496 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 2, FEBRUARY 2013

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1 496 IEEE TRANSACTIONS ON COUNICATIONS, VO 6, NO 2, FEBRUARY 23 Antenna Selection in Interference-Constrained Underlay Cognitive Radios: SEP-Optial Rule and Perforance Bencharking Rialapudi Sarvendranath, Student eber, IEEE, and Neelesh B ehta, Senior eber, IEEE Abstract In the underlay ode of cognitive radio, secondary users are allowed to transit when the priary is transitting, but under tight interference constraints that protect the priary However, these constraints liit the secondary syste perforance Antenna selection AS-based ultiple antenna techniques, which exploit spatial diversity with less hardware, help iprove secondary syste perforance We develop a novel and optial transit AS rule that iniizes the sybol error probability SEP of an average interference-constrained ultiple-input-single-output secondary syste that operates in the underlay ode We show that the optial rule is a non-linear function of the power gain of the channel fro the secondary transit antenna to the priary receiver and fro the secondary transit antenna to the secondary receive antenna We also propose a sipler, tractable variant of the optial rule that perfors as well as the optial rule We then analyze its SEP with transit antennas, and extensively benchark it with several heuristic selection rules proposed in the literature We also enhance these rules in order to provide a fair coparison, and derive new expressions for their SEPs The results bring out new inter-relationships between the various rules, and show that the optial rule can significantly reduce the SEP Index Ters Cognitive radio, underlay, antenna selection, diversity techniques, fading channels, sybol error probability, average interference constraint I INTRODUCTION RECENT studies reveal that electroagnetic spectru allocations are often underutilized [] This coupled with an increase in the nuber of users deanding high data rates has created a scarcity of spectru, and has led to the developent of cognitive radio CR technology to address the scarcity In one coon paradig of CR, two classes of users are defined, naely, priary users PU and secondary users SU A PU owns the license to use the spectru A SU can access the sae spectru as the priary, but is subject to constraints on the interference it causes to the PU so as to protect the PU Two types of CR access odels are coon in the literature, naely, overlay and underlay [2], [3] In overlay CR, anuscript received April 6, 22; revised August, 22 The associate editor coordinating the review of this paper and approving it for publication was H i The authors are with the Dept of Electrical Counication Eng, Indian Institute of Science IISc, Bangalore, India e-ail: sarvendranath@gailco, nbehta@eceiiscernetin A part of this paper has been accepted for presentation in the IEEE Global Counications Conf Globeco, USA, Dec 22 This research was partially supported by a grant fro ANRC Digital Object Identifier 9/TCO /3$3 c 23 IEEE which has also been referred to as interweave CR in [4], the SU transits only in the unused spectral regions Hence, the SU does not cause any interference to the PU, except when it senses the spectru incorrectly Whereas, in underlay CR, the SU can access the spectru even when the PU is transitting However, it is subject to tight constraints on the average or peak interference power that it can cause to the priary In order to liit the interference caused by the SU to the PU below a threshold, power allocation strategies were used in [5] to axiize SU capacity In [6], ultiple antennas were used to iprove the perforance of the SU; techniques such as transit beaforing were explored ultiple input ultiple output IO antenna techniques for CR were investigated in [7] [9] However, one drawback of a ultiple antenna syste is that each antenna eleent requires an expensive radio frequency RF chain to process the signal to or fro the antenna For exaple, at the transitter, the RF chain consists of a digital-to-analog converter, an upconverter, filters, and a power aplifier While antenna eleents are typically cheap, the RF chains constitute a significant portion of the total device cost To reduce the hardware costs of ultiple antenna systes, a technique called antenna selection AS has been extensively studied [], [] It uses fewer RF chains than the nuber of available antennas A subset of antennas is selected as a function of the channel conditions and connected to the RF chains Besides reducing hardware coplexity, cost, and size, AS effectively harnesses the diversity benefits of ultiple antennas [2] [4] Consequently, AS is now a part of next generation wireless standards such as the IEEE 82n and ong Ter Evolution TE [5] Given its proise, AS has also been considered in CR systes [6] [2], and has been shown to iprove secondary syste throughput In the overlay ode, since the SU does not interfere with the PU, the rule for selecting which antenna to transit fro reains the sae as for conventional AS systes, which are not subject to any interference constraint For exaple, in a ultiple input single output ISO syste in which the secondary transitter STx has transit antennas and the secondary receiver SRx has one receive antenna, the transit antenna with the strongest channel power gain to the SRx antenna should be selected We shall refer to this as the unconstrained AS rule henceforth However, in the underlay ode, the priary interference constraint fundaentally changes the criterion on the basis of which

