Research ArticleEnergy Efficient Power Allocation for OFDM-Based CognitiveRadio Systems with Partial Intersystem CSI

Yunli Chen1 Zhengguang Zheng2 Yibin Hou1 Yong Li1 and Shan Jin3

1 Institute of Embedded Software and Systems Beijing University of Technology Beijing 100022 China2Department of Vehicle Information Systems North Information Control Group Co Ltd Nanjing 211153 China3Department of National Key Laboratory of Science and Technology on CommunicationsUniversity of Electronic Science and Technology of China Chengdu 610054 China

Correspondence should be addressed to Zhengguang Zheng zhguangzhenggmailcom

Received 11 February 2014 Accepted 27 May 2014 Published 25 June 2014

Academic Editor Michael Vynnycky

Copyright copy 2014 Yunli Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates energy efficient power allocation for orthogonal frequency divisionmultiplexing- (OFDM-) based cognitiveradio (CR) systems with partial intersystem channel state information (CSI) available The goal is to maximize energy efficiency(EE) while ensuring the minimum rate of secondary user (SU) and keeping the average interference power (AIP) introduced toprimary user (PU) within a target probability level We propose a suboptimal algorithm to solve this optimization problem basedon classic water-filling (WF) technique Moreover we first address the relation between EE and water level In order to reducecomplexity a simplified algorithm with closed-form solution is also proposed Numerical results are provided to corroborate ourtheoretical analysis and to demonstrate the effectiveness of the proposed schemes

1 Introduction

Cognitive radio (CR) [1 2] which provides a flexible platformthat improves the spectral efficiency (SE) has become apromising technology to tackle the problem of spectrumscarcity In an underlay CR system a secondary user (SU)can be allowed access to the licensed spectrum band only ifthe resulting interference to the primary user (PU) is withina tolerable threshold [2]

Orthogonal frequency division multiplexing (OFDM)has been regarded as a potential multicarrier modulationtechnique for CR systems due to its inherent resistance tomultipath fading and the flexibility in resource allocation[3] In OFDM-based CR systems the interference powerconstraint to PU can be divided into two categories averageinterference power (AIP) constraint over all subcarriers andpeak interference power (PIP) constraint for each subcarrierFrom the perspective of SE the AIP constraint is morepractical than the PIP constraint since it is more flexiblefor power allocation [4] In this paper we focus on the AIPconstraint

In OFDM-based CR systems power allocation playsan important role in SE improvement while protecting PUfrom disturbance of SU Previous works on dynamic powerallocation for OFDM-based CR systems mainly focused onspectrum sensing spectrum sharing and SE optimization[5ndash9] However the methods to maximize SE may notbe suitable to the future green communications Energyefficiency (EE) [10ndash14] which is defined as the number oftransmission bits per unit energy can be used to evaluate theperformance Hence it is necessary to improve the energyutilization as efficiently as possible

Several recent works have been done to consider EEin CR systems In [11] both the optimal transmissionduration and energy efficient power allocation were inves-tigated to maximize EE Energy efficient power allocationand spectrum sharing in heterogeneous cognitive radionetworks with femtocells were studied in [12] Howeverneither of them considers the quality-of-service (QoS) ofSU In [13] a method named as water-filling factor aidedsearch (WFAS) was proposed tomaximize EE undermultipleconstraints with perfect channel state information (CSI) at

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 781528 7 pageshttpdxdoiorg1011552014781528

2 Mathematical Problems in Engineering

SU-transmitter (Tx) However compared to intrasystem CSIit would be difficult for the SU-Tx to obtain the perfectintersystem CSI due to its lack of cooperation and limitedfeedback overhead The global 120598-optimal power allocationsolution is achieved based on bisection search techniquewhose accuracy of the solution is dependent on the numberof iterations The high computational complexity also incursadditional energy consumption In [14] we have developedlow complexity algorithm to maximize the system EE formultiuser OFDM-based cognitive radio with PIP constraintIf we focus on the AIP constraint these algorithms cannotbe applied directly Motivated by the above discussions it isnecessary to study the energy efficient design under the AIPconstraint

In this paper we propose an energy efficient power allo-cation algorithm for OFDM-based CR systems with partialintersystem CSI It can be regarded as an improved version ofthe schemes proposed in [13] By analyzing the relationshipbetween water level and EE a closed-form power allocationsolution with low complexity and exact optimality is alsoderived

The remainder of this paper is organized as followsSection 2 introduces an OFDM-based CR system model andformulates the corresponding power allocation problem InSection 3 a bisection search aided energy efficient powerallocation algorithm is described To reduce complexity asimple method with closed-form expression is developed tomaximize EE In Sections 4 and 5 simulation results andconclusions are given respectively

2 System Model and Problem Formulation

21 System Model Consider an OFDM-based CR systemwhere a SU coexists with a PU over the same spectrum bandThe spectrum band is equally divided into 119873 subchannelswith bandwidth 119861 for each Let N ≜ 1 119899 119873

denote the set of these subchannels Define ℎ119904119904119899as the channel

response between SU-Tx and SU-receiver (Rx) on the 119899thsubcarrier In this model we assume that perfect intrasystemCSI ℎ119904119904

119899 is available at the SU-Tx However in a practical

system it is difficult or even infeasible to perfectly obtain ℎ119904119901119899

between SU-Tx and PU-Rx due to the lack of cooperation Itis reasonable to assume that only the statistics distribution ofℎ119904119901

119899is known To protect the communication links of licensed

users the probability that the AIP exceeds a certain thresholdshould be kept within a target level Using the uncorrelatedfading channel model [15] ℎ119904119904

119899and ℎ119904119901

119899are independent

Gaussian distributed random variables with CN(0 1205821) and

CN(0 1205822) respectively

22 Problem Formulation During each transmission theinterference caused by PU-Tx may degrade the performanceof SU-Rx If there are a large number of subchannels thisinterference introduced into SU-Rx can be approximated asadditive white Gaussian noise (AWGN) according to centrallimit theorem [6] Let 119901

119899be the transmit power on the 119899th

subcarrier the data rate is given by

119903119899= 119861 log

2(1 + 119892

119899119901119899) (1)

where 119892119899= |ℎ119904119904

119899|2

1205902 is the channel gain to noise ratio (CNR)

on the 119899th subcarrier and1205902 denotes the noise which includesthermal noise and interference

Consequently the overall throughput 119877 and transmitpower 119875 are

119877 = sum

119899isinN

119903119899 (2)

119875 = sum

119899isinN

119901119899 (3)

respectivelyReferring to [10] the circuit power consumption that is

incurred by signal processing and active circuit blocks suchas digital to analog converter (DAC) can be modeled as alinear function of throughput

119875119888= 119875119904+ 120573119877 (4)

where 119875119904is the static circuit power and 120573 is the dynamic

circuit power per unit data rateSince the total transmit power is limited we have

119875 le (5)

where is the maximum allowable transmit powerIn order to guarantee the user experience the transmis-

sion rate should be restricted by

119877 ge (6)

where is the minimum rate requirement related to the typeof traffic

To protect the activity of PU the power allocationproblemmust include a AIP outage constraintThus we have

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (7)

where 119868th is the predefined threshold and 120598 is the target levelOur objective is to determine the transmit power119901

119899(forall119899 isin

N) such that the EE is maximized under the data rate con-straint and AIP outage constraint Therefore the optimiza-tion problem P1 considered in this paper can be formulatedas follows

P1 max119901119899

119880 =

119877

120572119875 + 119875119888

(8a)

st 119875 le (8b)

119877 ge (8c)

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (8d)

where 120572 is a constant related to the efficiency of poweramplifier It is possible that problem in (8a)ndash(8d) does nothave any feasible solution when cannot be obtained subjectto the constraints (5) and (7) In this case the SU-Tx mayhave to decrease the minimum rate requirement to make thesolution feasible

Mathematical Problems in Engineering 3

3 Energy Efficient Design

In the following we first introduce two auxiliary optimizationproblems and then propose an optimal energy efficientpower allocation algorithm to achieve the maximum EEperformance

Define a rate maximization (Max119877) problem P2 as

P2 max119901119899

119877 (9a)

st 119875 le (9b)

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (9c)

Since the overall data rate is an increasing function oftransmit power 119875 it is shown that P2 has a unique solutionfor all 119875 ge 0

The other conventional optimization problem is transmitpower minimization (Min119875) problem P3 subject to data rateconstraint and AIP outage constraint It can be described asfollows

P3 min119901119899

119875 (10a)

st 119877 ge (10b)

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (10c)

Referring to [5] in a practical system with a suffi-ciently large number of subchannels random variable 119883 =

sum119899|ℎ119904119901

119899|2

119901119899can be modeled as a normal distributed variable

based on central limit theorem with mean 119898119900and variance

1205902

119900

119898119900= sum

119899isinN

1199011198991205822 (11)

1205902

119900= sum

119899isinN

(1199011198991205822)2

(12)

respectivelyThen the equivalent constraint of (7) can be expressed as

119875119900=

1

2

erfc(119873119868th minus 119898119900radic2120590119900

) le 120598 (13)

where erfc(119909) = (2radic120587) intinfin119909

119890minus1199052

119889119905If the transmit power 119875 is given the objective of problem

P1 is equivalent to problem P2 and the existing water-fillingpower allocation scheme can be used [16] However besidesadapting the power loading on all subcarriers the overalltransmit power can also be adapted according to the CSIand AIP level to maximize the EE Hence the existing powerallocation schemes cannot be applied directly

Without loss of generality assume that 1198921ge 1198922ge sdot sdot sdot ge

119892119873 It is noticeable that the exact power allocation solution

for P2 is too difficult to address However if we drop the

Step 1 Initialization 120583119900larr 0119873 larr 1 Slarr

Step 2 Calculate 120583119900according to (15)

Step 3 Update Slarr 119899 | 120583119900gt 1119892

119899 forall119899 isinN

Step 4 If |S| = 119873 then terminate the algorithm elseset119873 larr |S| and go to Step 2

Algorithm 1 Solving 120583119900

AIP constraint from P2 it can be solved by the water-fillingalgorithm and the power distribution is

119901119899= [120583 minus

1

119892119899

]

+

(14)

for all 119899 isinN where [119909]+ = max(119909 0) and 120583 is the water level

Lemma 1 The 119875119900is a monotonic increasing function of water

level 120583 for 120583 ge 11198921 and the maximum allowable water level

120583119900 for 119875

119900le 120598 is

120583119900=

minus119887 + radic1198872minus 4119886119888

2119886

(15)

where 119886 = (21198892

119873 minus 119873

2

)1205822

2 119887 = 2119873119868

119905ℎ1198731205822+ (2119873 minus

41198892

)11986611205822

2 119888 = 2119866

21205822

21198892

minus (119873119868119905ℎ+ 12058221198661)2 1198661

=

sum119873

119899=1(1119892119899) 1198662= sum119873

119899=1(11198922

119899) 119889 = erfcminus1(2120598) erfcminus1(sdot) is the

inverse function of erfc(sdot) and119873 is the number of subchannelswith transmit power 119901

119899gt 0

Proof Please see Appendix A

In order to protect PU the maximum allowable waterlevel is restricted which may lead the power of some subcar-riers equal to zero for special channel realizations Here wepresent an efficient algorithm to solve 120583

119900 See Algorithm 1

Lemma2 For problemP2without AIP constraint the optimalwater level 120583 to maximize system rate under the total transmitpower is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (16)

Proof Please see Appendix B

Lemma3 For problemP3without AIP constraint the optimalwater level to meet the minimum rate requirement is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(17)

Proof Please see Appendix C

With the help of Lemmas 1 to 3 the optimal water levels120583lowast and lowast for problems P2 and P3 are

120583lowast

= min (120583 120583119900) (18)

lowast

= min ( 120583119900) (19)

respectively

4 Mathematical Problems in Engineering

Step 1 Initialization 120583lowast larr 0Step 2 Calculate lowast and 120583lowast according to (18) and (19)Step 3 If 120583lowast le 120583

lowast then 120583mid larr 120583lowast and go to Step 7 else set 120583low larr

lowast and 120583high larr 120583lowast respectively

Step 4 Set 120583mid larr 12 (120583low + 120583high)Step 5 If its first derivative with respect to water level d119880d120583|

120583=120583midgt 0 then 120583low larr 120583mid else 120583high larr 120583mid

Step 6 Repeat Step 4 until 120583high minus 120583low le 120585 where 120585 is a small positive constant to control the convergence accuracyStep 7 Finish the optimal solution is 120583lowast larr 120583mid

Algorithm 2 Bisection search aided algorithm (BSAA)

Since the system rate 119877 is a strictly increasing function oftransmit power119875 it is shown that119877 has a unique value for all120583 ge 1119892

1 Hence the equivalent problem of P1 can be given

as

P4 max120583

119880 (120583) =

119877

120572119875 + 119875119888

(20a)

st lowast

le 120583 le 120583lowast

(20b)

Lemma 4 The system EE 119880 is a quasiconcave function ofwater level 120583 for 120583 ge 1119892

1

Proof Please see Appendix D

According to Lemma 4 a unique global optimal waterlevel 120583 always exists without any constraint in P4 Hence theoptimal water level 120583lowast is

120583lowast

= min (120583lowastmax (120583 lowast)) (21)

31 Bisection Search Aided Algorithm (BSAA) The remainingthing for P1 is to address 120583lowast and it can be solved by bisectionsearch technique Here we list the power allocation methodin Algorithm 2

