Performance Analysis of Cognitive Spectrum-Sharing Single-Carrier Systems With Relay Selection

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Performance Analysis of Cognitive Spectrum-Sharing Single-Carrier Systems With Relay Selection

Kyeong Jin Kim, Senior Member, IEEE, Trung Q. Duong, Member, IEEE, and Xuan-Nam Tran, Member, IEEE

Abstract—In this paper, we analyze the performance of co- operative spectrum sharing single-carrier (SC) relay systems.

Taking into account the peak interference power at the primary user (PU) and the maximum transmit power at the secondary user (SU) network, two separate power allocation constraints are formed. For a two-hop decode-and-forward (DF) relaying protocol and two power allocation constraints, two relay selection schemes, namely, a full-channel state information (CSI)-based best relay selection (BRS) and a partial CSI-based best relay se- lection (PBRS), are proposed. The distributions of the end-to-end signal-to-noise ratios (e2e-SNRs) for the four cases are derived first, and then their outage probabilities and asymptotic outage probabilities are derived in closed-form. The derived asymptotic outage probabilities are utilized to see different diversity gains.

Monte Carlo simulations have verified the derived diversity gains for the four different cases. We also present upper bounds on the ergodic capacities for two particular cases.

Index Terms—Asymptotic outage diversity, cooperative spec- trum sharing single-carrier (SC) relay system, end-to-end signal-to-noise ratio, ergodic capacity, maximum transmit power, multipath diversity, multiuser diversity, outage probability, peak interference power.

I. INTRODUCTION

R

ADIO frequency spectrum is the most sacred resource of wireless networks. However, recently, many measurement campaigns from all over the world have demonstrated that a large amount of licensed radio spectrum is inefficiently utilized, see e.g., in China [1], Germany [2], New Zealand [3], Spain [4], USA [5], and Vietnam [6]. The cognitive radio network invented by Mitola [7] is a promising solution to this short- coming in spectrum under-utilization. The basic concept is that a secondary user (SU) is allowed to use the spectrum which is assigned to a licensed user (primary user) when the channel is free (overlay spectrum sharing) [8] or the SU can co-occupy the spectrum as long as its interference to the primary user (PU) is under a threshold, i.e., does not cause any harmful interference to the PU (underlay spectrum sharing) [9]. In the overlay scheme, it is required that the secondary network must cor- rectly detect the spectrum if it is unoccupied and immediately

Manuscript received April 04, 2012; revised June 29, 2012; accepted September 02, 2012. Date of publication September 11, 2012; date of current version November 20, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Walaa Hamouda.

K. J. Kim is with Mitsubishi Electric Research Laboratories, Cambridge, MA 02139 USA (e-mail: [email protected]).

T. Q. Duong is with Blekinge Institute of Technology, Karlskrona 37179, Sweden (e-mail: [email protected]).

X.-N. Tran is with Le Quy Don Technical University, Hanoi 10000, Vietnam (e-mail: [email protected]).

Color versions of one or more of thefigures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2012.2218242

release the channel when the PU wants to take it back, which is a non-trivial operation [10]. As a result, the underlay scenario has been considered to be an efficient transmission scheme under such a limited spectrum environment [9].

Under limitations on the maximum transmit power as a form of peak interference power inflicted by the PU receivers, cooperative spectrum sharing systems have been proposed in [11]–[15]. In some of these previous works, multiple relays have been employed to exploit multiuser diversity in a distributed fashion. It has been shown that multiuser diversity can increase spectrum efficiency. Typical amplify-and-forward (AF) and decode-and-forward (DF) relaying protocols are employed in SU relay networks. In [11], the end-to-end signal-to-interference plus noise ratio (SINR) is used to select one relay among several relays. The authors in [12] propose a joint best relay selection (BRS) and power allocation to maximize the system throughput. In [13], the authors propose a selection cooperation DF relaying protocol. Most importantly, it is shown that the received signal-to-noise ratios (SNRs) for available links are not independent due to channel related interference at the PU. A similar question is also reported in [16]. A full channel state information (CSI)-based BRS and an adaptive cooperation with BRS are employed in [14]

and [15], respectively. In [17], closed-form expressions for approximated outage probability, symbol error probability, and ergodic capacity have been derived for non-identical Rayleigh fading channels. In the aforementioned works, the secondary outage probability, asymptotic outage probability, and average throughput are used to show the performance of the cooperative spectrum sharing system.

In contrast to the peak interference power at the PU receivers, more realistic maximum transmit power limitations on the SU-source and SU-relays owing to the nonlinearity of the power amplifier [18] are also considered to allocate the transmit power to the SU-source and SU-relays [9], [18]–[24].

