Globeco - Counication Theory Syposiu SEP-Optial Antenna Selection for Average Interference Constrained Underlay Cognitive Radios Rialapudi Sarvendranath, Student Meber, IEEE, Neelesh B. Mehta, Senior Meber, IEEE Abstract In the underlay ode of cognitive radio, secondary users can transit when the priary is transitting, but under tight interference constraints, which liit the secondary syste perforance. Antenna selection AS-based ultiple antenna techniques, which require less hardware and yet exploit spatial diversity, help iprove the secondary syste perforance. In this paper, we develop the optial transit AS rule that iniizes the sybol error probability SEP of an average interferenceconstrained secondary syste that operates in the underlay ode. We show that the optial rule is a non-linear function of the power gains of the channels fro secondary transit antenna to priary receiver and secondary transit antenna to secondary receive antenna. The optial rule is different fro the several ad hoc rules that have been proposed in the literature. We also propose a closed-for, tractable variant of the optial rule and analyze its SEP. Several results are presented to copare the perforance of the closed-for rule with the ad hoc rules, and interesting inter-relationships aong the are brought out. I. INTRODUCTION The increasing deand for high wireless data rates has increased the need for efficient spectru utilization techniques and has led to the developent of cognitive radio CR []. In one coon paradig of CR, two classes of users are defined, naely, priary users PU and secondary users SU. The PU is the owner of the spectru. A SU can use the sae spectru as the PU, but under constraints that are designed to protect the PU. For exaple, in the interweave ode of CR [], the SU transits only in the spectral regions not being used by PU. Hence, the SU does not cause any interference to the PU, except when it senses the spectru incorrectly. Whereas, in the underlay ode of CR [], the SU can access the spectru even when the PU is transitting, but under tight constraints on the average or peak interference that causes to the priary. These constraints liit the perforance of the secondary syste, when easured in ters of throughput or sybol error probability SEP. Eploying ultiple antennas at the SUs itigates the ipact of this liitation [3] [5]. However, one drawback of a ultiple antenna syste is that each antenna eleent requires a dedicated, expensive radio frequency RF chain to process its signals. In order to reduce the hardware costs, a technique called antenna selection AS has been extensively studied; see, for exaple, [6], [7] and references therein. In single transit AS, which is the focus of this paper, one of the transit antennas is selected as a function of the channel conditions and is connected to the one available R. Sarvendranath and N. B. Mehta are with the Dept. of Electrical Counication Eng. at the Indian Institute of Science IISc, Bangalore, India. Eails: {sarvendranath@gail.co, nbehta@ece.iisc.ernet.in} This work was partially supported by research grants fro ANRC and the Broadco Foundation, USA. RF chain in the transitter. Doing so reduces the hardware coplexity, cost, and size of the transitter. Yet, AS has been shown to harness the diversity benefits of ultiple antennas [6], [8], [9]. Given its proise and practical feasibility, AS has also been considered in CR systes [] [], and has been shown to iprove secondary syste perforance. In the interweave ode, since the SU does not interfere with the PU, the AS rule reains the sae as that in a conventional AS syste that is not subject to any interference constraint. For exaple, in a ultiple input single output MISO secondary syste, this involves selecting the secondary transitter STx antenna that has the strongest channel power gain to the secondary receiver SRx. We shall refer to this as the unconstrained AS rule henceforth. However, in the underlay ode, the priary interference constraint affects the choice of the transit antenna. Intuitively, even though an STx antenna has a strong link to the SRx, it should not get selected if it causes significant interference to the priary receiver PRx. Therefore, the AS rule ust take into consideration both the STx to SRx STx- SRx and STx to PRx STx-PRx channel power gains in the underlay ode. Related Work on AS in Underlay CR: Several interesting rules for selecting an antenna in a MISO CR, such as the iniu interference MI rule and the axiu signal power to leak interference power ratio MSLIR rule, were proposed in []. The MI rule selects the antenna that causes the least interference to the PRx. However, the