)

(

119873119903

sum119896=1

119891119896119894119899

119902(119898)

119896119894minus 1198751198681)]

+

120582(119898+1)

4119899= [120582(119898)

4119899+ 120583(119898)

(

119873119903

sum119896=1

119892119896119894119899

ℎ(119898)

119896119894minus 1198751198682)]

+

(41)


1 1205822 1205823119899 and 120582

4119899are



119894when




119896119894rsquos (119896 =

1 119873119903) that is ℎuni



119896119894= 1198751198681119873119904119891max119896119894

where 119891max119896119894

= max119899isinSPU

119891119896119894119899


119896119894=

min119902uni119894 119902max119896119894


119877 Similarly we


119896119894= 1198751198682119873119903119892max119896119894

where 119892max119896119894

= max119899isinSPU

119892119896119894119899


119896119894=

minℎuni119894 ℎmax119896119894


119877 Afterwards


119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903

(42)








119903+2(119873PU+1)119879)


5 Outage Analysis


1 PU TX





(119873PU(119873PU minus 119899))(120579120591)119899)minus1



119873119904

where I119873119904










y119889= H119904119889x119904+ n119889 (43)


119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889




119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898



120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)


(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904




119904= 119873119889= 2 [31] As such for a




119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)


119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)



y119889= H119903119889119894

x119904119894+ n119889 (49)


119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)


119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119894th SU Q119875119899




sum119899isinSPU

Tr(H119901119894119899

Q119901119899H119867119901119894119899


119903119894= 1205902119903 for all 119894 isin S



y119889= H119903119889119894

A119894y119903119894+ n119889

= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889

(2)



1205902119903119894= 12059020+ sum119899isinSPU

Tr(H119901119904119899

Q119901119899H119867119901119904119899

) where H119901119904119899


119877119894=1

2log2

100381610038161003816100381610038161003816I119873119889

+H119903119889119894

A119894H119904119903119894Q119894H119867119904119903119894A119867119894H119867119903119889119894

times(1205902

119889I119873119889

+ 1205902

119903H119903119889119894

A119894A119867119894H119867119903119889119894)minus1100381610038161003816100381610038161003816

(3)

whereQ119894= 119864x



119879 that is Tr(Q

119894) le 119875

119879


119877 that is Tr[A

119894(1205902119903I119873119903

+

H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875



119904119901119899Q119894H119867119904119901119899

) le 1198751198681

andTrH119894119901119899

[A119894(1205902119903I119873119903

+ H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682 for all


119894119901119899and H



1198681+ 1198751198682 One


119894and A


formulated as


QA119894

119877119894(Q119894A119894)

119894 = arg max119894

119877119894(Qlowast119894Alowast119894)

st Tr (Q119894) le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875119877

Tr (H119904119901119899

Q119894H119867119904119901119899

) le 1198751198681

Tr H119894119901119899

[A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682

A119894ge 0 Q

119894ge 0

(4)



Φ119894≜ Q119894| Tr (Q

119894) ⩽ 119875119879

Tr (H119904119901119899

Q119894H119867119904119901119899

) le 1198751198681 Q119894ge 0

Ψ119894≜ A119894| Tr [A

119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894] le 119875119877 A119894ge 0

Tr H119894119901119899

[A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]H119867119894119901119899

le 1198751198682

(5)





119877119894(Q119894A119894))) (6)


119894 for all 119894 isin S


119894


119894



119894andQ









position ofH119904119903119894H119867119904119903119894

andH119867119903119889119894

H119903119889119894

be

H119904119903119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

H119867119903119889119894

H119903119889119894

= V119903119889119894Σ119903119889119894

V119867119903119889119894

(7)

where U119904119903119894

and V119903119889119894


= diag1205721 1205722

120572119873119903

with 120572119897ge 0 and Σ

119903119889119894= diag120573

1 1205732 120573

119873119903

with120573119897ge 0




A119894 opt = V

119903119889119894ΛA119894

U119867119904119903119894 (8)

where U119904119903119894


H119867119904119903119894

and H119904119903119894

= H119904119903119894Q119894


119904119903119894

H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



andH119903119889119894

be

H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

H119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894

(9)


119877119894(Q119894A119894opt)

=1

2log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Σ119904119903119894(1205902

119889I119873119889

+ 1205902

119903Λ2

119903119889119894Λ2

A119894

)minus1100381610038161003816100381610038161003816

(10)


119904119903119894but not on U

119904119903119894 Then it can be


119904119903119894Q119894H119867119904119903119894

=




A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (11)



Q119894= V119904119903119894ΛQ119894

V119867119904119903119894 (12)

where ΛQ119894




H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

= [U1199041199031198941

U1199041199031198942

] [Σ119904119903119894

0] [U1199041199031198941 U1199041199031198942

]119867

(13)


satisfies H119904119903119894Q119894H119867119904119903119894




H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

= [U1199041199031198941 U1199041199031198942] [

Λ1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

(14)

where Λ1199041199031198941


isorthogonal to U

1199041199031198942 Moreover U


1199041199031198941



H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894 we have

H+119904119903119894H119904119903119894Q119894H119867119904119903119894H+119867119904119903119894

= [V1199041199031198941 V1199041199031198942] [

Λminus11199041199031198941

0] [U1199041199031198941 U1199041199031198942]119867

times [U1199041199031198941

U1199041199031198942

] [Σ1199041199031198941

0] [U1199041199031198941 U1199041199031198942

]119867

times [U1199041199031198941 U1199041199031198942] [

Λminus1

1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

= [V1199041199031198941 V1199041199031198942]

times [Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus11199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(15)


U1199041199031198941


119904119903119894U119904119903119894


119894(A)


then119872

sum119894=1


119872

sum119894=1

120582119894 (A) 120582119894 (B) (16)


