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Duan, Ruifeng; Zheng, Zhong; Jäntti, Riku; Hämäläinen, Jyri; Haas, Zygmunt. J.Asymptotic Analysis for Spectrum-sharing Systems with TAS/MRC Using Extreme ValueTheory

Published in:IEEE Access

DOI:10.1109/ACCESS.2019.2943083

Published: 23/09/2019

Document VersionPublisher's PDF, also known as Version of record

Please cite the original version:Duan, R., Zheng, Z., Jäntti, R., Hämäläinen, J., & Haas, Z. J. (2019). Asymptotic Analysis for Spectrum-sharingSystems with TAS/MRC Using Extreme Value Theory: An Overlooked Aspect. IEEE Access, 7, 138062-138078.https://doi.org/10.1109/ACCESS.2019.2943083

https://doi.org/10.1109/ACCESS.2019.2943083

https://doi.org/10.1109/ACCESS.2019.2943083

Received July 31, 2019, accepted September 11, 2019, date of publication September 23, 2019, date of current version October 3, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2943083

Asymptotic Analysis for Spectrum-SharingSystems With TAS/MRC Using ExtremeValue Theory: An Overlooked AspectRUIFENG DUAN 1, (Member, IEEE), ZHONG ZHENG2, (Member, IEEE),RIKU JÄNTTI 1, (Senior Member, IEEE), JYRI HÄMÄLÄINEN 1, (Senior Member, IEEE),AND ZYGMUNT J. HAAS3,4, (Fellow, IEEE)1Department of Communications and Networking, Aalto University, 02150 Espoo, Finland2School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China3Department of Computer Science, The University of Texas at Dallas, Richardson, TX 75080, USA4School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA

Corresponding author: Zhong Zheng ([email protected])

The work of R. Duan and R. Jäntti was supported in part by the Business Finland through the Project 5G Finnish Open ResearchCollaboration Ecosystem (5G-FORCE). The work of Z. Zheng was supported in part by the Beijing Institute of Technology Research FundProgram for Young Scholars. This work of J. Hämäläinen was supported in part by the Academy of Finland under Grant 287249 andGrant 311752. The work of Z. J. Haas was supported in part by the Army Research Laboratory (ARL) under Contract W911NF-18-1-0406,and in part by the National Science Foundation (NSF) under Grant CNS-1763627.

ABSTRACT We investigate the asymptotic behavior for an overlooked aspect of spectrum-sharing systemswhen the number of transmit antennas nt at the secondary transmitter (ST) grows to infinity. Consideringimperfect channel state information (CSI), we apply the transmit antenna selection and the maximal-ratiocombining techniques at the ST and the secondary receiver (SR), respectively. First, we obtain the signal-to-noise ratio (SNR) distributions received by the SR under perfect and imperfect CSI conditions. Then weshow that the SNR distributions are tail-equivalent in the sense that the right tails of the two distributionsdecay in the same rate as the number of transmit antennas nt grows to infinity. Based on the extreme valuetheory, when the transmit power of the ST is solely limited by the interference constraint, we show thatthe limiting SNR at the SR is Fréchet-distributed and the limiting rate scales as log(nt ). When the transmitpower of ST is determined by both the maximal transmit power and the interference power constraints,the limiting SNR is Gumbel-distributed and the limiting rate scales as log(log(nt )). We further show that theaverage rate can be estimated by the corresponding easier-to-obtain outage rate. Numerical results indicatethat the derived asymptotic rate expressions represent accurate approximations even when nt is ‘‘not-so-large’’. Finally, we study the robustness of the secondary transmissions by analyzing the correspondingaverage symbol error rates (SER) under general modulation and coding schemes. The findings indicate thatthe SER is Weibull distributed, when the maximal transmit power and interference power constraints arecomparable.

INDEX TERMS Spectrum sharing, extreme value theory, rate scaling law, symbol error rate.

I. INTRODUCTIONSpectrum sharing has been considered as a promisingtechnology to efficiently utilize the radio spectrum, and sig-nificant progress has been achieved in developing spectrum-sharing techniques, for instance, dynamic spectrum accessand 5G heterogeneous networks [1], [2], licensed and

The associate editor coordinating the review of this manuscript and

approving it for publication was Oussama Habachi .

unlicensed spectrum access [3], [4], drone networks [5], andco-primary spectrum sharing with applications in device-to-device communication [6]. In this work, we consider theunderlay paradigm that the secondary users (SUs) operateas underlay systems. Therein, the SUs are allowed to coexistwith the primary users (PUs)1 while the SUs need to control

1Although we call the communicating entities as SU and PU, the analysisin this paper can be applied in other communication systems that requireinterference control for transmission.

138062 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ VOLUME 7, 2019

https://orcid.org/0000-0002-6278-933X

https://orcid.org/0000-0002-5398-2381

https://orcid.org/0000-0002-3305-2961

https://orcid.org/0000-0001-7121-5760

R. Duan et al.: Asymptotic Analysis for Spectrum-Sharing Systems With TAS/MRC Using EVT

their transmit power in order to avoid harmful interferenceto the PUs [2], [7], [8]. The performance of such spectrum-sharing systems have been considered in literature, suchas [2], [7], [8] among others.

It is well-known that the multi-antenna techniques canbe used to improve the link-level performance of wirelesscommunications. The transmitter antenna selection (TAS)scheme, among other transceiver designs, significantlyreduces the hardware complexity and costs by simplifyingthe radio transmission units, as only a single RF chainis needed at the transmitter. In addition, the TAS schemedoes not require synchronization among transmit antennas.Hence, it is an effective transceiver design and has beenadopted in massive MIMO systems as well as the recentnon-orthogonal multiple access systems [9]–[11]. Comparedto other channel-dependent transmission schemes, TAS alsoreduces the required channel feedbacks [12], as only a singleindex of the transmit antenna to be activated is needed fromthe receiver to the transmitter. Moreover, TAS technique hasbeen adopted in [13] to improve the robustness of massiveMIMO systems against passive eavesdropping.

A combined TAS and receiver maximal ratio combining(MRC) scheme is analyzed in [14] in the traditional point-to-point communication systems. Therein, authors investigatedthe system performance in terms of the average bit-errorrate, and showed that the maximal diversity order is equal tothe product of the number of transmit and receive antennas.In [15] and [16], authors investigated the outage probabilityand the bit error rate of TAS/MRC for point-to-point com-munications in Rayleigh and Nakagami-m fading channels,respectively. Finally, in [17] authors deduce the expressionsfor the outage probability in a TAS/MRC system assumingmultiple users with independent and identically distributed(i.i.d.) Nakagami-m channels. The full diversity benefit isachieved at a relatively high SNR level.

A. RELATED WORKSThe ergodic capacity of the SU with TAS at the secondarytransmitter (ST) and MRC at the secondary receiver (SR) isinvestigated in [18] when there is no maximal transmit powerconstraint for the ST. In [19], on the other hand, the capacity-scaling law and diversity order for a cognitive radio systemwith MRC receiver is investigated. Therein, the asymptoticanalysis is conducted in the high SNR regime without TAS.Furthermore, authors in [20] analyze the multiuser diver-sity gain for single-antenna overlaid cognitive radio systems.We note that in [20] the interference towards the primaryreceiver (PR) is avoided by the spectrum sensing and onlythe maximum transmit power constraint is activated at thetransmitting ST.

When the instantaneous perfect channel state informa-tion (CSI) of both ST-to-PR and ST-to-SR links are availableat the ST, authors in [21] present a closed-form expression forthe cumulative distribution function (c.d.f.) of the secondarySNR applying the antenna selection, where the selectioncriterion is based on the ratio between channel gains of the

ST-to-SR and ST-to-PR links. This c.d.f. is applied to obtain,for instance, the outage probability, ergodic capacity, andbit error probability (BEP). Though a simple expression isprovided for high SNR regime, the general expression forthe BEP consists of the parabolic cylinder function. In addi-tion, the approximated expression of the ergodic capacityincludes the Laguerre polynomials, which do not explicitlydisplay the relation between the considered performancemet-rics and the number of transmit antennas. Moreover, as thenumber of transmit antennas grows, it is challenging for thesecondary receiver to feedback the full CSI of the ST-to-SRlink to the ST. In addition, TAS/MRC has been studied forunderlay spectrum sharing in [22], where asymptotic expres-sions of outage probability have been obtained as the transmitpower of the SU goes to infinity. This scenario can be deemedas a special case of the study in this work.

Authors in [23]–[25] have studied the impacts of the feed-back delay and the co-channel interference on the perfor-mance of the considered systems for multi-user multipleamplify-and-forward (AF) relaying networks and spectrum-sharing relaying networks. Results in [23] show that theperfect feedback is required to achieve the full diversityorder of the multi-user multiple AF relaying networks. TheTAS/MRC approach has also been considered in [24] forspectrum-sharing relaying networks. Therein, the exact andapproximate expressions for the outage probability of the sec-ondary network are obtained, which shows that interferencepower constraint is closely related to the achievable diversityorder and coding gain. Authors in [25] have considered atransmit beamforming and MRC-enabled AF relaying sys-tem with feedback delay and co-channel interference. Par-ticularly, the exact expressions for the c.d.f. of the SINRand the upper and lower bounds of the ergodic capacityhave been obtained. In [26], performance analysis of severalTAS schemes has been conducted for an energy harvest-ing decode-and-forward relaying network, where the asymp-totic outage probabilities have been obtained in high SNRregimes.

The previous works either provide complicated expres-sions for the performance metrics of the secondary systemsor conduct the performance analysis in high SNR regime,when the perfect CSI of the communication channels areaccessible. However, it is not a practical scenario for thesecondary users to operate in the high SNR regimes witherror-free channel estimations. In this work, we carry outthe performance analysis of the secondary users at genericSNR levels under imperfect CSI at the secondary receivers,which has been overlooked in literature. Furthermore, basedon the existing results, it is difficult to gain insight into theinterplay between the transmit power and interference powerconstraints of the secondary users and their impacts on theperformance of the considered TAS technique. For instance,for a moderate SNR value, how the two constraints directlyaffect the performance of the SU in terms of average rate,outage rate, and average symbol error rate as the number oftransmit antennas increases.

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Note that when the interference power constraint is notimposed, the limiting SNR distribution follows Gumbeldistribution [27]. However, to the best of our knowledge,when both the transmit power and the interference powerconstraints are active, the limiting SNR distribution andthe corresponding asymptotic performance metrics of theTAS/MRC secondary systems, as functions of the num-ber of transmit antennas, are unknown and are non-trivialto deduce such results on the existing works. In par-ticular, we show in this work that the additional inter-ference power constraint does change the limiting SNRdistribution.

