Full-Duplex Spectrum Sharing in Cooperative Single Carrier Systems

We propose cyclic prefix single carrier full-duplex transmission in amplify-and-forward cooperative spectrum sharing networks to achieve multipath diversity and full-duplex spectral efficiency. Integrating full-duplex transmission into cooperative spectrum sharing systems results in two intrinsic problems: 1) the residual loop interference occurs between the transmit and the receive antennas at the secondary relays and 2) the primary users simultaneously suffer interference from the secondary source (SS) and the secondary relays (SRs). Thus, examining the effects of residual loop interference under peak interference power constraint at the primary users and maximum transmit power constraints at the SS and the SRs is a particularly challenging problem in frequency selective fading channels. To do so, we derive and quantitatively compare the lower bounds on the outage probability and the corresponding asymptotic outage probability for max-min relay selection, partial relay selection, and maximum interference relay selection policies in frequency selective fading channels. To facilitate comparison, we provide the corresponding analysis for half-duplex. Our results show two complementary regions, named as the signal-to-noise ratio (SNR) dominant region and the residual loop interference dominant region, where the multipath diversity and spatial diversity can be achievable only in the SNR dominant region, however the diversity gain collapses to zero in the residual loop interference dominant region.

Abstract-We propose cyclic prefix single carrier full-duplex transmission in amplify-and-forward cooperative spectrum sharing networks to achieve multipath diversity and full-duplex spectral efficiency.Integrating full-duplex transmission into cooperative spectrum sharing systems results in two intrinsic problems: 1) the residual loop interference occurs between the transmit and the receive antennas at the secondary relays; and 2) the primary users simultaneously suffer interference from the secondary source (SS) and the secondary relays (SRs).Thus, examining the effects of residual loop interference under peak interference power constraint at the primary users and maximum transmit power constraints at the SS and the SRs is a particularly challenging problem in frequency selective fading channels.To do so, we derive and quantitatively compare the lower bounds on the outage probability and the corresponding asymptotic outage probability for max-min relay selection (MM), partial relay selection (PS), and maximum interference relay selection (MI) policies in frequency selective fading channels.To facilitate comparison, we provide the corresponding analysis for half-duplex.Our results show two complementary regions, named as the signal-to-noise ratio (SNR) dominant region and the residual loop interference dominant region, where the multipath diversity and spatial diversity can be achievable only in the SNR dominant region, however the diversity gain collapses to zero in the residual loop interference dominant region.
Index Terms-Cooperative transmission, cyclic prefix single carrier transmission, frequency selective fading, full-duplex transmission, residual loop interference, spectrum sharing.