selection is done only on the basis of the channel gains to the PRx. The MSLIR rule coproises between the MI and unconstrained rules, and selects the antenna with the highest ratio of STx- SRx and STx-PRx channel power gains. Note that the above rules do not consider the average interference constraint and ay not always be feasible. A difference selection DS rule was proposed in []. It selects that antenna that axiizes a weighted difference between the STx-SRx and STx-PRx channel power gains. It was shown to outperfor the MSLIR rule in any scenarios. All the above rules use fixed transit power. They are ad hoc as they do not provably optiize an end objective such as SEP or capacity. The optial AS rule for underlay CR even with fixed transit power is an open proble, and is the proble that this paper solves. Contributions: We systeatically develop the optial AS rule for a MISO underlay CR syste that iniizes the SEP at the SRx when the STx is subject to an average priary interference constraint. In our odel, the STx transits with a fixed power or with zero power depending on the channel 978--4673-9-9//$3. IEEE
conditions. The SEP-optial rule is shown to take a siple for; it is a linear cobination of the STx-PRx channel power gain and an exponentially decaying function of the STx-SRx gain. This rule turns out to be in the for of a single integral. In order to obtain a closed-for characterization for the AS rule, we then consider a variant of it that iniizes SEP upper bound instead of the exact SEP. For brevity we shall refer to this as the closed-for optial rule. We then analyze its SEP. To the best of our knowledge, such an SEP analysis is interesting even fro an AS perforance analysis point of view because it deals with a non-linear selection rule, while ost of the literature on the SEP analysis of AS in MISO systes focuses on linear rules [3], [4]. We also deterine the optial setting of the transit power. We then extensively benchark the perforance of the closedfor optial rule with several rules proposed in the literature. In order to provide a fair coparison, we copare against enhanced versions of the MI and MSLIR rules that can always adhere to the average interference constraint. The coparisons bring out interesting inter-relationships aong the rules. For exaple, we show that the closed-for optial rule, the enhanced MI rule, and the DS rule are equivalent only for large values of transit power, and that even the enhanced MSLIR rule is suboptial in ost scenarios of interest. Due to space constraints, we focus on the scenario where the STx has two transit antennas. The approach can be easily generalized to handle ore antennas at the STx [5]. The paper is organized as follows. Section II develops the syste odel. The optial selection rule and SEP analysis are developed in Section III. Nuerical results are presented in Section IV, and are followed by our conclusions in Section V. Matheatical derivations are relegated to the Appendix. II. SYSTEM MODEL AND PROBLEM STATEMENT We use the following notation henceforth. The absolute valueof a coplexnuberx is denotedby x. The probability of an event A and the conditional probability of A given B are denoted by Pr A and Pr A B, respectively. For a rando variable RV X, f X x denotes its probability density function PDF and E X [.] denotes its expectation. Scalar variables are shown in noral font and vector variables are shown in bold font. I {a} denotes the indicator function; it is ifais true and is otherwise. As shown in Figure, we consider an underlay CR syste in which an STx transits data to an SRx, and in the process interferes with a PRx. The SRx and STx constitute the secondary syste. The PRx and the SRx have one receive antenna each. The STx has two transit antennas and one RF chain; it, therefore, selects one of its antennas for transission. For i {, }, h i denotes the instantaneous channel power gain between the i th antenna of the STx and the SRx antenna, and g i denotes the instantaneous channel power gain between the i th antenna of the STx and the PRx antenna. All channels undergo Rayleigh fading. The STx-SRx channels are independent and identically distributed i.i.d. rando variables RVs, and so are the STx-PRx channels. The independence of the channel gains is justified when the antennas at the STx are spaced sufficiently apart in a rich scattering environent. Thus, h i and g i are i.i.d. exponential RVs with eans μ h and μ g, respectively. Let h [h, h ] and g [g, g ]. Fig.. Syste odel with one PRx and a secondary syste consisting of a STx with two transit antennas and one RF chain that counicates with a SRx with one receive