B. CONTRIBUTIONS AND PAPER ORGANIZATIONTo address the above discussed issues, at generic SNR levels,the extreme value theory (EVT) is applied to analyze variousperformance metrics of the secondary spectrum-sharing sys-tems subject to both the maximal transmit power constraintat ST and the maximal interference power constraint at PR.The performance of such secondary system is evaluated viaclosed-form expressions for the limiting SNR distributions,the average rate, the outage rate, and the average symbol errorrate (ASER) using general modulation and coding schemes.Based on these results, the scaling laws of the SU averagerate and outage rate with respect to the number of transmitantennas will be derived. We summarize the contributions ofthis work as follows:• Before applying TAS, the SNR distributions receivedby the secondary receiver are obtained under perfectand imperfect CSI conditions. In both cases, the SNRdistributions are shown to be tail-equivalent in the sensethat the right tails of the two distributions decay in thesame rate as the number of transmit antennas nt grows toinfinity. Therefore, under both perfect and imperfect CSIconditions, the limiting SNR distribution after apply-ing TAS can be obtained by using the same analyticalframework of EVT, which facilitates the following per-formance analysis.

• The limiting SNR distribution of the SU converges to theFréchet distribution when the maximal transmit powerconstraint dominates over the interference power con-straint, which is hereafter referred to as the interferencepower limited regime (IPLR); otherwise, the limitingSNR distribution converges to the Gumbel distribution,which is referred to as the non-dominant regime (NDR).

• The average rate scales as log(nt ) in IPLR, while scalesas log(log(nt )) in NDR, as nt increases. For large nt ,in IPLR there is a non-zero gap between the outage rateand the average rate, while in NDR the gap is vanishingas log(1 + 1/ log nt ). We show that the average ratecan be estimated by the corresponding easier-to-obtainoutage rate.

• The SER follows the Weibull distribution for generalmodulation and coding schemes with large nt . Basedon this result, a compact expression for the ASER isdeduced.

FIGURE 1. System Model: The i th antenna of the ST is selected.

The rest of the paper is organized as follows. In Section II,we outline the system model and present the c.d.f. of theSNR at the secondary receiver. In Section III, we brieflysummarize the principles of the EVT, and derive the limitingdistribution of the SNR at the secondary receiver. Using thisresult, the SU average rate and the corresponding rate scalinglaws are investigated in Section IV. The SU outage rate isanalysed in Section V. The average symbol error rate of theSU for general modulation types is deduced in Section VI.We present the simulation results in Section VII. Section VIIIconcludes this paper. Proofs of key technical results are pro-vided in the Appendices.

C. NOTATIONSWe adopt the following notations: Q denotes instantaneousinterference power constraint at the PR; Pm is the maximumtransmit power constraint at the ST; gi ∈ C1×nr represents thechannel vector between the ith antenna of the ST and theantenna array of the SR; hi ∈ C is the channel betweenthe ith antenna element of the ST and the PR; In denotesn× n identity matrix; γ (a, x) =

∫ x0 t

a−1e−t dt and 0(a, x) =∫∞

x ta−1e−t dt denote the lower and the upper incompleteGamma function, respectively; γ0 = 0.5772 . . . is the Euler-Mascheroni constant; The error function and complementaryerror function are denoted as erf(x) = 2

√π

∫ x0 e−t2 dt and

erfc(x) = 1− erf(x), respectively.

II. SYSTEM MODELConsider a spectrum-sharing system as shown in Fig. 1,which consists of a SU pair and a PU pair. The ST is equippedwith nt antenna elements, assuming that the transmit antennaselection is adopted. The secondary receiver (SR) has nrantenna elements, and the received signals are combinedaccording to theMRCprinciple. The primary transmitter (PT)and receiver (PR) are assumed to be equipped with oneantenna.2 To protect the PU communications, we consideran interference power constraint imposed by the PR, wherethe instantaneous interference from the ST to the PR is

2This corresponds to the legacy wireless systems, where the transceivershave limited hardware capability.

138064 VOLUME 7, 2019


constrained within a pre-specified power threshold due to theradio regulations. Meanwhile, the transmit power of the STis also limited by its maximum power constraint due to, forinstance, hardware limitations or by spectrum access policiesset by the regulators.

A. SIGNAL MODELWhen the ith transmit antenna of ST is used, the receivedsignal vector at the SR, y, reads

y =√µgP

(i)s gi x + v, (1)

where µg denotes the average path loss between the transmitantennas and the receive antennas, and P(i)s denotes the trans-mit power of the ST. Here, x is the transmitted symbol of ST,gi is the channel vector whose elements gi,j ∼ CN (0, 1),j = 1, · · · , nr , represent the complex channel coefficientbetween the ith transmit antenna and the jth receive antenna,and v ∼ CN (0, σ 2Inr ) is the complex white Gaussian noisevector. The channel vectors are i.i.d. standard complex Gaus-sian distributed, independent across transmit antennas. In thismodel, we adopt the same assumption for underlay spectrumsharing paradigm as in [7], [21], [28]–[31] that the interfer-ence from the PT is treated as Gaussian noise or via propercooperative schemes [32] such that we can obtain analyticresults with sufficient insights on the addressed problem.With multiple PTs, this can be justified by the central limittheorem and valid for certain network deployment topolo-gies, for instance when the SR and the PT are far away[7], [31].When the interference from the PU cannot be treatedas Gaussian noise, the SR receives a combination of thesignal, interference and noise, and the resulting SINRs of thereceived symbols sent by different transmit antennas becomecorrelated. The following EVT analysis of such correlatedrandom variables is an open problem in general.We leave thischallenging problem for our future work. For the analysis,we have adopted the following assumptions on the signalmodel:

(A1) The channels follow a block fading process, i.e., theentities of channel vectors remain constant overeach coding block and are i.i.d across from oneblock to another;

(A2) The SR has the imperfect CSI of gi, namely gi,to performmaximal ratio combining; the ST has theimperfect CSI of hi, namely hi;

(A3) The PT and ST apply Gaussian codebooks.We note that hi in (A2) can be obtained, when, for instance,the primary receiver feeds back the channel measurement ofhi to the secondary transmitter [33]. These assumptions areapplied also in previous works, for instance, [33]–[37] amongothers.

We consider the following imperfect CSI model, whichhas been widely applied in literature, where the pilot and thesignal are sent separately within a fading channel block, forinstance, [23]–[25], [38]–[40] among others. The imperfectCSIs of gi and hi are due to the Gaussian estimation errors

such that

gi = σggi +√1− σ 2

g gi, (2)

hi = σhhi +√1− σ 2

h hi, ∀i. (3)

where gi ∼ CN (0, Inr ) and hi ∼ CN (0, 1) are the normalizedchannel estimation errors of gi and hi, respectively. The terms0 ≤ σg, σh ≤ 1 denote correlations between the instanta-neous channel coefficients and their estimations. When σgand σh are equal to 1, the channel estimation is perfect; onthe other hand, when σg and σh are equal to 0, the channelestimations fail completely and the channels are unknown tothe transmitter.

Due to the imperfect CSI hi, the ST cannot strictly satisfythe interference power constraint set by the PR. Therefore,we consider the outage probability model for the interferencepower [24], where, in addition to the maximum transmissionpower Pm of the ST, it causes an interference towards the PRno larger than the power Q with a pre-defined outage proba-bility ν. That is, Pr

{µhP

(i)s |hi|2 ≤ Q

}= ν. Following [24],

the transmit power of the ST is given by

P(i)s = min{

κQ

µh |hi|2,Pm

},

where κ is a power margin, and µh denotes the average pathloss between the ith ST antenna and the receive antenna ofthe PR. Given the outage probability ν, the power margin κcan be numerically obtained by using [24, Eq. (15)]. In thispaper, we consider that κ is absorbed into Q, i.e., Q = κQ,so that the transmission power of the ST becomes

P(i)s = min{

Q

µh |hi|2,Pm

}. (4)

By using MRC at the SR, the received signals at the SRantenna array are weighted by the corresponding complexchannel coefficients and combined coherently [41]. The post-processed signal after the MRC at the SR can be written as

y = w†y =√µgP

(i)s

g†i||gi||F

gi x +g†i||gi||F

v, (5)

where w = gi||gi||F

is the MRC weight vector, || · ||F is

the Frobenius norm, and (·)† denotes the complex conjugatetranspose operator. Assuming perfect CSI first, the channelpower gain ||gi||2F follows a Chi-square distribution with 2nrdegrees of freedom, and the c.d.f. of ||gi||2F is given by [42]as:

F||gi||2F

(x) = 1−1

(nr − 1)!0

(nr ,

xµg

), x ≥ 0. (6)

With the imperfect CSI in (2), the c.d.f. of ||gi||2F is givenby [38], which has been applied in, e.g., [40], as:

F||gi||2F

(x) =nr∑m=1

Bnr−1m−1 (σ2g )

1−0(m, x

µg

)0(m)

, (7)

VOLUME 7, 2019 138065


where Bki (t) =(ki

)t i(1 − t)k−i denotes Bernstein basis poly-

nomial of degree k with(ki

)being a binomial coefficient.