119904119903119894H+119867119904119903119894

H+119904119903119894H119904119903119894


Tr (Q119894) ge Tr (H+

119904119903119894H119904119903119894Q119894H119867119904119903119894H119867+119904119903119894)

= Tr (Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus1

1199041199031198941)

(17)


Tr (Q119894) ge Tr (Σ

1199041199031198941Λminus2

1199041199031198941) (18)


Qopt119894 = [V1199041199031198941 V1199041199031198942] [

Σ1199041199031198941

Λminus21199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(19)


which satisfies

H119904119903119894Qopt119894H

119867

119904119903119894= U119904119903119894[Σ1199041199031198941

0]U119867

119904119903119894(20)



A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894

Q119894= V119904119903119894ΛQ119894

V119867119904119903119894

(21)

whereH119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

andH119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894


y119889= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889 (22)


y119889= U119903119889119894Λ119903119889119894ΛA119894

Λ119904119903119894V119867119904119903119894x119904119894+ U119903119889119894Λ119903119889119894ΛA119894

U119867119904119903119894n119903119894+ n119889

(23)

Suppose that y119889= U119867119903119889119894

y119889 x119904119894= V119867119904119903119894x119904119894 n119903119894= U119867119904119903119894n119903119894 and

n119889= U119867119903119889119894

n119889 Then

y119889= Λ119903119889119894ΛA119894

Λ119904119903119894x119904119894+ Λ119903119889119894ΛA119894

n119903119894+ n119889 (24)


119877119894= log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Λ2

119904119903119894ΛQ119894

(1205902

119903Λ2

119903119889119894Λ2

A119894

+ 1205902

119889I119873119889

)minus1100381610038161003816100381610038161003816

(25)


H119904119901119899

andH119867119894119901119899

H119894119901119899

is

H119867119904119901119899

H119904119901119899

= U119904119901119899

Λ119904119901119899

U119867119904119901119899

H119867119894119901119899

H119894119901119899

= U119894119901119899

Λ119894119901119899

U119867119894119901119899

(26)


Λ2

A119894

= diag 1198861119894 119886

119873119903119894

Λ2

119904119903119894= diag 119887

1119894 119887

119873119903119894

Λ2

119903119889119894= diag 119888

1119894 119888

119873119903119894

Λ119904119901119899

= diag 1198891119899 119889

119873119904119899 forall119899 isin SPU

Λ119894119901119899

= diag 1198901119894119899

119890119873119903119894119899 forall119899 isin SPU

ΛQ119894

= diag 1199021119894 119902

119873119904119894

(27)


119877119894=

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

) (28)


Tr (Q119894) le 119875119879997904rArr

119873119904

sum119896=1

119902119896119894le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]

= Tr (1205902119903Λ2

A119894

+ Λ2

119904119903119894ΛQ119894

Λ2

A119894

) le 119875119877

997904rArr

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

(29)


Tr (H119904119901119899

Q119894H119867119904119901119899

) = Tr (H119867119904119901119899

H119904119901119899

Q119894)

= Tr (U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

V119867119904119903119894)

= Tr (V119867119904119903119894U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

)

(30)



Tr (M119894119899Λ119904119901119899

M119867119894ΛQ119894

) =

119873119904

sum119896=1

(

119873119904

sum119897=1

100381610038161003816100381611989811989411989611989711989910038161003816100381610038162119889119897)119902119896119894 (31)

where119898119894119896119897119899


119894119899 Let 119891

119896119894119899= sum119873119904

119897=1|119898119894119896119897119899

|2119889119897119899 Therefore the


Tr (H119904119901119899

Q119894H119867119904119901119899

) =

119873119904

sum119896=1

119891119896119894119899

119902119896119894le 1198751198681 (32)


Tr H119894119901119899

A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894H119867119894119901119899

= Tr (V119867119903119889119894

U119894119901119899

Λ119894119901119899

U119867119894119901119899

V119903119889119894ΛA119894

times (1205902

119903I119873119903

+ Λ119904119903119894ΛQ119894

Λ119904119903119894)U119904119903119894ΛA119894

) le 1198751198682

(33)


119903119889119894U119894119901119899


119894119899is denoted by 119904

119894119896119897119899 Hence


sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894(

119873119903

sum119897=1

100381610038161003816100381611990411989411989611989711989910038161003816100381610038162119890119897119894119899) le 119875

1198682 (34)


Let 119892119896119894119899

= sum119873119903

119897=1|119904119894119896119897119899

|2119890119897119894119899


119873119903

sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894119892119896119894119899

le 1198751198682 (35)




119903 119873119904) instead of119873

119903 Let a

119894=

[1198861119894 119886

119873119903119894] and q

119894= [1199021119894 119902




q119894a119894

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

)

119894 = arg max119894

119873119903

sum119896=1

log2(1 +

119886lowast

119896119894119887119896119894119888119896119894119902lowast

119896119894

1205902119903119886lowast119896119894119888119896119894+ 1205902119889

)


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899

119886119896119894(1205902


(36)




q119894h119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

ℎ119896119894le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899


(37)




L (h119894 q119894 1205821 1205822 1205823119899 1205824119899)

= minus

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

+ 1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) + 120582

2(

119873119903

sum119896=1

ℎ119896119894minus 119875119877)

+

119873PU

sum119899=1

1205823119899(

119873119903

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681)

+

119873PU

sum119899=1

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682)

(38)

where 1205821 1205822 1205823119899 and 120582



1205821ge 0 120582

2ge 0 120582

3119899ge 0 120582

4119899ge 0

119902119897119894ge 0 ℎ

119896119894ge 0 119897 = 1 119873

119904 119896 = 1 119873

119903

1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) = 0

1205822(

119873119904

sum119896=1

ℎ119896119894minus 119875119877) = 0

1205823119899(

119873119904

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681) = 0

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682) = 0

120597L

120597119902119896119894

= 0 119896 = 1 119873119904

120597L

120597ℎ119896119894

= 0 119896 = 1 119873119903

(39)