antenna. A. Selection Options and Data Transission The STx transits a sybol x that is drawn with equal probability fro an M-ary PSK MPSK constellation. It can transit using one out of two antennas with a fixed sybol energy. Further, it ay choose to transit with zero power in order to avoid interfering with the priary. We shall represent the zero transit power option by, andshall define the corresponding channel power gains as zero, i.e., h and g. Note that the SRx cannot know when the STx has used the zero transit power option. When the STx transits a sybol with zero power, the SRx can correctly decode it with probability M. Thus, the SEP for this option is equal to M [6]. Transission using antenna i is represented by option i, fori,. Let s {,, } be the option selected. Then the signal r received by the SRx and the interference signal i p seen by the PRx are given by r hs e jθ hs x + n + wps, i p gs e jθgs x, where x, θ hs and θ gs are the phases of the coplex baseband STx-SRx and STx-PRx channel gains, respectively, for the selection option s, andn is circular syetric coplex additive white Gaussian noise at the SRx. The interference seen by the SRx due to priary transission is w ps,andis assued to be Gaussian. This corresponds to a worst case odel for the interference and akes the proble of finding the optial AS rule tractable. Therefore, n + w ps is a circular syetric coplex Gaussian RV, whose variance we denote by σ. We assue that the STx knows h and g, i.e., its channel power gains to the SRx and to the PRx, as has been assued The priary transitter PTx affects the secondary syste perforance entirely through σ. Therefore, no additional details about PTx are given. 3
in [], [], [7]. Note also that no knowledge of the phases of any coplex baseband channel gains is required at the STx. The SRx uses a coherent receiver, and is assued to know h s and θ hs. 3 No knowledge of g or any other channel gain is required at the SRx. B. Proble Stateent A selection rule φ is a apping φ :R + R + {,, } that selects one of the three options for every realization of h and g. Our goal is to develop the optial transit AS rule that iniizes the average SEP of the secondary syste and also ensures that the average interference caused to the PRx is below a threshold I ave.wefirst consider the case where is given. Let SEPh s denote the instantaneous SEP of the secondary syste as a function of the channel power gain h s of the selected option s. Using, the average interference at the PRx due to secondary transission is given by E h,g [g s ]. Terinology: We define a feasible selection rule to be a selection rule whose average interference is less than or equal to I ave.letz be the set of all feasible selection rules. Our problecan be atheatically stated as a iniization over the space of all selection rules Z: in φ Z E h,g [SEPh s ] such that E h,g [g s ] I ave, 3 s φh, g. III. OPTIMAL AS RULE AND SEP ANALYSIS We now derive and analyze the optial selection rule. A. Optial Selection Rule Let us first consider the unconstrained AS rule that iniizes the SEP at the SRx without taking into consideration the interference caused to the PRx. It selects the antenna with the highest channel power gain to the SRx. It is given by s argax i {,} {h i }. The average interference I un caused to the PRx by this rule is I un E h,g [g s ] μ g.the second equality follows because the choice of s does not depend on g. When I un > I ave, the unconstrained AS rule is not a feasible rule and, thus, cannot be optial. The following result copletely characterizes the optial AS rule. Theore : The optial selection rule φ,wheres φ h, g, that iniizes the SEP under the average interference constraint is given as follows: { s arg axi {,} {h i }, if I un I ave. arg in i {,,} {SEPh i +g i }, ifi un >I ave 4 In the tie division duplex TDD ode of operation, inforation about h and g can be obtained by the STx by exploiting reciprocity. Since phase inforation is not required, siple signal strength-based estiation techniques can be used. We note that these results also serve as bounds on the perforance of AS in average interference-constrained CR systes that have partial or iperfect knowledge of g. 3 In practice, this can be achieved by ebedding a pilot once in every coherence interval along with the data sybols since the channel does not change within a coherence interval. where > is chosen to satisfy the average interference constraint with equality: E h,g [g s ]I ave. Proof: The proof is given in Appendix A. The SEP as a function of h s for MPSK is given by [8, 4] SEPh s π hs exp π σ sin dθ, θ where csc π M. Thus, the SEP-optial AS rule is a nonlinear function of h s. This is unlike the MI, MSLIR, and the DS rules. Since the optial rule requires evaluation of an integral, we use the Chernoff upper bound of the SEP, i.e., SEPh s exp h s σ, in the selection rule in 4 in order to get an explicit and tractable characterization of the rule in ters of the channel power gains. Let hi y i exp σ In this case, for I un >I ave, 4 gets odified to, i {,, }. 