Amin, Osama; Abediseid, Walid; Alouini, Mohamed-Slim. Institute of Electrical and Electronics Engineers (IEEE)

Transcription

1 KAUST Reository Outage erformance of cognitive radio systems with Imroer Gaussian signaling Item tye Authors Erint version DOI Publisher Journal Rights Conference Paer Amin Osama; Abediseid Walid; Alouini Mohamed-Slim Post-rint.9/ISIT Institute of Electrical and Electronics Engineers IEEE 5 IEEE International Symosium on Information Theory ISIT c 5 IEEE. Personal use of this material is ermitted. Permission from IEEE must be obtained for all other users including rerinting/ reublishing this material for advertising or romotional uroses creating new collective works for resale or redistribution to servers or lists or reuse of any coyrighted comonents of this work in other works. Downloaded 8-Ar-8 9:36: Link to item htt://hdl.handle.net/754/579495

2 Outage Performance of Cognitive Radio Systems with Imroer Gaussian Signaling Osama Amin Walid Abediseid and Mohamed-Slim Alouini Comuter Electrical and Mathematical Sciences and Engineering CEMSE Division King Abdullah University of Science and Technology KAUST Thuwal Makkah Province Saudi Arabia. {osama.amin walid.abediseid Abstract Imroer Gaussian signaling has roved its ability to imrove the achievable rate of the systems that suffer from interference comared with roer Gaussian signaling. In this aer we first study imact of imroer Gaussian signaling on the erformance of the cognitive radio system by analyzing the outage robability of both the rimary user PU and the secondary user SU. We derive exact ression of the SU outage robability and uer and lower bounds for the PU outage robability. Then we design the SU signal by adjusting its transmitted ower and the circularity coefficient to minimize the SU outage robability while maintaining a certain PU quality-ofservice. Finally we evaluate the roosed bounds and adative algorithms by numerical results. I. INTRODUCTION The innovative rogress of wireless technology results in a roliferation of attractive wireless devices and diversity of services. In this era of the massive demand for data throughut and traffic the shortage of sectrum resources can limit significantly the wireless network erformance. Cognitive radio CR is a hierarchical dynamic sectrum access technique that can meet the market demand and solve the sectrum scarcity roblem. CR allows secondary users SU i.e. unlicensed users to access the sectrum as long as they do not affect the transmission quality of the rimary users PU i.e. licensed users. The cognitive techniques are achieved by either defining transmission eriods for SU as in overlay and interweave methods or by limiting the SU ower to avoid unaccetable interference levels at the PU as in the underlay method []. Adoting the underlay technique steers the research focus to mitigate the interference received at the PU end from the SU. Mitigating the interference in communication systems is a challenging research roblem and receives a lot of attention. Recently statistical signal characteristics are shown to affect significantly the maximum achievable rate. Different from the usual assumtion of roer Gaussian signaling which imose comlex signal with uncorrelated real and imaginary comonents and equal ower for each comonents imroer Gaussian signaling is shown to increase the achievable rate over interference channels systems []. Proer signaling term was introduced in the information theory field for the first time by Neeser and Massey in [3] where they defined new second order statistics quantity called seudo-covariance to fully describe the roertness of any comlex random variable besides the well known conventional The work of M.-S. Alouini was suorted by the Qatar National Research Fund a member of Qatar Foundation under NPRP Grant NPRP The statements made herein are solely the resonsibility of the authors. covariance. To study the imact of the imroer Gaussian signaling on communication systems Taubök investigated the influence of imroer Gaussian signaling on information theoretic quantities such as entroy divergence and caacity [4]. In cognitive radio research imroer Gaussian signaling is emloyed in [5] where the PU is assumed to have roer Gaussian signaling since there is no control uon it. On the other hand the SU is assumed to use imroer Gaussian signaling and have access on instantaneous channel state information of both the PU and SU communication channels. The