The post-processed SNR corresponding to the ith antennatransmission at the MRC combiner of the SR is of the form

γi = min

{SQ‖gi‖2F|hi|2

,SP‖gi‖2F

}. (8)

where SP = Pm/σ 2, and SQ = Q/µhσ 2. If TAS is appliedat ST, its i∗th transmit antenna corresponding to the largestreceive SNR γi∗ is selected to transmit the signals. That is,the SR feedbacks the antenna index i∗, which fulfills

i∗ = arg max1≤i≤nt

γi. (9)

Assuming an ideal feedback channel between the ST and theSR, the TAS scheme requires log2 nt bits of feedback infor-mation. The largest SNR is denoted by γmax = max1≤i≤nt γi.Proposition 1: The c.d.f. of the post-processed SNR γi

with imperfect CSI is given as:

Fγi (x) =nr∑m=1

Bnr−1m−1 (σ2g ){1−

10(m)

[0

(m,

xµgSP

)−

(x

µgSQ + x

)m0

(m,

xµgSP

+SQSP

)]}. (10)

Proof: The proof of Proposition 1 is in Appendix A.Note that by setting σ 2

g = 1, the c.d.f. of γi in (11) isobtained from (10) as:

Fγi (x) = 1−1

0(nr )

[0

(nr ,

xµgSP

)−

(x

µgSQ + x

)nr0

(nr ,

xµgSP

+SQSP

)], (11)

which is in line with the existing results in [18] and [19].Based on (9), the c.d.f. of γmax can be expressed as [43]

Fγmax (x) =[Fγi (x)

]nt . (12)

Considering a simplified scenario with Pm → ∞ andhence SP → ∞, the c.d.f. in (10) can be approximated,by applying 0(nr , 0) = 0(nr ) [50, eqn. (8.350-4)], as

Fγi (x) =nr∑m=1

Bnr−1m−1 (σ2g )(

xx + µgSQ

)m. (13)

With perfect CSI at the ST, (13) can be reduced to

Fγi (x) =(

xx + µgSQ

)nr. (14)

B. CONCEPT OF TAIL EQUIVALENCEIn this section, we prove that the limiting SNR distributionsunder the perfect and the imperfect CSI conditions are tail-equivalent so that the limiting behavior of the SNR γmaxremains the same as nt grows. For the sake of completeness,the definition of tail equivalence in [44] is stated next.Definition 1 (Tail Equivalence [44, Eq. (1.25)]): Let

F and G be two distribution functions, and set their

upper end-points as ω(F) = sup{y|F(y) < 1} andω(G) = sup{y|G(y) < 1}, respectively. Then F and Gare (right) tail equivalent for some ζ > 0 if, and only if,ω(F) = ω(G) and

limx→ω(F)

1− F(x)1− G(x)

= ζ. (15)

In the next proposition, we show that the SNR distribu-tions with and without channel estimation errors are tail-equivalent.Proposition 2: The distribution functions Fγi (x) in (10)

and Fγi (x) in (11) are right tail equivalent and

limx→ω(F)

1− Fγi (x)

1− Fγi (x)= ζ ≤ (σ 2

g )nr−1. (16)

Proof: The proof of Proposition 2 is in Appendix B.Obviously, with the perfect CSI, i.e., σ 2

g = 1, the ratiobetween the two tails is one as expected. In presence of imper-fect CSI, the right-hand-side of (2) reduces as the accuracy ofthe channel estimation decreases, which means the receiverwith imperfect CSI sees a degraded SNR compared to thecase with perfect CSI. The SNR degradation is proportionalto (σ 2

g )nr−1.

To the best of our knowledge, there is no explicit expres-sions for the performancemetrics of the secondary users, suchas the average rate and the average symbol error rate, obtainedwith the c.d.f. (10) or (12) due to their complexity. To enable atractable analysis and gain insight into the considered cogni-tive radio system, we deduce the asymptotic expressions forthe average rate, outage rate, and average symbol error rateusing EVT by increasing the number of transmit antennasnt → ∞ while keeping the number of receive antennas nrfixed.

III. EXTREME VALUE THEORYThe EVT provides elegant and powerful statistical tools wheninvestigating the asymptotic distributions of the maximum orthe minimum of a set of random variables [45]. Recently,EVT has proven useful in investigating the performance ofthe wireless communication networks. The authors in [46]conduct the asymptotic throughput analysis for the channel-aware scheduling in multi-user communications. The asymp-totic antenna selection gain and the capacity distribution fora multi-antenna system are studied in [27] and the so-calledcumulative distribution function based multi-user schedul-ing are investigated using EVT in [47] and [48]. Moreover,authors in [34] investigate a single-antenna spectrum-sharingmulti-hop relay system, where the EVT is applied to obtainthe limiting distribution functions of the lower and upperbounds on the end-to-end SNR of the relaying path.

A. PRELIMINARYLetX1,X2, ...,Xn denote a sequence of i.i.d. random variableswith distribution FX (x) and Mn = max{Xi : 1 ≤ i ≤ n}.Furthermore, assume that there exists sequences of real

138066 VOLUME 7, 2019


numbers an and bn such that

limn→∞

Mn − anbn

d−−→ G, (17)

whered−→ denotes the convergence in distribution and G is a

non-degenerate distribution. By the Fisher-Tippet-Gnedenkotheorem [49] it is known that G is one of the extreme valuedistributions: Gumbel, Fréchet, or Weibull, defined as:

Weibull: 9ξ (x) = e−(−x)−1/ξ

, x ≤ 0, ξ < 0 (18a)

Fréchet: 8ξ (x) = e−x−1/ξ

, x ≥ 0, ξ > 0 (18b)

Gumbel: 3ξ (x) = e−e−x, x ∈ R, ξ = 0, (18c)

where ξ denotes the extreme value index. Furthermore,according to the assumption (17), the distribution FX (x)belongs to the so-called Maximum Domain of Attrac-tion (MDA) of G.This EVT result forms a basis of the asymptotic approx-

imation for the SNR distribution (12) when the number oftransmit antenna nt is large. This asymptotic distributionrelies on the MDA of the underlying distribution FX (x).As we show in the following discussions, the SNR distribu-tion Fγi (x) for the perfect CSI scenario in (11) belongs to theMDA of the Gumbel or the Fréchet distribution, dependingon the dominating power constraints Q and Pm. Since thedistribution functions Fγi (x) and Fγi (x) are tail-equivalent asbeen shown in Section II-B, the SNR distribution Fγi (x) forthe imperfect CSI scenario in (10) lies on the same MDA,where Fγi (x) belongs [44]. Prior to using the extreme valuedistributions to approximate Fγmax(x) in (12), we first reviewtwo relevant results from [45], which state the sufficientconditions for a distribution to belong to the MDA of theGumbel and the Fréchet distributions.Lemma 1 [45, Theorem 2.7.2]: Let ω(FX ) = sup{x :

FX (x) < 1} as the right end-point of the support of FX (x).Assume that there is a real number x1, and for x1 ≤ x <ω(FX ), fX (x) = F ′X (x) 6= 0 and F ′′X (x) exists. If

limx→ω(FX )

ddx

[1− FX (x)fX (x)

]= 0, (19)

then FX (x) lies on the MDA of the Gumbel distribution withthe c.d.f. expressed as

3ξ (x) = exp(− exp

(−x − anbn

)), (20)

with the normalizing coefficients an and bn determined by

an = F−1X (1− 1/N ) , bn = F−1X (1− 1/Ne)− an. (21)

Here e is the base of the natural logarithm, andF−1X (·) denotesthe inverse of the distribution function.

A sufficient condition that a distribution lies on the MDAof the Fréchet distribution is given as follows:Lemma 2 [45, Theorem 2.1.1]: Let ω(FX ) = sup{x :

FX (x) < 1} = ∞. Assume that there is a constant ν > 0such that the following limit exists for all x > 0,

limt→∞

1− FX (tx)1− FX (t)

= x−ν, (22)

where ν = 1/ξ . Then, FX (x) lies on the MDA of the Fréchetdistribution with the c.d.f. expressed as

8ξ (x) = exp

[−

(xbn

)−1/ξ], ∀x > 0, (23)

with the coefficients determined by

an = 0, bn = F−1X (1− 1/N ) . (24)

We note that the choice of the normalizing coefficientsan and bn are not unique, and they can be selected as inother forms without affecting the convergence of (17) to thelimiting distributions (18) [45]. Nevertheless, we show inSection VII that the adopted normalizing coefficients (21)and (24) can be used to construst accurate approximationsfor the considered performance metrics when the number ofantennas nt is finite.

B. LIMITING SNR DISTRIBUTION OF THE SECONDARYUSERWe categorize the operational regimes of the consideredspectrum-sharing system into the following cases:

• Interference power limited regime (IPLR): Pm → ∞with Q/µh fixed.

• Transmit power limited regime (TPLR): Q/µh → ∞with Pm fixed.

• Otherwise, non-dominant regime (NDR).

Note that the system in TPLR can be reduced to the onestudied in [27]. We will, therefore, focus on the operationalregimes IPLR and NDR. Using Lemma 1 and Lemma 2,we prove in the next propositions that the c.d.f. of the SNRFγi (·) in (11) lies on theMDAof the Gumbel or Fréchet distri-butions for the two considered scenarios, i.e., IPLR andNDR,respectively. Recall that, as defined in the previous section,SP = Pm/σ 2, SQ = Q/µhσ 2, and limx→ω(F)

1−Fγi (x)1−Fγi (x)

=

ζ ≤ (σ 2g )nr−1.

Proposition 3: Given the transmit power (4), the distribu-tion Fγi (·) in (10) lies on the MDA of the Fréchet distributionin IPLR, and the SNR distribution of TAS/MRC secondaryuser is given by

Fγmax (x) = exp (−bn/x) , ∀x > 0 (25)

with the coefficient bn chosen as

bn =µgSQ

[nt/(nt − 1)]ζ/nr − 1. (26)

Proof: The proof of Proposition 3 is in Appendix C.Proposition 4: Given the transmit power (4), the distribu-

tion Fγi (·) in (10) lies on theMDA of the Gumbel distributionin NDR. The SNR distribution of the secondary user withTAS/MRC is given by

Fγmax (x) = exp(− exp

(−x − anbn

)), ∀x > 0 (27)

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FIGURE 2. Exact and approximate distributions of the SU SNR when thebest transmit antenna is selected in NDR (Gumbel), where SP = 5dB andSQ = −5dB. The number of receive antennas nr = 2. The solid and dashedcurves depict the exact and approximated distributions with perfect CSI,respectively. Markers ‘+’ and ‘o’, respectively, denote the exact andapproximated distributions with imperfect CSI, σ2

g = 0.9.

where the coefficients an and bn are

an = µgSP[log(nt )+ (nr − 1) log(log(nt ))

− log0(nr )+ log(ζ )+ log(1− e−SQ/SP

)],

bn = µgSP. (28)

Proof: The proof of Proposition 4 is in Appendix D.Remark: In the considered spectrum-sharing cognitive

radio system, the maximum transmit power Pm and the inter-ference power constraintQ play significant roles in determin-ing the limiting distribution of the SNR as nt → ∞, andthus, determining the asymptotic behavior of the secondarycognitive radio system. As the system operational regimeshifts away from NDR, the limiting distribution transitionsfrom the Gumbel distribution, in the form of double expo-nential function (18c), to the Fréchet distribution, in theform of single exponential function (18b). In other words,the dominance of the maximum transmit power determinesthe tail behavior of the SNR distribution, where the Fréchetdistribution has a heavy tail and the Gumbel distribution hasan exponential tail. Since the performance of the transmitantenna selection is determined by the tail behavior of theSNR distribution [27], Proposition 3 is vital to reveal theimpact of the power constraints on the performance of thecognitive radio system. In Fig. 2 and Fig. 3, we comparethe exact distributions (12) and the corresponding Gumbeland Fréchet distributions with perfect and imperfect CSI,respectively.