2 SARVENDRANATH and EHTA: ANTENNA SEECTION IN INTERFERENCE-CONSTRAINED UNDERAY COGNITIVE RADIOS: SEP-OPTIA RUE 497 the transit antenna is selected Intuitively, even though an 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 selection rule ust take into consideration both the STx to SRx STx-SRx and STx to PRx STx-PRx channel power gains Several rules for selecting an antenna in a ISO CR, such as the iniu interference I rule and the axiu signal power to leak interference power ratio SIR rule, are proposed in [7] for an STx that transits with a fixed power The I rule selects the antenna that causes the least interference to the priary However, since the selection is done entirely on the basis of the STx-PRx channel gains, the secondary syste does not benefit fro antenna diversity The SIR rule coproises between the I and unconstrained rules, and selects the antenna with the highest ratio of STx-SRx and STx-PRx channel power gains Note that all the above rules do not consider the average interference constraint and ay not always be adissible An AS rule siilar to SIR rule is proposed in [2], but STx uses variable power to transit A difference selection DS rule is proposed in [8], [2] It selects the antenna that axiizes a linear weighted difference of the STx-SRx and STx-PRx channel power gains It outperfors the SIR rule in any scenarios While the above rules are intuitive, they are ad hoc as they do not provably optiize an end objective such as sybol error probability SEP or capacity Contributions: We ake the following contributions We systeatically develop the optial AS rule that iniizes the SEP for a ISO secondary syste that is subject to an average priary interference constraint Given a transit power, we show that the SEP-optial AS rule is a linear cobination of the STx-PRx channel power gain and an exponentially decaying function of the STx-SRx gain The optial selection rule is, thus, non-linear in nature We also present a sipler variant of the optial rule called the upper bound-based optial rule that iniizes a tight Chernoff upper bound of the SEP instead Its appeal lies in its integral-free closed for We also show through our results that the SEPs of the SEPoptial rule and the upper bound-based optial rule are indistinguishable fro each other Another utility of the upper bound-based optial rule is that its SEP analysis is tractable, unlike that of the exact rule We derive its exact SEP and an SEP upper bound for the general case with transit antennas at STx We also show that the analytical expressions siplify further when the STx has =2antennas A key challenge that the analysis tackles is the non-linear for of the selection rule, which is unlike the linear selection rules that have been considered in the literature on single transit AS [], [22] An insightful asyptotic characterization of the upper bound-based optial rule is also developed It shows that an error floor occurs due to the average priary interference constraint, and that the error floor is an exponentially decreasing function of the nuber of transit antennas Extensive siulation results are presented to study the perforance of the upper bound-based optial AS rule and benchark its perforance with any rules that have been proposed in the literature In order to provide as fair a coparison as possible, we copare against enhanced versions of the I and SIR rules that always adhere to the average interference constraint This is achieved by allowing an extra zero-transit power option at the STx Intuitively, the latter option is beneficial when all the STx-PRx channel power gains are large, which akes the secondary transissions interfere considerably with the priary Finally, new analytical results for the SEPs of the enhanced I and SIR rules are also developed Hitherto, the I and SIR rules had only been studied using siulations These results lead to new insights about the optiality of the ad hoc rules and bring out new interrelationships aong the For exaple, we show that the upper bound-based optial rule, enhanced I rule, and the DS rule are equivalent only for large values of transit power, and that the enhanced SIR rule is suboptial in ost scenarios The paper is organized as follows Section II develops the syste odel and the proble stateent The optial selection rule and SEP analysis are in Section III Section IV analyzes the benchark selection rules Nuerical results in Section V are followed by our conclusions in Section VI Several atheatical derivations are relegated to the Appendix II SYSTE ODE AND PROBE STATEENT We use the following notation henceforth The absolute value of a coplex nuber x is denoted by 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 it expectation Scalar variables are written in noral font and vector variables are written in bold font I {a} denotes the indicator function; it is if a is true and is otherwise As shown in Figure, we consider an underlay CR syste in which an STx transits data to an SRx; its transissions cause interference at a PRx The SRx and the STx constitute the secondary syste The PRx and the SRx have one receive antenna each The STx has transit antennas and one RF chain; it, therefore, needs to select one of its antennas for transission For i {, 2,,}, 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 We assue Rayleigh fading The STx- SRx channels are assued to be independent and identically distributed iid rando variables RVs, and so are the STx- PRx channels This assuption is justified when the antennas at the STx are spaced sufficiently apart in a rich scattering environent Thus, the channel power gains h i and g i are iid exponential RVs with eans μ h and μ g, respectively et h [h,h 2,,h ] and g [,g 2,,g ]