In [9], the authors show the capacity gain of a non-cooperative spectrum sharing system by dynamic spectrum sharing. For a single relay, an optimal power allocation is proposed in [18] to achieve a higher ergodic and outage capacity under different types of power allocation constraints and fading models. The channel fading effects on the outage probability are investigated in [21]. For the cooperative spectrum sharing systems with multiple relays, multiuser diversity is exploited in [22]–[24] with different types of relaying protocols. A relay is selected out of a set of relays via either BRS [22], [23] or a partial-CSI-based BRS (PBRS) [24]. In particular, in [22], the authors provide the asymptotic outage probabilities of BRS-AF, BRS-DF, and PBRS-AF schemes. It is also verified that the diversity gain is the same as that of the non-spectrum sharing system.

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Single-carrier (SC) transmission [25] has been proposed in very high-speed wireless networks for short range broadband applications [26]. Since SC transmission requires a low peak-to- average power ratio (PAPR), low power-backing-off, and a less restrictive requirement for linear amplifiers having large dynamic ranges in contrast to orthogonal frequency division mul- tiplexing (OFDM) transmission [25], [27], we use SC transmission in the cooperative spectrum sharing relay system. Several works have been reported [28]–[32] in the SC-based cooperative relay networks. A space-time-block code (STBC) is proposed in [28] to provide transmit diversity in the cyclic-prefix SC (CP-SC) system. The AF and project-and-forward combined automatic repeat request (ARQ)-aided protocols are, respectively, proposed in [29] and [30]. A distributed space-frequency- block code (SFBC) is proposed in [31] for the SC system with the AF protocol. BRS and best terminal selection (BTS) are proposed to the SC system in [30] and [32], respectively. It turns out that BRS and BTS can exploit multipath diversity by SC transmission along with multiuser diversity. For the AF relaying protocol, minimum mean square error (MMSE)-based single carrier frequency-domain equalization (SC-FDE) is proposed in [33] over frequency selective channels. However, to the best of our knowledge, the multipath diversity analysis of the SC-based spectrum sharing relay system with different types of power allocation constraints and relay selection methods has not been investigated. Thus, in comparison with these existing works, our contributions are summarized as follows:

• We consider two types of power constraints: i) constraint 1 which considers the peak allowable interference power at the PU to determine transmit powers at the SU-source and SU-relays, and ii) constraint 2 which simultaneously considers the peak interference power at the PU and the maximum transmit power at the SU-source and SU-relays¹. For the use of multiple relays in the secondary network, we propose to use BRS and PBRS to achieve a better performance by virtue of multiuser diversity. For these two power allocation constraints and relay selection schemes, we derive exact outage probabilities and upper bounds on ergodic capacities².

• To provide additional insights into the impact of system parameters on the network performance, we also derived an asymptotic outage probability. We have shown that the diversity order is mainly determined by both multiuser diversity and multipath diversity. It is seen that an achievable diversity gain by either BRS or PBRS is irrespective of the constraints either in the region of high interference power or maximum transmit power . It is also verified that BRS achieves a better outage probability than PBRS due to a higher outage diversity gain³.

• It is important to note that in underlay spectrum sharing networks with multiple relays, due to the existence of

1It is shown from [18] that constraint is more efficient in protecting the receiver in the PU and maximizing the throughput of the SU network. Since we use SC transmission, a restrictive requirement of the use of linear amplifiers can be relaxed over OFDM transmission.

2To do this, the properties of the right circulant channel matrix [34] are used.

3Motivated by the works in [13] and [16], the statistical properties of interfering channels are also considered in deriving the expressions for the unconditional distributions of the effective signal-to-noise ratios (e-SNRs).

the link from the secondary source to the PU receiver, a common random variable (RV), i.e., the channel gain of this link, will appear in the individual e-SNRs. This results in the statistical dependence among RVs although the considered fading channel is assumed to be independent [13], [16]. As a result, the mathematical derivation will be troublesome. To address this difficulty, in this paper, we introduce a general framework for the computation of the conditional statistics of the e-SNR with respect to the common RV. This approach can enable us to obtain exact mathematical derivations for the conditional distributions. Using these distributions, exact and asymptotic performance criteria are obtained and further insights on the system performance can be revealed.

• For the two constraints and , we compare the performance of the proposed system. Depending on the ratio of peak interference to the maximum transmit power, either constraint or constraint provides a better performance. For different cases, different multipath diversity gains have been investigated and verified by simulations.

The rest of the paper is organized as follows. In Section II, we introduce the system and channel model for the cooperative spectrum sharing SC relaying system employing the two-hop DF relaying protocol. The outage probability of the proposed spectrum sharing system is derived in Section III. An upper bound on the ergodic capacity is analyzed in Section IV. Numerical results are provided in Section V. Conclusions are drawn in Section VI.

Notation: Superscripts , and denote complex conjugation, transposition and conjugate transposition, respectively. denotes expectation with respect to is an identity matrix; denotes an all-zero matrix of appro- priate dimensions; denotes the complex Gaussian distribution with mean and variance denotes the vector space of all complex matrices; a right circulant matrix is defined by a length- vector ; a zero-padded length- column vector from is denoted by denotes the Euclidean norm of a vector denotes the cumulative distribution function (CDF) of the RV . The probability density function (PDF) of is denoted by . The binomial coefficient is denoted

by .