In the subsequent sections, we use Proposition 3 andProposition 4 to obtain the scaling laws for the average rateand outage rate of the SU for the considered two scenarios,i.e. IPLR and NDR.

IV. ASYMPTOTIC AVERAGE RATEFor the considered secondary system with TAS/MRC config-uration, the instantaneous rate of the SU in nats/s/Hz can be

FIGURE 3. Exact and approximate distributions of the SU SNR when thebest transmit antenna is selected in IPLR (Fréchet), where SP = 25dB andSQ = −5dB. The number of receive antennas nr = 2. The solid and dashedcurves depict the exact and approximated distributions with perfect CSI,respectively. Markers ‘+’ and ‘o’, respectively, denote the exact andapproximated distributions with imperfect CSI, σ2

g = 0.9.

written as3

Rmax = max1≤i≤nt

Ri, (29)

where Ri = log(1+ γi

), i = 1, · · · , nt is the instantaneous

rate using the ith antenna. The average rate of SU can beobtained by averaging over the SNR distribution as

C =∫∞

0log (1+ γ )dFγmax (γ ), (30)

where the c.d.f. Fγmax (γ ) =[Fγi (γ )

]nt is given in (12) andFγi (γ ) is given in (10). To enable a tractable analysis forthe average rate, we apply the asymptotic SNR distributionobtained in Section III. The obtained results represent theasymptotic performance of the considered system when thenumber of transmit antenna is large. However, numericalresults in Section VII show that they serve as good approx-imations even for moderate number of transmit antennas.In addition, the asymptotic analysis enables insight into therate scaling of the secondary user.

A. ASYMPTOTIC AVERAGE RATE IN IPLRReplacing the limiting distribution in (23) into (30),the asymptotic SU average rate for large nt can be obtainedas

CIPLR =

∫∞

0log(1+ x)de

−

(xbn

)−1

= bn

[∫∞

0log(1+ t)e−tbndt −

∫∞

0log(t)e−tbndt

]= γ0 − ebnEi[−bn]+ log(bn), (31)

where bn is given in (26), γ0 = 0.5772 . . . is the Euler-Mascheroni constant, Ei[w] = −

∫∞

−we−tt dt,∀w < 0 denotes

the exponential integral function [50, Eq. (8.211)], the second

3In this paper, we use Shannon capacity to approximate the rate.

138068 VOLUME 7, 2019


equality is obtained by applying substitution t = 1/x, andthe third equality is obtained by applying [50, Eq. (4.331-1)and Eq. (4.337-2)]. Applying power series expansion for bnin (26) as nt →∞, we obtain

bn =(ntnr −

nr + 12

)µgSQζ+ O

(1nt

), (32)

where the notation f (x) = O(g(x)) is defined aslim supx→∞ |f (x)/g(x)| < ∞. Since ezEi[−z] → 0 as z →∞, the asymptotic, in nt , average rate CIPLR given in (31) canbe approximated as

CIPLR ≈ log (bn)+ γ0 ∝ log(ntnr ). (33)

Remark: This result indicates that the average rate scaleslogarithmically with the product of nt and nr for large nt . Forfixed nr , the scaling law in terms of nt for the average rateis log(nt ). However, this explicit and simple results cannot beseen directly from [18, Eq. (12)].

In the next subsection, we consider another operationalscenario of the spectrum sharing system, where the maximumpower Pm of ST is comparable to the interference powerconstraint Q/µh, i.e., operation in the NDR regime.

B. ASYMPTOTIC AVERAGE RATE IN NDRBased on Proposition 4, the SNR distribution of the secondarysystem in NDR lies on the MDA of Gumbel distribution. Thecorresponding average rate can be obtained by applying thelimiting throughput theorem (LTD) from [46], which is listedas follows.Lemma 3 [46]: Let 0i denote the received SNR at the

secondary receiver when ith transmit antenna is used. Assumethat0i, i = 1, · · · ,N , are i.i.d. with a common distributionF(γ ) and the supremum ω(F(γ ))→ ∞. The distribution ofthroughput lies on the MDA of the Gumbel distribution if thedistribution of the SNR belongs to the MDA of the Gumbeldistribution.

Due to Lemma 3, we have limnt→∞ (Rmax − an) /bnd−→

3ξ (x) and 3ξ (x) is expressed in (20) with the followingnormalizing coefficients obtained by (21) as

an = log[1+ F−1

γi(1− 1/nt)

],

bn = log[1+ F−1

γii(1− 1/nte)

]− an. (34)

Using [46, Lemma 2], the expectation of (Rmax − an) /bnconverges, and the average rate of the SU, C = E [Rmax], canbe approximated by integrating Rmax over (20) as

CNDR = an + bnγ0. (35)

The approximation is tight in the asymptotic regime nt →∞,since the approximation error is induced from Proposition 4.Observing the normalizing coefficients given in (34) for thelimiting throughput distribution and the ones in (21) forthe limiting SNR distribution (replacing N by nt ), we havean = log (1+ an) and bn = log (1+ bn/(1+ an)). Basedon Proposition 4, it is observed that an scales logarithmically

with nt , and hence an scales as log log(nt ) and bn vanishes asnt →∞. Therefore, the asymptotic average rate (35) in NDRcan be approximated as

CNDR ≈ an ∝ log log(nt ), (36)

which is in sharp contrast to the rate scaling law (33) obtainedfor the IPLR.

V. ASYMPTOTIC OUTAGE RATEIn this section, we investigate the SU outage rate in IPLRand NDR. Given a target rate r (nats/s/Hz), the SU ratedistribution after transmit antenna selection is defined as theprobability that the instantaneous rate Rmax is less than r :

Fout(r) = Pr {Rmax < r} , (37)

where Rmax is given by (29). The outage rate can be inter-preted as the maximum rate, r , such that Fout ≤ ε, whereε ∈ (0, 1) is the predefined outage threshold. Therefore,the outage rate is expressed as

Cε = max{r : Fout(r) ≤ ε}. (38)

Obtaining the above exact outage rate is algebraically com-plex. In order to have tractable analysis and gain insight intothe considered system, we obtain the asymptotic expressionsfor the outage rate.

A. OUTAGE RATE IN IPLRBy using the limiting distribution of the SNR in IPLR, the out-age rate of the SU is obtained in the following proposition.Proposition 5: Given the outage probability threshold ε in

(38), in IPLR, the asymptotic outage rate of the SU with TASand MRC is expressed by

C IPLRε = log

[1+ bn (log 1/ε)−1

], (39)

where bn is given in (26).Proof: Using the limiting distribution (23), and

Proposition 3, we obtain the SU asymptotic outage probabil-ity.

Fout(r) = Fγmax

(er − 1

)= e−

bner−1 , (40)

The proof is completed by inserting (40) into (38).Inserting (32) into (39), the outage rate C IPLR

ε shows thatthe SU outage rate scales as log (nt) for large nt in IPLR. Thecapacity gap between the average rate CIPLR and C IPLR

ε iscalculated as

1IPLRc = CIPLR − C IPLR

ε = γ0 + log log (1/ε)+1, (41)

where CIPLR is given in (31) and approximated in (33), and

1 = log(1+

log εbn − log ε

)∝ 1/bn. (42)

As bn ∝ nt , it is obvious that 1 < 0 vanishes at arate n−1t . Therefore, given the outage threshold ε, the gapbetween the average rate and the outage rate converges toγ0 + log log (1/ε) as nt → ∞. Hence, γ0 + log log (1/ε)

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TABLE 1. Summary of the asymptotic laws of the outage rate.

represents the bound of the rate gap. For instance, givenε = 10%, the rate gap between the outage rate and theaverage rate is bounded approximately by 1.41125 nats/s/Hz.

B. OUTAGE RATE IN NDRAccording to Lemma 3, the asymptotic distribution of Rmaxcan be approximated as (20) with the normalizing coefficientschosen as (34). The SU outage rate in NDR is given in thefollowing proposition.Proposition 6: With TAS andMRC, the asymptotic outage

rate in NDR of the SU is given by

CNDRε = an − bn log log (1/ε) , (43)

where the normalizing coefficients an and bn are given in (34).Proof: The proof of this proposition is straightforward

by substituting (20) into (38).Next, we study the scaling behavior for the SU outage

rate in the NDR. From (43) and (35), we can see that theSU outage rate CNDR

ε scales in the same manner as the SUaverage rate CNDR. In addition, the capacity gap in NDR,1NDRc = CNDR − CNDR

ε , is given by

1NDRc = bn

[γ0 + log log (1/ε)

]. (44)

where bn is given in (34), and vanishes as nt → ∞. Thus,the rate gap between the average rate and the outage rate inNDR diminishes as nt →∞.

We summarize the key results of the asymptotic laws of theoutage rate in Table 1.

VI. AVERAGE SYMBOL ERROR RATEIn this section we study the average symbol error rate (ASER)for the considered system. Denote SERi as the symbol errorrate when the ith transmit antenna is selected, which is givenby

SERi =α

2

[1− erf

(√βγi

)], (45)

where the SNR γi is distributed according to the CDF Fγi (γ ).A generic ASER expression for a wide range of modulationand coding schemes was provided in [51] as

ASER = mini

E [SERi] = E[α2

(1− erf

(√βγmax

))],

(46)

where α and β are modulation-related parameters, and γmaxdenotes the maximal SNR max1≤i≤nt (γi). As shown in [51],for binary phase-shift keying, α = 1 and β = 1; for binaryfrequency-shift keying with orthogonal signaling, α = 1

and β = 0.5; for M -ary pulse amplitude modulation,α = 2(M−1)/M and β = 3/(M2

−1); and forM -ary phase-shift keying, α = 2 and β = sin2(π/M ).

By applying the change of variables technique in (45),the CDF of SERi is given by

FSERi (x) = 1− Fγi

(1β

[erf−1

(1−

2xα

)]2). (47)

Based on (46), the average symbol error rate after trans-mit antenna selection requires the knowledge of the dis-tribution of the minimal symbol error rate among {SERi},i.e., min1≤i≤nt (SERi), and eventually the distribution ofγmax. To the best of our knowledge, it is problematic toobtain the ASER (46) with general modulation and codingschemes using the distribution function Fγmax (x) =

[Fγi (x)

]ntand (47), where Fγi (x) is given in (10). In the following,we present the results achieved by applying EVT.

First, we derive the ASER in IPLR.Proposition 7: The ASER for the SU of the considered

system in IPLR can be approximated as

ASERIPLR=α

2e−2√βbn , (48)

where bn = µgSQ/([nt/(nt − 1)]ζ/nr − 1

)is given in (26).