instantaneous achievable rate of both PU and SU are studied then the SU ower and the circularity coefficient are adjusted to maximize its rate while achieving the PU qualityof-service QoS. In this aer we study the outage robability of underlay CR with imroer Gaussian signaling assumtions at the SU. We derive a closed form ression for the SU outage robability and uer and lower bounds of the outage robability for the PU. Different from [5] where erfect CSI for all links are assumed to be known at the SU nodes we assume a ractical scenario where only average CSI is available at the SU nodes. Then we adjust the SU ower and circularity coefficient to maximize the SU rate while satisfying PU QoS. A. Preliminaries II. SYSTEM DESCRIPTION Consider a zero mean scalar random variable x whose conventional variance is defined as σx = E[ x ] and its seudo-variance is defined as σ x = E[x ] [3]. Definition : [3] [6] A comlex random variable is called roer if its seudo-variance is equal to zero otherwise it is called imroer. Definition : [7] The imroriety degree of x is measured by the circularity coefficient that is defined as C x = σ x /σ x where C x C x = denotes roer signal and C x = denotes maximally imroer signal. B. Underlay Cognitive Radio System We assume a sectrum sharing system consisting of SU air of transmitter and secondary receiver that coexist with another licensed communication air of the PU. The communication channels via all links are modelled as Rayleigh fading channels and the noise at the receivers end is modelled an additive white zero-mean circularly symmetric comlex Gaussian with variance σ. The SU transmitter needs to adjust its ower s without exceeding the maximum allowable interference level

3 at the PU receiver. The received signal at the PU is ressed as y = h x + s g s x s + n where is the PU transmitted ower x is the PU transmitted symbols which is assumed to be roer zero mean Gaussian signal with unity variance x s is the SU signal with circularity coefficient C x h is the fading channel of the PU transmission g s is the SU interfering channel to the PU and n is the noise at the PU receiver. Similarly the SU received signal is ressed as y s = s h s x s + g x + n s where h s is the SU direct link channel g is the PU interfering channel to the SU and n s is the noise at the SU. As a result of the imroer Gaussian signaling the achievable rate of the PU is ressed as [5] [8] R s C x = log + h σ + s g s + C log y CI 3 where C y and C I are the circularity coefficients of the received and interference-lus-noise signals at the PU resectively which are given by s g s C x C y = s g s + h + σ C I = s g s C x s g s + σ. 4 From 3 and 4 we observe that the PU rate in imroer Gaussian signaling is always higher than the roer Gaussian signaling case i.e. C x =. In the later the second term in 3 vanishes while in the former it gives always a ositive value which increases the rate of the PU. After some maniulations R s C x can be written as R s C x = s g s log + h + σ s g s 4 Cx σ + s g s s g s 4 Cx 5 where the circularity coefficient of the interference-lus-noise terms equals zero. As for the SU the circularity coefficient of the interference term equals zero thus the SU achievable rate is ressed as R s s C x = log s h s 4 Cx g + σ + s hs g + σ +. III. OUTAGE PROBABILITY ANALYSIS The error erformance achieved by the otimal coding and decoding strategies is limited by the so-called outage robability. In this section the overall outage robability of our system is analyzed in details. A. Secondary User Outage Probability Let R s be defined as the target rate of the SU channel. The outage robability of the SU P outs is defined as 6 P outs s C x = Pr [R s s C x < R s ]. 7 Substituting 6 in 7 we get [ ] s C x γ s P outs s C x = Pr + I + sγs Γ s < + I 8 where Γ s = Rs γ s = h s /σ is an onential random variable that reresents the direct-channel-to-noiseratio of the SU with mean E{γ s } = γ s and I = g /σ is an onential random variable with mean E{I } = I that reresents the interference-channel-to-noise-ratio of the PU on the SU. By solving the inequality that aears inside the robability of 8 one can show that the conditional SU outage robability conditioned on I is given by P outs s C x I = γso γ s xγs dx 9 where γ so = I+ s C + x Γs Cx reresents the non-negative root that satisfies the inequality in 8. By evaluating 9 we get P outs s C x I = I + Cx Ψ s C x where Ψ s C x = + Γs Cx / s γ s. By averaging over the onential statistics of I we obtain P outs s C x = E I {P outs s C x I } = C x C x + I Ψ s C x Ψ s C x C x For C x = the above outage robability reduces to P outs s =. + I Ψ s Ψ s On the other extreme when C x yields Γs P outs s = lim Poutss Cx = sγ s. 3 C x + IΓs sγ s B. Primary User Outage Probability Similar to the above subsection our goal here is to find a closed form for the PU outage robability in terms of the signal and channels arameters. Let R be defined as the target rate of the SU channel. The outage robability of the PU P out s C x is defined as P out s C x = Pr [R s C x < R ]. 