5 s arg in {y i + g i }. 6 i {,,} As entioned, we shall refer to this as the closed-for optial rule for brevity. Note that optial rule depends on which has to be coputed nuerically. This is typical for optiization probles that handle average power constraints, e.g., rate adaptation and water filling in space, tie, or frequency [9]. We also see that akes it equivalent to unconstrained rule; its SEP is given in [9, 36]. B. Perforance Analysis of Closed-for Optial AS Rule We now analyze the SEP of the rule in 6 for >. Theore : The average SEP of the SU of the closed-for optial rule is given by Ω e SEP Ω, + Ω π [ e y π Ω, y e y Ω, +e y γ Ω, y γ Ω, ] y csc θ y Ω dy dθ + π sin 4 θ π Ω+sin θ Ω+sin dθ, 7 θ where Ω Etμ h σ and s, x x ts e t dt. Proof: The proof is given in Appendix B. Note that.,. is a odified version of the lower incoplete gaa function [, 6.5.] and can be evaluated using standard routines available for the latter. The expression in 7 is in the for of a double integral. The Chernoff bound for the SEP, which is denoted by SEP UB, is obtained by using the inequality sin θ. Itisinthe for of a single integral, and is not shown here to conserve 4
Sybol error probabilty 3 Siulation Analysis 4 6 8 4 6 8 Sybol energy, db.5..5 Sybol error probability 3 Exact SEP Upperbound.4.3. 3 4 5 6 Sybol energy, db Fig.. SEP as a function of for different values of, each of which corresponds to a different value of Iave QPSK. Fig. 3. SEP as a function of large for different 8PSK. space. Using Gauss-Legendre quadrature [], SEP UB can be evaluated accurately as the following su of a few ters: SEP UB k e [ N w k z Ω k Ω Ω Ω, e z k γ Ω, z k + Ω + Ω, ] + e Ω, + + e z k Ω, z ] k + [ γ Ω, Ω + Ω+. Here, Z k x k + and x k and w k are the N Gauss- Legendre abscissas and weights, respectively. As N increases, the approxiation becoes tighter. We have found that N 3 ters are sufficient for., N 5ters are sufficient for.5 <<.. For.5 ore ters are required. IV. PERFORMANCE EVALUATION AND BENCHMARKING We now present Monte Carlo siulations that use 6 saples to verify our analytical results and quantitatively understand the behavior of the optial selection rule under different conditions. The ean channel powers and noise variance are set as unity: μ h μ g σ. Figure plots the SEP as a function of for a fixed value of on each curve. Fro the average interference constraint in 3, a fixed iplies that the ratio Iave is kept constant. The curve corresponds to the unconstrained AS rule. As increases, the SEP increases due to a tighter average interference constraint. Notice that the analysis and siulation results atch each other very well. Given the good atch between the two, we no longer distinguish between the in the results that follow below. Figure 3 plots the SEP and its upper bound for large for different. Atlarge, when the STx transits, the SEP is close to zero, but the SEP due to the zero transit power option is always. Thus, at higher, the SEP saturates to e.thus,anerrorfloor occurs due to the average interference constraint. As expected, the SEP increases as increases. Notice also that the gap between the exact SEP and its upper bound disappears at larger. A. Benchark Selection Rules We now state the MI, MSLIR, and DS rules, which have been proposed in the literature. We shall enhance the MI and MSLIR rules in order to ake the feasible and serve as useful perforance bencharks for all average interference thresholds. MI Rule: The MI rule always selects the transit antenna with the sallest channel power gain to the PRx []. It selects the antenna s i argin i {,} {g i }. If the average interference I i causedtotheprxbythemiruleisgreater than I ave, then the MI rule is infeasible. To overcoe this we introduce the zero transit power option; the STx transits with zero power in case g i τ, i.e., when the power gains of all its channels that interfere with the PRx are large. Thus, the enhanced MI EMI rule selects the option s ei as follows: {, if g τ,g s ei τ. 8 arg in i {,} {g i }, otherwise Note that when I i <I ave, τ. Thus, the EMI and MI rules are equivalent whenever the latter is feasible. However, τ is finite when I i > I ave, in which case the MI rule is infeasible but the EMI is still feasible. MSLIR Rule: The MSLIR rule selects the antenna with the highest ratio of the STx-SRx and the STx-PRx channel power gains []. { } It, therefore, selects the option s slir argax hi i {,} g i. As before, we introduce the zero transit power option in order to ake the MSLIR rule feasible for all I ave. The enhanced MSLIR EMSLIR rule is as follows: {, if h g s eslir η, h g η. arg ax i {,} { hi g i }, otherwise 3 Difference AS Rule: The difference selection DS rule is given by [] s ds argax {,} {δh i δg i }, 9 5