Proof: By substituting (25) into (46), the symbol errorrate can be obtained as

llASERIPLR=

α2

∫∞

0 erfc(√βx)de−

bnx =

α2 e−2√βbn ,

where in the last step we apply [52, eq. 7.4.20].This compact result cannot be obtained usingFγmax (x) =

[Fγi (x)

]nt , where Fγi (x) is given in (13). Then,using the EVT, we prove in the following proposition that thedistribution of the SER (45) in NDR converges to the Weibulldistribution, which enables a simple approximation for theASER of the secondary system with TAS.Proposition 8: As the number of transmit antenna nt

increases, the distribution of the SER after transmit antennaselection in NDR converges in distribution to the Weibulldistribution, given by

FSERmin(x) = 1− exp

[−

(2xαbn

)1/βµgSP]

(49)

for 0 ≤ x ≤ α/2, and FSERmin(x) = 1 for x > α/2, wherethe normalizing coefficients can be chosen as

bn = an − erf(√βF−1

γi(1− 1/nt)

). (50)

The ASER for the secondary user of the considered systemcan be approximated as

ASER =12bn β α µgSP γ

(β µgSP, b

−1/β µgSPn

), (51)

where bn is given in (50), and γ (a, x) =∫ x0 e−t ta−1d t is the

lower incomplete Gamma function [50, eq. 8.350-1].Proof: The proof of Proposition 8 is in Appendix E.

138070 VOLUME 7, 2019


FIGURE 4. SU average rate versus nt with nr = 1.

FIGURE 5. SU average rate versus nt with nr = 5.

VII. SIMULATION RESULTSThis section presents the simulation results of the SU averagerate, the outage rate, and the average symbol error rate ofthe considered system. The ST employs the transmit antennaselection, while the SR employs the maximal ratio combiningtechnique. The transmission of the ST is constrained by thepeak interference constraint and the maximal transmit powerconstraint shown in (4). The average channel gains are setto 1. For each simulated curve, the considered performancemetric is averaged over 1 × 106 channel realizations. Underimperfect CSI condition, the channel correlation coefficentbetween the channel coefficient and the channel estimation ischosen to be 0.9, i.e. σ 2

g = 0.9.Fig. 4 and Fig. 5 depict the average rate of the SU as a

function of nt assuming 1 and 5 receive antennas, respec-tively. The average rates are plotted for the IPLR and NDRscenarios. We also plot log(nt ) and log log(nt ) to show thecorresponding scaling laws. We can see that even for thecase with less than 5 transmit antennas the asymptotic resultsusing EVT are quite accurate. We can also observe thatthe SU average rate scales logarithmically in terms of ntin IPLR. In NDR, the figures illustrate that the curves ofthe SU average rate become more flat as predicted by thescaling law in (36), i.e., the average rate scales as log(log(nt )).

FIGURE 6. SU outage rate versus nt with ε = 10% and nr = 1.

FIGURE 7. SU outage rate versus nt with ε = 10% and nr = 5.

In Fig. 5, the curves with marker � illustrate the average ratefor δ2g = 0.9. Compared to the curves of the perfect CSIscenario, they have the same scaling behavior in NDR andIPLR as nt grows to infinity. This confirms the analysis resultsin previous sections that the SNR distributions for the perfectand the imperfect scenarios are tail equivalent.

In Fig. 6 and in Fig. 7, we depict the SU outage rateas a function of nt assuming 1 and 5 receive antennas,respectively. The outage threshold ε is set to 10%. The EVTapproximations (39) in NDR and (43) in IPLR agree withthe simulation results. In addition, the outage rates in eachoperational regime scale as the number of transmit antenna nt ,following the same scaling law as the corresponding averagerate, i.e., in NDR Cε ∼ log log nt and in IPLR Cε ∼ log nt .In Fig. 7, the curves with marker � show the outage rate forδ2g = 0.9. Compared to the curves of the perfect CSI scenario,they have the same scaling behavior in NDR and IPLR as ntgrows to infinity. This confirms the analysis results for SUoutage rate.

Fig. 8 shows the average rate and the outage rate of the SUas a function of the ratio ρ = SP/SQ assuming one receiveantenna, where SP = Pm/σ 2 and SQ = Q/µhσ 2. We havethe following key observations: (a) the rates approach theIPLR scenario as ρ increases; (b) given nt , the gap between

VOLUME 7, 2019 138071


FIGURE 8. SU average rate and outage rate versus ρ = SP/SQ withε = 10%, SQ = 0dB, and nr = 1.

FIGURE 9. Rate gap versus nt with nr = 1 and ε = 10%. In NDR SQ = 0dB,and SP →∞ in IPLR.

the average rate and the corresponding outage rate firstincreases (NDR regime) and then becomes constant (IPLRregime) as ρ increases; (c) the values of nt impacts thespeed at which the rates transition from NDR to IPLR asρ increases; for instance, when nt = 20 the outage ratealmost saturates at ρ = 12, while for nt = 100 the outagerate almost saturates at ρ = 20; (d) given nt , the outagerate approaches to the IPLR scenario faster than the aver-age rate does. These observations confirm that the asymp-totic behaviors of the SU rate shift from the NDR regime,governed by Gumbel distribution, to the IPLR regime, gov-erned by the Fréchet distribution, as SP becomes dominantover SQ.The rate gap of the SU between the outage rate and

the average rate is depicted in Fig. 9. In addition, we plot therate ratio for the IPLR and the NDR, which is defined as theratio of the outage rate and the corresponding average rate.We set SP → ∞ in IPLR, and SQ = 0dB in NDR. First,the rate gap in IPLR is approaching to a constant value γ0 +log log(1/ε) as indicated in (41). Second, the rate gap in NDRis vanishing slowly as suggested by (44). This shows thatthe average rate can be estimated by the outage rate, whichis, in practice, easier to obtain than the average rate in (30).

FIGURE 10. SU average symbol error rate versus nt . The number ofreceive antennas nr = 1. The markers show the simulation results. Thevalues of Sp and SQ are in dB.

For instance, given ε = 10%, the rate gap between the outagerate and the average rate is always bounded approximately by1.41 nats/s/Hz. Moreover, the rate ratio curves confirm theseobservations.

Fig. 10 shows the SU average symbol error rate for BPSKand 4-PSK modulation schemes as a function of nt . For theIPLR scenario, we assume that SQ = −5dB and SP = 20dB.The results using EVT in IPLR and NDR are obtained fromProposition 7 and Proposition 8, respectively. For each nt ,the simulation results are obtained by averaging over 106

channel realizations. Obviously, the ASER decreases as ntincreases. The figure explicitly illustrates that the ASERcurves in IPLR have larger slops than the ones in NDR.Moreover, it shows that when nt is small the interferencepower constraint SQ has more influence on ASER than SP.Assuming 4-PSK modulation, when nt < 50, the ASER forSP = 20dB and SQ = −5dB has higher value than in thecase SP = 5dB and SQ = 0dB. However, when nt > 50,the IPLR provides lower ASER than in NDR. Moreover,the observations clearly illustrate that the ASER as a functionof nt decreases faster in IPLR than that in NDR, which hasbeen predicted in Section VI.

VIII. CONCLUSIONApplying Extreme Value Theory, we revealed a fundamental,yet overlooked property of the SNR distribution of the sec-ondary user in a spectrum-sharing system provided imperfectCSI, where the limiting SNR distributions are governed eitherby Fréchet or Gumbel distributions. In addition, we showedthat the SNR distributions for the perfect and the imperfectscenarios are tail-equivalent. The obtained results enable acompact, yet accurate approximation for the average rate,the outage rate, and the average symbol error rate of theSU for general modulation types. These results, which wereoverlooked in literature, have been utilized to gain insightsinto the behavior of the SU average rate, outage rate, and theirscaling characteristics in the considered spectrum-sharingsystem.

138072 VOLUME 7, 2019


Particularly, the SU average rate and outage rate havetwo scaling behaviors as the number of transmit antennasnt increases: When the transmit power of the secondarytransmitter is mainly restricted by the maximal interferenceconstraint imposed by the primary receiver, i.e. the IPLRregime, the rates of the secondary user scale as log(nt );when such interference power constraint is comparable tothe maximal transmit power of the secondary transmitter, i.e.the NDR regime, the rates scale as log(log(nt )), implyingrapidly diminishing gain on addingmore antennas.Moreover,the observations of the rate gap between the outage rateand the average rate allows us to estimate the average ratefrom the corresponding easier-to-obtain outage rate.

APPENDIX ADERIVATION OF PROPOSITION 1Let Ph = Pr

(|hi|2 ≤

SQSP

)= 1 − e−SQ/SP . Recall that

SP = Pm/σ 2 and SQ = Q/µhσ 2.

Fγi (x)

= Pr

(min

{SQ‖gi‖2F|hi|2

,SP‖gi‖2F

}≤ x

)

= 1− Pr

(‖gi‖2F >

|hi|2 xSQ

, ‖gi‖2F >xSP

)

= 1−∫∞

SQSP

Pr(‖gi‖2F >

xtSQ

)dF|hi|2

(t)

−

∫ SQSP

0Pr(‖gi‖2F >

xSP

)dF|hi|2

(t)

=

∫∞

SQSP

F||gi||2F

(xtSQ

)dF|hi|2

(t)+ PhF||gi||2F

(xSP

)

=

nr∑m=1

Bnr−1m−1 (σ2g )∫∞

SQSP

1− 0(m, xt

µgSQ

)0(m)

dF|hi|2

(t)

︸︷︷︸F (1)γi

(x)

+Phnr∑m=1


1−0(m, x

SPµg

)0(m)

,where

F (1)γi

(x) =∫∞

SQSP

1− 0(m, xt

µgSQ

)0(m)

dF|hi|2

(t)

= 1−

1− 0(m, x

µgSP

)0(m)

Ph

−

∫∞

SQSP

F|hi|2

(t)d

1− 0(m, xt

µgSQ

)0(m)

︸︷︷︸

F (2)γi

(x)

with

F (2)γi

(x) =∫∞

SQSP

F|hi|2

(t)d

1− 0(m, xt

µgSQ

)0(m)

=

10(m)

[0

(m,

xµgSP

)−

(x

µgSQ + x

)m0

(m,

xµgSP

+SQSP

)].

Hence, F||gi||2F

(x) given in Proposition 1 is obtained.