4 Substituting 5 in 4 we get [ P out s C x = Pr γ + s + γ ] Γ [ si s + s Cx ] < 5 where Γ = R γ = h /σ is an onential random variable that reresents the direct-channel-to-noiseratio of the PU with mean E{γ } = γ and = g s /σ is an onential random variable with mean E{ } = that reresents the interference-channel-to-noise-ratio of the SU on the PU. By solving the inequality that aears inside the robability of 5 one can show that the conditional SU outage robability conditioned on is given by P out s C x = γ o /γ 6

4 [ where γ o = sis+ + Γ sc x/ s + ] reresents the root that satisfies the inequality in 5. Averaging 6 over the statistics of we get P out s C x = E Is {P out s C x } = z sz + Ψ Cx sz + sz dz 7 where Ψ x = + Γ [ x ] / γ. Obtaining a closed form ression for the aforementioned integral is very difficult excet for C x = where it reduces to P out s = Ψ + s Ψ. 8 For all values of C x i.e. for imroer Gaussian signaling we will resort to deriving lower and uer bounds of the PU outage robability to rovide us with PU outage robability behavior limits. Lower Bound of the PU Outage Probability: One way to lower bound the PU outage robability is by using the fact that for z we have s z/ + s z. In this case we may lower bound P out s C x for any value of C x as P out s C x = z sz + Ψ C x dz Ψ Cx = Pout LB s C x. 9 + sψ C x Uer Bound of the PU Outage Probability: An uer bound can be obtained for z where we have that s z/ + s z. However the drawback of using the above bound is that the deendency of the outage ression on the imortant imroer arameter C x will vanish and result in a loose PU outage robability uer bound. To obtain a tighter bound we slit the integral in 7 as follows P out s C x = zis P out s C x = zdz a + zis P out s C = zdz zis sz + Ψ dz zis Cx s sz + Ψ dz + s = P UB out s C x where a follows from using the fact that for z we have sz + sz in the first integral and the fact that for z we have sz + s sz + s in the second integral. Evaluating both integrals we get + Ψ Pout UB I s s s C x = + Ψ + sψ + sψ + Ψ I s s + sψ C x + C x +. The best uer bound can be obtained by finding the value of that minimizes P UB out s C x P UB out s C x = min which can equivalently found from [ ] Cx + Ψ Ψ + sis Cx sψ + P UB out s C x + γ + γ Ψ Cx + sis C γ Γ x Cx sψ + =. 3 Assuming the we obtain the aroximate ression Ψ I s s P out s C x = + + sψ Ψ Ψ + sψ I C s s x. 4 + sψ C x IV. SU ADAPTIVE SIGNAL ADJUSTMENT In this section we aim to adjust the SU signal arameters s and C x to imrove the SU erformance measured by the outage robability while maintaining redetermined PU QoS reresented in an outage robability threshold of a target rate. According to the PU ersective the system is designed to achieve a target QoS considering an accetable interference margin. In this case P outth is ressed in term of the maximum interference ower P int as ] P outth = Pr [log + h < R σ + P int = + Imax + Γ 5 γ where I max = P int /σ is the maximum allowable interference-to-noise ratio at the PU receiver end. Thus the PU should adjust its ower according to = + I max γ log P outth + Γ. 6 As a result the SU has to consider the PU design conditions while adjusting its signal arameters. For the design urose we consider different cases in the following subsections. A. Proer Gaussian signaling Design For the roer signaling case the SU system adjusts s to suress its interference to the PU to be within the accetable margin. In other words s needs to be comuted such that P out s satisfy a redefined outage threshold P outth for a given rate R which is ressed as P outth = + s Ψ Ψ. 7 After some simlifications s is found to be s = Ψ P outth Ψ P outth. 8 From 8 and 7 the SU can oerate while satisfying the PU QoS requirements if Ψ > P outth which is valid as long as the maximum marginal interference-to-noise ratio does not equal zero otherwise the SU remains silent.