where δ [, ]. Notice that the DS rule behaves as the unconstrained AS rule when δ and as the MI rule when δ. The values of τ, η and δ in the above rules are chosen such that interference constraint 3 is et with equality. Sybol error probabilty 3 Unconstrained 6 7 8 9 5 Sybol energy, db Closed for EMI DS EMSLIR Constrained Fig. 4. Coparison of the SEPs of the closed-for optial AS rule and several benchark rules I ave 6dB and QPSK. Coparison with other rules and optiization of : Figure 4 plots the SEP of the closed-for optial rule and the benchark rules as a function of,fori ave 6dB. We see that there are three regions of operation for the closed-for optial rule: i 6 db: In this case, the interference constraint is not active. Hence,. In this regie, the closed-for optial rule, the DS rule, and the unconstrained rule are equivalent. However, the EMI and EMSLIR rules are sub-optial. ii 6 db < 8 db: In this case is non-zero, but sall. The SEP of the closed-for optial rule decreases as increases. In this regie, the DS rule perfors worse than the closed-for optial rule. iii > 8 db: The SEP of the closed-for optial rule increases as increases. This is because the probability that the SU does not transit increases as increases. The SEPs of the closedfor optial, EMI, and DS rules atch at 9 db. Beyond this the DS rule becoes infeasible and the closedfor optial rule is equivalent to the EMI rule. Further, the EMSLIR rule is again sub-optial. Thus, 8dB is the optial transit sybol energy for the closed-for optial rule when I ave 6dB. At the optial, the SEP of the closed-for optial rule is lower by a factor of 6.5, 3.5, and.4 than the iniu SEPs of the EMI, EMSLIR, and DS rules. V. CONCLUSIONS We developed the optial AS rule that iniizes the SEP of a secondary syste that operates in the underlay CR ode under an average interference constraint. The STx can transit at a fixed power or with zero power. For the closed-for variant of the optial AS rule, we derived expressions for the SEP and its upper bound. The optial rule turns out to be functionally different fro the any ad hoc rules that have been proposed in the literature, and is inherently non-linear in nature. A coparison of the closed-for optial rule with enhanced versions of the ad hoc rules showed that it behaves as the unconstrained rule for sall values of and as the EMI rule for very large values of. We saw that for a fixed Iave an error floor arises. While this paper focuses on on-off power control, in which the STx transits with a fixed power or with zero power, a natural generalization is to allow continuous power control along with AS at the STx. APPENDIX A. Proof of Theore When I un I ave, the unconstrained rule is feasible. Therefore, it ust be the SEP-optial rule. Now, consider the case when I un >I ave. A selection rule that always chooses the zero transit power option causes zero interference to the PRx. It is, therefore, feasible for any I ave. Therefore, the set of all feasible selection rules Z is a non-epty set. Let φ Z be a feasible rule. For a given >, let L φ E h,g [SEPh s +g s ], where s φ h, g. Fro the definition of φ in 4, it follows that L φ L φ. Therefore, E h,g [SEPh s ]+E h,g [g s ] E h,g [SEPh s ]+E h,g [g s ], where s φ h, g. Choose such that E h,g [g s ] Iave. 4 Thus, φ is also feasible. By rearranging the ters in the above inequality we get E h,g [SEPh s ] E h,g [SEPh s ] + E h,g [g s ] I ave. However, since φ is feasible, we know that E h,g [g s ] I ave. Hence, for any feasible rule φ, E h,g [SEPh s ] E h,g [SEPh s ]. Thus, φ is the SEP-optial selection rule. B. Proof of Theore The probability of error conditioned on h and g can be written in ters of all the possible selection options as follows: Pr Err h, g Pr s i h, g Pr Err h, g,s i. i When option s is selected, the conditional error probability depends only on h s. Further, for the zero transit power option, Pr Err h. Averaging over h and g, weget SEP E h,g [Pr s h, g] [ +E h,g Pr s h, g π π ] h exp σ sin dθ, θ where the factor arises because the channel gains of the two transit antennas are i.i.d. Fro the definition of y i in 5, the above expression for the SEP can be recast as SEPE h,g [Pr s h, g] + π π E y,g [Pr s y, g y ] csc θ dθ, 4 Such a unique choice of is possible since I ave < I un and the average interference decreases onotonically as increases. 6