Our goal aims to maximize the system EE under theconstraints on transmit power the minimum rate require-ment and AIP outage The water-filling based algorithm isproposed to obtain the optimal solution The key work ofAlgorithm 2 is to determine the sign of d119880d120583 It can bedetermined by calculating the value of 119880(120583 + Δ120583) minus 119880(120583)where Δ120583 is an infinitely small positive constant

32 Simplified Energy Efficient Power Allocation (SiPA)Although the suboptimal power allocation solution is desir-able the cost to pay is the computational complexity initerative search It takes at most lceillog

2((120583lowast

minus lowast

)120585)rceil iterationsto convergence where lceil119909rceil denotes the smallest integer notless than 119909 Thus this straightforward approach is clearly notpractical andwe need a low complexity approach to solve thisproblem

Since EE119880 is a quasiconcave function of water level 120583 theoptimal unconstrained water level always exists Differentiat-ing (20a) with respect to 120583 and setting the derivative to zerowe can obtain

120597119880

120597120583

= 119865(1 +

119875119904

120572120583

minus

1198663

120583

minus ln (1205831198664)) = 0 (22)

or

1198751199041198664

120572119890

minus

11986631198664

119890

=

1205831198664

119890

ln(1205831198664

119890

) (23)

where 119890 is the base of the natural logarithm 119865 is a nonzerovalue independent of 120583 and 119866

3and 119866

4are defined as

1198663=

1

sum

119899=1

1

119892119899

(24)

1198664= (

prod

119899=1

119892119899)

1

(25)

respectively Let 119883 = ln(1205831198664119890) and 119884 = ((119875

1199041198664)(120572119890)) minus

((11986631198664)119890) then (23) can be expressed as

119883119890119883

= 119884 (26)

Its solution is 119883 = 1198820(119884) where 119882

0(sdot) denotes the

real branch of the Lambert function [13] Substituting thissolution into (26) the global optimal water level 120583 withoutany constraint is

120583 =

119890

1198664

exp [1198820(

1198751199041198664

120572119890

minus

11986631198664

119890

)] (27)

According to Lemma 4 the optimal water level 120583lowast corre-sponding to P1 can be obtained by substituting 120583 into (21)

4 Numerical Results

Simulations have been performed to evaluate the perfor-mance of our proposed algorithms The channel gains |ℎ119904119904

119899|2

and |ℎ119904119901119899|2 (119899 isin N) are independent chi-square distributed

random variables Simulation parameters are set as follows119861 = 15 kHz119873 = 512 120582 = 120582

11205822(we fix 120582

2= 1 and vary 120582

1)

= 3W 1205902 = 001W 119868th = 1205881205902 120598 = 005 119875

119904= 08W 120572 = 2

= 23Mbps and 120573 = 02WMbps The simulation curvesare obtained by averaging over 1000 channel realizations

In Figure 1 we compare the average EE of various powerallocation algorithms with 120588 = 05 Compared with otheralgorithms our proposed algorithms can achieve significantperformance gain in terms of EE The BSAA and SiPA havethe same performance and both of them always performbetter than the simplified WFAS algorithm in [13] at the cost

Mathematical Problems in Engineering 5

4

3

2

1

00 5 10 15 20

120582 (dB)

EE (M

bits

J)

MinPMaxRBSAA

SiPASimplified WFAS in [13]

Figure 1 System EE versus average CNR 120588 = 05

MinPMaxRBSAA

SiPASimplified WFAS in [13]

25

2

15

1

05

00 01 02 03 04

120588

EE (M

bits

J)

Figure 2 System EE versus average CNR 120582 = 10 dB

of serious interference to PU In the low 120582 region the transmitpower dominates the total power consumption while in thehigh 120582 region the circuit power does for the energy efficientdesign However the gap between proposed algorithms andconventional ones becomes larger as 120582 keeps increasing

In Figure 2 the average EE dramatically increases as 120588increases in the low region However when 120588 is greaterthan a saturation point the EE does not further increasefor energy efficient power allocation In addition it canbe observed that all schemes may have to operate exactlyat the maximum allowable transmit power to meet the

minimum rate requirement when 120588 is much smaller than thethreshold

5 Conclusion

In this paper we considered the energy efficient design forOFDM-based CR systems under the AIP outage probabilityconstraint and minimum rate requirement constraint Dueto the loose cooperation the intersystem CSI is partiallyknown to the SU-Tx To solve this problem we proposed asuboptimal algorithm based on bisection search techniqueFurthermore in order to reduce the computational com-plexity we also derived a closed-form solution for simplifiedalgorithm without performance loss We demonstrate thesuperior EE performance of the proposed algorithms byextensive simulations

Appendices

A Proof of Lemma 1

From (11) (12) and (14) (13) can be written as

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583 minus (1119892

119899))

1205822

radic2sum

119899=1(120583 minus (1119892

119899))2

) le 120598 (A1)

where 120583 gt 1119892119899 for 119899 = 1 2 and 120583 le 1119892

119899 for 119899 =

+ 1 119873It can be easily proved that 119875

119900is a monotonic increasing

function of 120583 Let 120583119900be the maximum allowable power level

corresponding to AIP constraint we have

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583119900minus (1119892

119899))

1205822

radic2sum

119899=1(120583119900minus (1119892

119899))2

) = 120598 (A2)

After somemathematical manipulation a single equationwith the variable 120583

119900can be derived as

119886120583119900

2

+ 119887120583119900+ 119888 = 0 (A3)

where 119886 = (21198892

119873 minus 119873

2

)1205822

2 119887 = 2119873119868th1198731205822 + (2119873 minus

41198892

)11986611205822

2 119888 = 2119866

21205822

21198892

minus (119873119868th + 12058221198661)2 1198661= sum119873

119899=1(1119892119899)

1198662= sum119873

119899=1(11198922

119899) 119889 = erfcminus1(2120598) and erfcminus1(sdot) is the

inverse function of erfc(sdot) When solving the above equationthe maximum allowable power level corresponding to AIPconstraint is

120583119900=

minus119887 + radic1198872minus 4119886119888

2119886

(A4)

Here Lemma 1 is proved

6 Mathematical Problems in Engineering

B Proof of Lemma 2

For a given allowable transmit power the optimal powerallocation can be achieved by classical water-filling methodThe true power level is assumed to be 120583 we have

=

119873

sum

119899=1

119901119899=

119872

sum

119899=1

(120583 minus

1

119892119899

) = 119872120583 minus

119872

sum

119899=1

1

119892119899

(B1)

or

120583 =

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B2)

where 119872 le 119873 119901119899= 120583 minus (1119892

119899) gt 0 119899 = 1 2 119872 and

120583 le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

(1119872119894)( + sum

119872119894

119899=1(1119892119899)) 119894 = 1 2 Because 120583 gt (1119892

119899) 119899 =

1 119872 and 120583 le (1119892119899) 119899 = 119872+1 119873 in (B2) we derive

the results that 1205831gt 120583 and 120583

2gt 120583 as follows

1205831minus 120583 =

1

1198721

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198721

sum

119899=1

1

119892119899

) minus 120583

=

1

1198721

[(119872 minus1198721) 120583 minus

119872

sum

119899=1198721+1

1

119892119899

]

=

1

1198721

119872

sum

119899=1198721+1

(120583 minus

1

119892119899

)

gt 0

1205832minus 120583 =

1

1198722

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198722

sum

119899=1

1

119892119899

) minus 120583

=

1

1198722

[minus (1198722minus119872)120583 +

1198722

sum

119899=119872+1

1

119892119899

]

= minus

1

1198722

1198722

sum

119899=119872+1

(120583 minus

1

119892119899

)

ge 0

(B3)

Hence the true power level 120583 corresponding to themaximum allowable transmit power is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B4)

Here Lemma 2 is proved

C Proof of Lemma 3

Let be the optimal water level corresponding to theminimum rate requirement without AIP constraint wehave

= 119861

119873

sum

119899=1

log2(1 + 119892

119899119901119899)

= 119861

119872

sum

119899=1

log2(1 + ( minus

1

119892119899

)119892119899)

= 119861log2(119872

119872

prod

119899=1

119892119899)

(C1)

or

= 2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C2)

where 119872 le 119873 119901119899= minus (1119892

119899) gt 0 119899 = 1 2 119872 and

le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

2(119861119872

119894)

(prod119872119894

119899=1119892119899)minus1119872

119894 119894 = 1 2 It can be seen that 119892119899gt 1

119899 = 1 119872 and 119892119899le 1 119899 = 119872 + 1 119873 We derive the

results that 1205831gt and 120583

2gt as follows

1205831

=

2(119861119872

1)

(prod1198721

119899=1119892119899)

minus11198721

=

(119872

prod119872

119899=1119892119899prod1198721

119899=11119892119899)

11198721

= (

119872

prod

119899=1198721+1

119892119899)

11198721

gt 1

1205832

=

2(119861119872

2)

(prod1198722

119899=1119892119899)

minus11198722

=

(119872

prod119872

119899=1119892119899prod1198722

119899=11119892119899)

11198722

= (

1198722

prod

119899=119872+1

1

119892119899

)

11198722

ge 1

(C3)

Since 120583119894gt 0 (119894 = 1 2) the optimal power level that

satisfies 119877 = without the AIP constraint is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C4)

Here Lemma 3 is proved

Mathematical Problems in Engineering 7

D Proof of Lemma 4

From problem (8a) the objective of EE optimization ismodeled as

max119901119899

119880 (119875) =

1

((120572119875 + 119875119904) 119877 (119875)) + 120573

(D1)

where 119875 is the total transmit power and 119877(119901) is the corre-sponding data rate

Since 120572 gt 0 119877 gt 0 119875 gt 0 and 119875119904gt 0 the EE 119880 is a

strictly monotone decreasing function of 120572119875+119875119904119877Then the

equivalent optimization problem is reformulated as

max119901119899

119880 (119875) =

119877 (119875)

120572119875 + 119875119904

(D2)

For arbitrary 120579 isin (0 1) since 119877(119875) is concave function of119875 we have

119880 (1205791199091+ (1 minus 120579) 119909

2) ge

120579119877 (1199091) + (1 minus 120579) 119877 (119909

2)

120579 (1205721199091+ 119875119904) + (1 minus 120579) (120572119909

2+ 119875119904)

ge min (119880 (1199091) 119880 (119909

2))

(D3)

According to [13 14] we can conclude that119880 is quasicon-cave for 119875 ge 0 Since 119875 is a monotonic increasing function ofwater level for 120583 gt (1max119892

119899) Hence EE is a quasiconcave

function of water level

Conflict of Interests

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

Acknowledgment

The work in this paper has been supported by fundingfrom Beijing Natural Science Foundation under Grant no4122009

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] S Srinivasa and S A Jafar ldquoThe throughput potential of cog-nitive radio a theoretical perspectiverdquo IEEE CommunicationsMagazine vol 45 no 5 pp 73ndash79 2007

[3] T A Weiss and F K Jondral ldquoSpectrum pooling an innovativestrategy for the enhancement of spectrum efficiencyrdquo IEEECommunications Magazine vol 42 no 3 pp S8ndashS14 2004

[4] R Zhang ldquoOn peak versus average interference power con-straints for protecting primary users in cognitive radio net-worksrdquo IEEE Transactions on Wireless Communications vol 8no 4 pp 2112ndash2120 2009

[5] K Son B C Jung S Chong and D K Sung ldquoPower allocationfor OFDM-based cognitive radio systems under outage con-straintsrdquo in Proceedings of the IEEE International Conference onCommunications (ICC rsquo10) pp 1ndash5 May 2010

[6] X Zhou B Wu P-H Ho and X Ling ldquoAn efficient powerallocation algorithm for OFDM based underlay cognitive radionetworksrdquo in Proceedings of the 54th Annual IEEE GlobalTelecommunications Conference (GLOBECOM rsquo11) pp 1ndash5December 2011

[7] 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

[8] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[9] G Ding Q Wu N Min and G Zhou ldquoAdaptive resourceallocation in multi-user OFDM-based cognitive radio systemsrdquoin Proceedings of the IEEE International Conference on WirelessCommunications and Signal Processing (WCSP rsquo09) November2009

[10] D Feng C Jiang G Lim L J Cimini Jr G Feng and G Y LildquoA survey of energy-efficient wireless communicationsrdquo IEEECommunications Surveys and Tutorials vol 15 no 1 pp 167ndash178 2013

[11] Y Li X Zhou H Xu G Y Li D Wang and A SoongldquoEnergy-efficient transmission in cognitive radio networksrdquo inProceedings of the 7th IEEE Consumer Communications andNetworking Conference (CCNC rsquo10) January 2010

[12] M L Treust and S Lasaulce ldquoA repeated game formulationof energy-efficient decentralized power controlrdquo IEEE Transac-tions on Wireless Communications vol 9 no 9 pp 2860ndash28692010

[13] J Mao G Xie J Gao and Y Liu ldquoEnergy efficiency optimiza-tion for ofdm-based cognitive radio systems a water-fillingfactor aided search methodrdquo IEEE Transactions on WirelessCommunications vol 12 no 5 pp 2366ndash2375 2013

[14] Y Chen Z Zheng Y Hou and Y Li ldquoEnergy efficient designfor OFDM-based underlay cognitive radio networksrdquo Mathe-matical Problems in Engineering vol 2014 Article ID 431878 7pages 2014

[15] K Fazel and S Kaiser Multi-Carrier and Spread SpectrumSystems John Wiley amp Sons Chichester UK 2003

[16] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert W functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