3 498 IEEE TRANSACTIONS ON COUNICATIONS, VO 6, NO 2, FEBRUARY 23 a coherent receiver, and is assued to know h s and θ hs 2 No knowledge of g or the channel gains of any other antenna is required at the SRx Fig Syste odel with one PRx and a secondary syste consisting of an STx with transit antennas and one RF chain that counicates with an 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 -ary phase shift keying PSK constellation It can transit using one of the antennas with fixed sybol energy Transission using Antenna i is represented by option i, fori =,, Further, it ay decide to transit with zero power in order to not interfere with the priary We shall represent the zero-transit power option by and shall define the corresponding channel power gains as zero, ie, h and g When the STx uses the zero-transit power option, the SEP is, since the optial receiver in this case just chooses any one of the sybols as its decoded sybol with equal probability [23] et s {,,,} be the option selected 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, 2 where x 2 =, θ hs and θ gs are the phases of the coplex baseband STx-SRx and STx-PRx channel gains, respectively, and n is circular syetric coplex additive white Gaussian noise at the SRx The interference seen by the SRx due to priary transissions is w ps, and is 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 is denoted by σ 2 We assue that the STx knows h and g, ie, its channel power gains to the SRx and to the PRx This has also been assued in the literature on AS in CR, eg, [7], [8], [2] Note also that no knowledge of the phases of any coplex baseband channel gains is required at the STx The SRx uses In the tie division duplex TDD ode of operation, inforation about h and g can be obtained by the STx by exploiting reciprocity The STx uses the signals it receives fro the SRx and PRx when they transit in order to estiate h and g Since phase inforation is not required, siple signal strength-based techniques can be used for estiation We note that these results also serve as bounds on the perforance of AS in CR systes that have access to either partial or iperfect knowledge of g B Proble Stateent Terinology: A selection rule φ is a apping φ :R + R + {,,,} that selects one of the +options for every realization of h and g Wedefineafeasible selection rule to be a rule whose average interference is less than or equal to I ave etz be the set of all feasible selection rules Our goal is to find the optial transit AS rule φ,which iniizes the average SEP of the secondary syste while ensuring that the average interference caused to the PRx is below a threshold I ave We first consider the case where is given The optiization of is handled in Section V et SEPh s denote the instantaneous SEP given channel power gain h s of the selected option s Using 2, the average interference caused to the PRx is given by E h,g [g s ] Our proble can be atheatically stated as follows: in φ E h,g [SEPh s ] subject to E h,g [g s ] I ave, 3 s = φh, g III OPTIA ANTENNA SEECTION RUE AND SEP ANAYSIS We now derive the optial selection rule We then analyze its SEP A Optial Selection Rule et us first consider the selection rule that iniizes the SEP at the SRx when the average interference constraint in 3 is not active Clearly, in this case, the optial rule is the unconstrained rule, which selects the antenna with the highest channel power gain fro the STx to the SRx It is given by s =argax i {,,} {h i } Therefore, the average interference caused to the PRx by the unconstrained rule, I un, is I un = E h,g [g s ]= μ g The second equality follows because the unconstrained rule does not take into account the STx-PRx channel power gain However, when I un >I ave, the unconstrained AS rule is not a feasible rule Therefore, it 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 as follows: { s arg axi {,,} {h = i }, I un I ave arg in i {,,,} {SEPh i +g i }, I un >I ave 4 When I un >I ave,wehave> Thevalueof is such that the STx satisfies the average interference constraint with equality, ie, E h,g [g s ]=I ave Proof: The proof is given in Appendix A 2 In practice this can be achieved by ebedding a pilot along with the data sybols once in every coherence interval, since the channel does not change within a coherence interval