5 B. Imroer Gaussian signaling Design For the imroer Gaussian signaling case we have an additional design arameter i.e. C x which controls the signal imroriety. Thus we ect to have an infinite s C x set that satisfy the PU QoS. This design flexibility gives us the oortunity to consider another objective to imrove in addition to achieving the PU erformance requirement. In this work the SU signal arameters s and C x is adatably comuted to minimize the SU outage robability while satisfying the PU QoS i.e. P out s C x P outth. For this urose we formulate the following otimization roblem min P outs s C x s C x s.t. P out s C x P outth s smax C x where smax is the maximum SU transmitter ower budget. To simlify this roblem we use the bounds derived in the revious section and obtain the following simlified roblems. Lower-bound-based design: Thanks to the simlified PU lower bound ression obtained in 9 we can obtain s that can satisfy the PU QoS i.e. Pout LB s C x = P outth in terms of C x and P outth as s C x = Ψ C x P outth Ψ C x P outth. 9 In this case we observe that Ψ C x > P outth is always valid which means that the SU may always transmit. In the same time increasing the signal imroriety results in a large increase of s which should be considered carefully to meet the maximum SU ower budget. Based on 9 and the outage robability of the SU can be ressed in terms of C x. Therefore we obtain the following simlified otimization roblem min C x P outs C x subject to C x < s C x smax 3 to comute C x then we can find s from 9. To solve 3 first we can rove that P outs C x is monotonically decreasing in C x while s C x is monotonically increasing in C x. Thus the SU can achieve lower outage if it uses maximally imroer signal with the maximum ossible transmitted ower. As a result the lower bound ower solution is smax and the corresonding C x is comuted from Algorithm I where C o is comuted from C o = + [ { } /Is s + γ Γ Γ W γ ] P outth s s and W {.} is the Lambert-W function [9]. Algorithm I : Inut smax γ Γ and P outth. : if smax then 3: C x 4: else 5: C x C o 6: end if Uer-bound-based design: Uer bound ression can be used to simlify the design roblem by imosing the PU outage constraint on the derived uer bound in 4 i.e. Pout UB s C x = P outth. As a result C x can be ressed in terms of s as + C x = + Γ [ γ + γ { }] + /Is s Γ s + W P outth Λ s s 3 where Λ s is defined as Ψ + Λ s = + sψ sis Ψ. 3 + sψ Algorithm II : Inut smax γ Γ and P outth. : if smax then 3: C x 4: s smax 5: else 6: if C x smax > then 7: C x 8: s C x s 9: else : s smax : C x C x smax : end if 3: end if Alternatively if we wish to use the PU outage robability aroximate ression to obtain another simlified otimization roblem we follow similar stes to that of the uer bound and obtain C x in terms of s as C x s = + + γ / I s s Γ Γ W P outth µ s where µ s is defined as µ s = Ψ + sψ Ψ s + sψ γ s ] Then the SU signal arameters can be obtained using the aroximate formula 33 in Algorithm II. V. NUMERICAL RESULTS In this section we comare the PU outage robability bounds with the exact ressions. Then we use these bounds to design the SU signal arameters to minimize the SU outage while satisfying the PU QoS. Examle : This examle assumes that the PU is working to achieve R = b/sec with = W. Fig. lots the PU outage robability versus γ based on the exact value comuted from 7 lower bound 9 uer bound and the aroximate ression 4. The SU is assumed to have s = W C x =.5 and = 5 db. is assumed to be comuted from 3. We observe that the bounds is tight