SU-transmitter (Tx) However compared to intrasystem CSIit would be difficult for the SU-Tx to obtain the perfectintersystem CSI due to its lack of cooperation and limitedfeedback overhead The global 120598-optimal power allocationsolution is achieved based on bisection search techniquewhose accuracy of the solution is dependent on the numberof iterations The high computational complexity also incursadditional energy consumption In [14] we have developedlow complexity algorithm to maximize the system EE formultiuser OFDM-based cognitive radio with PIP constraintIf we focus on the AIP constraint these algorithms cannotbe applied directly Motivated by the above discussions it isnecessary to study the energy efficient design under the AIPconstraint

In this paper we propose an energy efficient power allo-cation algorithm for OFDM-based CR systems with partialintersystem CSI It can be regarded as an improved version ofthe schemes proposed in [13] By analyzing the relationshipbetween water level and EE a closed-form power allocationsolution with low complexity and exact optimality is alsoderived

The remainder of this paper is organized as followsSection 2 introduces an OFDM-based CR system model andformulates the corresponding power allocation problem InSection 3 a bisection search aided energy efficient powerallocation algorithm is described To reduce complexity asimple method with closed-form expression is developed tomaximize EE In Sections 4 and 5 simulation results andconclusions are given respectively

2 System Model and Problem Formulation

21 System Model Consider an OFDM-based CR systemwhere a SU coexists with a PU over the same spectrum bandThe spectrum band is equally divided into 119873 subchannelswith bandwidth 119861 for each Let N ≜ 1 119899 119873

denote the set of these subchannels Define ℎ119904119904119899as the channel

response between SU-Tx and SU-receiver (Rx) on the 119899thsubcarrier In this model we assume that perfect intrasystemCSI ℎ119904119904

119899 is available at the SU-Tx However in a practical

system it is difficult or even infeasible to perfectly obtain ℎ119904119901119899

between SU-Tx and PU-Rx due to the lack of cooperation Itis reasonable to assume that only the statistics distribution ofℎ119904119901

119899is known To protect the communication links of licensed

users the probability that the AIP exceeds a certain thresholdshould be kept within a target level Using the uncorrelatedfading channel model [15] ℎ119904119904

119899and ℎ119904119901

119899are independent

Gaussian distributed random variables with CN(0 1205821) and

CN(0 1205822) respectively

22 Problem Formulation During each transmission theinterference caused by PU-Tx may degrade the performanceof SU-Rx If there are a large number of subchannels thisinterference introduced into SU-Rx can be approximated asadditive white Gaussian noise (AWGN) according to centrallimit theorem [6] Let 119901

119899be the transmit power on the 119899th

subcarrier the data rate is given by

119903119899= 119861 log

2(1 + 119892

119899119901119899) (1)

where 119892119899= |ℎ119904119904

119899|2

1205902 is the channel gain to noise ratio (CNR)

on the 119899th subcarrier and1205902 denotes the noise which includesthermal noise and interference

Consequently the overall throughput 119877 and transmitpower 119875 are

119877 = sum

119899isinN

119903119899 (2)

119875 = sum

119899isinN

119901119899 (3)

respectivelyReferring to [10] the circuit power consumption that is

incurred by signal processing and active circuit blocks suchas digital to analog converter (DAC) can be modeled as alinear function of throughput

119875119888= 119875119904+ 120573119877 (4)

where 119875119904is the static circuit power and 120573 is the dynamic

circuit power per unit data rateSince the total transmit power is limited we have

119875 le (5)

where is the maximum allowable transmit powerIn order to guarantee the user experience the transmis-

sion rate should be restricted by

119877 ge (6)

where is the minimum rate requirement related to the typeof traffic

To protect the activity of PU the power allocationproblemmust include a AIP outage constraintThus we have

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (7)

where 119868th is the predefined threshold and 120598 is the target levelOur objective is to determine the transmit power119901

119899(forall119899 isin

N) such that the EE is maximized under the data rate con-straint and AIP outage constraint Therefore the optimiza-tion problem P1 considered in this paper can be formulatedas follows

P1 max119901119899

119880 =

119877

120572119875 + 119875119888

(8a)

st 119875 le (8b)

119877 ge (8c)

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (8d)

where 120572 is a constant related to the efficiency of poweramplifier It is possible that problem in (8a)ndash(8d) does nothave any feasible solution when cannot be obtained subjectto the constraints (5) and (7) In this case the SU-Tx mayhave to decrease the minimum rate requirement to make thesolution feasible

Mathematical Problems in Engineering 3

3 Energy Efficient Design

In the following we first introduce two auxiliary optimizationproblems and then propose an optimal energy efficientpower allocation algorithm to achieve the maximum EEperformance

Define a rate maximization (Max119877) problem P2 as

P2 max119901119899

119877 (9a)

st 119875 le (9b)

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (9c)

Since the overall data rate is an increasing function oftransmit power 119875 it is shown that P2 has a unique solutionfor all 119875 ge 0

The other conventional optimization problem is transmitpower minimization (Min119875) problem P3 subject to data rateconstraint and AIP outage constraint It can be described asfollows

P3 min119901119899

119875 (10a)

st 119877 ge (10b)

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (10c)

Referring to [5] in a practical system with a suffi-ciently large number of subchannels random variable 119883 =

sum119899|ℎ119904119901

119899|2

119901119899can be modeled as a normal distributed variable

based on central limit theorem with mean 119898119900and variance

1205902

119900

119898119900= sum

119899isinN

1199011198991205822 (11)

1205902

119900= sum

119899isinN

(1199011198991205822)2

(12)

respectivelyThen the equivalent constraint of (7) can be expressed as

119875119900=

1

2

erfc(119873119868th minus 119898119900radic2120590119900

) le 120598 (13)

where erfc(119909) = (2radic120587) intinfin119909

119890minus1199052

119889119905If the transmit power 119875 is given the objective of problem

P1 is equivalent to problem P2 and the existing water-fillingpower allocation scheme can be used [16] However besidesadapting the power loading on all subcarriers the overalltransmit power can also be adapted according to the CSIand AIP level to maximize the EE Hence the existing powerallocation schemes cannot be applied directly

Without loss of generality assume that 1198921ge 1198922ge sdot sdot sdot ge

119892119873 It is noticeable that the exact power allocation solution

for P2 is too difficult to address However if we drop the

Step 1 Initialization 120583119900larr 0119873 larr 1 Slarr

Step 2 Calculate 120583119900according to (15)

Step 3 Update Slarr 119899 | 120583119900gt 1119892

119899 forall119899 isinN

Step 4 If |S| = 119873 then terminate the algorithm elseset119873 larr |S| and go to Step 2

Algorithm 1 Solving 120583119900

AIP constraint from P2 it can be solved by the water-fillingalgorithm and the power distribution is

119901119899= [120583 minus

1

119892119899

]

+

(14)

for all 119899 isinN where [119909]+ = max(119909 0) and 120583 is the water level

Lemma 1 The 119875119900is a monotonic increasing function of water

level 120583 for 120583 ge 11198921 and the maximum allowable water level

120583119900 for 119875

119900le 120598 is

120583119900=

minus119887 + radic1198872minus 4119886119888

2119886

(15)

where 119886 = (21198892

119873 minus 119873

2

)1205822

2 119887 = 2119873119868

119905ℎ1198731205822+ (2119873 minus

41198892

)11986611205822

2 119888 = 2119866

21205822

21198892

minus (119873119868119905ℎ+ 12058221198661)2 1198661

=

sum119873

119899=1(1119892119899) 1198662= sum119873

119899=1(11198922

119899) 119889 = erfcminus1(2120598) erfcminus1(sdot) is the

inverse function of erfc(sdot) and119873 is the number of subchannelswith transmit power 119901

119899gt 0

Proof Please see Appendix A

In order to protect PU the maximum allowable waterlevel is restricted which may lead the power of some subcar-riers equal to zero for special channel realizations Here wepresent an efficient algorithm to solve 120583

119900 See Algorithm 1

Lemma2 For problemP2without AIP constraint the optimalwater level 120583 to maximize system rate under the total transmitpower is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (16)

Proof Please see Appendix B

Lemma3 For problemP3without AIP constraint the optimalwater level to meet the minimum rate requirement is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(17)

Proof Please see Appendix C

With the help of Lemmas 1 to 3 the optimal water levels120583lowast and lowast for problems P2 and P3 are

120583lowast

= min (120583 120583119900) (18)

lowast

= min ( 120583119900) (19)

respectively

4 Mathematical Problems in Engineering

Step 1 Initialization 120583lowast larr 0Step 2 Calculate lowast and 120583lowast according to (18) and (19)Step 3 If 120583lowast le 120583

lowast then 120583mid larr 120583lowast and go to Step 7 else set 120583low larr

lowast and 120583high larr 120583lowast respectively

Step 4 Set 120583mid larr 12 (120583low + 120583high)Step 5 If its first derivative with respect to water level d119880d120583|

120583=120583midgt 0 then 120583low larr 120583mid else 120583high larr 120583mid

Step 6 Repeat Step 4 until 120583high minus 120583low le 120585 where 120585 is a small positive constant to control the convergence accuracyStep 7 Finish the optimal solution is 120583lowast larr 120583mid

Algorithm 2 Bisection search aided algorithm (BSAA)

Since the system rate 119877 is a strictly increasing function oftransmit power119875 it is shown that119877 has a unique value for all120583 ge 1119892

1 Hence the equivalent problem of P1 can be given

as

P4 max120583

119880 (120583) =

119877

120572119875 + 119875119888

(20a)

st lowast

le 120583 le 120583lowast

(20b)

Lemma 4 The system EE 119880 is a quasiconcave function ofwater level 120583 for 120583 ge 1119892

1

Proof Please see Appendix D

According to Lemma 4 a unique global optimal waterlevel 120583 always exists without any constraint in P4 Hence theoptimal water level 120583lowast is

120583lowast

= min (120583lowastmax (120583 lowast)) (21)

31 Bisection Search Aided Algorithm (BSAA) The remainingthing for P1 is to address 120583lowast and it can be solved by bisectionsearch technique Here we list the power allocation methodin Algorithm 2

Our goal aims to maximize the system EE under theconstraints on transmit power the minimum rate require-ment and AIP outage The water-filling based algorithm isproposed to obtain the optimal solution The key work ofAlgorithm 2 is to determine the sign of d119880d120583 It can bedetermined by calculating the value of 119880(120583 + Δ120583) minus 119880(120583)where Δ120583 is an infinitely small positive constant

32 Simplified Energy Efficient Power Allocation (SiPA)Although the suboptimal power allocation solution is desir-able the cost to pay is the computational complexity initerative search It takes at most lceillog

2((120583lowast

minus lowast

)120585)rceil iterationsto convergence where lceil119909rceil denotes the smallest integer notless than 119909 Thus this straightforward approach is clearly notpractical andwe need a low complexity approach to solve thisproblem

Since EE119880 is a quasiconcave function of water level 120583 theoptimal unconstrained water level always exists Differentiat-ing (20a) with respect to 120583 and setting the derivative to zerowe can obtain

120597119880

120597120583

= 119865(1 +

119875119904

120572120583

minus

1198663

120583

minus ln (1205831198664)) = 0 (22)

or

1198751199041198664

120572119890

minus

11986631198664

119890

=

1205831198664

119890

ln(1205831198664

119890

) (23)

where 119890 is the base of the natural logarithm 119865 is a nonzerovalue independent of 120583 and 119866

3and 119866

4are defined as

1198663=

1

sum

119899=1

1

119892119899

(24)

1198664= (

prod

119899=1

119892119899)

1

(25)

respectively Let 119883 = ln(1205831198664119890) and 119884 = ((119875

1199041198664)(120572119890)) minus

((11986631198664)119890) then (23) can be expressed as

119883119890119883

= 119884 (26)

Its solution is 119883 = 1198820(119884) where 119882

0(sdot) denotes the

real branch of the Lambert function [13] Substituting thissolution into (26) the global optimal water level 120583 withoutany constraint is

120583 =

119890

1198664

exp [1198820(

1198751199041198664

120572119890

minus

11986631198664

119890

)] (27)

According to Lemma 4 the optimal water level 120583lowast corre-sponding to P1 can be obtained by substituting 120583 into (21)

4 Numerical Results

Simulations have been performed to evaluate the perfor-mance of our proposed algorithms The channel gains |ℎ119904119904

119899|2

and |ℎ119904119901119899|2 (119899 isin N) are independent chi-square distributed

random variables Simulation parameters are set as follows119861 = 15 kHz119873 = 512 120582 = 120582

11205822(we fix 120582

2= 1 and vary 120582

1)

= 3W 1205902 = 001W 119868th = 1205881205902 120598 = 005 119875

119904= 08W 120572 = 2

= 23Mbps and 120573 = 02WMbps The simulation curvesare obtained by averaging over 1000 channel realizations

In Figure 1 we compare the average EE of various powerallocation algorithms with 120588 = 05 Compared with otheralgorithms our proposed algorithms can achieve significantperformance gain in terms of EE The BSAA and SiPA havethe same performance and both of them always performbetter than the simplified WFAS algorithm in [13] at the cost

Mathematical Problems in Engineering 5

4

3

2

1

00 5 10 15 20

120582 (dB)

EE (M

bits

J)

MinPMaxRBSAA

SiPASimplified WFAS in [13]

Figure 1 System EE versus average CNR 120588 = 05

MinPMaxRBSAA

SiPASimplified WFAS in [13]