Definition 1: Let the right circulant channel matrix

be defined by , then we have

. Recall that is zero-padded length- channel vector from . Thus, when is composed of independent and identically distributed (i.i.d.) complex Gaussian RVs with zero means and unit variances, the RV has a chi-squared distribution with degrees of freedom [34], and we express the distribution of as . In addition, we express the distribution for

as . The PDF and CDF of are given

by, respectively, and

, where

denotes the unit step function, ,

and .

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II. SYSTEM ANDCHANNELMODEL

The spectrum sharing system under consideration employs two-hop DF relays and SC transmission. Its block diagram is shown in Fig. 1. We use the following channel models:

• All channels are assumed to be known exactly in the system [9], [11]–[15], [18], [19], [21], [23], [24].

Perfect CSI for SU-PU channels can be obtained at SU transmitters (source and relay) through several mechanisms [35]: i) direct feedback from the PU receiver (PU-Rx), indirect feedback of the band manager [36], and ii) channel reciprocity [37]. The interfering channel from the SU-source is defined by . The path loss component over this channel is denoted by . A channel over a link in the secondary network is denoted by , where the channel length over the link is given

by .

For example, denotes a channel from the SU-source to the th SU-relay, whereas a channel from the th SU-relay to the SU-destination is denoted by . The path loss component over the link is given by , respectively. Note that and are defined for the th channel from the SU-source to SU-relays, whereas and are defined for the channel from the th SU-relay to the SU-destination. An interfering channel from the th SU-relay is defined by , so that for a link

, a channel is defined by

with the channel length .

The corresponding path loss component over a link is

defined by . Due to deep

fading from the SU-source to the SU-destination, we assume no direct channel between them.

• All channels are composed of i.i.d. complex Gaussian RVs with zero means and unit variances. We assume that these channels are mutually independent.

• The maximum channel length is denoted by .

A. SC Transmission With DF Relays and the Constraint In the first hop, the SU-source prefixes symbols to the front of the symbol block to elim- inate intersymbol interference (ISI). The symbol block com- prising modulated symbols is assumed to be independent with zero mean and unit variance. Subject to the constraint , the transmit power at the SU-source is given by . Recall that denotes the peak interference power at the PU.

The e-SNR over the channel from the SU-source to the SU- is defined by [34]

(1)

where , and . An

additive noise vector on this SU-source to the SU- channel

Fig. 1. System model of a cooperative spectrum sharing SC relay system with the DF relaying protocol and multiple relays.

is assumed to be . Similarly, the e-SNR over the channel from the th SU-relay to the SU-destination is defined as follows:

(2) Note that accounts for an additive noise over the channel from the th SU-relay to the SU-destination. Obeying the constraint , the transmit power at the th SU-relay is given by . Having allocated this power to the th SU-relay, we can evaluate (2) as:

(3)

where , and . The

e2e-SNR achieved by the DF relaying protocol is given

by . It is worth-

while to notice that from (1) and (3), the e-SNRs depend on the interfering channels in the forms of and [13], [16].

B. SC Transmission With DF Relays and the Constraint Under the constraint , the transmit power at the SU-source

will be modified by , where

with the maximum transmit power . With this power allocation, the e-SNRs over the channels from the SU-source to the SU- and from the SU- to the SU-destination are given, respectively, as follows:

and

(4) Thus, the e2e-SNR achieved by the DF relaying protocol is

given by . Notice that also relates

to and depending on the ratio of to .

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Based onDefinition 1, we express

, and

for the channel-related RVs. They all represent the e-SNRs for all available channels in the considered system.

Now the conditional distributions of and given are given, respectively, as follows:

(5) and

(6) Notice that is independent of , which is the e-SNR of the interfering channel from the SU-source to the PU. Using ([38], (3.351.3)), and the statistical properties of and

is evaluated as follows:

(7) where

(8) Based on (5) and (6), we obtain the following conditional distribution of

(9) Further, we canfind the following distributions with the constraint

(10) and

(11)

where , and . From

(10) and (11), it is easy to show that

(12) C. Relay Selection Scheme

To select a desired SU-relay out of available ones, BRS [30] is employed with SC transmission. For other non-SC transmissions, BRS has been proposed to increase the throughput exploiting multiuser diversity. Similar to BRS used in [19], [22], and [23], the best SU-relay is selected as follows:

and

(13) Then, the maximum achievable e2e-SNR by BRS is given by

either or depending

on the employed power allocation constraint. Owing to the order statistics, BRS gives the distributions shown in (14) and (15) at the bottom of the page.

Compared with BRS, PBRS selects the desired SU-relay as follows:

and

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(14) and

(15)

(5)

from which wefind that only channel information for the channels from the SU-source to SU-relays are used in computing the relay selection metrics. The maximum achievable e-SNRs by PBRS are denoted by and , respectively, depending on the power allocation constraints and . Their conditional distributions are, respectively, given by:

(17) and

(18) Upon applying the DF relaying protocol, the e2e-SNRs are

given by and

, from which we can compute their conditional distributions as

(19) with the power allocation constraint . Notice that according to (14), (15), and (19), the exact conditional distributions of the e2e-SNRs are available.