APPENDIX BDERIVATION OF PROPOSITION 2The distribution functions Fγi (x) and Fγi (x) are given in (10)and (11), respectively. We define

Uk (x) ≡10(k)

[0

(k,

xµgSP

)−

(x

µgSQ + x

)k0

(k,

xµgSP

+SQSP

)],

where k is an integer and 0 ≤ Uk (x) ≤ 1. When k isgiven, Uk (x) is a decreasing function of x. Therefore, we haveFγi (x) = 1−

∑nrm=1 B

nr−1m−1 (σ

2g )Um(x) and Fγi (x) = 1−Unr (x).

Recall that 0(k + 1) = k0(k), 0(k, z) = 0(k)e−z∑k−1

l=0zll! ,

and 0(k+ 1, z) = k0(k, z)+ zke−z. We first prove in (52), asshown at the top of the next page that Uk (x) is an increasingfunction of k when other parameters are given.We verify that

limx→ω(Fγi )

1− Fγi (x)

1− Fγi (x)

= limx→∞

∑nrm=1 B

nr−1m−1 (σ

2g )Um(x)

Unr (x)

= limx→∞

nr∑m=1


Um(x)Unr (x)

≤ limx→∞

nr∑m=1


≤ (σ 2g )nr−1

This shows that there exists some 0 < ζ ≤ (σ 2g )nr−1.

Hence, according to Definition 1, Fγi (x) and Fγi (x) are tailequivalent. Consequently, Proposition 2 holds.

APPENDIX CDERIVATION OF PROPOSITION 3As SP → ∞ or SP � SQ, the c.d.f. is given in (14). Thus,it is easy to verify that

limt→ω(Fγi )

[1− Fγi (tx)1− Fγi (t)

]= lim

t→∞

1−(

txtx+µgSQ

)nr1−

(t

t+µgSQ

)nr

=nrµgSQ/ (tx)+O

(1/t2

)nrµgSQ/t +O

(1/t2

)= x−1,

VOLUME 7, 2019 138073


Uk+1(x) =1

0(k + 1)

[0

(k + 1,

xµgSP

)−

(x

µgSQ + x

)k+10

(k + 1,

xµgSP

+SQSP

)]

=1

0(k + 1)

[k0(k,

xµgSP

)+

(x

µgSP

)ke−

xµgSP −

(x

µgSQ + x

)k+1 ( xµgSP

+SQSP

)ke−

xµgSP−

SQSP

− k(

xµgSQ + x

)k+10

(k,

xµgSP

+SQSP

)]

=1

0(k + 1)

[k0 (k)Uk (x)+ k

(x

µgSQ + x

)k0

(k,

xµgSP

+SQSP

)+

(x

µgSP

)ke−

xµgSP

−

(x

µgSQ + x

)(x

µgSP

)ke−

xµgSP−

SQSP − k

(x

µgSQ + x

)k+10

(k,

xµgSP

+SQSP

)].

1U = Uk+1(x)− Uk (x)

=1

0(k + 1)

[k(

xµgSQ + x

)k0

(k,

xµgSP

+SQSP

)− k

(x

µgSQ + x

)k+10

(k,

xµgSP

+SQSP

)

+

(x

µgSP

)ke−

xµgSP −

(x

µgSQ + x

)(x

µgSP

)ke−

xµgSP−

SQSP

]≥ 0. (52)

where O(tn) represents a term of order tn. Therefore,according to Lemma 2 the distribution Fγi (x) lies on theMDA of Fréchet distribution. Since that Fγi (x) and Fγi (x)are tail equivalent as shown in Proposition 2, the distri-bution Fγi lies on the MDA of the Fréchet distribution.This suggests that we can work on an easy tail equiva-lent distribution and compute the normalizing coefficientsfor it.

NORMALIZING COEFFICIENTS FOR FRÉCHETGiven the c.d.f. in (14), the corresponding inverse c.d.f. reads

F−1γi (y) =µgSQ

y−1/nr − 1. (53)

As x → ω(F), Fγi (x) and Fγi (x) are tail equivalent,i.e., nt (1− Fγi (bnx + an)) ∼ nt (1− Fγi (bnx + an))ζ

−1 such

that Fntγi(bnx + an) →

[Fntγi (bnx + an)

]ζ−1=

(x

x+µgSQ

)nr/ζ[44, Proposition 1.19]. Hence, given the c.d.f. in (13), the cor-responding inverse c.d.f. can be approximately as

F−1γi

(y) =µgSQ

y−ζ/nr − 1. (54)

Then, using Proposition 3 and (24), the normalizing coef-ficients (17) for the Fréchet distribution is obtained as

an = 0, bn =µgSQ

[nt/(nt − 1)]ζ/nr − 1.

Using series expansion to (26) in terms of nt , we can directlysee that bn ∝ nt as nt → ∞ given other parameters in (26).This suggests that bn is unbounded in IPLR.

APPENDIX DDERIVATION OF PROPOSITION 4We first show that the distribution Fγi (x) lies on the MDAof the Gumbel distribution, and then apply the property thatFγi (x) and Fγi (x) are tail equivalent.The p.d.f. of γi is obtained as the derivative of Fγi (·):

fγi (x) =e−

xσ2Pmµg xnr−1

0(nr )A(x).

where

A(x) =(

σ 2

Pmµg

)nr−

(σ 2

Pmµg

)xe−

QPmµh

x + Qµgµhσ 2

+nrQµg/(µhσ 2)(x + Qµg

µhσ 2

)nr+10(nr ,

xσ 2

Pmµg+

QPmµh

). (55)

Then we have

f ′γi (x)

fγi (x)= −

σ 2

Pmµg+nr − 1x+A′(x)A(x)

, (56)

and

limx→∞

A′(x)A(x)

= 0. (57)

Thus,

limx→∞

fγi (x)f ′γi (x)

= −Pmµgσ 2 . (58)

It is obvious that limx→∞ fγi (x) = 0 and limx→∞[−f ′γi (x)

]= 0. We need to show, after taking the derivative

138074 VOLUME 7, 2019


according to (19), that

limx→∞

[(1− Fγi (x))f

′γi(x)

f 2γi (x)

]= −1. (59)

Since Fγi (x) is the c.d.f. of γi, thus we have

limx→∞

[Fγi (x)− 1

]= 0. (60)

Therefore we obtain, according to L’Hospital’s rule,

limx→∞

[1− Fγi (x)fγi (x)

]= lim

x→∞

[(1− Fγi (x)

)′f ′γi (x)

]

= limx→∞

[fγi (x)−f ′γi (x)

]=Pmµgσ 2 . (61)

First, we assume Pm < ∞. Both of the limitslimx→∞

[1−Fγi (x)fγi (x)

]and limx→∞

[fγi (x)−f ′γi (x)

]are finite. There-

fore, we have

limx→∞

[(1− Fγi (x))f

′γi(x)

f 2γi (x)

]

= limx→∞

(1− Fγi (x)fγi (x)

)limx→∞

(f ′γi (x)

fγi (x)

)= −1, (62)

which indicates that in NDR Fγi (x) is a von Mises functionsuch that it belongs to the MDA of the Gumbel distribu-tion [49]. As a consequence, the distribution Fγi also lies onthe MDA of the Gumbel distribution since the two distribu-tions are tail equivalent.

NORMALIZING COEFFICIENTS FOR GUMBELIn NDR, we apply tail equivalence of distribution function toobtain the normalizing coefficients, which provide insightsinto the scaling laws.Lemma 4: [53, Proposition 1]. Let F be a common distri-

bution function of i.i.d. random variables Xi, i = 1, · · · ,N ,which lies on the MDA of Gumbel distribution. Suppose thatthere exist α > 0, β ∈ R, c > 0, and d > 0 such that

limx→∞

1− F(x)

αxβe−cxd= 1.

Then, the normalizing coefficients in (17) can be chosen as

an = (log(N )/c)1d

+(β/d)

[log(log(N ))− log(c)

]+ log(α)

(log(N )/c)1−1d d c

,

bn =(log(N )/c)

1d−1

d c. (63)

Proposition 9: In NDR, the asymptotic normalizing coef-ficients, using the tail equivalence with the common distribu-tion given in (11), can be chosen as

an = µgSP[log(nt )+ (nr − 1) log(log(nt ))

− log0(nr )+ log(ζ )+ log(1− e−SQ/SP

)],

bn = µgSP,

where, as defined in previous section, SP = Pm/σ 2, andSQ = Q/µhσ 2.

Proof:

limx→∞

1− Fγi (x)

αxβe−cxd

= limx→∞

1− Fγi (x)

1− Fγi (x)1− Fγi (x)

αxβe−cxd

= ζ

10(nr )

(1− e−

SQSP

) (µgSP

)1−nr xnr−1e− xµgSP

αxβe−cxd.

Applying tail equivalence, we obtain

α =ζ

0(nr )

(1− e−

SQSP

) (µgSP

)1−nr , β = nr − 1,

c =(µgSP

)−1, d = 1, (64)

Then by substituting (64) into (63) and replacing N by nt ,we obtain (28).

Remark: As SQ →∞, the considered model is equivalentto the point-to-point TAS/MRC model investigated in [27],and (28) is identical to the results in [27, Lemma 2].

APPENDIX EDERIVATION OF PROPOSITION 8We introduce a lemma on the MDA of the Weibull distribu-tion.Lemma 5: [49, Theorem 1.1.13] Let x∗ denote the right

end of the random variable X , and FX (t) be its distributionfunction. Suppose that x∗ is finite, and the first derivative ofthe distribution F ′X (t) exists for X < x∗ < ∞. We say thatFX (t) belongs to the MDA of the Weibull distribution, if

limt→x∗

(x∗ − t)F ′X (t)1− FX (t)

= −ξ−1, ξ < 0,

where ξ is given in (18).Let Yi = erf

(√βγi

), ∀i = 1, · · · , nt which are random

variables with common distribution function FY (y). The c.d.f.FY (y), 0 < y < 1, can be obtained as

FY (y) = Pr {Y ≤ y} = Pr{γi ≤

1β

[erf−1(y)

]2}= Fγi

(1β

[erf−1(y)

]2),

and then

F−1Y (z) = erf(√

βFγi−1(z)

). (65)

Let H (y) denote[erf−1(y)

]2/β for simplicity, and hence

H (y)→∞ as y→ 1. Then we have

H ′(y) =dH (y)d y

=

√πH (y)β

eβH (y).