25

2

15

1

05

00 01 02 03 04

120588

EE (M

bits

J)

Figure 2 System EE versus average CNR 120582 = 10 dB

of serious interference to PU In the low 120582 region the transmitpower dominates the total power consumption while in thehigh 120582 region the circuit power does for the energy efficientdesign However the gap between proposed algorithms andconventional ones becomes larger as 120582 keeps increasing

In Figure 2 the average EE dramatically increases as 120588increases in the low region However when 120588 is greaterthan a saturation point the EE does not further increasefor energy efficient power allocation In addition it canbe observed that all schemes may have to operate exactlyat the maximum allowable transmit power to meet the

minimum rate requirement when 120588 is much smaller than thethreshold

5 Conclusion

In this paper we considered the energy efficient design forOFDM-based CR systems under the AIP outage probabilityconstraint and minimum rate requirement constraint Dueto the loose cooperation the intersystem CSI is partiallyknown to the SU-Tx To solve this problem we proposed asuboptimal algorithm based on bisection search techniqueFurthermore in order to reduce the computational com-plexity we also derived a closed-form solution for simplifiedalgorithm without performance loss We demonstrate thesuperior EE performance of the proposed algorithms byextensive simulations

Appendices

A Proof of Lemma 1

From (11) (12) and (14) (13) can be written as

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583 minus (1119892

119899))

1205822

radic2sum

119899=1(120583 minus (1119892

119899))2

) le 120598 (A1)

where 120583 gt 1119892119899 for 119899 = 1 2 and 120583 le 1119892

119899 for 119899 =

+ 1 119873It can be easily proved that 119875

119900is a monotonic increasing

function of 120583 Let 120583119900be the maximum allowable power level

corresponding to AIP constraint we have

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583119900minus (1119892

119899))

1205822

radic2sum

119899=1(120583119900minus (1119892

119899))2

) = 120598 (A2)

After somemathematical manipulation a single equationwith the variable 120583

119900can be derived as

119886120583119900

2

+ 119887120583119900+ 119888 = 0 (A3)

where 119886 = (21198892

119873 minus 119873

2

)1205822

2 119887 = 2119873119868th1198731205822 + (2119873 minus

41198892

)11986611205822

2 119888 = 2119866

21205822

21198892

minus (119873119868th + 12058221198661)2 1198661= sum119873

119899=1(1119892119899)

1198662= sum119873

119899=1(11198922

119899) 119889 = erfcminus1(2120598) and erfcminus1(sdot) is the

inverse function of erfc(sdot) When solving the above equationthe maximum allowable power level corresponding to AIPconstraint is

120583119900=

minus119887 + radic1198872minus 4119886119888

2119886

(A4)

Here Lemma 1 is proved

6 Mathematical Problems in Engineering

B Proof of Lemma 2

For a given allowable transmit power the optimal powerallocation can be achieved by classical water-filling methodThe true power level is assumed to be 120583 we have

=

119873

sum

119899=1

119901119899=

119872

sum

119899=1

(120583 minus

1

119892119899

) = 119872120583 minus

119872

sum

119899=1

1

119892119899

(B1)

or

120583 =

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B2)

where 119872 le 119873 119901119899= 120583 minus (1119892

119899) gt 0 119899 = 1 2 119872 and

120583 le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

(1119872119894)( + sum

119872119894

119899=1(1119892119899)) 119894 = 1 2 Because 120583 gt (1119892

119899) 119899 =

1 119872 and 120583 le (1119892119899) 119899 = 119872+1 119873 in (B2) we derive

the results that 1205831gt 120583 and 120583

2gt 120583 as follows

1205831minus 120583 =

1

1198721

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198721

sum

119899=1

1

119892119899

) minus 120583

=

1

1198721

[(119872 minus1198721) 120583 minus

119872

sum

119899=1198721+1

1

119892119899

]

=

1

1198721

119872

sum

119899=1198721+1

(120583 minus

1

119892119899

)

gt 0

1205832minus 120583 =

1

1198722

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198722

sum

119899=1

1

119892119899

) minus 120583

=

1

1198722

[minus (1198722minus119872)120583 +

1198722

sum

119899=119872+1

1

119892119899

]

= minus

1

1198722

1198722

sum

119899=119872+1

(120583 minus

1

119892119899

)

ge 0

(B3)

Hence the true power level 120583 corresponding to themaximum allowable transmit power is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B4)

Here Lemma 2 is proved

C Proof of Lemma 3

Let be the optimal water level corresponding to theminimum rate requirement without AIP constraint wehave

= 119861

119873

sum

119899=1

log2(1 + 119892

119899119901119899)

= 119861

119872

sum

119899=1

log2(1 + ( minus

1

119892119899

)119892119899)

= 119861log2(119872

119872

prod

119899=1

119892119899)

(C1)

or

= 2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C2)

where 119872 le 119873 119901119899= minus (1119892

119899) gt 0 119899 = 1 2 119872 and

le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

2(119861119872

119894)

(prod119872119894

119899=1119892119899)minus1119872

119894 119894 = 1 2 It can be seen that 119892119899gt 1

119899 = 1 119872 and 119892119899le 1 119899 = 119872 + 1 119873 We derive the

results that 1205831gt and 120583

2gt as follows

1205831

=

2(119861119872

1)

(prod1198721

119899=1119892119899)

minus11198721

=

(119872

prod119872

119899=1119892119899prod1198721

119899=11119892119899)

11198721

= (

119872

prod

119899=1198721+1

119892119899)

11198721

gt 1

1205832

=

2(119861119872

2)

(prod1198722

119899=1119892119899)

minus11198722

=

(119872

prod119872

119899=1119892119899prod1198722

119899=11119892119899)

11198722

= (

1198722

prod

119899=119872+1

1

119892119899

)

11198722

ge 1

(C3)

Since 120583119894gt 0 (119894 = 1 2) the optimal power level that

satisfies 119877 = without the AIP constraint is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C4)

Here Lemma 3 is proved

Mathematical Problems in Engineering 7

D Proof of Lemma 4

From problem (8a) the objective of EE optimization ismodeled as

max119901119899

119880 (119875) =

1

((120572119875 + 119875119904) 119877 (119875)) + 120573

(D1)

where 119875 is the total transmit power and 119877(119901) is the corre-sponding data rate

Since 120572 gt 0 119877 gt 0 119875 gt 0 and 119875119904gt 0 the EE 119880 is a

strictly monotone decreasing function of 120572119875+119875119904119877Then the

equivalent optimization problem is reformulated as

max119901119899

119880 (119875) =

119877 (119875)

120572119875 + 119875119904

(D2)

For arbitrary 120579 isin (0 1) since 119877(119875) is concave function of119875 we have

119880 (1205791199091+ (1 minus 120579) 119909

2) ge

120579119877 (1199091) + (1 minus 120579) 119877 (119909

2)

120579 (1205721199091+ 119875119904) + (1 minus 120579) (120572119909

2+ 119875119904)

ge min (119880 (1199091) 119880 (119909

2))

(D3)

According to [13 14] we can conclude that119880 is quasicon-cave for 119875 ge 0 Since 119875 is a monotonic increasing function ofwater level for 120583 gt (1max119892

119899) Hence EE is a quasiconcave

function of water level

Conflict of Interests

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

Acknowledgment

The work in this paper has been supported by fundingfrom Beijing Natural Science Foundation under Grant no4122009

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] S Srinivasa and S A Jafar ldquoThe throughput potential of cog-nitive radio a theoretical perspectiverdquo IEEE CommunicationsMagazine vol 45 no 5 pp 73ndash79 2007

[3] T A Weiss and F K Jondral ldquoSpectrum pooling an innovativestrategy for the enhancement of spectrum efficiencyrdquo IEEECommunications Magazine vol 42 no 3 pp S8ndashS14 2004

[4] R Zhang ldquoOn peak versus average interference power con-straints for protecting primary users in cognitive radio net-worksrdquo IEEE Transactions on Wireless Communications vol 8no 4 pp 2112ndash2120 2009

[5] K Son B C Jung S Chong and D K Sung ldquoPower allocationfor OFDM-based cognitive radio systems under outage con-straintsrdquo in Proceedings of the IEEE International Conference onCommunications (ICC rsquo10) pp 1ndash5 May 2010

[6] X Zhou B Wu P-H Ho and X Ling ldquoAn efficient powerallocation algorithm for OFDM based underlay cognitive radionetworksrdquo in Proceedings of the 54th Annual IEEE GlobalTelecommunications Conference (GLOBECOM rsquo11) pp 1ndash5December 2011

[7] 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

[8] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[9] G Ding Q Wu N Min and G Zhou ldquoAdaptive resourceallocation in multi-user OFDM-based cognitive radio systemsrdquoin Proceedings of the IEEE International Conference on WirelessCommunications and Signal Processing (WCSP rsquo09) November2009

[10] D Feng C Jiang G Lim L J Cimini Jr G Feng and G Y LildquoA survey of energy-efficient wireless communicationsrdquo IEEECommunications Surveys and Tutorials vol 15 no 1 pp 167ndash178 2013

[11] Y Li X Zhou H Xu G Y Li D Wang and A SoongldquoEnergy-efficient transmission in cognitive radio networksrdquo inProceedings of the 7th IEEE Consumer Communications andNetworking Conference (CCNC rsquo10) January 2010

[12] M L Treust and S Lasaulce ldquoA repeated game formulationof energy-efficient decentralized power controlrdquo IEEE Transac-tions on Wireless Communications vol 9 no 9 pp 2860ndash28692010

[13] J Mao G Xie J Gao and Y Liu ldquoEnergy efficiency optimiza-tion for ofdm-based cognitive radio systems a water-fillingfactor aided search methodrdquo IEEE Transactions on WirelessCommunications vol 12 no 5 pp 2366ndash2375 2013

[14] Y Chen Z Zheng Y Hou and Y Li ldquoEnergy efficient designfor OFDM-based underlay cognitive radio networksrdquo Mathe-matical Problems in Engineering vol 2014 Article ID 431878 7pages 2014

[15] K Fazel and S Kaiser Multi-Carrier and Spread SpectrumSystems John Wiley amp Sons Chichester UK 2003

[16] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert W functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

3 Energy Efficient Design

In the following we first introduce two auxiliary optimizationproblems and then propose an optimal energy efficientpower allocation algorithm to achieve the maximum EEperformance

Define a rate maximization (Max119877) problem P2 as

P2 max119901119899

119877 (9a)

st 119875 le (9b)

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (9c)

Since the overall data rate is an increasing function oftransmit power 119875 it is shown that P2 has a unique solutionfor all 119875 ge 0

The other conventional optimization problem is transmitpower minimization (Min119875) problem P3 subject to data rateconstraint and AIP outage constraint It can be described asfollows

P3 min119901119899

119875 (10a)

st 119877 ge (10b)

119875119900= Pr[ 1

119873

sum

119899isinN

1003816100381610038161003816ℎ119904119901

119899

1003816100381610038161003816

2

119901119899ge 119868th] le 120598 (10c)

Referring to [5] in a practical system with a suffi-ciently large number of subchannels random variable 119883 =

sum119899|ℎ119904119901

119899|2

119901119899can be modeled as a normal distributed variable

based on central limit theorem with mean 119898119900and variance

1205902

119900

119898119900= sum

119899isinN

1199011198991205822 (11)

1205902

119900= sum

119899isinN

(1199011198991205822)2

(12)

respectivelyThen the equivalent constraint of (7) can be expressed as

119875119900=

1

2

erfc(119873119868th minus 119898119900radic2120590119900

) le 120598 (13)

where erfc(119909) = (2radic120587) intinfin119909

119890minus1199052

119889119905If the transmit power 119875 is given the objective of problem

P1 is equivalent to problem P2 and the existing water-fillingpower allocation scheme can be used [16] However besidesadapting the power loading on all subcarriers the overalltransmit power can also be adapted according to the CSIand AIP level to maximize the EE Hence the existing powerallocation schemes cannot be applied directly

Without loss of generality assume that 1198921ge 1198922ge sdot sdot sdot ge

119892119873 It is noticeable that the exact power allocation solution

for P2 is too difficult to address However if we drop the

Step 1 Initialization 120583119900larr 0119873 larr 1 Slarr

Step 2 Calculate 120583119900according to (15)

Step 3 Update Slarr 119899 | 120583119900gt 1119892

119899 forall119899 isinN

Step 4 If |S| = 119873 then terminate the algorithm elseset119873 larr |S| and go to Step 2

Algorithm 1 Solving 120583119900

AIP constraint from P2 it can be solved by the water-fillingalgorithm and the power distribution is

119901119899= [120583 minus

1

119892119899

]

+

(14)

for all 119899 isinN where [119909]+ = max(119909 0) and 120583 is the water level

Lemma 1 The 119875119900is a monotonic increasing function of water

level 120583 for 120583 ge 11198921 and the maximum allowable water level

120583119900 for 119875

119900le 120598 is

120583119900=

minus119887 + radic1198872minus 4119886119888

2119886

(15)

where 119886 = (21198892

119873 minus 119873

2

)1205822

2 119887 = 2119873119868

119905ℎ1198731205822+ (2119873 minus

41198892

)11986611205822

2 119888 = 2119866

21205822

21198892

minus (119873119868119905ℎ+ 12058221198661)2 1198661

=

sum119873

119899=1(1119892119899) 1198662= sum119873

119899=1(11198922

119899) 119889 = erfcminus1(2120598) erfcminus1(sdot) is the

inverse function of erfc(sdot) and119873 is the number of subchannelswith transmit power 119901