6 Exact Lower bound Uer bound Aroximate = db - Lower bound Uer bound Aroximate Proer Gaussian - Pout =5dB Pouts =db - =db - =5dB γ γ s Fig.. A comarison between the exact PU outage robability lower bound uer bound and aroximate ressions versus γ s for = 5 db. Fig.. SU outage robability for roer and imroer Gaussian signaling versus γ s for = 5 db. to the exact PU outage robability for different ranges of γ and γ. Similar results are observed at different target rates. Examle : To examine the SU imroer Gaussian signal design we first assume the PU ower is adjusted to achieve a rate of b/sec with outage of. considering I max = db. The communication channels are assumed to have γ = db I = 3 db and = db and 5 db. The SU roer Gaussian design adjusts the ower according to 8 without violating the SU ower budget smax = W and the PU QoS requirements. The imroer SU signal design is based on either Algorithm I for lower bound or Algorithm II uer bound and aroximate ression. Fig. shows the SU outage robability versus γ s for different values. For = db all bounds reduces to the roer Gaussian signaling design because the SU interference signal ower is comarable to maximum allowable interference margin at the PU hence roer signaling tends to use the maximum ower. Therefore the imroer Gaussian signaling system cannot increase the imroriety degree because it will be at the cost of increasing the transmitted ower which cannot be achieved. On the other hand when the SU interference channel to the PU is strong the roer signaling design tends to use less ower to meet the PU QoS while imroer signaling design uses more ower to imrove its outage erformance and comensate its effect on the PU by increasing the signal imroriety. Fig. shows a -3 db imrovement of imroer Gaussian signaling over the roer Gaussian signaling design. VI. CONCLUSION In this aer we studied the outage robability of underlay cognitive radio system with imroer Gaussian signaling. We derived closed form ression for the SU outage robability and tight bounds for the PU. Based on the derived ressions and using the average CSI we adjusted the SU ower and circularity coefficient to imrove its erformance measured in terms of the outage robability while satisfying the PU QoS and meeting the SU ower budget. The simulation results show that the benefit of imroer Gaussian signaling system over the roer Gaussian signaling increases as the interference-tonoise ratio of the SU to the PU increases. REFERENCES [] Q. Zhao and B. M. Sadler A survey of dynamic sectrum access IEEE Signal Process. Mag. vol. 4 no May 7. [] Y. Zeng C. M. Yetis E. Gunawan Y. L. Guan and R. Zhang Transmit otimization with imroer Gaussian signaling for interference channels IEEE Trans. Signal Process. vol. 6 no Jun. 3. [3] F. D. Neeser and J. L. Massey Proer comlex random rocesses with alications to information theory IEEE Trans. Inf. Theory vol. 39 no Jul [4] G. Tauböck Comlex-valued random vectors and channels: entroy divergence and caacity IEEE Trans. Inf. Theory vol. 58 no May. [5] C. Lameiro I. Santamaria and P. Schreier Benefits of imroer signaling for underlay cognitive radio IEE Wireless Commun. Lett. vol. 4 no.. 5 Feb. 5. [6] P. J. Schreier and L. L. Scharf Statistical signal rocessing of comlexvalued data: the theory of imroer and noncircular signals. Cambridge University Press. [7] S. Lagen A. Agustin and J. Vidal Imroer Gaussian signaling for the Z-interferece channel in in Proc. IEEE International Conference on Acoustics Seech and Signal Processing ICASSP [8] Y. Zeng R. Zhang E. Gunawan and Y. L. Guan Otimized transmission with imroer Gaussian signaling in the K-user MISO interference channel IEEE Trans. Wireless Commun. vol. no Dec. 3. [9] R. M. Corless G. H. Gonnet D. E. Hare D. J. Jeffrey and D. E. Knuth On the lambertw function Advances in Comutational Mathematics vol. 5 no Dec. 996.