As studied in [30], two phases are employed in implementing BRS. A training block symbol is used in thefirst pilot phase to select a relay that has the maximum achievable e2e-SNR. In the following data phase, the block symbol is transmitted from the SU-source to the selected SU-relay. To make BRS work, channels are assumed to be quasi-static for the two phases, and we assume no delay in feeding back a selected SU-relay index in the system. Similar assumptions are also made to PBRS for the proper operation.

III. OUTAGEPROBABILITYANALYSIS

In this section, we willfirst derive the closed-form expression for the outage probability. For this, the unconditional distributions of the e2e-SNRs will be obtained. Then, we will provide the asymptotic outage probabilities to see the diversity gains in the four possible cases.

Let us denote by a predefined threshold e2e-SNR re- sulting in the outage event. Thus, the outage probabilities are, respectively, defined by

and

(20) for different types of power allocation constraints and relay selection schemes. To simplify our derivations, we assume identical fading in the sequel, so that we have

, and .

A. Closed-Form Expressions for the Outage Probabilities With identical fading, (14) and (15) can be simplified as follows:

(21) and

(22) where

and

In particular, (19) becomes as follows with PBRS and the constraint

(23) The conditional distribution with PBRS and the constraint is then obtained from (19) as follows:

(24) Based on the conditional distributions derived in (21)–(24), the outage probabilities with identical fading are derived in The- orem 1.

Theorem 1: When the channel vectors are composed of i.i.d.

complex Gaussian RVs with zero means and unit variances, the outage probabilities are given by (25) and (26)

(25)

(6)

and

(26)

In (25) and (26), we define

, and .

Similarly the outage probabilities with PBRS (27) and (28) can be obtained from (23) and (24) as follows:

(27) and

(28)

Proof: A proof of this theorem is provided in Appendix A.

Although the derived outage probabilities enable us to evaluate the performance of the proposed cooperative spectrum sharing system, their complex forms do not allow us to gain valuable insights on how the multiuser and multipath diversities affect the overall outage diversity. Thus, we perform an asymptotic analysis of the outage probability in the following subsection.

B. Asymptotic Outage Probability Analysis With Identical Fading

To develop a feasible asymptotic performance analysis, we first approximate the conditional distributions in the high interference region, i.e.,

and

(29)

A similar approximation is also used in [21]. Further, the approximated distributions of and as are given as follows:

(30) and

(31)

where , and .

Proposition 1: Using (29), (30), and (31), the approximated conditional distributions with the constraint and two relay selection schemes are given by:

(32)

and

(33)

where , and .

Similarly, we can also derive the approximated conditional distributions with the constraint as shown in (34) at the top of the next page. In (34), we define ,

and .

Proof: A proof of these results is provided in Appendix B.

Based on the asymptotic distributions of the e2e-SNRs, the asymptotic outage diversities are derived in Theorem 2.

complex Gaussian RVs with zero means and unit variances, the asymptotic outage probabilities are given by (35)-(36) at the

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bottom of the page. In contrast to the constraint , the asymptotic outage probabilities with the constraint are given by

(37)-(38) shown at the bottom of the page. In (38), we define .

and

(34)

(35)

and

(36)

(37)

and

(38)

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By the definition of the asymptotic outage di-

versity: and

with the relay selection , (35)–(38) reveal that

and

(39) That is, the asymptotic outage diversity is independent of the power allocation constraints at the SU-source and SU-relays, whereas the relay selection scheme mainly determines the asymptotic outage diversity. For BRS, multiuser diversity and multipath diversity which is provided by the DF relaying protocol, simultaneously determine the asymptotic outage diversity. In particular, (39)

shows that PBRS provides and

, so that compared with the full-CSI BRS, there will be an outage diversity gain loss with

PBRS when .

Proof: We provide the proof of this theorem in Appendix C.

Since SC transmission limits and

with the power allocation constraint X to avoid ISI, we have

and .

IV. ERGODICCAPACITYANALYSIS

In this section, we investigate the ergodic capacity of the proposed spectrum sharing system. With the available e2e-SNRs for different relay selection schemes (BRS, PBRS) and power allocation constraints , the ergodic capacity is defined in [20] and [39] as:

(40) With some algebraic manipulations, we canfind an alterna- tive expression for [20] as follows:

(41) Using Jensen’s inequality, an upper bound on the ergodic capacity is given by [40]

(42) Fortunately, using the derived distributions, we can readilyfind the two particular cases of as

(43)

where

(44)

and , where

(45) Although we can obtain the corresponding expressions for and , it is infeasible to find exact and . Thus, we do not include the expressions for them. Instead, the exact ergodic capacity will be obtained numerically by the simulations.

complex Gaussian RVs with zero means and unit variances, an upper bound on the ergodic capacity is given by (46),

(46) where

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Fig. 2. Outage probability on a plot for various numbers of with .

and

In addition, and denote the beta function and the Gauss hypergeometric function, respectively. Further, we have (47).