4 SARVENDRANATH and EHTA: ANTENNA SEECTION IN INTERFERENCE-CONSTRAINED UNDERAY COGNITIVE RADIOS: SEP-OPTIA RUE 499 The paraeter is coputed nuerically, as is typical of several optiization probles in wireless counications that are subject to an average constraint, eg, optial link adaptation [24] and water-filling in tie, space, or frequency [25] The SEP as a function of h s for PSK is given by [26, 4] SEPh s = hs sin 2 exp σ 2 sin 2 dθ, 5 θ where, as entioned, = Substituting 5 in 4 we see that the SEP-optial AS rule is a non-linear function of h i This is unlike the I, SIR, and the DS rules B Sipler Upper Bound-based Optial Rule Since the rule in 4 is in the for of a single integral, it is desirable to siplify it This is achieved by instead iniizing the Chernoff upper bound of the SEP, as we show below The upper bound on the SEP of PSK is given by hs sin 2 SEPh s exp 6 The optiization proble that iniizes the above bound can be written as [ hs sin 2 ] in E h,g exp φ subject to E h,g [g s ] I ave, 7 s = φh, g Along lines of Appendix A, it can be shown that the optial rule, which we shall refer to as the upper bound-based optial rule, isgivenby { s arg axi {,,} {h = i }, I un I ave, 8 arg in i {,,,} {y i + g i }, I un >I ave where hi sin 2 y i exp, i {,,,} 9 σ 2 Note that =akes the rules in 4 and 8 equivalent to the unconstrained rule, whose SEP is given in [3, 36] C SEP Analysis General Case of Transit Antennas: We now analyze the SEP of the upper bound-based rule in 8 when > The average SEP for transit antennas is denoted by SEP Theore 2: The average SEP of the secondary syste for the upper bound-based optial rule is given by e SEP =, + μ g e y + y y σ 2 σ 2 +, y + csc 2 θ y e g d dy dθ, where csc 2, Etμ h σ,ands, x x 2 ts e t dt Proof: The proof is given in Appendix B Note that, is a odified version of the lower incoplete gaa function [27, 835], and can be evaluated using standard routines available for the latter The SEP expression in is in the for of a triple integral It is a function of and, which depends on I ave / The second ter in can be siplified further by using the inequality sin 2 θ to get the following upper bound SEP UB : SEP SEP UB = μ g e y + + y +, y + e y e d dy, Given the non-linear nature of the optial selection rule, further siplifications are not possible to the best of our knowledge The double integral above is evaluated nuerically Note that even this result is a significant iproveent copared to onte Carlo siulations In a siilar anner, the average interference I opt caused to the PRx when the upper bound-based optial rule in 8 is used can be shown to be equal to y I opt = μ g e y + y e g +, y + d dy This result is useful in nuerically deterining when it is non-zero, as it is the solution of the equation I opt = I ave Approxiate SEP Analysis: To derive the upper bound in, we used the Chernoff bound Very siilar expressions also arise when the integral-free SEP approxiations for PSK that were proposed in [24] are used instead of the exact SEP expression These are also otivated by the Chernoff bound The only difference lies in the constants 2 = 2 Transit Antennas: We now investigate the special case of an STx with two transit antennas, and show that the SEP expressions siplify further Corollary : The average SEP of the secondary syste for the upper bound-based optial rule for =2transit antennas is given by SEP 2 = e [, 2 + e y, y e y 2, ] + e y γ, y γ, y csc 2 θ y dy dθ sin 4 θ +sin 2 θ +2sin 2 dθ 2 θ Proof: The proof is given in Appendix C 2

5 5 IEEE TRANSACTIONS ON COUNICATIONS, VO 6, NO 2, FEBRUARY 23 The expression in 2 is in the for of a sipler double integral, unlike the expression for the general case transit antennas The Chernoff upper bound for SEP 2, which is denoted by SEP 2 UB, will be in the for of a single integral Using Gauss-egendre quadrature [28], SEP 2 UB can be evaluated accurately as a su of a few ters as follows: SEP 2 UB = e [ N k=, w k z k e z k γ, z k , ] + e 2, ] + e z k, z k [ γ, , 3 where z k 2 x k + and x k and w k are the N Gauss- egendre abscissas and weights, respectively As N increases, the approxiation becoes tighter We have found that N = 3 ters are sufficient for 2 and N =5ters are sufficient for 5 <<2 For 5 ore ters are required D Asyptotic Behavior of the Selection Rule Now we analyze the regie in which is large in order to gain further insights about the perforance of optial selection As increases, SEPh i, for i {,,}, becoes negligible copared to g i Hence, the SEP due to the zero-transit power option, which is, becoes the doinant contributor to the SEP In this case, the optial rule in 4 for > can be shown to reduce to { s, if g =,,g arg in i {,,} {g i }, otherwise 4 Fro 4, we get probability of s =as Pr s = = Pr,,g = e Thus, the SEP in the asyptotic regie, which we denote by SEP asy,issiply SEP asy li E SEP = e 5 t We, thus, see that an error floor occurs when the interference constraint is active As expected, the error floor increases as increases, which corresponds to a tighter interference constraint However, it decreases exponentially when the nuber of transit antennas increases IV BENCHARK SEECTION RUES We now state the I, SIR, and DS rules, which have been proposed in the literature We shall enhance the I and SIR rules in order to ake the feasible for all interference threshold values so that a fair perforance coparison becoes possible We then analyze these enhanced rules A I Rule The I rule proposed in [7] always selects the transit antenna with the sallest channel power gain to the PRx It is given by s i =arg in {g i} 6 i {,,} et I i denote the average interference caused by the I