119896119894and 119902119896119894


119902119896119894=

1205902119903

2119887119896119894

[[

[

radic1198882119896119894

1205904119889

ℎ2119896119894minus

4119887119896119894119888119896119894

(1205821+ 119891119896119894119899

1205823) 12059021199031205902119889ln 2

ℎ119896119894

minus(2 +119888119896119894

1205902119889

ℎ119896119894)]]

]

+

ℎ119896119894=

1205902

119889

2119888119896119894

[[

[

radic1198872119896119894

1205904119903

1199022119896minus

4119887119896119894119888119896119894

(1205822+ 119892119896119894119899

1205824) 12059021199031205902119889ln 2

119902119896

minus(2 +119887119896119894

1205902119903

119902119896119894)]]

]

+

(40)


1 1205822 1205823119899 and 120582

4119899

through iteration

120582(119898+1)

1= [120582(119898)

1+ 120583(119898)

(

119873119904

sum119896=1

119902(119898)

119896119894minus 119875119879)]

+

120582(119898+1)

2= [120582(119898)

2+ 120583(119898)

(

119873119903

sum119896=1

ℎ(119898)

119896119894minus 119875119877)]

+

120582(119898+1)

3119899= [120582(119898)

3119899+ 120583(119898)

(

119873119903

sum119896=1

119891119896119894119899

119902(119898)

119896119894minus 1198751198681)]

+

120582(119898+1)

4119899= [120582(119898)

4119899+ 120583(119898)

(

119873119903

sum119896=1

119892119896119894119899

ℎ(119898)

119896119894minus 1198751198682)]

+

(41)


1 1205822 1205823119899 and 120582

4119899are



119894when




119896119894rsquos (119896 =

1 119873119903) that is ℎuni



119896119894= 1198751198681119873119904119891max119896119894

where 119891max119896119894

= max119899isinSPU

119891119896119894119899


119896119894=

min119902uni119894 119902max119896119894


119877 Similarly we


119896119894= 1198751198682119873119903119892max119896119894

where 119892max119896119894

= max119899isinSPU

119892119896119894119899


119896119894=

minℎuni119894 ℎmax119896119894


119877 Afterwards


119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903

(42)








119903+2(119873PU+1)119879)


5 Outage Analysis


1 PU TX





(119873PU(119873PU minus 119899))(120579120591)119899)minus1



119873119904

where I119873119904










y119889= H119904119889x119904+ n119889 (43)


119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889




119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898



120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)


(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904




119904= 119873119889= 2 [31] As such for a




119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)


119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)



y119889= H119903119889119894

x119904119894+ n119889 (49)


119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)


119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












A119894 opt = V

119903119889119894ΛA119894

U119867119904119903119894 (8)

where U119904119903119894


H119867119904119903119894

and H119904119903119894

= H119904119903119894Q119894


119904119903119894

H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



andH119903119889119894

be

H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

H119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894

(9)


119877119894(Q119894A119894opt)

=1

2log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Σ119904119903119894(1205902

119889I119873119889

+ 1205902

119903Λ2

119903119889119894Λ2

A119894

)minus1100381610038161003816100381610038161003816

(10)


119904119903119894but not on U

119904119903119894 Then it can be


119904119903119894Q119894H119867119904119903119894

=




A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (11)



Q119894= V119904119903119894ΛQ119894

V119867119904119903119894 (12)

where ΛQ119894




H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894

= [U1199041199031198941

U1199041199031198942

] [Σ119904119903119894

0] [U1199041199031198941 U1199041199031198942

]119867

(13)


satisfies H119904119903119894Q119894H119867119904119903119894




H119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

= [U1199041199031198941 U1199041199031198942] [

Λ1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

(14)

where Λ1199041199031198941


isorthogonal to U

1199041199031198942 Moreover U


1199041199031198941



H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894 we have

H+119904119903119894H119904119903119894Q119894H119867119904119903119894H+119867119904119903119894

= [V1199041199031198941 V1199041199031198942] [

Λminus11199041199031198941

0] [U1199041199031198941 U1199041199031198942]119867

times [U1199041199031198941

U1199041199031198942

] [Σ1199041199031198941

0] [U1199041199031198941 U1199041199031198942

]119867

times [U1199041199031198941 U1199041199031198942] [

Λminus1

1199041199031198941

0] [V1199041199031198941 V1199041199031198942]119867

= [V1199041199031198941 V1199041199031198942]

times [Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus11199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(15)


U1199041199031198941


119904119903119894U119904119903119894


119894(A)


then119872

sum119894=1


119872

sum119894=1

120582119894 (A) 120582119894 (B) (16)


119904119903119894H+119867119904119903119894

H+119904119903119894H119904119903119894


Tr (Q119894) ge Tr (H+

119904119903119894H119904119903119894Q119894H119867119904119903119894H119867+119904119903119894)

= Tr (Λminus11199041199031198941

U1198671199041199031198941

U1199041199031198942

Σ1199041199031198941

U1199041199031198941

U1199041199031198941

Λminus1

1199041199031198941)

(17)


Tr (Q119894) ge Tr (Σ

1199041199031198941Λminus2

1199041199031198941) (18)


Qopt119894 = [V1199041199031198941 V1199041199031198942] [

Σ1199041199031198941

Λminus21199041199031198941

0]

times [V1199041199031198941 V1199041199031198942]119867

(19)


which satisfies

H119904119903119894Qopt119894H

119867

119904119903119894= U119904119903119894[Σ1199041199031198941

0]U119867

119904119903119894(20)