(47) Proof: A proof of this theorem is provided in Appendix D.

V. NUMERICALRESULTS

For the numerical results, we use fixed and for the symbol block size and the length of the CP, respectively. Quadrature phase-shift keying (QPSK) modula- tion is employed for the data symbols. In all simulations, we use afixed dB to determine the outage probability. All nodes are located in the two-dimensional plane. For example, the SU-source and SU-destination are placed in a straight line with the following coordinators and , respectively.

The PU is located at , whereas all SU-relays are located

Fig. 3. Outage probability on a plot for various numbers of with .

between the SU-source and SU-destination. The pathloss component for the channel between two nodes at C and D is exponentially decaying as , where is the distance between C and D and is the path loss exponent. We assume in all simulations. The curves obtained via actual link simulations are denoted bySim., whereas analytically derived curves are denoted byAn. We place the PU at . A. Performance With the Constraint

In Fig. 2, we use ,

and we investigate the analytically derived outage probabilities as a function of in comparison with the outage probabilities obtained via simulations for various values of . We can observe good agreement between analytically derived and simulated outage probabilities for a different channel tap size . Compared with PBRS, BRS shows a better outage probability due to a higher diversity gain. Since

as , the slope on a plot with BRS will not change as a function of when . For

, and , the diver-

sity gains are , and , respectively, in the considered range of . Notice that the slopes of PBRS with

and are the same

as those of BRSs since . Fig. 3

shows the effects of on the outage probability with

and . For BRS, the slopes are

approximately measured as 2, 4, 6 for , and 3 on a plot, whereas the slopes are approximately 2, 4, and 5 with PBRS. Thus, as , the diversity gains can be ex-

pected to be and

. In Fig. 4, we can observe an influence of the location of the PU on the outage probability. At a different location of the PU, we experience a different outage event. As the PU located near the secondary network, a worse outage probability can be obtained. Thus, a near-far effect can be observed in the outage probability. Since the asymptotic diversity is independent of the location of the PU, which is verified in Theorem 3, the slopes are same with BRS and PBRS.

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Fig. 4. Outage probability for a function of the location of the PU at with .

Fig. 5. Outage probability on a plot for various numbers of

with and .

B. Performance With the Constraint

Fig. 5 shows the outage probability for various values of channel sizes and the numbers of relays. We first observe the validity of the analytically derived outage

probabilities and compared with the

simulated ones. For various numbers of ,

we can expect that and

as . Further, for

and are

shown to have the same slope. In Fig. 6, we numerically verify the analytically derived asymptotic outage probabilities with the constraint. This figure shows good matches between the asymptotic outage probabilities for BRS and PBRS and the analytical results as . Moreover, as , BRS shows a better outage probability over PBRS with

.

Fig. 6. Outage probability for various numbers of and with .

Fig. 7. Outage probability with constraints and with various values of

. We use .

C. Performance Comparison With Different Constraints In Fig. 7, we compare the outage probability with constraints

and . Wefind that if , the constraint will result

in the same outage probability as that of the constraint , since is much larger than and is dominated by . As a result, the constraint will be reduced to the constraint

. Similar explanation can be applied for the cases and . As derived in Theorem 3, BRS and PBRS with the constraint can have the same diversity gain as those with the constraint .

In Fig. 8, we use and

. Thisfigure shows the tightness of the derived upper bound on the ergodic capacity. For , we verified them for BRS and PBRS by comparing with those obtained from simulations (denoted bySim., bound). It is seen

that leads us to achieve a better

tight upper bound on the erogodic capacity over

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Fig. 8. Ergodic capacity with the constraint and for various numbers of

. We use .

Fig. 9. Ergodic capacity with the constraint and various values of

. We use .

. In other words, the diversity gain affects the tightness on the derived upper bound. In Fig. 9, we show the simulated ergodic capacities for BRS and PBRS with different values of for the constraint . It is seen that BRS leads to a higher ergodic capacity over PBRS irrespective of the value of .

VI. CONCLUSION

In this paper, we have investigated the diversity effects on the outage probability and the ergodic capacity of a cooperative spectrum sharing SC system with multiple relays. To see the overall diversity in the proposed system, an asymptotic outage probability analysis has been conducted. For two relay selection schemes, namely BRS and PBRS, and two different power allocation constraints, the overall diversity can be seen as func- tions of multiuser diversity and multipath diversity either in the region of high interference power or maximum transmit power.

The simulations have verified the derived outage probability and two upper bounds on ergodic capacities.

APPENDIXA PROOF OFTHEOREM1 The binomial identity [41] converts (21) into

(A.1)

Now using the multinomial identity to expand the power terms [41], we can simplify (A.1) as

(A.2)

Applying the distribution for is evaluated as

(A.3)

Using the binomial and multinomial identities, we first rewrite (22) as follows:

(A.4)

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Since , we obtain as

(A.5) Using again ([38], (8.351.3)) in (A.5), (A.5) is evaluated as:

(A.6) For PBRS, wefirst compute the unconditional distribution from (23) as

(A.7)

Similar to previous developments, (24) is equivalent to

(A.8) Applying the binomial identity to the last term in the above equation, we can obtain

(A.9) which is equivalent to

(A.10) From the definition of the outage probability, we can obtain (25)–(28).