119899gt 0

Proof Please see Appendix A

In order to protect PU the maximum allowable waterlevel is restricted which may lead the power of some subcar-riers equal to zero for special channel realizations Here wepresent an efficient algorithm to solve 120583

119900 See Algorithm 1

Lemma2 For problemP2without AIP constraint the optimalwater level 120583 to maximize system rate under the total transmitpower is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (16)

Proof Please see Appendix B

Lemma3 For problemP3without AIP constraint the optimalwater level to meet the minimum rate requirement is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(17)

Proof Please see Appendix C

With the help of Lemmas 1 to 3 the optimal water levels120583lowast and lowast for problems P2 and P3 are

120583lowast

= min (120583 120583119900) (18)

lowast

= min ( 120583119900) (19)

respectively

4 Mathematical Problems in Engineering

Step 1 Initialization 120583lowast larr 0Step 2 Calculate lowast and 120583lowast according to (18) and (19)Step 3 If 120583lowast le 120583

lowast then 120583mid larr 120583lowast and go to Step 7 else set 120583low larr

lowast and 120583high larr 120583lowast respectively

Step 4 Set 120583mid larr 12 (120583low + 120583high)Step 5 If its first derivative with respect to water level d119880d120583|

120583=120583midgt 0 then 120583low larr 120583mid else 120583high larr 120583mid

Step 6 Repeat Step 4 until 120583high minus 120583low le 120585 where 120585 is a small positive constant to control the convergence accuracyStep 7 Finish the optimal solution is 120583lowast larr 120583mid

Algorithm 2 Bisection search aided algorithm (BSAA)

Since the system rate 119877 is a strictly increasing function oftransmit power119875 it is shown that119877 has a unique value for all120583 ge 1119892

1 Hence the equivalent problem of P1 can be given

as

P4 max120583

119880 (120583) =

119877

120572119875 + 119875119888

(20a)

st lowast

le 120583 le 120583lowast

(20b)

Lemma 4 The system EE 119880 is a quasiconcave function ofwater level 120583 for 120583 ge 1119892

1

Proof Please see Appendix D

According to Lemma 4 a unique global optimal waterlevel 120583 always exists without any constraint in P4 Hence theoptimal water level 120583lowast is

120583lowast

= min (120583lowastmax (120583 lowast)) (21)

31 Bisection Search Aided Algorithm (BSAA) The remainingthing for P1 is to address 120583lowast and it can be solved by bisectionsearch technique Here we list the power allocation methodin Algorithm 2

Our goal aims to maximize the system EE under theconstraints on transmit power the minimum rate require-ment and AIP outage The water-filling based algorithm isproposed to obtain the optimal solution The key work ofAlgorithm 2 is to determine the sign of d119880d120583 It can bedetermined by calculating the value of 119880(120583 + Δ120583) minus 119880(120583)where Δ120583 is an infinitely small positive constant

32 Simplified Energy Efficient Power Allocation (SiPA)Although the suboptimal power allocation solution is desir-able the cost to pay is the computational complexity initerative search It takes at most lceillog

2((120583lowast

minus lowast

)120585)rceil iterationsto convergence where lceil119909rceil denotes the smallest integer notless than 119909 Thus this straightforward approach is clearly notpractical andwe need a low complexity approach to solve thisproblem

Since EE119880 is a quasiconcave function of water level 120583 theoptimal unconstrained water level always exists Differentiat-ing (20a) with respect to 120583 and setting the derivative to zerowe can obtain

120597119880

120597120583

= 119865(1 +

119875119904

120572120583

minus

1198663

120583

minus ln (1205831198664)) = 0 (22)

or

1198751199041198664

120572119890

minus

11986631198664

119890

=

1205831198664

119890

ln(1205831198664

119890

) (23)

where 119890 is the base of the natural logarithm 119865 is a nonzerovalue independent of 120583 and 119866

3and 119866

4are defined as

1198663=

1

sum

119899=1

1

119892119899

(24)

1198664= (

prod

119899=1

119892119899)

1

(25)

respectively Let 119883 = ln(1205831198664119890) and 119884 = ((119875

1199041198664)(120572119890)) minus

((11986631198664)119890) then (23) can be expressed as

119883119890119883

= 119884 (26)

Its solution is 119883 = 1198820(119884) where 119882

0(sdot) denotes the

real branch of the Lambert function [13] Substituting thissolution into (26) the global optimal water level 120583 withoutany constraint is

120583 =

119890

1198664

exp [1198820(

1198751199041198664

120572119890

minus

11986631198664

119890

)] (27)

According to Lemma 4 the optimal water level 120583lowast corre-sponding to P1 can be obtained by substituting 120583 into (21)

4 Numerical Results

Simulations have been performed to evaluate the perfor-mance of our proposed algorithms The channel gains |ℎ119904119904

119899|2

and |ℎ119904119901119899|2 (119899 isin N) are independent chi-square distributed

random variables Simulation parameters are set as follows119861 = 15 kHz119873 = 512 120582 = 120582

11205822(we fix 120582

2= 1 and vary 120582

1)

= 3W 1205902 = 001W 119868th = 1205881205902 120598 = 005 119875

119904= 08W 120572 = 2

= 23Mbps and 120573 = 02WMbps The simulation curvesare obtained by averaging over 1000 channel realizations

In Figure 1 we compare the average EE of various powerallocation algorithms with 120588 = 05 Compared with otheralgorithms our proposed algorithms can achieve significantperformance gain in terms of EE The BSAA and SiPA havethe same performance and both of them always performbetter than the simplified WFAS algorithm in [13] at the cost

Mathematical Problems in Engineering 5

4

3

2

1

00 5 10 15 20

120582 (dB)

EE (M

bits

J)

MinPMaxRBSAA

SiPASimplified WFAS in [13]

Figure 1 System EE versus average CNR 120588 = 05

MinPMaxRBSAA

SiPASimplified WFAS in [13]

25

2

15

1

05

00 01 02 03 04

120588

EE (M

bits

J)

Figure 2 System EE versus average CNR 120582 = 10 dB

of serious interference to PU In the low 120582 region the transmitpower dominates the total power consumption while in thehigh 120582 region the circuit power does for the energy efficientdesign However the gap between proposed algorithms andconventional ones becomes larger as 120582 keeps increasing

In Figure 2 the average EE dramatically increases as 120588increases in the low region However when 120588 is greaterthan a saturation point the EE does not further increasefor energy efficient power allocation In addition it canbe observed that all schemes may have to operate exactlyat the maximum allowable transmit power to meet the

minimum rate requirement when 120588 is much smaller than thethreshold

5 Conclusion

In this paper we considered the energy efficient design forOFDM-based CR systems under the AIP outage probabilityconstraint and minimum rate requirement constraint Dueto the loose cooperation the intersystem CSI is partiallyknown to the SU-Tx To solve this problem we proposed asuboptimal algorithm based on bisection search techniqueFurthermore in order to reduce the computational com-plexity we also derived a closed-form solution for simplifiedalgorithm without performance loss We demonstrate thesuperior EE performance of the proposed algorithms byextensive simulations

Appendices

A Proof of Lemma 1

From (11) (12) and (14) (13) can be written as

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583 minus (1119892

119899))

1205822

radic2sum

119899=1(120583 minus (1119892

119899))2

) le 120598 (A1)

where 120583 gt 1119892119899 for 119899 = 1 2 and 120583 le 1119892

119899 for 119899 =

+ 1 119873It can be easily proved that 119875

119900is a monotonic increasing

function of 120583 Let 120583119900be the maximum allowable power level

corresponding to AIP constraint we have

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583119900minus (1119892

119899))

1205822

radic2sum

119899=1(120583119900minus (1119892

119899))2

) = 120598 (A2)

After somemathematical manipulation a single equationwith the variable 120583

119900can be derived as

119886120583119900

2

+ 119887120583119900+ 119888 = 0 (A3)

where 119886 = (21198892

119873 minus 119873

2

)1205822

2 119887 = 2119873119868th1198731205822 + (2119873 minus

41198892

)11986611205822

2 119888 = 2119866

21205822

21198892

minus (119873119868th + 12058221198661)2 1198661= sum119873

119899=1(1119892119899)

1198662= sum119873

119899=1(11198922

119899) 119889 = erfcminus1(2120598) and erfcminus1(sdot) is the

inverse function of erfc(sdot) When solving the above equationthe maximum allowable power level corresponding to AIPconstraint is

120583119900=

minus119887 + radic1198872minus 4119886119888

2119886

(A4)

Here Lemma 1 is proved

6 Mathematical Problems in Engineering

B Proof of Lemma 2

For a given allowable transmit power the optimal powerallocation can be achieved by classical water-filling methodThe true power level is assumed to be 120583 we have

=

119873

sum

119899=1

119901119899=

119872

sum

119899=1

(120583 minus

1

119892119899

) = 119872120583 minus

119872

sum

119899=1

1

119892119899

(B1)

or

120583 =

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B2)

where 119872 le 119873 119901119899= 120583 minus (1119892

119899) gt 0 119899 = 1 2 119872 and

120583 le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

(1119872119894)( + sum

119872119894

119899=1(1119892119899)) 119894 = 1 2 Because 120583 gt (1119892

119899) 119899 =

1 119872 and 120583 le (1119892119899) 119899 = 119872+1 119873 in (B2) we derive

the results that 1205831gt 120583 and 120583

2gt 120583 as follows

1205831minus 120583 =

1

1198721

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198721

sum

119899=1

1

119892119899

) minus 120583

=

1

1198721

[(119872 minus1198721) 120583 minus

119872

sum

119899=1198721+1

1

119892119899

]

=

1

1198721

119872

sum

119899=1198721+1

(120583 minus

1

119892119899

)

gt 0

1205832minus 120583 =

1

1198722

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198722

sum

119899=1

1

119892119899

) minus 120583

=

1

1198722

[minus (1198722minus119872)120583 +

1198722

sum

119899=119872+1

1

119892119899

]

= minus

1

1198722

1198722

sum

119899=119872+1

(120583 minus

1

119892119899

)

ge 0

(B3)

Hence the true power level 120583 corresponding to themaximum allowable transmit power is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B4)

Here Lemma 2 is proved

C Proof of Lemma 3

Let be the optimal water level corresponding to theminimum rate requirement without AIP constraint wehave

= 119861

119873

sum

119899=1

log2(1 + 119892

119899119901119899)

= 119861

119872

sum

119899=1

log2(1 + ( minus

1

119892119899

)119892119899)

= 119861log2(119872

119872

prod

119899=1

119892119899)

(C1)

or

= 2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C2)

where 119872 le 119873 119901119899= minus (1119892

119899) gt 0 119899 = 1 2 119872 and

le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

2(119861119872

119894)

(prod119872119894

119899=1119892119899)minus1119872

119894 119894 = 1 2 It can be seen that 119892119899gt 1

119899 = 1 119872 and 119892119899le 1 119899 = 119872 + 1 119873 We derive the

results that 1205831gt and 120583

2gt as follows

1205831

=

2(119861119872

1)

(prod1198721

119899=1119892119899)

minus11198721

=

(119872

prod119872

119899=1119892119899prod1198721

119899=11119892119899)

11198721

= (

119872

prod

119899=1198721+1

119892119899)

11198721

gt 1

1205832

=

2(119861119872

2)

(prod1198722

119899=1119892119899)

minus11198722

=

(119872

prod119872

119899=1119892119899prod1198722

119899=11119892119899)

11198722

= (

1198722

prod

119899=119872+1

1

119892119899

)

11198722

ge 1

(C3)

Since 120583119894gt 0 (119894 = 1 2) the optimal power level that

satisfies 119877 = without the AIP constraint is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C4)

Here Lemma 3 is proved

Mathematical Problems in Engineering 7

D Proof of Lemma 4

From problem (8a) the objective of EE optimization ismodeled as

max119901119899

119880 (119875) =

1

((120572119875 + 119875119904) 119877 (119875)) + 120573

(D1)

where 119875 is the total transmit power and 119877(119901) is the corre-sponding data rate

Since 120572 gt 0 119877 gt 0 119875 gt 0 and 119875119904gt 0 the EE 119880 is a

strictly monotone decreasing function of 120572119875+119875119904119877Then the

equivalent optimization problem is reformulated as

max119901119899

119880 (119875) =

119877 (119875)

120572119875 + 119875119904

(D2)

For arbitrary 120579 isin (0 1) since 119877(119875) is concave function of119875 we have

119880 (1205791199091+ (1 minus 120579) 119909

2) ge

120579119877 (1199091) + (1 minus 120579) 119877 (119909

2)

120579 (1205721199091+ 119875119904) + (1 minus 120579) (120572119909

2+ 119875119904)

ge min (119880 (1199091) 119880 (119909

2))

(D3)

According to [13 14] we can conclude that119880 is quasicon-cave for 119875 ge 0 Since 119875 is a monotonic increasing function ofwater level for 120583 gt (1max119892

119899) Hence EE is a quasiconcave

function of water level

Conflict of Interests

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

Acknowledgment

The work in this paper has been supported by fundingfrom Beijing Natural Science Foundation under Grant no4122009

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] S Srinivasa and S A Jafar ldquoThe throughput potential of cog-nitive radio a theoretical perspectiverdquo IEEE CommunicationsMagazine vol 45 no 5 pp 73ndash79 2007

[3] T A Weiss and F K Jondral ldquoSpectrum pooling an innovativestrategy for the enhancement of spectrum efficiencyrdquo IEEECommunications Magazine vol 42 no 3 pp S8ndashS14 2004