A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894

Q119894= V119904119903119894ΛQ119894

V119867119904119903119894

(21)

whereH119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

andH119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894


y119889= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889 (22)


y119889= U119903119889119894Λ119903119889119894ΛA119894

Λ119904119903119894V119867119904119903119894x119904119894+ U119903119889119894Λ119903119889119894ΛA119894

U119867119904119903119894n119903119894+ n119889

(23)

Suppose that y119889= U119867119903119889119894

y119889 x119904119894= V119867119904119903119894x119904119894 n119903119894= U119867119904119903119894n119903119894 and

n119889= U119867119903119889119894

n119889 Then

y119889= Λ119903119889119894ΛA119894

Λ119904119903119894x119904119894+ Λ119903119889119894ΛA119894

n119903119894+ n119889 (24)


119877119894= log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Λ2

119904119903119894ΛQ119894

(1205902

119903Λ2

119903119889119894Λ2

A119894

+ 1205902

119889I119873119889

)minus1100381610038161003816100381610038161003816

(25)


H119904119901119899

andH119867119894119901119899

H119894119901119899

is

H119867119904119901119899

H119904119901119899

= U119904119901119899

Λ119904119901119899

U119867119904119901119899

H119867119894119901119899

H119894119901119899

= U119894119901119899

Λ119894119901119899

U119867119894119901119899

(26)


Λ2

A119894

= diag 1198861119894 119886

119873119903119894

Λ2

119904119903119894= diag 119887

1119894 119887

119873119903119894

Λ2

119903119889119894= diag 119888

1119894 119888

119873119903119894

Λ119904119901119899

= diag 1198891119899 119889

119873119904119899 forall119899 isin SPU

Λ119894119901119899

= diag 1198901119894119899

119890119873119903119894119899 forall119899 isin SPU

ΛQ119894

= diag 1199021119894 119902

119873119904119894

(27)


119877119894=

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

) (28)


Tr (Q119894) le 119875119879997904rArr

119873119904

sum119896=1

119902119896119894le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]

= Tr (1205902119903Λ2

A119894

+ Λ2

119904119903119894ΛQ119894

Λ2

A119894

) le 119875119877

997904rArr

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

(29)


Tr (H119904119901119899

Q119894H119867119904119901119899

) = Tr (H119867119904119901119899

H119904119901119899

Q119894)

= Tr (U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

V119867119904119903119894)

= Tr (V119867119904119903119894U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

)

(30)



Tr (M119894119899Λ119904119901119899

M119867119894ΛQ119894

) =

119873119904

sum119896=1

(

119873119904

sum119897=1

100381610038161003816100381611989811989411989611989711989910038161003816100381610038162119889119897)119902119896119894 (31)

where119898119894119896119897119899


119894119899 Let 119891

119896119894119899= sum119873119904

119897=1|119898119894119896119897119899

|2119889119897119899 Therefore the


Tr (H119904119901119899

Q119894H119867119904119901119899

) =

119873119904

sum119896=1

119891119896119894119899

119902119896119894le 1198751198681 (32)


Tr H119894119901119899

A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894H119867119894119901119899

= Tr (V119867119903119889119894

U119894119901119899

Λ119894119901119899

U119867119894119901119899

V119903119889119894ΛA119894

times (1205902

119903I119873119903

+ Λ119904119903119894ΛQ119894

Λ119904119903119894)U119904119903119894ΛA119894

) le 1198751198682

(33)


119903119889119894U119894119901119899


119894119899is denoted by 119904

119894119896119897119899 Hence


sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894(

119873119903

sum119897=1

100381610038161003816100381611990411989411989611989711989910038161003816100381610038162119890119897119894119899) le 119875

1198682 (34)


Let 119892119896119894119899

= sum119873119903

119897=1|119904119894119896119897119899

|2119890119897119894119899


119873119903

sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894119892119896119894119899

le 1198751198682 (35)




119903 119873119904) instead of119873

119903 Let a

119894=

[1198861119894 119886

119873119903119894] and q

119894= [1199021119894 119902




q119894a119894

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

)

119894 = arg max119894

119873119903

sum119896=1

log2(1 +

119886lowast

119896119894119887119896119894119888119896119894119902lowast

119896119894

1205902119903119886lowast119896119894119888119896119894+ 1205902119889

)


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899

119886119896119894(1205902


(36)




q119894h119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

ℎ119896119894le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899


(37)




L (h119894 q119894 1205821 1205822 1205823119899 1205824119899)

= minus

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

+ 1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) + 120582

2(

119873119903

sum119896=1

ℎ119896119894minus 119875119877)

+

119873PU

sum119899=1

1205823119899(

119873119903

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681)

+

119873PU

sum119899=1

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682)

(38)

where 1205821 1205822 1205823119899 and 120582



1205821ge 0 120582

2ge 0 120582

3119899ge 0 120582

4119899ge 0

119902119897119894ge 0 ℎ

119896119894ge 0 119897 = 1 119873

119904 119896 = 1 119873

119903

1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) = 0

1205822(

119873119904

sum119896=1

ℎ119896119894minus 119875119877) = 0

1205823119899(

119873119904

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681) = 0

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682) = 0

120597L

120597119902119896119894

= 0 119896 = 1 119873119904

120597L

120597ℎ119896119894

= 0 119896 = 1 119873119903

(39)



119896119894and 119902119896119894


119902119896119894=

1205902119903

2119887119896119894

[[

[

radic1198882119896119894

1205904119889

ℎ2119896119894minus

4119887119896119894119888119896119894

(1205821+ 119891119896119894119899

1205823) 12059021199031205902119889ln 2

ℎ119896119894

minus(2 +119888119896119894

1205902119889

ℎ119896119894)]]

]