APPENDIXB PROOF OF(32)–(34)

Using an expression for in (29), substituting in (31), and manipulating the integral in (31), yields

(B.1)

We now approximate as

, which can be further approximated as follows:

(B.2)

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As a consequence of (B.2), can be devised

by , which shows (32).

Using (29), (30), (31), and (B.1), PBRS yields the asymptotic distribution (33).

For the constraint , the conditional distribution of the e2e-SNR can be approximated by

(B.3)

Substituting the PDF of into the above expression, and after some algebraic manipulations, we can obtain (34).

APPENDIXC PROOF OFTHEOREM2 Using (32), we can compute

which becomes

(C.1) where the last term in the above equation can be obtained by applying the binomial identity first. That is,

we first express as

, then

(C.2)

We can readily derive the last term in (37) as follows:

(C.3)

APPENDIXD PROOF OFTHEOREM3

Wefirst rewrite an equivalent expression for in the

following form:

(D.1)

where . Applying the multinomial identity to (D.1), the following equation can be obtained

(D.2) Notice that the dependence on the power of is more apparent in (D.2) than the original form of . Substituting (D.2) into (44), yields

(D.3) Applying ([38], (3.259.3)) in (42), (46) can be obtained.

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To verify (47), wefirst rewrite as follows:

(D.4) so that we have

(D.5) where the equivalent equation (D.1) is used for . Again using ([38], (3.259.3)) in (42), (47) can be derived.

REFERENCES

[1] D. Chen, S. Yin, Q. Zhang, M. Liu, and S. Li, “Mining spectrum usage data: A large-scale spectrum measurement study,” inProc. Int. Conf.

Mobile Comput. Netw., Beijing, China, Sep. 2009, pp. 13–24.

[2] M. Wellens, J. Wu, and P. Mahonen, “Evaluation of spectrum occupancy in indoor and outdoor scenario in the context of cognitive radio,”

inProc. Int. Conf. Cognit. Radio Orient.Wireless Netw. Commun., Or- lando, FL, Jul. 2007, pp. 420–427.

[3] R. I. C. Chiang, G. B. Rowe, and K. W. Sowerby, “A quantitative analysis of spectral occupancy measurements for cognitive radio,”

in Proc. IEEE Veh. Techol. Conf., Dublin, Ireland, Apr. 2007, pp.

3016–3020.

[4] M. Lopez-Benitez, A. Umbert, and F. Casadevall, “Evaluation of spectrum occupancy in Spain for cognitive radio applications,” inProc.

IEEE Veh. Techol. Conf., Barcelona, Spain, Apr. 2009, pp. 1–5.

[5] M. A. McHenry, “NSF Spectrum Occupancy Measurements Project Summary,” Shared Spectrum Co., Vienna, VA, Tech. Rep., Aug. 2005.

[6] V. N. Q. Bao, L. Q. Cuong, L. Q. Phu, T. D. Thuan, L. M. Trung, and N. T. Quy, “Spectrum survey in Vietnam: Occupancy measurements and analysis for cognitive radio applications,” inProc. Int. Conf. Adv.

Technol. Commun., Da Nang, Vietnam, Aug. 2011, pp. 135–143.

[7] I. J. Mitola and J. G. Q. Maguire, “Cognitive radio: Making software radios more personal,”IEEE Pers. Commun., vol. 6, no. 4, pp. 13–18, Aug. 1999.

[8] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,”IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb.

2005.

[9] A. Ghasemi and E. S. Sousa, “Fundamental limits of spectrum-sharing in fading environments,”IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 649–658, Feb. 2007.

[10] Y.-C. Liang, Y. Zeng, E. Peh, and A. T. Hoang, “Sensing-throughput tradeoff for cognitive radio networks,”IEEE Trans. Wireless Commun., vol. 7, no. 4, pp. 1326–1337, Apr. 2008.

[11] J. Si, Z. Li, X. Chen, B. Hao, and Z. Liu, “On the performance of cognitive relay networks under primary user’s outage constrain,”IEEE Commun. Lett., vol. 15, pp. 422–424, Apr. 2011.

[12] L. Li, X. Zhou, H. Xu, G. Y. Li, D. Wang, and A. Soong, “Simplified relay selection and power allocation in cooperative cognitive radio systems,”IEEE Trans. Wireless Commun., vol. 10, no. 1, pp. 33–36, Jan.

2011.

[13] L. Luo, P. Zhang, G. Zhang, and J. Qin, “Outage performance for cognitive relay networks with underlay spectrum sharing,”IEEE Commun.

Lett., vol. 15, no. 7, pp. 710–712, Jul. 2011.

[14] Y. Guo, N. Zhang, W. Zhou, and P. Zhang, “Outage performance of relay-assisted cognitive-radio system under spectrum-sharing constraints,”Electron. Lett., vol. 46, pp. 182–184, Jan. 21, 2010.