I. INTRODUCTION
Cognitive radio (CR) has emerged as a revolutionary approach to ease the spectrum utilization inefficiency [2].In underlay CR networks, the secondary users (SUs) are permitted to access the spectrum of the primary users (PUs), only when the peak interference power constraint at the PUs is satisfied [3].One drawback of this approach is the constrained transmit power at the SU, which typically results in unstable transmission and restricted coverage [4,5].To overcome this challenge, cognitive relaying was proposed as a solution for reliable communication and coverage extension at the secondary network, and interference reduction at the primary network [6][7][8][9][10][11][12].In [6,7], the generalized selection combining is proposed for spectrum sharing cooperative relay networks.In [8], the performance of cognitive relaying with max-min relay selection was evaluated.In [12], the partial relay selection was proposed in underlay CR networks.
Full-duplex transmission has been initiated as a new technology for the future Wireless Local Area Network (WLAN) [13], WiFi network [14], and the Full-Duplex Radios for Local Access (DUPLO) projects, which aims at developing new technology and system solutions for future generations of mobile data networks [15], 3GPP Long-Term Evolution (LTE), and Worldwide Interoperability for Microwave Access (WiMAX) systems [16].Recent advances in radio frequency integrated circuit design and complementary metal oxide semiconductor processing have enabled the suppression of residual loop interference.For example, advanced time-domain interference cancellation [17], physical isolation between antennas [18], and antenna directivity [19] have been proposed in existing works.However, these techniques can not enable perfect isolation [20,21].Thus, the residual loop interference is still inevitable and significantly deteriorates the performance.Recent research and development on full-duplex relaying (FDR) without utilizing residual loop interference mitigation has attracted increasing attention, considering that FDR offers high spectral efficiency compared to half-duplex relaying (HDR) by transmitting and receiving signals simultaneously using the same channel [22][23][24][25][26].In [25], FDR was first applied in underlay cognitive relay networks with single PU, the optimal power allocation is studied to minimize the outage probability.
The main objective of this paper is to consider the fullduplex spectrum sharing cooperative system with limited transmit power in the transmitter over frequency selective fading environment.We can convert the frequency selective fading channels into flat fading channels via Orthogonal Frequency-Division Multiplexing (OFDM) transmission.However, the peak-to-average power ratio (PAPR) is an intrinsic problem in the OFDM-based system.Also, in general, development of the channel equalizer is a big burden to the receiver of single carrier (SC) transmission [27] in the frequency selective fading channels.Thus, to jointly reduce PAPR and channel equalization burden in the practical system, we consider SC with the cyclic prefix (CP).Single carrier (SC) transmission [27] is currently under consideration for IEEE 802.11ad [28] and LTE [29], owing to the fact that SC can provide lower peak-to-average power ratio and power amplifier back-off [30,31] compared to Orthogonal Frequency-Division Multiplexing (OFDM).In addition, by adding the cyclic prefix (CP) to the front of the transmission symbol block, the multipath diversity gain can be obtained [32].
Different from the aforementioned works, we introduce FDR and amplify and forward (AF) relay selection in SC spectrum sharing systems to obtain spatial diversity and spectral efficiency.The full-duplex relaying proposed in this paper is a promising approach to prevent capacity degradation due to additional use of time slots, even though additional design innovations are needed before it is used in operational networks.We consider three relay selection policies, namely max-min relay selection (MM), partial relay selection (PS), and maximum interference relay selection (MI), each with a different channel state information (CSI) requirement.We consider a realistic scenario where transmissions from the secondary source (SS) and the selected secondary relay (SR) are conducted simultaneously in the presence of multiple PU receivers.Unlike the cognitive half-duplex relay network (CogHRN), in the cognitive full-duplex relay network (CogFRN) the concurrent reception and transmission entails two intrinsic problems: 1) the peak interference power constraint at the PUs are concurrently inflicted on the transmit power at the SS and the SRs; and 2) the residual loop interference due to signal leakage is introduced between the transmit and the receive antennas at each SR.Against this background, the preeminent objective of this paper is to characterize the feasibility of fullduplex relaying in the presence of residual loop interference by comparing with half-duplex systems.The impact of frequency selectivity in fading channels is another important dimension far from trivial.For purpose of comparison, we provide the corresponding analysis for cooperative CP-SC CogHRN.
Our main contributions are summarized as follows.
1) Taking into account the residual loop interference, we derive new expressions for the probability density function (PDF) and the cumulative distribution function (CDF) of the signal-to-noise ratio (SNR) of the SS to the kth SR link under frequency selective fading channels.2) We then derive the expressions for the lower bound on the outage probability.We establish that outage probability floors occur in the residual loop interference dominant region with high SNRs for all the policies in CogFDR.We show that irrespective of the SNR, the MM policy outperforms the PS and the MI policies.We also show that the PS policy outperforms the MI policy.3) To understand the impact of the system parameters, we derive the asymptotic outage probability and characterize the diversity gain.For FDR, in the residual loop interference dominant region, we see that the asymptotic diversity gain is zero regardless of the spatial diversity might be offered by the relay selection policy, and the multipath diversity might be offered by the single carrier system.However, the full diversity gain of HDR is achievable.
4) We verify our new expressions for lower bound on the outage probabilities and their corresponding asymptotic diversity gains via simulations.We showcase the impact of the number of SRs and the number of PUs on the outage probability.We conclude that the outage probability of CogFDR decreases with increasing number of SRs, and increases with increasing the number of PUs.Interestingly, we notice that the outage probability of CogFDR decreases as the ratio of the maximum transmit power constraint at the SR to the maximum transmit power at the SS decreases.Notations: The superscript (•) H denotes complex conjugate transposition, E{•} denotes expectation, and CN µ, σ 2 denotes the complex Gaussian distribution with mean µ and variance σ 2 .The F ϕ (•) and F ϕ (•) denote the CDF of the random variable (RV) ϕ for FDR and HDR, respectively.Also, f ϕ (•) and f ϕ (•) denote the PDF of ϕ for FDR and HDR, respectively.The binomial coefficient is denoted by