+

ℎ119896119894=

1205902

119889

2119888119896119894

[[

[

radic1198872119896119894

1205904119903

1199022119896minus

4119887119896119894119888119896119894

(1205822+ 119892119896119894119899

1205824) 12059021199031205902119889ln 2

119902119896

minus(2 +119887119896119894

1205902119903

119902119896119894)]]

]

+

(40)


1 1205822 1205823119899 and 120582

4119899

through iteration

120582(119898+1)

1= [120582(119898)

1+ 120583(119898)

(

119873119904

sum119896=1

119902(119898)

119896119894minus 119875119879)]

+

120582(119898+1)

2= [120582(119898)

2+ 120583(119898)

(

119873119903

sum119896=1

ℎ(119898)

119896119894minus 119875119877)]

+

120582(119898+1)

3119899= [120582(119898)

3119899+ 120583(119898)

(

119873119903

sum119896=1

119891119896119894119899

119902(119898)

119896119894minus 1198751198681)]

+

120582(119898+1)

4119899= [120582(119898)

4119899+ 120583(119898)

(

119873119903

sum119896=1

119892119896119894119899

ℎ(119898)

119896119894minus 1198751198682)]

+

(41)


1 1205822 1205823119899 and 120582

4119899are



119894when




119896119894rsquos (119896 =

1 119873119903) that is ℎuni



119896119894= 1198751198681119873119904119891max119896119894

where 119891max119896119894

= max119899isinSPU

119891119896119894119899


119896119894=

min119902uni119894 119902max119896119894


119877 Similarly we


119896119894= 1198751198682119873119903119892max119896119894

where 119892max119896119894

= max119899isinSPU

119892119896119894119899


119896119894=

minℎuni119894 ℎmax119896119894


119877 Afterwards


119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903

(42)








119903+2(119873PU+1)119879)


5 Outage Analysis


1 PU TX





(119873PU(119873PU minus 119899))(120579120591)119899)minus1



119873119904

where I119873119904










y119889= H119904119889x119904+ n119889 (43)


119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889




119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898



120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)


(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904




119904= 119873119889= 2 [31] As such for a




119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)


119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)



y119889= H119903119889119894

x119904119894+ n119889 (49)


119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)


119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










which satisfies

H119904119903119894Qopt119894H

119867

119904119903119894= U119904119903119894[Σ1199041199031198941

0]U119867

119904119903119894(20)



A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894

Q119894= V119904119903119894ΛQ119894

V119867119904119903119894

(21)

whereH119904119903119894

= U119904119903119894Λ119904119903119894V119867119904119903119894

andH119903119889119894

= U119903119889119894Λ119903119889119894

V119867119903119889119894


y119889= H119903119889119894

A119894H119904119903119894x119904119894+H119903119889119894

A119894n119903119894+ n119889 (22)


y119889= U119903119889119894Λ119903119889119894ΛA119894

Λ119904119903119894V119867119904119903119894x119904119894+ U119903119889119894Λ119903119889119894ΛA119894

U119867119904119903119894n119903119894+ n119889

(23)

Suppose that y119889= U119867119903119889119894

y119889 x119904119894= V119867119904119903119894x119904119894 n119903119894= U119867119904119903119894n119903119894 and

n119889= U119867119903119889119894

n119889 Then

y119889= Λ119903119889119894ΛA119894

Λ119904119903119894x119904119894+ Λ119903119889119894ΛA119894

n119903119894+ n119889 (24)


119877119894= log2

100381610038161003816100381610038161003816I119873119889

+ Λ2

119903119889119894Λ2

A119894

Λ2

119904119903119894ΛQ119894

(1205902

119903Λ2

119903119889119894Λ2

A119894

+ 1205902

119889I119873119889

)minus1100381610038161003816100381610038161003816

(25)


H119904119901119899

andH119867119894119901119899

H119894119901119899

is

H119867119904119901119899

H119904119901119899

= U119904119901119899

Λ119904119901119899

U119867119904119901119899

H119867119894119901119899

H119894119901119899

= U119894119901119899

Λ119894119901119899

U119867119894119901119899

(26)


Λ2

A119894

= diag 1198861119894 119886

119873119903119894

Λ2

119904119903119894= diag 119887

1119894 119887

119873119903119894

Λ2

119903119889119894= diag 119888

1119894 119888

119873119903119894

Λ119904119901119899

= diag 1198891119899 119889

119873119904119899 forall119899 isin SPU

Λ119894119901119899

= diag 1198901119894119899

119890119873119903119894119899 forall119899 isin SPU

ΛQ119894

= diag 1199021119894 119902

119873119904119894

(27)


119877119894=

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

) (28)


Tr (Q119894) le 119875119879997904rArr

119873119904

sum119896=1

119902119896119894le 119875119879

Tr [A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894]

= Tr (1205902119903Λ2

A119894

+ Λ2

119904119903119894ΛQ119894

Λ2

A119894

) le 119875119877

997904rArr

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

(29)


Tr (H119904119901119899

Q119894H119867119904119901119899

) = Tr (H119867119904119901119899

H119904119901119899

Q119894)

= Tr (U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

V119867119904119903119894)

= Tr (V119867119904119903119894U119904119901119899

Λ119904119901119899

U119867119904119901119899

V119904119903119894ΛQ119894

)

(30)



Tr (M119894119899Λ119904119901119899

M119867119894ΛQ119894

) =

119873119904

sum119896=1

(

119873119904

sum119897=1

100381610038161003816100381611989811989411989611989711989910038161003816100381610038162119889119897)119902119896119894 (31)

where119898119894119896119897119899


119894119899 Let 119891

119896119894119899= sum119873119904

119897=1|119898119894119896119897119899

|2119889119897119899 Therefore the


Tr (H119904119901119899

Q119894H119867119904119901119899

) =

119873119904

sum119896=1

119891119896119894119899

119902119896119894le 1198751198681 (32)