[15] Y. Zou, J. Zhu, B. Zheng, and Y.-D. Yao, “An adaptive cooperating diversity scheme with best-relay selection in cognitive radio networks,”

IEEE Trans. Signal Process., vol. 58, pp. 5438–5445, Oct. 2010.

[16] T. Q. Duong, V. N. Q. Bao, G. C. Alexandropoulos, and H.-J. Zeper- nick, “Cooperative spectrum sharing networks with AF relay and selection diversity,”Electron. Lett., vol. 47, pp. 1149–1151, Sep. 20, 2011.

[17] T. Q. Duong, D. B. da Costa, G. C. Alexandropoulos, V. N. Q. Bao, and A. Nallanathan, “Cognitive amplify-and-forward relaying with best relay selection in non-identical Rayleigh fading,” IEEE Commun.

Lett. 2012, under review.

[18] X. Kang, Y.-C. Liang, A. Nallanathan, H. K. Garg, and R. Zhang,

“Optimal power allocation for fading channels in cognitive radio networks: Ergodic capacity and outage capacity,”IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 940–950, Feb. 2009.

[19] T. W. Ban, W. Choi, B. C. Jung, and D. K. Sung, “Multi-user diversity in a spectrum sharing system,”IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 102–106, Jan. 2009.

[20] H. A. Suraweera, P. J. Smith, and M. Shafi, “Capacity limits and performance analysis of cognitive radio with imperfect channel knowledge,”

IEEE Trans. Veh. Technol., vol. 59, pp. 1811–1822, May 2010.

[21] C. Zhong, T. Patnarajah, and K.-K. Wong, “Outage analysis of decode- and-forward cognitive dual-hop systems with the interference constrain in Nakagami-m fading channels,”IEEE Trans. Veh. Technol., vol. 60, pp. 2875–2879, Jul. 2011.

[22] H. Ding, J. Ge, D. B. da Costa, and Z. Jiang, “Asymptotic analysis of cooperative diversity systems with relay selection in a spectrum- sharing scenario,”IEEE Trans. Veh. Technol., vol. 60, pp. 457–472, Feb. 2011.

[23] R. Zhang and Y.-C. Liang, “Investigating on multiuser diversity in spectrum sharing based cognitive radio networks,”IEEE Commun.

Lett., vol. 14, no. 2, pp. 133–135, Feb. 2010.

[24] V. Asghari and S. Aissa, “Cooperative relay communication performance under spectrum-sharing resource requirements,” in Proc. IEEE Int. Conf. Commun., Cape Town, South Africa, May 2010, pp. 1–5.

[25] S. Kato, H. Harada, R. Funada, T. Baykas, C. S. Sum, J. Wang, and M. A. Rahman, “Single carrier transmission for multi-gigabit 60-GHz WPAN systems,”IEEE J. Sel. Areas Commun., vol. 27, pp. 1466–1478, Oct. 2009.

[26] Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: Enhancements for Very High Throughput in the 60 GHz Band, IEEE P802.11ad/D0.1, Jun. 2010.

[27] P. Smulders, “Exploiting the 60 GHz band for local wireless multi- media access: Prospects and future directions,”IEEE Commun. Mag., no. 9, pp. 140–147, Jan. 2002.

[28] D.-Y. Seol, U.-K. Kwon, and G.-H. Im, “Performance of single carrier transmission with cooperative diversity over fast fading channels,”

IEEE Trans. Commun., vol. 57, pp. 2799–2807, Sep. 2009.

[29] H. Chergui, T. Ait-Idir, M. Benjillali, and S. Saoudi, “Joint-over-transmissions project and forward relaying for single carrier broadband MIMO ARQ systems,” inProc. IEEE Veh. Techol. Conf., Yokohama, Japan, May 2011, pp. 1–5.

[30] K. J. Kim and T. A. Tsiftsis, “On the performance of cyclic prefix-based single-carrier cooperative diversity systems with best relay selection,”

IEEE Trans. Wireless Commun., vol. 10, no. 4, pp. 1269–1279, Apr.

2011.

[31] H. Mheidat, M. Uysal, and N. Al-Dhahir, “Equalization techniques for distributed space-time block codes with amplify-and-forward relaying,”IEEE Trans. Signal Process., vol. 55, no. 5, pp. 1839–1852, May 2007.

[32] K. J. Kim, T. A. Tsiftsis, and H. V. Poor, “Power allocation in cyclic prefixed single-carrier relaying systems,”IEEE Trans. Wireless Commun., vol. 10, no. 7, pp. 2297–2305, Jul. 2011.

[33] H. Eghbali, S. Muhaidat, and N. Al-Dhahir, “A novel receiver design for single-carrier frequency domain equalization in broadband wireless networks with amplify-and-forward relaying,”IEEE Trans. Wireless Commun., vol. 10, no. 3, pp. 721–727, Mar. 2011.