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Next we evaluate limy→y∗ (y∗ − y) fY (y)/(1 − FY (y)), wherey∗ denotes the upper end, i.e., y∗ = 1.

limy→1

(1− y) fY (y)1− FY (y)

= limy→1

(1− y) fγi (H (y))H ′(y)

1− Fγi (H (y))

= limy→1

[1− erf(

√βH (y))

]fγi (H (y))

1− Fγi (H (y))

√πH (y)β

eβH (y)

= limt→∞

fγi (t)

1− Fγi (t)︸︷︷︸C1

√π tβeβt

[1− erf(

√βt)]

︸︷︷︸C2

,

where we replace H (y) by t in the last step, and

limt→∞

C1 = limt→∞

fγi (t)

fγi (t)fγi (t)

1− Fγi (t)1− Fγi (t)1− Fγi (t)

= limt→∞

fγi (t)1− Fγi (t)

= 1/µgSP.

The last step, we have applied (61) and that Fγi (x) andFγi (x) are tail equivalent. Hence, C1 has finite limit, i.e.limt→∞ C1 = 1/µgSP.Using the identity [52, 7.1.23]

1− erf(x) =e−x

2

x√π

(1−

12x2− · · ·

)(66)

for C2 yields

C2 =1β

(1−

12βt− · · ·

).

It follows that

limt→∞

C2 =1β<∞.

Consequently, we obtain

limy→1

(1− y) fY (y)1− FY (y)

=

(limt→∞

C1

) (limt→∞

C2

)=

1βµgSP

.

Following Lemma 5, we have that FY (y) lies in the MDAof the Weibull distribution. In consequence, according to theFisher-Tippet-Gnedenko theorem given in (18), the limitingdistribution of Znt = max(Yi, · · · ,Ynt ) can be written as

FZnt (z) = exp

[−

(an − zbn

)1/βµgSP]

(67)

with the possible normalizing coefficients chosen as

an = 1 and bn = an − F−1Y

(1−

1nt

).

Note that error function erf(x),∀x ≥ 0, is invertible and non-decreasing, and hence from (65) bn reads

bn = 1− F−1Y

(1−

1nt

)= 1− erf

(√βFγi

−1(1−

1nt

)).

Recall that SERmin =[1− erf

(√βγmax

)]α/2 defined in

Section VI. It follows that SERmin =(1− Znt

)α/2. The

distribution of SERmin can be deduced as follows:

FSERmin (x) = 1− FZnt (1− 2x/α)

= 1− exp

[−

(2xαbn

)1/βµgSP]

(68)

for 0 ≤ x ≤ α/2, and FSERmin (x) = 1 for x > α/2.Substituting (67) into (46) yields the ASER as follows

ASER =∫ α/2

0

[1− FSERmin(x)

]d x

=

∫ α/2

0exp

[−

(2xαbn

)1/βµgSP]d x

=12bn β α µgSP γ

(β µgSP, b

−1/β µgSPn

),

where in the last step we apply [50, eq. 3.381-8].

REFERENCES[1] S. Rajendran, B. Van den Bergh, S. Pollin, and T. Vermeulen, ‘‘IEEE

5G spectrum sharing challenge: A practical evaluation of learning andfeedback,’’ IEEE Commun. Mag., vol. 54, no. 11, pp. 210–216, Nov. 2016.

[2] F. Hu, B. Chen, and K. Zhu, ‘‘Full spectrum sharing in cognitive radionetworks toward 5G: A survey,’’ IEEE Access, vol. 6, pp. 15754–15776,2018.

[3] R. H. Tehrani, S. Vahid, D. Triantafyllopoulou, H. Lee, and K. Moessner,‘‘Licensed spectrum sharing schemes for mobile operators: A survey andoutlook,’’ IEEE Commun. Surveys Tuts., vol. 18, no. 4, pp. 2591–2623,4th Quart., 2016.

[4] F. Teng, D. Guo, and M. L. Honig, ‘‘Sharing of unlicensed spectrumby strategic operators,’’ IEEE J. Sel. Areas Commun., vol. 35, no. 3,pp. 668–679, Mar. 2017.

[5] C. Zhang and W. Zhang, ‘‘Spectrum sharing for drone networks,’’ IEEEJ. Sel. Areas Commun., vol. 35, no. 1, pp. 136–144, Jan. 2017.

[6] B. Cho, K.Koufos, R. Jäntti, and S.-L. Kim, ‘‘Co-primary spectrum sharingfor inter-operator device-to-device communication,’’ IEEE J. Sel. AreasCommun., vol. 35, no. 1, pp. 91–105, Jan. 2017.

[7] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, ‘‘Breaking spectrumgridlock with cognitive radios: An information theoretic perspective,’’Proc. IEEE, vol. 97, no. 5, pp. 894–914, Apr. 2009.

[8] M. E. Tanab andW.Hamouda, ‘‘Resource allocation for underlay cognitiveradio networks: A survey,’’ IEEE Commun. Surveys Tuts., vol. 19, no. 2,pp. 1249–1276, 2nd Quart., 2017.

[9] N. B. Mehta, S. Kashyap, and A. F. Molisch, ‘‘Antenna selection in LTE:From motivation to specification,’’ IEEE Commun. Mag., vol. 50, no. 10,pp. 144–150, Oct. 2012.

[10] S. Asaad, A. M. Rabiei, and R. R. Müller, ‘‘Massive MIMO with antennaselection: Fundamental limits and applications,’’ IEEE Trans. WirelessCommun., vol. 17, no. 12, pp. 8502–8516, Dec. 2018.

[11] Y. Yu, H. Chen, Y. Li, Z. Ding, L. Song, and B. Vucetic, ‘‘Antenna selectionfor MIMO nonorthogonal multiple access systems,’’ IEEE Trans. Veh.Technol., vol. 67, no. 4, pp. 3158–3171, Apr. 2018.

[12] S. Sanayei and A. Nosratinia, ‘‘Capacity of MIMO channels with antennaselection,’’ IEEE Trans. Inf. Theory, vol. 53, no. 11, pp. 4356–4362,Nov. 2007.

[13] A. Bereyhi, S. Asaad, and R. R. Müller, R. F. Schaefer, and A. M. Rabiei,‘‘On robustness of massive MIMO systems against passive eavesdroppingunder antenna selection,’’ in Proc. IEEEGLOBECOM, Dec. 2018, pp. 1–7.

[14] S. Thoen, L. Van der Perre, B. Gyselinckx, and M. Engels, ‘‘Performanceanalysis of combined transmit-SC/receive-MRC,’’ IEEE Trans. Commun.,vol. 49, no. 1, pp. 5–8, Jan. 2001.

[15] Z. Chen, J. Yuan, and B. Vucetic, ‘‘Analysis of transmit antennaselection/maximal-ratio combining in Rayleigh fading channels,’’ IEEETrans. Veh. Technol., vol. 54, no. 4, pp. 1312–1321, Jul. 2005.

[16] Z. Chen, Z. Chi, Y. Li, and B. Vucetic, ‘‘Error performance of maximal-ratio combining with transmit antenna selection in flat Nakagami-m fadingchannels,’’ IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 424–431,Jan. 2009.

[17] C.-C. Hung, C.-T. Chiang, N.-Y. Yen, and R.-C. Wu, ‘‘Outage probabilityof multiuser transmit antenna selection/maximal-ratio combining systemsover arbitrary Nakagami-m fading channels,’’ IET Commun., vol. 4, no. 1,pp. 63–68, Jan. 2010.

[18] V. Blagojevic and P. Ivanis, ‘‘Ergodic capacity for TAS/MRC spec-trum sharing cognitive radio,’’ IEEE Commun. Lett., vol. 16, no. 3,pp. 321–323, Mar. 2012.

138076 VOLUME 7, 2019


[19] D. Li, ‘‘Performance analysis of MRC diversity for cognitive radio sys-tems,’’ IEEE Trans. Veh. Technol., vol. 61, no. 2, pp. 849–853, Feb. 2012.

[20] J.-P. Hong andW. Choi, ‘‘Throughput characteristics bymultiuser diversityin a cognitive radio system,’’ IEEE Trans. Signal Process., vol. 59, no. 8,pp. 3749–3763, Aug. 2011.

[21] K. Tourki, F. A. Khan, K. A. Qaraqe, H.-C. Yang, and M.-S. Alouini,‘‘Exact performance analysis of MIMO cognitive radio systems usingtransmit antenna selection,’’ IEEE J. Sel. Areas Commun., vol. 32, no. 3,pp. 425–438, Mar. 2014.

[22] P. L. Yeoh, M. Elkashlan, K. J. Kim, T. Q. Duong, and G. K. Karagiannidis,‘‘Transmit antenna selection in cognitiveMIMO relayingwithmultiple pri-mary transceivers,’’ IEEE Trans. Veh. Technol., vol. 65, no. 1, pp. 483–489,Jan. 2016.

[23] Y. Huang, F. Al-Qahtani, C. Zhong, Q. Wu, J. Wang, and H. Alnuweiri,‘‘Performance analysis of multiuser multiple antenna relaying net-works with co-channel interference and feedback delay,’’ IEEE Trans.Commun., vol. 62, no. 1, pp. 59–73, Jan. 2014.

[24] Y. Huang, F. S. Al-Qahtani, C. Zhong, Q. Wu, J. Wang, andH. M. Alnuweiri, ‘‘Cognitive MIMO relaying networks with primaryuser’s interference and outdated channel state information,’’ IEEE Trans.Commun., vol. 62, no. 12, pp. 4241–4254, Dec. 2014.

[25] Y. Huang, C. Li, C. Zhong, J.Wang, Y. Cheng, andQ.Wu, ‘‘On the capacityof dual-hop multiple antenna AF relaying systems with feedback delay andCCI,’’ IEEE Commun. Lett., vol. 17, no. 6, pp. 1200–1203, Jun. 2013.

[26] Y. Huang, J. Wang, P. Zhang, and Q. Wu, ‘‘Performance analysis of energyharvesting multi-antenna relay networks with different antenna selectionschemes,’’ IEEE ACCESS, vol. 6, pp. 5654–5665, 2018.

[27] D. Bai, P. Mitran, S. S. Ghassemzadeh, R. R. Miller, and V. Tarokh,‘‘Rate of channel hardening of antenna selection diversity schemes andits implication on scheduling,’’ IEEE Trans. Inf. Theory, vol. 55, no. 10,pp. 4353–4365, Oct. 2009.

[28] J. Lee, H. Wang, J. G. Andrews, and D. Hong, ‘‘Outage probabilityof cognitive relay networks with interference constraints,’’ IEEE Trans.Wireless Commun., vol. 10, no. 2, pp. 390–395, Feb. 2011.