Tr H119894119901119899

A119894(1205902

119903I119873119903

+H119904119903119894Q119894H119867119904119903119894)A119867119894H119867119894119901119899

= Tr (V119867119903119889119894

U119894119901119899

Λ119894119901119899

U119867119894119901119899

V119903119889119894ΛA119894

times (1205902

119903I119873119903

+ Λ119904119903119894ΛQ119894

Λ119904119903119894)U119904119903119894ΛA119894

) le 1198751198682

(33)


119903119889119894U119894119901119899


119894119899is denoted by 119904

119894119896119897119899 Hence


sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894(

119873119903

sum119897=1

100381610038161003816100381611990411989411989611989711989910038161003816100381610038162119890119897119894119899) le 119875

1198682 (34)


Let 119892119896119894119899

= sum119873119903

119897=1|119904119894119896119897119899

|2119890119897119894119899


119873119903

sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894119892119896119894119899

le 1198751198682 (35)




119903 119873119904) instead of119873

119903 Let a

119894=

[1198861119894 119886

119873119903119894] and q

119894= [1199021119894 119902




q119894a119894

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

)

119894 = arg max119894

119873119903

sum119896=1

log2(1 +

119886lowast

119896119894119887119896119894119888119896119894119902lowast

119896119894

1205902119903119886lowast119896119894119888119896119894+ 1205902119889

)


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899

119886119896119894(1205902


(36)




q119894h119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

ℎ119896119894le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899


(37)




L (h119894 q119894 1205821 1205822 1205823119899 1205824119899)

= minus

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

+ 1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) + 120582

2(

119873119903

sum119896=1

ℎ119896119894minus 119875119877)

+

119873PU

sum119899=1

1205823119899(

119873119903

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681)

+

119873PU

sum119899=1

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682)

(38)

where 1205821 1205822 1205823119899 and 120582



1205821ge 0 120582

2ge 0 120582

3119899ge 0 120582

4119899ge 0

119902119897119894ge 0 ℎ

119896119894ge 0 119897 = 1 119873

119904 119896 = 1 119873

119903

1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) = 0

1205822(

119873119904

sum119896=1

ℎ119896119894minus 119875119877) = 0

1205823119899(

119873119904

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681) = 0

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682) = 0

120597L

120597119902119896119894

= 0 119896 = 1 119873119904

120597L

120597ℎ119896119894

= 0 119896 = 1 119873119903

(39)



119896119894and 119902119896119894


119902119896119894=

1205902119903

2119887119896119894

[[

[

radic1198882119896119894

1205904119889

ℎ2119896119894minus

4119887119896119894119888119896119894

(1205821+ 119891119896119894119899

1205823) 12059021199031205902119889ln 2

ℎ119896119894

minus(2 +119888119896119894

1205902119889

ℎ119896119894)]]

]

+

ℎ119896119894=

1205902

119889

2119888119896119894

[[

[

radic1198872119896119894

1205904119903

1199022119896minus

4119887119896119894119888119896119894

(1205822+ 119892119896119894119899

1205824) 12059021199031205902119889ln 2

119902119896

minus(2 +119887119896119894

1205902119903

119902119896119894)]]

]

+

(40)


1 1205822 1205823119899 and 120582

4119899

through iteration

120582(119898+1)

1= [120582(119898)

1+ 120583(119898)

(

119873119904

sum119896=1

119902(119898)

119896119894minus 119875119879)]

+

120582(119898+1)

2= [120582(119898)

2+ 120583(119898)

(

119873119903

sum119896=1

ℎ(119898)

119896119894minus 119875119877)]

+

120582(119898+1)

3119899= [120582(119898)

3119899+ 120583(119898)

(

119873119903

sum119896=1

119891119896119894119899

119902(119898)

119896119894minus 1198751198681)]

+

120582(119898+1)

4119899= [120582(119898)

4119899+ 120583(119898)

(

119873119903

sum119896=1

119892119896119894119899

ℎ(119898)

119896119894minus 1198751198682)]

+

(41)


1 1205822 1205823119899 and 120582

4119899are



119894when




119896119894rsquos (119896 =

1 119873119903) that is ℎuni



119896119894= 1198751198681119873119904119891max119896119894

where 119891max119896119894

= max119899isinSPU

119891119896119894119899


119896119894=

min119902uni119894 119902max119896119894


119877 Similarly we


119896119894= 1198751198682119873119903119892max119896119894

where 119892max119896119894

= max119899isinSPU

119892119896119894119899


119896119894=

minℎuni119894 ℎmax119896119894


119877 Afterwards


119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903

(42)








119903+2(119873PU+1)119879)


5 Outage Analysis


1 PU TX





(119873PU(119873PU minus 119899))(120579120591)119899)minus1



119873119904

where I119873119904










y119889= H119904119889x119904+ n119889 (43)


119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889




119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898



120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)


(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904




119904= 119873119889= 2 [31] As such for a




119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)


119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)



y119889= H119903119889119894

x119904119894+ n119889 (49)


119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)


119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Let 119892119896119894119899

= sum119873119903

119897=1|119904119894119896119897119899

|2119890119897119894119899


119873119903

sum119896=1

(1205902

119903+ 119887119896119894119902119896119894) 119886119896119894119892119896119894119899

le 1198751198682 (35)




119903 119873119904) instead of119873

119903 Let a

119894=

[1198861119894 119886

119873119903119894] and q

119894= [1199021119894 119902




q119894a119894

119873119903

sum119896=1

log2(1 +

119886119896119894119887119896119894119888119896119894119902119896119894

1205902119903119886119896119894119888119896119894+ 1205902119889

)

119894 = arg max119894

119873119903

sum119896=1

log2(1 +

119886lowast

119896119894119887119896119894119888119896119894119902lowast

119896119894

1205902119903119886lowast119896119894119888119896119894+ 1205902119889

)