(15)

[34] K. J. Kim and T. A. Tsiftsis, “Performance analysis of QRD-based cyclically prefixed single-carrier transmissions with opportunistic scheduling,”IEEE Trans. Veh. Technol., vol. 60, pp. 328–333, Jan.

2011.

[35] J.-P. Hong, B. Hong, T. W. Ban, and W. Choi, “On the cooperative diversity gain in underlay cognitive radio systems,” IEEE Trans.

Commun., vol. 60, no. 1, pp. 209–219, Jan. 2012.

[36] J. Peha, “Approaches to spectrum sharing,”IEEE Commun. Mag., vol.

43, no. 2, pp. 10–12, Feb. 2005.

[37] Q. Zhao, S. Geirhofer, L. Tong, and B. Sadler, “Opportunistic spectrum access via periodic channel sensing,”IEEE Trans. Signal Process., vol.

56, no. 2, pp. 785–796, Feb. 2008.

[38] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Prod- ucts. New York: Academic, 2007.

[39] O. Waqar, D. C. McLernon, and M. Ghogho, “Exact evaluation of ergodic capacity for multihop variable-gain relay networks: A uni- fied framework for generalized fading channels,”IEEE Trans. Veh.

Technol., vol. 59, pp. 4181–4187, Oct. 2010.

[40] S. S. Ikki and S. Aissa, “Multihop wireless relaying systems in the presence of cochannel interferences: Performance analysis and design optimization,”IEEE Trans. Veh. Technol., vol. 61, pp. 566–573, Feb.

2012.

[41] M. Abramovitz and I. A. Stegun, Handbook of Mathematical Func- tions With Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover, 1972.

Kyeong J. Kim(SM’11) received the M.S. degree from the Korea Advanced Institute of Science and Technology, Daejeon, South Korea, in 1991, and the M.S. and Ph.D. degrees in electrical and computer engineering from the Uni- versity of California Santa Barbara, CA, in 2000.

During 1991–1995, he was a Research Engineer in the Video Research Center of Daewoo Electronics, Ltd., Korea. In 1997, he joined the Data Transmission and Networking Laboratory at the University of California Santa Barbara. After receiving his degrees, he joined the Nokia Research Center, Nokia Inc., Dallas, TX, as a Senior Research Engineer, where from 2005 to 2009 he was an L1 Specialist. During 2010–2011, he was an Invited Professor at Inha University, Korea. Since 2012, he has been a member of the Senior Principal Research Staff at Mitsubishi Electric Research Laboratories, Cambridge, MA. He has published and presented more than 50 technical papers in scientific journals and international conferences. His research has been focused on the transceiver design, resource management, scheduling in the cooperative wireless communications systems, cooperative spectrum sharing system, and device-to-device communications.

Dr. Kim currently serves as an Editor for the IEEE COMMUNICATIONS LETTERS.

Trung Q. Duong(S’05–M’12) was born in HoiAn, Quang Nam Province, Vietnam, in 1979. He received the B.S. degree in electrical engineering from Ho Chi Minh City University of Technology, Vietnam, in 2002, and the M.Sc. degree in computer engineering from Kyung Hee University, South Korea, in 2005.

In April 2004, he joined the electrical engineering faculty of the Ho Chi Minh City University of Transport, Vietnam. He was a recipient of the Korean Government IT Scholarship Program for International Graduate Students from 2003 to 2007.

In December 2007, he joined the Radio Communication Group, Blekinge Institute of Technology, Sweden, as a research staff member working toward his Ph.D. degree. He was a Visiting Scholar at Polytechnic Institute of New York University, NY, from December 2009 to January 2010 and then at Sin- gapore University of Technology and Design from July 2012 to August 2012.

He received his Ph.D. degree in telecommunication systems from Blekinge Institute of Technology, Sweden, in 2012. His current research interests include cross-layer design, cooperative communications, and cognitive radio networks.

Dr. Duong is a frequent reviewer for numerous journals/conferences and a TPC member for many conferences including ICC, WCNC, VTC, PIMRC. He is a TPC co-chair of the International Conference on Computing, Managements, and Telecommunications 2013 (ComManTel). He was awarded the Best Paper Award of IEEE Student Paper Contest-IEEE Seoul Section in December 2006 and was afinalist for the best paper award at the IEEE Radio and Wireless Sym- posium, San Diego, CA, in 2009.

Xuan-Nam Tran(M’03) received the M.E. degree in telecommunications engineering from University of Technology, Sydney, Australia, in 1998, and Doctor of Engineering degree in electronic engineering from The University of Electro-Communications, Japan, in 2003.

He is currently an Associate Professor in the De- partment of Communications Engineering, Le Quy Don Technical University, Vietnam. From November 2003 to March 2006 he was a Research Associate in the Information and Communication Systems Group, Department of Information and Communication Engineering, The University of Electro-Communications, Tokyo, Japan. His research interests are in the areas of adaptive antennas, space-time processing, space-time coding, and MIMO systems.

Dr. Tran is a recipient of the 2003 IEEE AP-S Japan Chapter Young Engi- neer Award. He is a member of IEICE and the Radio-Electronics Association of Vietnam.