[4] R Zhang ldquoOn peak versus average interference power con-straints for protecting primary users in cognitive radio net-worksrdquo IEEE Transactions on Wireless Communications vol 8no 4 pp 2112ndash2120 2009

[5] K Son B C Jung S Chong and D K Sung ldquoPower allocationfor OFDM-based cognitive radio systems under outage con-straintsrdquo in Proceedings of the IEEE International Conference onCommunications (ICC rsquo10) pp 1ndash5 May 2010

[6] X Zhou B Wu P-H Ho and X Ling ldquoAn efficient powerallocation algorithm for OFDM based underlay cognitive radionetworksrdquo in Proceedings of the 54th Annual IEEE GlobalTelecommunications Conference (GLOBECOM rsquo11) pp 1ndash5December 2011

[7] 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

[8] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[9] G Ding Q Wu N Min and G Zhou ldquoAdaptive resourceallocation in multi-user OFDM-based cognitive radio systemsrdquoin Proceedings of the IEEE International Conference on WirelessCommunications and Signal Processing (WCSP rsquo09) November2009

[10] D Feng C Jiang G Lim L J Cimini Jr G Feng and G Y LildquoA survey of energy-efficient wireless communicationsrdquo IEEECommunications Surveys and Tutorials vol 15 no 1 pp 167ndash178 2013

[11] Y Li X Zhou H Xu G Y Li D Wang and A SoongldquoEnergy-efficient transmission in cognitive radio networksrdquo inProceedings of the 7th IEEE Consumer Communications andNetworking Conference (CCNC rsquo10) January 2010

[12] M L Treust and S Lasaulce ldquoA repeated game formulationof energy-efficient decentralized power controlrdquo IEEE Transac-tions on Wireless Communications vol 9 no 9 pp 2860ndash28692010

[13] J Mao G Xie J Gao and Y Liu ldquoEnergy efficiency optimiza-tion for ofdm-based cognitive radio systems a water-fillingfactor aided search methodrdquo IEEE Transactions on WirelessCommunications vol 12 no 5 pp 2366ndash2375 2013

[14] Y Chen Z Zheng Y Hou and Y Li ldquoEnergy efficient designfor OFDM-based underlay cognitive radio networksrdquo Mathe-matical Problems in Engineering vol 2014 Article ID 431878 7pages 2014

[15] K Fazel and S Kaiser Multi-Carrier and Spread SpectrumSystems John Wiley amp Sons Chichester UK 2003

[16] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert W functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Step 1 Initialization 120583lowast larr 0Step 2 Calculate lowast and 120583lowast according to (18) and (19)Step 3 If 120583lowast le 120583

lowast then 120583mid larr 120583lowast and go to Step 7 else set 120583low larr

lowast and 120583high larr 120583lowast respectively

Step 4 Set 120583mid larr 12 (120583low + 120583high)Step 5 If its first derivative with respect to water level d119880d120583|

120583=120583midgt 0 then 120583low larr 120583mid else 120583high larr 120583mid

Step 6 Repeat Step 4 until 120583high minus 120583low le 120585 where 120585 is a small positive constant to control the convergence accuracyStep 7 Finish the optimal solution is 120583lowast larr 120583mid

Algorithm 2 Bisection search aided algorithm (BSAA)

Since the system rate 119877 is a strictly increasing function oftransmit power119875 it is shown that119877 has a unique value for all120583 ge 1119892

1 Hence the equivalent problem of P1 can be given

as

P4 max120583

119880 (120583) =

119877

120572119875 + 119875119888

(20a)

st lowast

le 120583 le 120583lowast

(20b)

Lemma 4 The system EE 119880 is a quasiconcave function ofwater level 120583 for 120583 ge 1119892

1

Proof Please see Appendix D

According to Lemma 4 a unique global optimal waterlevel 120583 always exists without any constraint in P4 Hence theoptimal water level 120583lowast is

120583lowast

= min (120583lowastmax (120583 lowast)) (21)

31 Bisection Search Aided Algorithm (BSAA) The remainingthing for P1 is to address 120583lowast and it can be solved by bisectionsearch technique Here we list the power allocation methodin Algorithm 2

Our goal aims to maximize the system EE under theconstraints on transmit power the minimum rate require-ment and AIP outage The water-filling based algorithm isproposed to obtain the optimal solution The key work ofAlgorithm 2 is to determine the sign of d119880d120583 It can bedetermined by calculating the value of 119880(120583 + Δ120583) minus 119880(120583)where Δ120583 is an infinitely small positive constant

32 Simplified Energy Efficient Power Allocation (SiPA)Although the suboptimal power allocation solution is desir-able the cost to pay is the computational complexity initerative search It takes at most lceillog

2((120583lowast

minus lowast

)120585)rceil iterationsto convergence where lceil119909rceil denotes the smallest integer notless than 119909 Thus this straightforward approach is clearly notpractical andwe need a low complexity approach to solve thisproblem

Since EE119880 is a quasiconcave function of water level 120583 theoptimal unconstrained water level always exists Differentiat-ing (20a) with respect to 120583 and setting the derivative to zerowe can obtain

120597119880

120597120583

= 119865(1 +

119875119904

120572120583

minus

1198663

120583

minus ln (1205831198664)) = 0 (22)

or

1198751199041198664

120572119890

minus

11986631198664

119890

=

1205831198664

119890

ln(1205831198664

119890

) (23)

where 119890 is the base of the natural logarithm 119865 is a nonzerovalue independent of 120583 and 119866

3and 119866

4are defined as

1198663=

1

sum

119899=1

1

119892119899

(24)

1198664= (

prod

119899=1

119892119899)

1

(25)

respectively Let 119883 = ln(1205831198664119890) and 119884 = ((119875

1199041198664)(120572119890)) minus

((11986631198664)119890) then (23) can be expressed as

119883119890119883

= 119884 (26)

Its solution is 119883 = 1198820(119884) where 119882

0(sdot) denotes the

real branch of the Lambert function [13] Substituting thissolution into (26) the global optimal water level 120583 withoutany constraint is

120583 =

119890

1198664

exp [1198820(

1198751199041198664

120572119890

minus

11986631198664

119890

)] (27)

According to Lemma 4 the optimal water level 120583lowast corre-sponding to P1 can be obtained by substituting 120583 into (21)

4 Numerical Results

Simulations have been performed to evaluate the perfor-mance of our proposed algorithms The channel gains |ℎ119904119904

119899|2

and |ℎ119904119901119899|2 (119899 isin N) are independent chi-square distributed

random variables Simulation parameters are set as follows119861 = 15 kHz119873 = 512 120582 = 120582

11205822(we fix 120582

2= 1 and vary 120582

1)

= 3W 1205902 = 001W 119868th = 1205881205902 120598 = 005 119875

119904= 08W 120572 = 2

= 23Mbps and 120573 = 02WMbps The simulation curvesare obtained by averaging over 1000 channel realizations

In Figure 1 we compare the average EE of various powerallocation algorithms with 120588 = 05 Compared with otheralgorithms our proposed algorithms can achieve significantperformance gain in terms of EE The BSAA and SiPA havethe same performance and both of them always performbetter than the simplified WFAS algorithm in [13] at the cost

Mathematical Problems in Engineering 5

4

3

2

1

00 5 10 15 20

120582 (dB)

EE (M

bits

J)

MinPMaxRBSAA

SiPASimplified WFAS in [13]

Figure 1 System EE versus average CNR 120588 = 05

MinPMaxRBSAA

SiPASimplified WFAS in [13]

25

2

15

1

05

00 01 02 03 04

120588

EE (M

bits

J)

Figure 2 System EE versus average CNR 120582 = 10 dB

of serious interference to PU In the low 120582 region the transmitpower dominates the total power consumption while in thehigh 120582 region the circuit power does for the energy efficientdesign However the gap between proposed algorithms andconventional ones becomes larger as 120582 keeps increasing

In Figure 2 the average EE dramatically increases as 120588increases in the low region However when 120588 is greaterthan a saturation point the EE does not further increasefor energy efficient power allocation In addition it canbe observed that all schemes may have to operate exactlyat the maximum allowable transmit power to meet the

minimum rate requirement when 120588 is much smaller than thethreshold

5 Conclusion

In this paper we considered the energy efficient design forOFDM-based CR systems under the AIP outage probabilityconstraint and minimum rate requirement constraint Dueto the loose cooperation the intersystem CSI is partiallyknown to the SU-Tx To solve this problem we proposed asuboptimal algorithm based on bisection search techniqueFurthermore in order to reduce the computational com-plexity we also derived a closed-form solution for simplifiedalgorithm without performance loss We demonstrate thesuperior EE performance of the proposed algorithms byextensive simulations

Appendices

A Proof of Lemma 1

From (11) (12) and (14) (13) can be written as

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583 minus (1119892

119899))

1205822

radic2sum

119899=1(120583 minus (1119892

119899))2

) le 120598 (A1)

where 120583 gt 1119892119899 for 119899 = 1 2 and 120583 le 1119892

119899 for 119899 =

+ 1 119873It can be easily proved that 119875

119900is a monotonic increasing

function of 120583 Let 120583119900be the maximum allowable power level

corresponding to AIP constraint we have

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583119900minus (1119892

119899))

1205822

radic2sum

119899=1(120583119900minus (1119892

119899))2

) = 120598 (A2)

After somemathematical manipulation a single equationwith the variable 120583

119900can be derived as

119886120583119900

2

+ 119887120583119900+ 119888 = 0 (A3)

where 119886 = (21198892

119873 minus 119873

2

)1205822

2 119887 = 2119873119868th1198731205822 + (2119873 minus

41198892

)11986611205822

2 119888 = 2119866

21205822

21198892

minus (119873119868th + 12058221198661)2 1198661= sum119873

119899=1(1119892119899)

1198662= sum119873

119899=1(11198922

119899) 119889 = erfcminus1(2120598) and erfcminus1(sdot) is the

inverse function of erfc(sdot) When solving the above equationthe maximum allowable power level corresponding to AIPconstraint is

120583119900=

minus119887 + radic1198872minus 4119886119888

2119886

(A4)

Here Lemma 1 is proved

6 Mathematical Problems in Engineering

B Proof of Lemma 2

For a given allowable transmit power the optimal powerallocation can be achieved by classical water-filling methodThe true power level is assumed to be 120583 we have

=

119873

sum

119899=1

119901119899=

119872

sum

119899=1

(120583 minus

1

119892119899

) = 119872120583 minus

119872

sum

119899=1

1

119892119899

(B1)

or

120583 =

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B2)

where 119872 le 119873 119901119899= 120583 minus (1119892

119899) gt 0 119899 = 1 2 119872 and

120583 le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

(1119872119894)( + sum

119872119894

119899=1(1119892119899)) 119894 = 1 2 Because 120583 gt (1119892

119899) 119899 =

1 119872 and 120583 le (1119892119899) 119899 = 119872+1 119873 in (B2) we derive

the results that 1205831gt 120583 and 120583

2gt 120583 as follows

1205831minus 120583 =

1

1198721

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198721

sum

119899=1

1

119892119899

) minus 120583

=

1

1198721

[(119872 minus1198721) 120583 minus

119872

sum

119899=1198721+1

1

119892119899

]

=

1

1198721

119872

sum

119899=1198721+1

(120583 minus

1

119892119899

)

gt 0

1205832minus 120583 =

1

1198722

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198722

sum

119899=1

1

119892119899

) minus 120583

=

1

1198722

[minus (1198722minus119872)120583 +

1198722

sum

119899=119872+1

1

119892119899

]

= minus

1

1198722

1198722

sum

119899=119872+1

(120583 minus

1

119892119899

)

ge 0

(B3)

Hence the true power level 120583 corresponding to themaximum allowable transmit power is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B4)

Here Lemma 2 is proved

C Proof of Lemma 3

Let be the optimal water level corresponding to theminimum rate requirement without AIP constraint wehave

= 119861

119873

sum

119899=1

log2(1 + 119892

119899119901119899)

= 119861

119872

sum

119899=1

log2(1 + ( minus

1

119892119899

)119892119899)

= 119861log2(119872

119872

prod

119899=1

119892119899)

(C1)

or

= 2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C2)

where 119872 le 119873 119901119899= minus (1119892

119899) gt 0 119899 = 1 2 119872 and

le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

2(119861119872

119894)

(prod119872119894

119899=1119892119899)minus1119872

119894 119894 = 1 2 It can be seen that 119892119899gt 1

119899 = 1 119872 and 119892119899le 1 119899 = 119872 + 1 119873 We derive the

results that 1205831gt and 120583

2gt as follows

1205831

=

2(119861119872

1)

(prod1198721

119899=1119892119899)

minus11198721

=

(119872

prod119872

119899=1119892119899prod1198721

119899=11119892119899)

11198721

= (

119872

prod

119899=1198721+1

119892119899)

11198721

gt 1

1205832

=

2(119861119872

2)

(prod1198722

119899=1119892119899)

minus11198722

=

(119872

prod119872

119899=1119892119899prod1198722

119899=11119892119899)

11198722

= (

1198722

prod

119899=119872+1

1

119892119899

)

11198722

ge 1

(C3)

Since 120583119894gt 0 (119894 = 1 2) the optimal power level that

satisfies 119877 = without the AIP constraint is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C4)