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

119886119896119894(1205902

119903+ 119887119896119894119902119896119894) le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899

119886119896119894(1205902


(36)




q119894h119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903


sum119896=1

119902119896119894le 119875119879

119873119903

sum119896=1

ℎ119896119894le 119875119877

119873119904

sum119896=1

119891119896119894119899


119873119903

sum119896=1

119892119896119894119899


(37)




L (h119894 q119894 1205821 1205822 1205823119899 1205824119899)

= minus

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎ1198961198941205902119889) (1 + 119887

1198961198941199021198961198941205902119903)

1 + 119888119896119894ℎ1198961198941205902119889+ 1198871198961198941199021198961198941205902119903

+ 1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) + 120582

2(

119873119903

sum119896=1

ℎ119896119894minus 119875119877)

+

119873PU

sum119899=1

1205823119899(

119873119903

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681)

+

119873PU

sum119899=1

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682)

(38)

where 1205821 1205822 1205823119899 and 120582



1205821ge 0 120582

2ge 0 120582

3119899ge 0 120582

4119899ge 0

119902119897119894ge 0 ℎ

119896119894ge 0 119897 = 1 119873

119904 119896 = 1 119873

119903

1205821(

119873119904

sum119896=1

119902119896119894minus 119875119879) = 0

1205822(

119873119904

sum119896=1

ℎ119896119894minus 119875119877) = 0

1205823119899(

119873119904

sum119896=1

119891119896119894119899

119902119896119894minus 1198751198681) = 0

1205824119899(

119873119903

sum119896=1

119892119896119894119899

ℎ119896119894minus 1198751198682) = 0

120597L

120597119902119896119894

= 0 119896 = 1 119873119904

120597L

120597ℎ119896119894

= 0 119896 = 1 119873119903

(39)



119896119894and 119902119896119894


119902119896119894=

1205902119903

2119887119896119894

[[

[

radic1198882119896119894

1205904119889

ℎ2119896119894minus

4119887119896119894119888119896119894

(1205821+ 119891119896119894119899

1205823) 12059021199031205902119889ln 2

ℎ119896119894

minus(2 +119888119896119894

1205902119889

ℎ119896119894)]]

]

+

ℎ119896119894=

1205902

119889

2119888119896119894

[[

[

radic1198872119896119894

1205904119903

1199022119896minus

4119887119896119894119888119896119894

(1205822+ 119892119896119894119899

1205824) 12059021199031205902119889ln 2

119902119896

minus(2 +119887119896119894

1205902119903

119902119896119894)]]

]

+

(40)


1 1205822 1205823119899 and 120582

4119899

through iteration

120582(119898+1)

1= [120582(119898)

1+ 120583(119898)

(

119873119904

sum119896=1

119902(119898)

119896119894minus 119875119879)]

+

120582(119898+1)

2= [120582(119898)

2+ 120583(119898)

(

119873119903

sum119896=1

ℎ(119898)

119896119894minus 119875119877)]

+

120582(119898+1)

3119899= [120582(119898)

3119899+ 120583(119898)

(

119873119903

sum119896=1

119891119896119894119899

119902(119898)

119896119894minus 1198751198681)]

+

120582(119898+1)

4119899= [120582(119898)

4119899+ 120583(119898)

(

119873119903

sum119896=1

119892119896119894119899

ℎ(119898)

119896119894minus 1198751198682)]

+

(41)


1 1205822 1205823119899 and 120582

4119899are



119894when




119896119894rsquos (119896 =

1 119873119903) that is ℎuni



119896119894= 1198751198681119873119904119891max119896119894

where 119891max119896119894

= max119899isinSPU

119891119896119894119899


119896119894=

min119902uni119894 119902max119896119894


119877 Similarly we


119896119894= 1198751198682119873119903119892max119896119894

where 119892max119896119894

= max119899isinSPU

119892119896119894119899


119896119894=

minℎuni119894 ℎmax119896119894


119877 Afterwards


119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903

(42)








119903+2(119873PU+1)119879)


5 Outage Analysis


1 PU TX





(119873PU(119873PU minus 119899))(120579120591)119899)minus1



119873119904

where I119873119904










y119889= H119904119889x119904+ n119889 (43)


119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889




119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898



120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)


(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904




119904= 119873119889= 2 [31] As such for a




119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)


119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)



y119889= H119903119889119894

x119904119894+ n119889 (49)


119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)


119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896119894and 119902119896119894


119902119896119894=

1205902119903

2119887119896119894

[[

[

radic1198882119896119894

1205904119889

ℎ2119896119894minus

4119887119896119894119888119896119894

(1205821+ 119891119896119894119899

1205823) 12059021199031205902119889ln 2

ℎ119896119894

minus(2 +119888119896119894

1205902119889

ℎ119896119894)]]

]

+

ℎ119896119894=

1205902

119889

2119888119896119894

[[

[

radic1198872119896119894

1205904119903

1199022119896minus

4119887119896119894119888119896119894

(1205822+ 119892119896119894119899

1205824) 12059021199031205902119889ln 2

119902119896

minus(2 +119887119896119894

1205902119903

119902119896119894)]]

]

+

(40)


1 1205822 1205823119899 and 120582

4119899

through iteration

120582(119898+1)

1= [120582(119898)

1+ 120583(119898)

(

119873119904

sum119896=1

119902(119898)

119896119894minus 119875119879)]

+

120582(119898+1)

2= [120582(119898)

2+ 120583(119898)

(

119873119903

sum119896=1

ℎ(119898)

119896119894minus 119875119877)]

+

120582(119898+1)

3119899= [120582(119898)

3119899+ 120583(119898)

(

119873119903

sum119896=1

119891119896119894119899

119902(119898)