[29] E. Biglieri, A. J. Goldsmith, L. J. Greenstein, N. B. Mandayam, and H. V.Poor, Principles of Cognitive Radio. Cambridge, U.K.: Cambridge Univ.Press, 2012.

[30] Z. Yan, X. Zhang, H.-L. Liu, and Y.-C. Liang, ‘‘An efficient transmit powercontrol strategy for underlay spectrum sharing networks with spatiallyrandom primary users,’’ IEEE Trans. Wireless Commun., vol. 17, no. 7,pp. 4341–4351, Jul. 2018.

[31] R. Sarvendranath and N. B. Mehta, ‘‘Transmit antenna selection forinterference-outage constrained underlay CR,’’ IEEE Trans. Commun.,vol. 66, no. 9, pp. 3772–3783, Sep. 2018.

[32] R. F. Manna, F. S. Al-Qahtani, and S. A. Zummo, ‘‘A full diversitycooperative spectrum sharing scheme for cognitive radio networks,’’ IEEEAccess, vol. 5, pp. 17722–17732, 2017.

[33] J. M. Peha, ‘‘Sharing spectrum through spectrum policy reform and cog-nitive radio,’’ Proc. IEEE, vol. 97, no. 4, pp. 708–719, Apr. 2009.

[34] M. Xia and S. Aissa, ‘‘Spectrum-sharing multi-hop cooperative relaying:Performance analysis using extreme value theory,’’ IEEE Trans. WirelessCommun., vol. 13, no. 1, pp. 234–245, Jan. 2014.

[35] H. Ding, J. Ge, D. B. da Costa, and Z. Jiang, ‘‘Asymptotic analysis of coop-erative diversity systems with relay selection in a spectrum-sharing sce-nario,’’ IEEE Trans. Veh. Technol., vol. 60, no. 2, pp. 457–472, Feb. 2011.

[36] C. Zhong, T. Ratnarajah, and K.-K. Wong, ‘‘Outage analysis of decode-and-forward cognitive dual-hop systems with the interference constraint inNakagami-m fading channels,’’ IEEE Trans. Veh. Technol., vol. 60, no. 6,pp. 2875–2879, Jul. 2011.

[37] L. Musavian and S. Aissa, ‘‘Fundamental capacity limits of cognitive radioin fading environments with imperfect channel information,’’ IEEE Trans.Commun., vol. 57, no. 11, pp. 3472–3480, Nov. 2009.

[38] M. Gans, ‘‘The effect of Gaussian error in maximal ratio combiners,’’ IEEETrans. Commun. Technol., vol. COM-19, no. 4, pp. 492–500, Aug. 1971.

[39] J. L. Zhang and G. F. Pan, ‘‘Outage analysis of wireless-powered relayingMIMO systems with non-linear energy harvesters and imperfect CSI,’’IEEE Access, vol. 4, pp. 7046–7053, 2016.

[40] D. Lee, ‘‘Performance analysis of dual selection with maximal ratio com-bining over nonidentical imperfect channel estimation,’’ IEEE Trans. Veh.Technol., vol. 67, no. 3, pp. 2819–2823, Mar. 2018.

[41] G. L. Stüber, Principles of Mobile Communication, 2nd ed. Norwell, MA,USA: Kluwer, 2001.

[42] J. Proakis and M. Salehi, Digital Communications, 5th ed. New York, NY,USA: McGraw-Hill, 2008.

[43] H. A. David and H. N. Nagaraja, Order Statistics. Hoboken, NJ, USA:Wiley, 2003.

[44] S. I. Resnick, Extreme Values, Regular Variation and Point Processes. NewYork, NY, USA: Springer-Verlag, 2008.

[45] J. Galambos, The Asymptotic Theory of Extreme Order Statistics, 1st ed.New York, NY, USA: Wiley, 1978.

[46] G. Song and Y. Li, ‘‘Asymptotic throughput analysis for channel-awarescheduling,’’ IEEE Trans. Commun., vol. 54, no. 10, pp. 1827–1834,Oct. 2006.

[47] Y. Huang and B. D. Rao, ‘‘An analytical framework for heterogeneouspartial feedback design in heterogeneous multicell OFDMA networks,’’IEEE Trans. Signal Process., vol. 61, no. 3, pp. 753–769, Feb. 2013.

[48] Z. Zheng, J. Hämäläinen, and Z. Ding, ‘‘On the sum rate of fair resourceallocation with selective feedback,’’ IEEE Trans. Wireless Commun.,vol. 15, no. 8, pp. 5193–5205, Aug. 2016.

[49] L. de Haan and A. Ferreira, Extreme Value Theory: An Introduction.New York, NY, USA: Springer, 2006.

[50] I. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products,7th ed. New York, NY, USA: Academic, 2007.

[51] M. R. McKay, A. J. Grant, and I. B. Collings, ‘‘Performance analysis ofMIMO-MRC in double-correlated Rayleigh environments,’’ IEEE Trans.Commun., vol. 55, no. 3, pp. 497–507, Mar. 2007.

[52] M. Abramowitz and I. A. Stegun, Eds., Handbook of MathematicalFunctions: With Formulas, Graphs, and Mathematical Tables, 10th ed.New York, NY, USA: Dover, 1972.

[53] R. Takahashi, ‘‘Normalizing constants of a distribution which belongs tothe domain of attraction of the Gumbel distribution,’’ Stat. Probab. Lett.,vol. 5, pp. 197–200, Apr. 1987.

RUIFENG DUAN (S’09–M’14) received theM.Sc. (Tech.) and D.Sc. (Tech.) degrees intelecommunications engineering from the Uni-versity of Vaasa, Finland, in 2008 and 2014,respectively.

He held a visiting position at the Departmentof Electrical and Computer Engineering, PrairieView A&M University (a member of the TexasA& M University System). Since August 2014,he has been with the Department of Communi-

cations and Networking, Aalto University, Finland. His current researchinterests include random matrix theory, extreme value theory, ambientbackscatter communication, cognitive radio, and ultra-reliable low-latencycommunications. He was a recipient of the Vaasa University FoundationScholarship for the academic years of 2010 to 2011 and 2011 to 2012, the Ellaand Georg Ehrnrooth Foundation Scholarship, in 2010, the IEEE GLOBE-COM 2010 Early-Bird Student Award, and the Best Paper Award at TheInternational Symposium on Networks, Computers and Communications,June 2018.

ZHONG ZHENG (S’10–M’15) received theB.Eng. degree from the Beijing University ofTechnology, Beijing, China, in 2007, the M.Sc.degree from the Helsinki University of Technol-ogy, Espoo, Finland, in 2010, and the D.Sc. degreefrom Aalto University, Espoo, Finland, in 2015.From 2015 to 2018, he held visiting positions atThe University of Texas at Dallas and the NationalInstitute of Standards and Technology. In 2019,he joined the School of Information and Electron-

ics, Beijing Institute of Technology, Beijing, as an Associate Professor. Hisresearch interests include massive MIMO, secure communications, millime-ter wave communications, randommatrix theory, and free probability theory.

VOLUME 7, 2019 138077


RIKU JÄNTTI (M’02–SM’07) received the M.Sc.degree (Hons.) in electrical engineering and theD.Sc. degree (Hons.) in automation and systemstechnology from the Helsinki University of Tech-nology (TKK), in 1997 and 2001, respectively.In 2006, he joined the Aalto University School ofElectrical Engineering, Finland (formerly knownas TKK), where he is currently a Full Professor ofcommunications engineering and the Head of theDepartment of Communications and Networking.

Prior to joining Aalto (TKK) in August 2006, he was a Pro-Term Professorwith the Department of Computer Science, University of Vaasa. He currentlyholds docentship at the University of Vaasa. He is an Associate Editor of theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY and a Distinguished Lecturerof the IEEE Vehicular Technology Society (Class 2016). His research inter-ests include radio resource control and optimization for machine type com-munications, cloud-based radio access networks, spectrum and co-existencemanagement, RF Inference, ambient backscatter communication, and quan-tum communications.

JYRI HÄMÄLÄINEN received the M.Sc. andPh.D. degrees from the University of Oulu,Finland, in 1992 and 1998, respectively. From1999 to 2007, he was a Senior Specialist and Pro-gram Manager with Nokia, where he worked withvarious aspects of mobile communication systemsand 3GPP standardization. He was nominated toProfessor of Aalto University, in 2008, and he wastenured, in 2013. Since 2015, he has been servingas the Dean of the Aalto University School of Elec-

trical Engineering. He has authored or coauthored more than 190 scientificpublications and holds more tha 36 US patents or patent applications. Hisresearch interests include mobile and wireless systems, special focus beingin dynamic and heterogeneous networks, including 4–6G, small cells, multi-antenna transmission and reception techniques, scheduling, relays, and manyother topics. He is an Expert Member of the Highest Administrative Courtof Finland, nominated by the President of Finland.

ZYGMUNT J. HAAS (S’84–M’88–SM’90–F’07)received the Ph.D. degree in electrical engineer-ing from Stanford University, Stanford, CA, USA,in 1988.

In 1988, he joined the Network ResearchArea, AT&T Bell Laboratories, where he pursuedresearch in wireless communications, mobilitymanagement, fast protocols, optical networks, andoptical switching. In 1995, he joined the Facultywith the School of Electrical and Computer Engi-

neering, Cornell University, Ithaca, NY, USA, where he is currently a Pro-fessor Emeritus. He is also a Professor and a Distinguished Chair with theComputer Science Department, The University of Texas at Dallas, Richard-son, TX, USA, since 2013. He heads the Wireless Network Laboratory,a research groupwith extensive contributions and international recognition inthe area of ad hoc networks and sensor networks. He has authored more than200 technical conference and journal articles. He holds 18 patents in the areasof wireless networks and wireless communications, optical switching andoptical networks, and high-speed networking protocols. His research inter-ests include protocols for mobile and wireless communication and networks,secure communications, and the modeling and performance evaluation oflarge and complex systems. He was a recipient of the number of awardsand distinctions, including best paper awards, the 2012 IEEE ComSoc WTCRecognitionAward for outstanding achievements and contribution in the areaof wireless communications systems and networks, and the 2016 IEEE Com-Soc AHSN Recognition Award for outstanding contributions to securing adhoc and sensor networks. He has organized several workshops, deliverednumerous tutorials at major IEEE and ACM conferences. He served asthe Chair for the IEEE Technical Committee on Personal Communica-tions. He served as an Editor for several journals and magazines, includingthe IEEE/ACM Transactions on Networking, the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS, the IEEE Communications Magazine, and theWireless Networks (Springer). He has been a Guest Editor of the IEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS ISSUES.

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