Here Lemma 3 is proved

Mathematical Problems in Engineering 7

D Proof of Lemma 4

From problem (8a) the objective of EE optimization ismodeled as

max119901119899

119880 (119875) =

1

((120572119875 + 119875119904) 119877 (119875)) + 120573

(D1)

where 119875 is the total transmit power and 119877(119901) is the corre-sponding data rate

Since 120572 gt 0 119877 gt 0 119875 gt 0 and 119875119904gt 0 the EE 119880 is a

strictly monotone decreasing function of 120572119875+119875119904119877Then the

equivalent optimization problem is reformulated as

max119901119899

119880 (119875) =

119877 (119875)

120572119875 + 119875119904

(D2)

For arbitrary 120579 isin (0 1) since 119877(119875) is concave function of119875 we have

119880 (1205791199091+ (1 minus 120579) 119909

2) ge

120579119877 (1199091) + (1 minus 120579) 119877 (119909

2)

120579 (1205721199091+ 119875119904) + (1 minus 120579) (120572119909

2+ 119875119904)

ge min (119880 (1199091) 119880 (119909

2))

(D3)

According to [13 14] we can conclude that119880 is quasicon-cave for 119875 ge 0 Since 119875 is a monotonic increasing function ofwater level for 120583 gt (1max119892

119899) Hence EE is a quasiconcave

function of water level

Conflict of Interests

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

Acknowledgment

The work in this paper has been supported by fundingfrom Beijing Natural Science Foundation under Grant no4122009

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] S Srinivasa and S A Jafar ldquoThe throughput potential of cog-nitive radio a theoretical perspectiverdquo IEEE CommunicationsMagazine vol 45 no 5 pp 73ndash79 2007

[3] T A Weiss and F K Jondral ldquoSpectrum pooling an innovativestrategy for the enhancement of spectrum efficiencyrdquo IEEECommunications Magazine vol 42 no 3 pp S8ndashS14 2004

[4] R Zhang ldquoOn peak versus average interference power con-straints for protecting primary users in cognitive radio net-worksrdquo IEEE Transactions on Wireless Communications vol 8no 4 pp 2112ndash2120 2009

[5] K Son B C Jung S Chong and D K Sung ldquoPower allocationfor OFDM-based cognitive radio systems under outage con-straintsrdquo in Proceedings of the IEEE International Conference onCommunications (ICC rsquo10) pp 1ndash5 May 2010

[6] X Zhou B Wu P-H Ho and X Ling ldquoAn efficient powerallocation algorithm for OFDM based underlay cognitive radionetworksrdquo in Proceedings of the 54th Annual IEEE GlobalTelecommunications Conference (GLOBECOM rsquo11) pp 1ndash5December 2011

[7] 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

[8] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[9] G Ding Q Wu N Min and G Zhou ldquoAdaptive resourceallocation in multi-user OFDM-based cognitive radio systemsrdquoin Proceedings of the IEEE International Conference on WirelessCommunications and Signal Processing (WCSP rsquo09) November2009

[10] D Feng C Jiang G Lim L J Cimini Jr G Feng and G Y LildquoA survey of energy-efficient wireless communicationsrdquo IEEECommunications Surveys and Tutorials vol 15 no 1 pp 167ndash178 2013

[11] Y Li X Zhou H Xu G Y Li D Wang and A SoongldquoEnergy-efficient transmission in cognitive radio networksrdquo inProceedings of the 7th IEEE Consumer Communications andNetworking Conference (CCNC rsquo10) January 2010

[12] M L Treust and S Lasaulce ldquoA repeated game formulationof energy-efficient decentralized power controlrdquo IEEE Transac-tions on Wireless Communications vol 9 no 9 pp 2860ndash28692010

[13] J Mao G Xie J Gao and Y Liu ldquoEnergy efficiency optimiza-tion for ofdm-based cognitive radio systems a water-fillingfactor aided search methodrdquo IEEE Transactions on WirelessCommunications vol 12 no 5 pp 2366ndash2375 2013

[14] Y Chen Z Zheng Y Hou and Y Li ldquoEnergy efficient designfor OFDM-based underlay cognitive radio networksrdquo Mathe-matical Problems in Engineering vol 2014 Article ID 431878 7pages 2014

[15] K Fazel and S Kaiser Multi-Carrier and Spread SpectrumSystems John Wiley amp Sons Chichester UK 2003

[16] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert W functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

4

3

2

1

00 5 10 15 20

120582 (dB)

EE (M

bits

J)

MinPMaxRBSAA

SiPASimplified WFAS in [13]

Figure 1 System EE versus average CNR 120588 = 05

MinPMaxRBSAA

SiPASimplified WFAS in [13]

25

2

15

1

05

00 01 02 03 04

120588

EE (M

bits

J)

Figure 2 System EE versus average CNR 120582 = 10 dB

of serious interference to PU In the low 120582 region the transmitpower dominates the total power consumption while in thehigh 120582 region the circuit power does for the energy efficientdesign However the gap between proposed algorithms andconventional ones becomes larger as 120582 keeps increasing

In Figure 2 the average EE dramatically increases as 120588increases in the low region However when 120588 is greaterthan a saturation point the EE does not further increasefor energy efficient power allocation In addition it canbe observed that all schemes may have to operate exactlyat the maximum allowable transmit power to meet the

minimum rate requirement when 120588 is much smaller than thethreshold

5 Conclusion

In this paper we considered the energy efficient design forOFDM-based CR systems under the AIP outage probabilityconstraint and minimum rate requirement constraint Dueto the loose cooperation the intersystem CSI is partiallyknown to the SU-Tx To solve this problem we proposed asuboptimal algorithm based on bisection search techniqueFurthermore in order to reduce the computational com-plexity we also derived a closed-form solution for simplifiedalgorithm without performance loss We demonstrate thesuperior EE performance of the proposed algorithms byextensive simulations

Appendices

A Proof of Lemma 1

From (11) (12) and (14) (13) can be written as

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583 minus (1119892

119899))

1205822

radic2sum

119899=1(120583 minus (1119892

119899))2

) le 120598 (A1)

where 120583 gt 1119892119899 for 119899 = 1 2 and 120583 le 1119892

119899 for 119899 =

+ 1 119873It can be easily proved that 119875

119900is a monotonic increasing

function of 120583 Let 120583119900be the maximum allowable power level

corresponding to AIP constraint we have

119875119900=

1

2

erfc(119873119868th minus 1205822sum

119899=1(120583119900minus (1119892

119899))

1205822

radic2sum

119899=1(120583119900minus (1119892

119899))2

) = 120598 (A2)

After somemathematical manipulation a single equationwith the variable 120583

119900can be derived as

119886120583119900

2

+ 119887120583119900+ 119888 = 0 (A3)

where 119886 = (21198892

119873 minus 119873

2

)1205822

2 119887 = 2119873119868th1198731205822 + (2119873 minus

41198892

)11986611205822

2 119888 = 2119866

21205822

21198892

minus (119873119868th + 12058221198661)2 1198661= sum119873

119899=1(1119892119899)

1198662= sum119873

119899=1(11198922

119899) 119889 = erfcminus1(2120598) and erfcminus1(sdot) is the

inverse function of erfc(sdot) When solving the above equationthe maximum allowable power level corresponding to AIPconstraint is

120583119900=

minus119887 + radic1198872minus 4119886119888

2119886

(A4)

Here Lemma 1 is proved

6 Mathematical Problems in Engineering

B Proof of Lemma 2

For a given allowable transmit power the optimal powerallocation can be achieved by classical water-filling methodThe true power level is assumed to be 120583 we have

=

119873

sum

119899=1

119901119899=

119872

sum

119899=1

(120583 minus

1

119892119899

) = 119872120583 minus

119872

sum

119899=1

1

119892119899

(B1)

or

120583 =

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B2)

where 119872 le 119873 119901119899= 120583 minus (1119892

119899) gt 0 119899 = 1 2 119872 and

120583 le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

(1119872119894)( + sum

119872119894

119899=1(1119892119899)) 119894 = 1 2 Because 120583 gt (1119892

119899) 119899 =

1 119872 and 120583 le (1119892119899) 119899 = 119872+1 119873 in (B2) we derive

the results that 1205831gt 120583 and 120583

2gt 120583 as follows

1205831minus 120583 =

1

1198721

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198721

sum

119899=1

1

119892119899

) minus 120583

=

1

1198721

[(119872 minus1198721) 120583 minus

119872

sum

119899=1198721+1

1

119892119899

]

=

1

1198721

119872

sum

119899=1198721+1

(120583 minus

1

119892119899

)

gt 0

1205832minus 120583 =

1

1198722

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198722

sum

119899=1

1

119892119899

) minus 120583

=

1

1198722

[minus (1198722minus119872)120583 +

1198722

sum

119899=119872+1

1

119892119899

]

= minus

1

1198722

1198722

sum

119899=119872+1

(120583 minus

1

119892119899

)

ge 0

(B3)

Hence the true power level 120583 corresponding to themaximum allowable transmit power is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B4)

Here Lemma 2 is proved

C Proof of Lemma 3

Let be the optimal water level corresponding to theminimum rate requirement without AIP constraint wehave

= 119861

119873

sum

119899=1

log2(1 + 119892

119899119901119899)

= 119861

119872

sum

119899=1

log2(1 + ( minus

1

119892119899

)119892119899)

= 119861log2(119872

119872

prod

119899=1

119892119899)

(C1)

or

= 2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C2)

where 119872 le 119873 119901119899= minus (1119892

119899) gt 0 119899 = 1 2 119872 and

le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

2(119861119872

119894)

(prod119872119894

119899=1119892119899)minus1119872

119894 119894 = 1 2 It can be seen that 119892119899gt 1

119899 = 1 119872 and 119892119899le 1 119899 = 119872 + 1 119873 We derive the

results that 1205831gt and 120583

2gt as follows

1205831

=

2(119861119872

1)

(prod1198721

119899=1119892119899)

minus11198721

=

(119872

prod119872

119899=1119892119899prod1198721

119899=11119892119899)

11198721

= (

119872

prod

119899=1198721+1

119892119899)

11198721

gt 1

1205832

=

2(119861119872

2)

(prod1198722

119899=1119892119899)

minus11198722

=

(119872

prod119872

119899=1119892119899prod1198722

119899=11119892119899)

11198722

= (

1198722

prod

119899=119872+1

1

119892119899

)

11198722

ge 1

(C3)

Since 120583119894gt 0 (119894 = 1 2) the optimal power level that

satisfies 119877 = without the AIP constraint is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C4)

Here Lemma 3 is proved

Mathematical Problems in Engineering 7

D Proof of Lemma 4

From problem (8a) the objective of EE optimization ismodeled as

max119901119899

119880 (119875) =

1

((120572119875 + 119875119904) 119877 (119875)) + 120573

(D1)

where 119875 is the total transmit power and 119877(119901) is the corre-sponding data rate

Since 120572 gt 0 119877 gt 0 119875 gt 0 and 119875119904gt 0 the EE 119880 is a

strictly monotone decreasing function of 120572119875+119875119904119877Then the

equivalent optimization problem is reformulated as

max119901119899

119880 (119875) =

119877 (119875)

120572119875 + 119875119904

(D2)

For arbitrary 120579 isin (0 1) since 119877(119875) is concave function of119875 we have

119880 (1205791199091+ (1 minus 120579) 119909

2) ge

120579119877 (1199091) + (1 minus 120579) 119877 (119909

2)

120579 (1205721199091+ 119875119904) + (1 minus 120579) (120572119909

2+ 119875119904)

ge min (119880 (1199091) 119880 (119909

2))

(D3)

According to [13 14] we can conclude that119880 is quasicon-cave for 119875 ge 0 Since 119875 is a monotonic increasing function ofwater level for 120583 gt (1max119892

119899) Hence EE is a quasiconcave

function of water level

Conflict of Interests

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

Acknowledgment

The work in this paper has been supported by fundingfrom Beijing Natural Science Foundation under Grant no4122009

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] S Srinivasa and S A Jafar ldquoThe throughput potential of cog-nitive radio a theoretical perspectiverdquo IEEE CommunicationsMagazine vol 45 no 5 pp 73ndash79 2007

[3] T A Weiss and F K Jondral ldquoSpectrum pooling an innovativestrategy for the enhancement of spectrum efficiencyrdquo IEEECommunications Magazine vol 42 no 3 pp S8ndashS14 2004

[4] R Zhang ldquoOn peak versus average interference power con-straints for protecting primary users in cognitive radio net-worksrdquo IEEE Transactions on Wireless Communications vol 8no 4 pp 2112ndash2120 2009

[5] K Son B C Jung S Chong and D K Sung ldquoPower allocationfor OFDM-based cognitive radio systems under outage con-straintsrdquo in Proceedings of the IEEE International Conference onCommunications (ICC rsquo10) pp 1ndash5 May 2010

[6] X Zhou B Wu P-H Ho and X Ling ldquoAn efficient powerallocation algorithm for OFDM based underlay cognitive radionetworksrdquo in Proceedings of the 54th Annual IEEE GlobalTelecommunications Conference (GLOBECOM rsquo11) pp 1ndash5December 2011

[7] 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

[8] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[9] G Ding Q Wu N Min and G Zhou ldquoAdaptive resourceallocation in multi-user OFDM-based cognitive radio systemsrdquoin Proceedings of the IEEE International Conference on WirelessCommunications and Signal Processing (WCSP rsquo09) November2009

[10] D Feng C Jiang G Lim L J Cimini Jr G Feng and G Y LildquoA survey of energy-efficient wireless communicationsrdquo IEEECommunications Surveys and Tutorials vol 15 no 1 pp 167ndash178 2013