rule to the priary Thus, if I i > I ave, the I rule above is infeasible even though its goal is to iniize the interference caused to the priary To overcoe this we introduce the zero-transit power option in the I rule; the STx transits with zero power in case all the channel power gains to the PRx exceed a threshold τ Thus, the enhanced I EI rule that we use is given by {, if g τ,,g s ei = τ arg in i {,,} {g i }, otherwise 7 The threshold τ is chosen to satisfy the average interference constraint Clearly, τ = akes the EI rule equivalent to the I rule Notice also that τ = corresponds to the asyptotic version of the optial selection rule in 4 Thus, for large, the optial selection rule reduces to the EI rule We now derive the SEP and average interference of the EI rule Theore 3: The SEP of the EI rule with transit antennas is given by SEP = e τ + + tan e τ + tan 8 Etμ h Recall that = csc 2 and = σ The average 2 interference I ei causedtotheprxbytheeiruleis I ei = E [ tμ g + τ ] e τ 9 μ g Proof: The proof is given in Appendix D Taking the liit τ in 9 yields I i = Et Equating 9 with I ave yields the following explicit characterization of τ in ters of I ave : τ = μ g [ Iave W ], 2 e μ g e where W x is the lower branch of the abert-w function, which is defined as the inverse of the function fx = xe x [29] B SIR Rule The SIR rule proposed in [7] selects the antenna with the highest ratio of the STx-SRx and STx-PRx gains It is given by { } hi s slir =arg ax 2 i {,,} et the average interference caused by this rule be denoted by I slir Therefore, the SIR rule is infeasible if I slir >I ave g i

6 SARVENDRANATH and EHTA: ANTENNA SEECTION IN INTERFERENCE-CONSTRAINED UNDERAY COGNITIVE RADIOS: SEP-OPTIA RUE 5 As before, we introduce the zero-transit power option in the rule The enhanced SIR ESIR rule is then as follows: {, if h g s eslir = η,, h g η arg ax i {,,} { hi g i }, otherwise 22 The threshold η is chosen such that the interference constraint is satisfied with equality, ie, E [g seslir ]=I ave Note that η =akes the ESIR rule equivalent to the SIR rule The expressions for the SEP and average interference of the ESIR rule are as follows Theore 4: The average SEP of the ESIR rule with transit antennas is given by SEP η = + h ημ g + μ h μ e h σ 2 sin 2 θ h γ 2, h + h γ 2, h dh dθ, 23 μ h ημ g μ h where γ, is the incoplete gaa function [27] The average interference I eslir caused to the PRx is equal to I eslir = 2E + tμ g η η ημ g + μ h ημ g + μ h 24 Proof: The proof is given in Appendix E Equating 24 with I ave yields η Substituting η =in 24 yields I slir = 2Et + The SEP expression in 23 is in the for of a double integral Its upper bound can be expressed in a single integral for using sin 2 θ, and can be coputed accurately as a su of a few ters using Gauss-aguerre quadrature [28] The details are oitted C DS Rule The difference AS rule for δ [, ] is given by [8] s ds =arg ax {δh i δg i } 25 {,,} Note that δ =corresponds to the unconstrained selection rule and δ = corresponds to the I rule in 6 Thus, the DS rule can control the average interference caused to the priary by choosing an appropriate δ However, the iniu interference caused by the DS rule is the sae as that of the I rule in 6 Therefore, the DS rule is infeasible when I i >I ave Introducing the zero-transit power option can ake it feasible for all I ave We do not delve into it further due to space constraints The SEP of this rule is given in [2, 6] for BPSK, and can be generalized to PSK The average interference caused to the priary is given in [8, 22] V NUERICA RESUTS AND PERFORANCE BENCHARKING We now present onte Carlo siulations that use 6 saples to verify our analytical results and benchark the behavior of the upper bound-based optial AS rule under different conditions The ean channel powers and noise variance are set as unity, ie, μ h = μ g = σ 2 = Figure 2 copares the SEPs of the optial selection rule in 4 and the upper bound-based optial rule in 8 as a Sybol error probability SEP optial rule Upper bound based optial rule Sybol energy, E db t Fig 2 Coparison of the SEPs of the optial AS rule in 4 and the upper bound-based optial rule in 8 =2, =4,andI ave =6dB Sybol error probability Siulation Analysis = =2 =4 = Sybol energy, db Fig 3 SEP as a function of of the upper bound-based optial rule for different nubers of secondary transit antennas = and = 4 function of the sybol energy We see that the two are indistinguishable fro each other This is because the Chernoff bound for the SEP of PSK is tight As a result, the sae antenna gets selected by the two rules with high probability Figure 3 plots the SEP of the upper bound-based optial rule as a function of for different nubers of STx antennas with = Fro the average interference constraint in 3, afixed iplies that the ratio Iave is kept constant on each curve We observe that the analytical and siulation results atch very well As expected, the SEP decreases significantly as increases Figure 4 studies the SEP of the upper bound-based optial rule as a function of for =2antennas for different values of The =curve corresponds to the scenario where the interference constraint is not active and the unconstrained rule is optial As increases, the SEP increases due to a tighter average interference constraint Notice that the analytical and siulation results again atch each other very well We, therefore, no longer distinguish between the two henceforth Figure 5 plots the SEP and its upper bound for a larger range of for =5 and different values of The figure verifies the result in Section III-D about the occurrence of an error floor occurs at larger Notice that the error floor drops significantly as increases Notice also that the gap between the exact SEP and its upper bound disappears at larger Figure 6 plots the SEP as a function of when the average interference threshold is fixed at I ave =db We see that