119896119894minus 1198751198681)]

+

120582(119898+1)

4119899= [120582(119898)

4119899+ 120583(119898)

(

119873119903

sum119896=1

119892119896119894119899

ℎ(119898)

119896119894minus 1198751198682)]

+

(41)


1 1205822 1205823119899 and 120582

4119899are



119894when




119896119894rsquos (119896 =

1 119873119903) that is ℎuni



119896119894= 1198751198681119873119904119891max119896119894

where 119891max119896119894

= max119899isinSPU

119891119896119894119899


119896119894=

min119902uni119894 119902max119896119894


119877 Similarly we


119896119894= 1198751198682119873119903119892max119896119894

where 119892max119896119894

= max119899isinSPU

119892119896119894119899


119896119894=

minℎuni119894 ℎmax119896119894


119877 Afterwards


119894 = arg max119894

119873119903

sum119896=1

log2

(1 + 119888119896119894ℎlowast1198961198941205902119889) (1 + 119887

119896119894119902lowast1198961198941205902119903)

1 + 119888119896119894ℎlowast1198961198941205902119889+ 119887119896119894119902lowast1198961198941205902119903

(42)








119903+2(119873PU+1)119879)


5 Outage Analysis


1 PU TX





(119873PU(119873PU minus 119899))(120579120591)119899)minus1



119873119904

where I119873119904










y119889= H119904119889x119904+ n119889 (43)


119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889




119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898



120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)


(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904




119904= 119873119889= 2 [31] As such for a




119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)


119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)



y119889= H119903119889119894

x119904119894+ n119889 (49)


119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)


119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















y119889= H119904119889x119904+ n119889 (43)


119877119863= log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119904119889H119867119904119889

100381610038161003816100381610038161003816100381610038161003816 (44)

where H119904119889




119877119863=

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119904119889119898

) (45)

where 120582119904119889119898



120582119904119889119898

119898 = 1 119873119889is given by [29]

119901 (1205821199041198891 120582

119904119889119873119889

) = (119873119889119870119873119889119873119904

)minus1

exp(minus119873119889

sum119898=1

120582119904119889119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119904119889119898)


(120582119904119889119898

minus 120582119904119889119899

)2)

(46)

where 119870119873119889119873119904




119904= 119873119889= 2 [31] As such for a




119877119863997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

(ln 2)2119873119904(1 + 120588)2) (47)


119875119863

out = 119875 (119877119863lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic1198731198891205882(ln 2)2119873119903(1 + 120588)

2

)

(48)



y119889= H119903119889119894

x119904119894+ n119889 (49)


119877119862

119894=1

2log2

100381610038161003816100381610038161003816100381610038161003816I119873119889

+120588

1205902119889

H119903119889119894

H119867119903119889119894

100381610038161003816100381610038161003816100381610038161003816 (50)


119877119862

119894=1

2

119873119889

sum119898=1

log2(1 +

120588

1205902119889

120582119903119889119894119898

) (51)


where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 120582119903119889119894119898



H119867119903119889119894

The joint pdf of120582119903119889119894119898

119898 = 1 119873119889is given by [29]

119901 (1205821199031198891198941

120582119903119889119894119873

119889

) = (N119889119870119873119889119873119903

)minus1

exp(minus119873119889

sum119898=1

120582119903119889119894119898

)

times (

119873119889

prod119898=1

120582119873119903minus119873119889

119903119889119894119898)


(120582119903119889119894119898

minus 120582119903119889119894119899

)2)

(52)

where119870119873119889119873119903


119877119862

119894997888rarrN(119873

119889log2(1 + 120588)

1198731198891205882

4(ln 2)2119873119903(1 + 120588)2) (53)


119875119862119894

out = 119875 (119877119862

119894lt 119877min)

= 119876(119873119889log2(1 + 120588) minus 119877min

radic11987311988912058824(ln 2)2119873119903(1 + 120588)

2

)

(54)


119904in order not







119875120601

out = 119875 (Δ = 120601) =

119873SU

prod119898=1

119875119877119898

out (55)



119875119877119898

out = 119876(119873119903log2(1 + 1205881015840) minus 119877min

radic11987311990312058810158402(ln 2)2119873119904(1 + 1205881015840)

2

) (56)



119875out = (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(57)


119906is





119875 (Δ = Δ119906) = ( prod

119898isinΔ119906

(1 minus 119875119877119898

out ))(prod

119898isinΔ119906

119875R119898out ) = 120573

119906

(58)




119875Δ119906


119875119862119894

out (59)


119875119860

out = 119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out (60)


119875out = 119875 (119860)119875119860

out + 119875 (119860) 119875119860

out

= (1 minus 120572)(119875120601


sum119906=1

119875 (Δ = Δ119906) 119875Δ119906

out) + 120572119875119863

out

(61)





0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)


119879)




= 1198751198682








119903= 1205902119889= 10minus6WHz


S119877 is 119875



0

2

4

6

8

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)


OA (N = 3)

LCA (N = 3)

OA (N = 2)

LCA (N = 2)

RS-OPA (N = 2)

RS-EPA (N = 2)



119877)







119879is small


119877) is







2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 4 6 8 10 12 14 16 18 208

10

12

14

16

18

Achi

evab

le d

ata r

ate (

bps

Hz)





7 Conclusions


Appendix



A119894= V119903119889119894ΛA119894

U119867119904119903119894 (A1)

where ΛA119894




H119904119903119894

= H119904119903119894Qminus12119894


119894and Q






A119894opt = V

119903119889119894ΛA119894

U119867119904119903119894 (A2)

where U119904119903119894


119904119903119894H119867119904119903119894

= H119904119903119894Q119894H119867119904119903119894

= U119904119903119894Σ119904119903119894U119867119904119903119894



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Optimal and Suboptimal Resource...

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