[11] Y Li X Zhou H Xu G Y Li D Wang and A SoongldquoEnergy-efficient transmission in cognitive radio networksrdquo inProceedings of the 7th IEEE Consumer Communications andNetworking Conference (CCNC rsquo10) January 2010

[12] M L Treust and S Lasaulce ldquoA repeated game formulationof energy-efficient decentralized power controlrdquo IEEE Transac-tions on Wireless Communications vol 9 no 9 pp 2860ndash28692010

[13] J Mao G Xie J Gao and Y Liu ldquoEnergy efficiency optimiza-tion for ofdm-based cognitive radio systems a water-fillingfactor aided search methodrdquo IEEE Transactions on WirelessCommunications vol 12 no 5 pp 2366ndash2375 2013

[14] Y Chen Z Zheng Y Hou and Y Li ldquoEnergy efficient designfor OFDM-based underlay cognitive radio networksrdquo Mathe-matical Problems in Engineering vol 2014 Article ID 431878 7pages 2014

[15] K Fazel and S Kaiser Multi-Carrier and Spread SpectrumSystems John Wiley amp Sons Chichester UK 2003

[16] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert W functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

B Proof of Lemma 2

For a given allowable transmit power the optimal powerallocation can be achieved by classical water-filling methodThe true power level is assumed to be 120583 we have

=

119873

sum

119899=1

119901119899=

119872

sum

119899=1

(120583 minus

1

119892119899

) = 119872120583 minus

119872

sum

119899=1

1

119892119899

(B1)

or

120583 =

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B2)

where 119872 le 119873 119901119899= 120583 minus (1119892

119899) gt 0 119899 = 1 2 119872 and

120583 le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

(1119872119894)( + sum

119872119894

119899=1(1119892119899)) 119894 = 1 2 Because 120583 gt (1119892

119899) 119899 =

1 119872 and 120583 le (1119892119899) 119899 = 119872+1 119873 in (B2) we derive

the results that 1205831gt 120583 and 120583

2gt 120583 as follows

1205831minus 120583 =

1

1198721

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198721

sum

119899=1

1

119892119899

) minus 120583

=

1

1198721

[(119872 minus1198721) 120583 minus

119872

sum

119899=1198721+1

1

119892119899

]

=

1

1198721

119872

sum

119899=1198721+1

(120583 minus

1

119892119899

)

gt 0

1205832minus 120583 =

1

1198722

(119872120583 minus

119872

sum

119899=1

1

119892119899

+

1198722

sum

119899=1

1

119892119899

) minus 120583

=

1

1198722

[minus (1198722minus119872)120583 +

1198722

sum

119899=119872+1

1

119892119899

]

= minus

1

1198722

1198722

sum

119899=119872+1

(120583 minus

1

119892119899

)

ge 0

(B3)

Hence the true power level 120583 corresponding to themaximum allowable transmit power is

120583 = min1le119872le119873

1

119872

( +

119872

sum

119899=1

1

119892119899

) (B4)

Here Lemma 2 is proved

C Proof of Lemma 3

Let be the optimal water level corresponding to theminimum rate requirement without AIP constraint wehave

= 119861

119873

sum

119899=1

log2(1 + 119892

119899119901119899)

= 119861

119872

sum

119899=1

log2(1 + ( minus

1

119892119899

)119892119899)

= 119861log2(119872

119872

prod

119899=1

119892119899)

(C1)

or

= 2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C2)

where 119872 le 119873 119901119899= minus (1119892

119899) gt 0 119899 = 1 2 119872 and

le (1119892119899) 119899 = 119872 + 1 119873

For arbitrary 1 le 1198721lt 119872 lt 119872

2le 119873 we have 120583

119894=

2(119861119872

119894)

(prod119872119894

119899=1119892119899)minus1119872

119894 119894 = 1 2 It can be seen that 119892119899gt 1

119899 = 1 119872 and 119892119899le 1 119899 = 119872 + 1 119873 We derive the

results that 1205831gt and 120583

2gt as follows

1205831

=

2(119861119872

1)

(prod1198721

119899=1119892119899)

minus11198721

=

(119872

prod119872

119899=1119892119899prod1198721

119899=11119892119899)

11198721

= (

119872

prod

119899=1198721+1

119892119899)

11198721

gt 1

1205832

=

2(119861119872

2)

(prod1198722

119899=1119892119899)

minus11198722

=

(119872

prod119872

119899=1119892119899prod1198722

119899=11119892119899)

11198722

= (

1198722

prod

119899=119872+1

1

119892119899

)

11198722

ge 1

(C3)

Since 120583119894gt 0 (119894 = 1 2) the optimal power level that

satisfies 119877 = without the AIP constraint is

= min1le119872le119873

2(119861119872)

(

119872

prod

119899=1

119892119899)

minus1119872

(C4)

Here Lemma 3 is proved

Mathematical Problems in Engineering 7

D Proof of Lemma 4

From problem (8a) the objective of EE optimization ismodeled as

max119901119899

119880 (119875) =

1

((120572119875 + 119875119904) 119877 (119875)) + 120573

(D1)

where 119875 is the total transmit power and 119877(119901) is the corre-sponding data rate

Since 120572 gt 0 119877 gt 0 119875 gt 0 and 119875119904gt 0 the EE 119880 is a

strictly monotone decreasing function of 120572119875+119875119904119877Then the

equivalent optimization problem is reformulated as

max119901119899

119880 (119875) =

119877 (119875)

120572119875 + 119875119904

(D2)

For arbitrary 120579 isin (0 1) since 119877(119875) is concave function of119875 we have

119880 (1205791199091+ (1 minus 120579) 119909

2) ge

120579119877 (1199091) + (1 minus 120579) 119877 (119909

2)

120579 (1205721199091+ 119875119904) + (1 minus 120579) (120572119909

2+ 119875119904)

ge min (119880 (1199091) 119880 (119909

2))

(D3)

According to [13 14] we can conclude that119880 is quasicon-cave for 119875 ge 0 Since 119875 is a monotonic increasing function ofwater level for 120583 gt (1max119892

119899) Hence EE is a quasiconcave

function of water level

Conflict of Interests

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

Acknowledgment

The work in this paper has been supported by fundingfrom Beijing Natural Science Foundation under Grant no4122009

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] S Srinivasa and S A Jafar ldquoThe throughput potential of cog-nitive radio a theoretical perspectiverdquo IEEE CommunicationsMagazine vol 45 no 5 pp 73ndash79 2007

[3] T A Weiss and F K Jondral ldquoSpectrum pooling an innovativestrategy for the enhancement of spectrum efficiencyrdquo IEEECommunications Magazine vol 42 no 3 pp S8ndashS14 2004

[4] R Zhang ldquoOn peak versus average interference power con-straints for protecting primary users in cognitive radio net-worksrdquo IEEE Transactions on Wireless Communications vol 8no 4 pp 2112ndash2120 2009

[5] K Son B C Jung S Chong and D K Sung ldquoPower allocationfor OFDM-based cognitive radio systems under outage con-straintsrdquo in Proceedings of the IEEE International Conference onCommunications (ICC rsquo10) pp 1ndash5 May 2010

[6] X Zhou B Wu P-H Ho and X Ling ldquoAn efficient powerallocation algorithm for OFDM based underlay cognitive radionetworksrdquo in Proceedings of the 54th Annual IEEE GlobalTelecommunications Conference (GLOBECOM rsquo11) pp 1ndash5December 2011

[7] 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

[8] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[9] G Ding Q Wu N Min and G Zhou ldquoAdaptive resourceallocation in multi-user OFDM-based cognitive radio systemsrdquoin Proceedings of the IEEE International Conference on WirelessCommunications and Signal Processing (WCSP rsquo09) November2009

[10] D Feng C Jiang G Lim L J Cimini Jr G Feng and G Y LildquoA survey of energy-efficient wireless communicationsrdquo IEEECommunications Surveys and Tutorials vol 15 no 1 pp 167ndash178 2013

[11] Y Li X Zhou H Xu G Y Li D Wang and A SoongldquoEnergy-efficient transmission in cognitive radio networksrdquo inProceedings of the 7th IEEE Consumer Communications andNetworking Conference (CCNC rsquo10) January 2010

[12] M L Treust and S Lasaulce ldquoA repeated game formulationof energy-efficient decentralized power controlrdquo IEEE Transac-tions on Wireless Communications vol 9 no 9 pp 2860ndash28692010

[13] J Mao G Xie J Gao and Y Liu ldquoEnergy efficiency optimiza-tion for ofdm-based cognitive radio systems a water-fillingfactor aided search methodrdquo IEEE Transactions on WirelessCommunications vol 12 no 5 pp 2366ndash2375 2013

[14] Y Chen Z Zheng Y Hou and Y Li ldquoEnergy efficient designfor OFDM-based underlay cognitive radio networksrdquo Mathe-matical Problems in Engineering vol 2014 Article ID 431878 7pages 2014

[15] K Fazel and S Kaiser Multi-Carrier and Spread SpectrumSystems John Wiley amp Sons Chichester UK 2003

[16] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert W functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

D Proof of Lemma 4

From problem (8a) the objective of EE optimization ismodeled as

max119901119899

119880 (119875) =

1

((120572119875 + 119875119904) 119877 (119875)) + 120573

(D1)

where 119875 is the total transmit power and 119877(119901) is the corre-sponding data rate

Since 120572 gt 0 119877 gt 0 119875 gt 0 and 119875119904gt 0 the EE 119880 is a

strictly monotone decreasing function of 120572119875+119875119904119877Then the

equivalent optimization problem is reformulated as

max119901119899

119880 (119875) =

119877 (119875)

120572119875 + 119875119904

(D2)

For arbitrary 120579 isin (0 1) since 119877(119875) is concave function of119875 we have

119880 (1205791199091+ (1 minus 120579) 119909

2) ge

120579119877 (1199091) + (1 minus 120579) 119877 (119909

2)

120579 (1205721199091+ 119875119904) + (1 minus 120579) (120572119909

2+ 119875119904)

ge min (119880 (1199091) 119880 (119909

2))

(D3)

According to [13 14] we can conclude that119880 is quasicon-cave for 119875 ge 0 Since 119875 is a monotonic increasing function ofwater level for 120583 gt (1max119892

119899) Hence EE is a quasiconcave

function of water level

Conflict of Interests

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

Acknowledgment

The work in this paper has been supported by fundingfrom Beijing Natural Science Foundation under Grant no4122009

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] S Srinivasa and S A Jafar ldquoThe throughput potential of cog-nitive radio a theoretical perspectiverdquo IEEE CommunicationsMagazine vol 45 no 5 pp 73ndash79 2007

[3] T A Weiss and F K Jondral ldquoSpectrum pooling an innovativestrategy for the enhancement of spectrum efficiencyrdquo IEEECommunications Magazine vol 42 no 3 pp S8ndashS14 2004

[4] R Zhang ldquoOn peak versus average interference power con-straints for protecting primary users in cognitive radio net-worksrdquo IEEE Transactions on Wireless Communications vol 8no 4 pp 2112ndash2120 2009

[5] K Son B C Jung S Chong and D K Sung ldquoPower allocationfor OFDM-based cognitive radio systems under outage con-straintsrdquo in Proceedings of the IEEE International Conference onCommunications (ICC rsquo10) pp 1ndash5 May 2010

[6] X Zhou B Wu P-H Ho and X Ling ldquoAn efficient powerallocation algorithm for OFDM based underlay cognitive radionetworksrdquo in Proceedings of the 54th Annual IEEE GlobalTelecommunications Conference (GLOBECOM rsquo11) pp 1ndash5December 2011

[7] 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

[8] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[9] G Ding Q Wu N Min and G Zhou ldquoAdaptive resourceallocation in multi-user OFDM-based cognitive radio systemsrdquoin Proceedings of the IEEE International Conference on WirelessCommunications and Signal Processing (WCSP rsquo09) November2009

[10] D Feng C Jiang G Lim L J Cimini Jr G Feng and G Y LildquoA survey of energy-efficient wireless communicationsrdquo IEEECommunications Surveys and Tutorials vol 15 no 1 pp 167ndash178 2013

[11] Y Li X Zhou H Xu G Y Li D Wang and A SoongldquoEnergy-efficient transmission in cognitive radio networksrdquo inProceedings of the 7th IEEE Consumer Communications andNetworking Conference (CCNC rsquo10) January 2010

[12] M L Treust and S Lasaulce ldquoA repeated game formulationof energy-efficient decentralized power controlrdquo IEEE Transac-tions on Wireless Communications vol 9 no 9 pp 2860ndash28692010

[13] J Mao G Xie J Gao and Y Liu ldquoEnergy efficiency optimiza-tion for ofdm-based cognitive radio systems a water-fillingfactor aided search methodrdquo IEEE Transactions on WirelessCommunications vol 12 no 5 pp 2366ndash2375 2013

[14] Y Chen Z Zheng Y Hou and Y Li ldquoEnergy efficient designfor OFDM-based underlay cognitive radio networksrdquo Mathe-matical Problems in Engineering vol 2014 Article ID 431878 7pages 2014

[15] K Fazel and S Kaiser Multi-Carrier and Spread SpectrumSystems John Wiley amp Sons Chichester UK 2003

[16] R M Corless G H Gonnet D E G Hare D J Jeffreyand D E Knuth ldquoOn the Lambert W functionrdquo Advances inComputational Mathematics vol 5 no 4 pp 329ndash359 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of