7 52 IEEE TRANSACTIONS ON COUNICATIONS, VO 6, NO 2, FEBRUARY 23 Sybol error probabilty 2 3 Siulation Analysis =5 =2 =5 = Sybol energy, db Sybol error probability 2 3 = I ave = = 2 = 4 > > I ave Sybol energy, E db t Fig 4 SEP as a function of of the SEP upper bound-based optial rule for different values of, different values of which correspond to different values for Iave =2and =4 Fig 6 SEP of the upper bound-based optial rule as a function of for different nubers of secondary transit antennas I ave =db and =4 Sybol error probability 2 4 = =2 =4 =8 Upper bound Exact SEP Sybol energy, E db t Fig 5 arge : SEP and its upper bound of the upper bound-based optial rule as a function of for different nuber of STx antennas =5 and =4 there are three regions of operation of the optial rule for =2:i db: In this case the interference constraint is not active Hence, = and the optial rule selects the antenna with the highest STx-SRx channel power gain ii db < 2 db: In this case becoes non-zero, but is very sall The SEP continues to decrease as increases iii > 2 db: The SEP now increases as increases This is because the probability that the STx does not transit increases so as to adhere to the average interference constraint Thus, =2dBis the optial transit sybol energy when I ave =db and =2 Siilarly =4dB is optial for =4 3 The optial value of is I ave itself when = Thus, the optial value of increases with Figure 7 plots the SEPs of the optial rule, the SEP Chernoff upper bound SEP UB in, and the SEPs obtained by using the approxiate SEP expressions naed odel 2 and odel 3 in [24, 2, 3] We see that these approxiations track the exact SEP well for saller values of while the SEP UB curve atches the exact SEP curve for larger Figure 8 plots the SEP as a function of when the average interference threshold I ave is fixed at 2 db This is 3 It is difficult to analytically characterize the optial in closed-for because the SEP expression and its bound depend on, which itself depends on Sybol error probability 2 3 odel 3 odel 2 Exact Chernoff bound Sybol energy, E db t Fig 7 Coparison of SEPs obtained using the approxiate SEP expressions proposed in [24] and the exact SEP and its upper bound derived in and, respectively =2, =4,andI ave =6dB done for different PSK constellation sizes As expected, the SEP increases as the constellation size increases Further, the optial sybol energy is a function of the constellation size Figure 9 plots the cuulative distribution function CDF of the instantaneous interference seen by the PRx Fro this, other perforance etrics of interest such as the priary outage probability, which is the probability that the interference at the PRx exceeds a threshold, can be read off Perforance bencharking: Figure copares the SEP of the optial rule with those of the EI, ESIR, and DS rules, for I ave =6dB and =2For <I ave =6dB, the SEP of the optial rule is the sae as that of unconstrained rule and the DS rule δ = For 9 db, the optial rule becoes equivalent to the EI rule cf Section IV-A The DS rule perfors worse than the optial rule when 6 db < < 9 db, and is infeasible beyond 9 db The ESIR rule is sub-optial for all values of We observe that the iniu SEP of the optial rule is lower by a factor of 65, 35, and 24 than the iniu SEPs of the EI, ESIR, and DS rules VI CONCUSIONS We considered the proble of antenna selection at a secondary transitter that operates under an average interference constraint iposed by the underlay ode of operation of a

8 SARVENDRANATH and EHTA: ANTENNA SEECTION IN INTERFERENCE-CONSTRAINED UNDERAY COGNITIVE RADIOS: SEP-OPTIA RUE 53 Sybol error probability 2 4 = 4 = 8 = Sybol energy, db Sybol error probabilty Unconstrained Upper bound EI DS ESIR Constrained Sybol energy, db Fig 8 SEP of the upper bound-based optial rule as a function of for different constellation sizes I ave =2dB and =4 Fig Coparison of the SEPs of the upper bound-based optial rule and several benchark rules =2, =4,andI ave =6dB CDF = db = 2 db Interference at PRx db Fig 9 CDF of interference seen by PRx I ave = 6 db, = 2,and =4 cognitive radio We developed the optial selection rule that iniizes the SEP, and saw that it is non-linear in nature It is functionally quite different fro the any ad hoc rules that have been proposed in the literature We then analyzed the SEP of the upper bound-based optial rule for the general case of transit antennas These expressions siplified further for =2antennas We also analyzed the SEPs of the enhanced I and SIR rules We saw that the SEPs of the optial rule and the upper bound-based optial rule are indistinguishable fro each other The unconstrained rule and the DS rule behave as the optial rule for sall,andthe optial rule becoes equivalent to