where y [y, y ]. Using the fundaental theore of expectation, we get E h,g [Pr s h, g] Pr s and y E y,g [Pr ] [ csc θ y ] csc θ s y E y Pr s y. We evaluate the above two ters separately below. Using the AS rule in 6, the first ter of, denoted by T, can be written as T Pr s Pr y + g >,y + g >, Pr y + g >. 3 Here, 3 follows because the channel power gains of antennas and are i.i.d. Further, the PDF of y can be shown to be f y y y Ω,fory,]. Hence, we get Pr y + g > e e y y e g μg μ g y Ω Ω y dg dy, dy Ω, Ω Ω Ω,. 4 Here, the last equality follows fro the definition of.,. in the theore stateent. Substituting Pr y + g > in 3 yields T e Ω Ω,. 5 Let the integrand in the second ter of be denoted by T. It can be written as y T E y [Pr ] csc θ s y y csc θ Pr s yy y Ω dy dy. 6 For the closed-for optial antenna selection rule in 6, Pr s y Pr y + g <y + g,y + g < y. By rearranging ters and suing over the utually exclusive events y <y and y >y,weget Prs ypr y <y,g g < y y + Pr y >y,g g < y y,g < y,g < y y. Since g and g are i.i.d. exponential RVs, we can show that y Pr s y e y y e y +y I {y<y } + e y y e y +y I {y>y }. 7 Recall that I {.} denotes the indicator function. Substituting 7 in 6 and updating the integration liits of y yields T [ y e y y y +y Ω e y Ω dy Ω ] + e y y y +y y e y Ω csc θ dy y Ω dy. y Substituting T and T in and writing the integral over y in ters of incoplete gaa functions yields the desired result in 7. REFERENCES [] Spectru policy task force, Tech. Rep. 35, Federal Counications Coission, Nov.. [] A. Giorgetti, M. Varrella, and M. Chiani, Analysis and perforance coparison of different cognitive radio algoriths, in Proc. Cognitive Radio and Advanced Spectru Manageent, pp. 7 3, May 9. [3] R. Zhang and Y. C. Liang, Exploiting ulti-antennas for opportunistic spectru sharing in cognitive radio networks, IEEE J. Sel. Topics Signal Process., vol., pp. 88, Feb. 8. [4] G. Scutari, D. Paloar, and S. Barbarossa, Cognitive MIMO radio, IEEE Signal Process. Mag., vol. 5, pp. 46 59, Nov. 8. [5] S. Sridharan and S. Vishwanath, On the capacity of a class of MIMO cognitive radios, IEEE J. Sel. Topics Signal Process., vol., pp. 3 7, Feb. 8. [6] A. F. Molisch and M. Win, MIMO systes with antenna selection, IEEE Microwave Mag., vol. 5, pp. 46 56, Mar. 4. [7] S. Sanayei and A. Nosratinia, Antenna selection in MIMO syste, IEEE Coun. Mag., vol. 4, pp. 68 73, Oct. 4. [8] A. Ghrayeb and T. Duan, Perforance analysis of MIMO systes with antenna selection over quasi-static fading channels, IEEE Trans. Veh. Technol., vol. 5, pp. 8 88, Mar. 3. [9] M. Win and J. Winters, Virtual branch analysis of sybol error probability for hybrid selection/axial-ratio cobining in rayleigh fading, IEEE Trans. Coun., vol. 49, pp. 96 934, Nov.. [] J. Zhou and J. Thopson, Single-antenna selection for MISO cognitive radio, in Proc. IET, pp. 5, Sep. 8. [] Y. Wang and J. Coon, Difference antenna selection and power allocation for wireless cognitive systes, IEEE Trans. Coun., vol. 59, pp. 3494 353, Dec.. [] P. A. Dochowski, P. J. Sith, M. Shafi, and H. A. Suraweera, Ipact of antenna selection on cognitive radio syste capacity, in Proc. CROWNCOM, pp. 5, Jun.. [3] V. Kriste, N. B. Mehta, and A. F. Molisch, Optial receive antenna selection in tie-varying fading channels with practical training constraints, IEEE Trans. Coun., vol. 58, pp. 3 34, Jul.. [4] W. Gifford, M. Win, and M. Chiani, Antenna subset diversity with non-ideal channel estiation, IEEE Trans. Wireless Coun., vol. 7, pp. 57 539, May 8. [5] R. Sarvendranath and N. B. Mehta, Antenna selection in interferenceconstrained underlay cognitive radios: SEP-optial rule and perforance bencharking, Subitted to IEEE Trans. Coun.,. [6] L. Li and M. Pesavento, Link reliability of underlay cognitive radio: sybol error rate analysis and optial power allocation, in Proc. Cognitive Radio and Advanced Spectru Manageent, pp. 7: 7:5,. [7] H. Wang, J. Lee, S. Ki, and D. Hong, Capacity enhanceent of secondary links through spatial diversity in spectru sharing, IEEE Trans. Wireless Coun., vol. 9, pp. 494 499, Feb.. [8] M. S. Alouini and A. Goldsith, A unified approach for calculating error rates of linearly odulated signals over generalized fading channels, IEEE Trans. Coun., vol. 47, pp. 34 334, Sep. 999. [9] A. J. Goldsith, Wireless Counications. Cabridge Univ. Press, 5. [] M. Abraowitz and I. Stegun, Handbook of atheatical functions with forulas, graphs, and atheatical tables. Dover, 9 ed., 97. 7