the EI rule for larger While an error floor is unavoidable, it decreases exponentially as increases The analytical techniques presented in this paper open the door to developing optial subset selection rules when the transitter has ore than one RF chain Further, it is of interest to investigate the optial selection rule when the STx is subject to a constraint on the outage probability it causes at the PRx Another interesting proble is characterizing the joint and optial power control and antenna selection policy 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 zerotransit 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 et φ Z be a feasible rule For a given >, define φ E h,g [SEPh s +g s ], 26 where s = φ h, g Fro the definition of φ for I un >I ave in 4, it follows that φ φ Therefore, E h,g [SEPh s ]+E h,g [g s ] E h,g [SEPh s ]+E h,g [g s ], 27 where s = φ h, g Choose such that E h,g [g s ]= Iave Such a unique choice of is possible since I ave <I un and the average interference decreases onotonically as increases Thus, φ is also a feasible selection rule Rearranging the ters in 27, we get E h,g [SEPh s ] E h,g [SEPh s ] + E h,g [g s ] I ave 28 However, since φ is a feasible rule, we know that E h,g [g s ] I ave Hence, 28 iplies that for any feasible antenna selection rule φ, E h,g [SEPh s ] E h,g [SEPh s ] Thus, φ ust be the optial rule B Proof of Theore 2 The SEP conditioned on h and g, which we denote by Pr Err h, g, can be written as Pr Err h, g =Pr s =, Err h, g+ Pr s = i, Err h, g i= Averaging over h and g and using the chain rule, we get the following expression for the SEP: SEP = E h,g [Pr s = h, g Pr Err h, g,s=] + E h,g [Pr s = i h, g Pr Err h, g,s= i] 29 i= Using syetry and the fact that the SEP conditioned on option s depends only on h s,weget SEP = E h,g [Pr s = h, g Pr Err h ] + E h,g [Pr s = h, g Pr Err h ] 3

9 54 IEEE TRANSACTIONS ON COUNICATIONS, VO 6, NO 2, FEBRUARY 23 Substituting 5 and using the fact that Pr Err h =, we get SEP = E h,g [Pr s = h, g] + h sin E h,g [Pr 2 ] s = h, gexp σ 2 sin 2 dθ θ 3 Fro the definition of y i in 9 and y [y,,y ],the above expression for the SEP can be recast as SEP = E h,g [Pr s = h, g] + y E y,g [Pr ] csc 2 θ s = y, g dθ 32 Fro the law of total expectation we know that Siilarly, E h,g [Pr s = h, g] = Pr s = 33 y E y,g [Pr ] csc 2 θ s = y, g y = E y, [Pr ] csc 2 θ s = y, We evaluate the two ters in 32 separately below First ter: Recall that the selection rule in 8 is s = arg in i {,,,} {y i + g i } Therefore, the first ter can be written as Pr s ==Pr y + >,,y + g >, = Pr y + > 34 Here, the second equality follows fro the independence of the channel power gains of the antennas Further, using the PDF of y, which is given by f y y = y,y,], weget = Pr y + >= e y y y e μ g y dy = e d dy, Here, the last equality follows fro the definition of, in the theore stateent Substituting the above equation in 34 copletes the evaluation of the first ter, which we denote by T, and yields e T =Pr s ==, 35 Second ter: In a siilar anner, fro 8, we get Pr s = y, = Pr y 2 +g 2 >y +,,y +g >y +, >y + y,, =Pr y 2 +g 2 >y +,y + < y, 36 Siplifying Pr y 2 + g 2 >y +,y + < y, is siilar to siplifying Pr y + > in the first ter This yields Pr s = y, = e y + +, y + I {y+<} et the second ter of 32 be denoted by T 2 Substituting the above equation in T 2 and changing the integration liits of to ensure that y + <,weget T 2 = μ g e y + y y +, y + csc 2 θ y e g d dy dθ Cobining the expressions for T and T 2 gives the desired result in C Proof of Corollary Starting fro 32 and substituting =2, weget SEP 2 = E h,g [Pr s = h, g] + 2 y E y,g [Pr ] csc 2 θ s = y, g dθ 37 The first ter can be obtained directly fro 35 However, the expectation in the second ter can be siplified differently Fro the law of total expectation, we know that E y,g [Pr s = y, g Fro 8, we know that y ] csc 2 θ = E y [Pr s = y y ] csc 2 θ 38 Pr s = y =Pr y + <y 2 + g 2,y + < y Rearranging the ters and suing over the utually exclusive events y 2 <y and y 2 >y,weget Pr s = y = Pr y 2 <y, g 2 < y 2 y, < y y + Pr y 2 >y, g 2 < y 2 y, < y y Since and g 2 are iid exponential RVs, we can show that Pr s = y = e y 2 y e y +y 2 2 I {y2<y } 2 e y y 2 e y +y 2 2 I {y2>y }

10 SARVENDRANATH and EHTA: ANTENNA SEECTION IN INTERFERENCE-CONSTRAINED UNDERAY COGNITIVE RADIOS: SEP-OPTIA RUE 55 Thus, the expectation in the second ter of 37, which we denote by Q, can be written as Q = 2 [ y e y 2 y 2 e y +y 2 2 y 2 dy 2 ] + 2 e y y 2 e y +y 2 2 y 2 dy 2 y y csc 2 θ y dy Siplifying Q in ters of incoplete gaa functions yields the desired result in 2 D Proof of Theore 3 SEP Analysis: We start fro 3 In the EI rule, selection of an option s depends only on g Therefore, the SEP can be written as SEP = E g [Pr s = g] +E g [Pr s = g] h sin E h [exp 2 ] σ 2 sin 2 dθ θ Since E g [Pr s = i g] = Pr s = i, fori {, }, andh is an exponential RV, we get SEP =Pr s =+Pr s = sin 2 θ sin 2 θ+ dθ 39 For the EI rule, Pr s ==Pr >τ,,g >τ= Pr >τ = e τ By syetry, the probability of selecting any one of the antennas is the sae Therefore, Pr s = Pr s == = e τ 4 Substituting Pr s =and Pr s =in 39, we get SEP = e τ + e τ sin 2 θ sin 2 θ+ dθ 4 The single integral in the above equation can be siplified by using [27, 2562], and leads to the desired result 2 Average Interference Analysis: The average interference caused to the PRx when we eploy the EI rule is given by I ei = i= E g [g i Pr s = i g], = E g [ Pr s = g], 42 = E g [ Pr s = ] 43 Here, 42 follows fro syetry, and 43 follows fro the law of total expectation Hence, I ei = E t Pr s = e dg 44 μ g For the EI rule, Antenna is chosen if g 2 >,,g > and <τ Hence, Pr s = =Pr g 2 >,,g >, <τ, =Pr g 2 > I {g<τ} 45 Using the result Pr g 2 > = e further yields 9 and siplifying E Proof of Theore 4 SEP Analysis: Starting fro 3 and proceeding along lines siilar to Appendix B, we get SEP = Pr s = + h sin E h,g [Pr 2 ] s = h, exp σ 2 sin 2 dθ θ The ESIR rule selects the zero-transit power option when h < η,, h g < η Thus, we have Pr s = = Pr h <η,, h g <η = Pr h Furtherore, <η h Pr <η = = Pr h <η e d, μ g e η μ e g h Siilarly, Antenna is selected when h2 g 2 and h >η Thus, μ g d = ημ g ημ g + μ h 46 < h,, h g < h Pr s = h, h2 = Pr < h,, h < h, h >η h, 47 g 2 g Conditioned on h and,theevents {2,,}, and h the RVs h2 g 2,, h g { } hi g i < h, for i >ηare utually independent Further are identically distributed Thus, h2 Pr s = h, =Pr < h h, g 2 Using 46, we get Pr h2 g 2 I { h >η } 48 < h μ h, = gh μ h +μ gh Hence, SEP = h η h sin 2 e e h μ h exp σ 2 sin 2 θ μ g μ h Ḷ μ g h η d dh dθ + 49 μ h + μ g h ημ g + μ h Using the variable substitution q = g μ g + h μ h and siplifying further yields 23 2 Average Interference Analysis: Carrying out the sae steps as in 43 we get I eslir = E h, [ Pr s = h, ] 5 Substituting the expression for Pr s = h, fro 48 and siplifying further yields the desired result in 24 REFERENCES [] Spectru Policy Task Force, Tech Rep 235, Federal Counications Coission, Nov 22 [2] V D Chakravarthy, Z Wu, A Shaw, A Teple, R Kannan, and F Garber, A general overlay/underlay analytic expression representing cognitive radio wavefor, in Proc 27 Int Conf Wavefor Diversity Design, pp [3] W ee, H ee, and D-H Cho, Cognition based sealess transissions by using underlay-overlay switching ethod in future wireless counication syste, in Proc 27 CROWNCO, pp

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radio: sybol error rate analysis and optial power allocation, in Proc 2 Cognitive Radio Advanced Spectru anageent, pp 7: 7:5 [24] S T Chung and A J Goldsith, Degrees of freedo in adaptive odulation: a unified view, IEEE Trans Coun, vol 49, pp 56 57, Jan 2 [25] D Tse and P Vishwanath, Fundaentals of Wireless Counications Cabridge University Press, 25 [26] 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 , Sept 999 [27] S Gradshteyn and Ryzhik, Tables of Integrals, Series and Products Acadeic Press, 2 [28] Abraowitz and I Stegun, Handbook of atheatical Functions with Forulas, Graphs, and atheatical Tables, 9th edition Dover, 972 [29] F Chapeau-Blondeau and A onir, Nuerical evaluation of the abert W function and application to generation of generalized Gaussian noise with exponent /2, IEEE Trans Signal Process, vol 5, pp , Sept 22 Rialapudi Sarvendranth received his Bachelor of Engineering degree in Electrical and Electronics Engineering fro the National Institute of Technology Karnataka, Surathkal in 29 He received his aster of Engineering degree fro the Dept of Electrical Counication Engineering, Indian Institute of Science, Bangalore, India in 22 He is currently with Broadco Counications Technologies Pvt td, Bangalore, India, working on the ipleentation of the TE standard Fro 29 2, he was in the Dept of Instruentation, Indian Institute of Science, Bangalore, India, where he was involved in the developent of iage processing algoriths His research interests include wireless counication, ultiple antenna techniques, and next generation wireless standards Neelesh B ehta S 98- -S 6 received his Bachelor of Technology degree in Electronics and Counications Eng fro the Indian Institute of Technology IIT, adras in 996, and his S and PhD degrees in Electrical Engineering fro the California Institute of Technology, Pasadena, CA, USA in 997 and 2, respectively He is now an Associate Professor in the Dept of Electrical Counication Eng, Indian Institute of Science IISc, Bangalore, India Prior to joining IISc, he was a research scientist in the Wireless Systes Research group in AT&T aboratories, iddletown, NJ, USA fro 2 to 22, Broadco Corp, atawan, NJ, USA fro 22 to 23, and itsubishi Electric Research aboratories ER, Cabridge, A, USA fro 23 to 27 His research includes work on link adaptation, ultiple access protocols, WCDA downlinks, cellular syste design, IO and antenna selection, cooperative counications, and cognitive radio He was also actively involved in the Radio Access Network RAN standardization activities in 3GPP fro 23 to 27 He has served on several TPCs He was a TPC co-chair for Wireless Counications Syposiu of ICC 23, WISARD 2 and 2, National Conference on Counications NCC 2, the Transission Technologies track of VTC 29 Fall, and the Frontiers of Networking and Counications syposiu of Chinaco 28 He was the tutorials co-chair for SPCO 2 and publications co-chair for SPCO 22 He has co-authored 35 IEEransactions papers, 6+ conference papers, and three book chapters, and is a co-inventor in 2 issued US patents He is an Editor of IEEE WIREESS COUNICATIONSETTERS and serves as the Director of Conference Publications on the Board of Governors of the IEEE Counications Society