II. SYSTEM AND CHANNEL MODEL
We consider a cooperative spectrum sharing network consisting of L PU-receivers (PU 1 , . . ., PU L ), a single SS, a single secondary destination (SD), and a cluster of K SRs (SR 1 , . . ., SR K ) as shown in Fig. 1,where the solid and the dashed lines represent the secondary channel and the interference channel, respectively.The CP-SC transmission is used in this network.Among the K SRs, the best SR which fulfills the relay selection criterion is selected to forward the transmission to the SD using the AF relaying protocol.Similar to the model used in [8], [33], and [34], we focus on the coexistence of long-range primary system such as IEEE 802.22, and short range CR networks, such as WLANs, D2D networks and sensor networks.In this case, the primary to secondary link is severely attenuated to neglect the interference from the PU transmitters to the SU receivers.We also assume there is no direct link between the SS and the SRs due to long distance and deep fades.In this network, we make the following assumptions for the channel models, which are practically valid in cooperative spectrum sharing networks.
Assumption 1.For the secondary channel, the instantaneous sets of channel impulse responses (CIRs) from the SS to the kth SR and from the kth SR to the SD composing of N 1,k and N 2,k multipath channels, are denoted as , respectively1 .For the primary channel, we assume perfect CSI from the SS to the lth PU link and from the kth SR to the lth PU link, which can be obtained through direct feedback from the PU [35], indirect feedback from a third party, and periodic sensing of pilot signal from the PU [36].The instantaneous sets of CIRs from the SS to the lth PU (PU l ) and from the kth SR to the lth PU l composing of N 3,l and N 4,k,l multipath channels, are denoted as f s,l N 3,l = f s,l 0 , . . ., f s,l N 3,l −1 All channels are composed of independent and identically distributed (i.i.d.) complex Gaussian RVs with zero means and unit variances.The maximum channel length } is assumed to be shorter than the CP length, denoted by N CP , to restrain the interblock symbol interference (IBSI) and intersymbol interference (ISI) in single carrier transmission [31].Accordingly, the path loss components from the SS to the kth SR, from the kth SR to the SD, from the SS to the PU l , and from the kth SR to the PU l are defined as α 1,k , α 2,k , α 3,l , and α 4,k,l , respectively.
Assumption 2. For underlay spectrum sharing, the peak interference power constraint at the lth PU is denoted as I th .Also due to hardware limitations, the transmit power at the SS and the SRs are restricted by the maximum transmit power constraints P T and P R , respectively.

A. CogFRN
In the full-duplex mode, each SR is equipped with a single transmit and a single receive antenna, which enable full-duplex transmission in the same frequency band at the expense of introducing residual loop interference.The SS and the SR transmit to the SD in the same time slot.As such, the PUs suffer interference from the SS and the SRs concurrently.Similar as [25], we simply assume that the maximum interference inflicted on the PUs by the SS or the SRs are set to be a half of the total peak interference power constraint at the PUs ( 1 2 I th = Q), where Q is the peak interference constraint2 .Therefore, the transmit power at the SS and the kth SR are given by where and Note that although the peak interference power constraint demands a higher feedback overhead than the average interference power constraint, it is an excellent fit to real-time systems.Let x s ∈ C Ns×1 denote the transmit block symbol after applying digital modulation.We assume that E{x s } = 0 and E{x s x H s } = I Ns .After appending the CP with N CP symbols at the beginning of x s , the augmented transmit block symbol is transmitted over the frequency selective channels {g s,k N 1,k }.After the removal of the CP-related received signal part, the received signal at the kth SR is given by where G s,k N 1,k is the right circulant matrix determined by the channel vector The residual loop interference channel is denoted as , which is a diagonal channel matrix between the transmit and receive antennas at the kth SR.Due to the existence of many weak multipath components, the overall residual loop interference channel power gain is presumed to follow exponential distribution based on the central limit theorem.In (5), x r,k denotes the residual block symbol.Note that {x r,k } K k=1 have the same statistical properties as those of x s .It is assumed that the thermal noise received at the kth relay is modeled as a complex Gaussian random variable with zero mean and variance σ 2 n , i.e., n s,k ∼ CN (0, σ 2 n I Ns ).In AF relaying, the SRs are unable to distinguish between the signal from the SS and the residual loop interference signals at the SRs.Thus, both signals are amplified and forwarded to the SD.The received signal at the SD via the kth SR is given by where G k,d N 2,k is the right circulant matrix formed by Ns is the relay gain matrix for the kth SR, and n r,d ∼ CN (0, σ 2 n I Ns ) 3 .The relay gain g F k is given by where h k = {h k,n } Ns n=1 .Inserting ( 5) and ( 7) into ( 6), the end-to-end SINR (e2e-SINR) at the SD is derived as where , and

B. CogHRN
In the half-duplex mode, the SS and the SRs transmit signals in different channels and time slots.The maximum interference imposed on the PUs by the SS or the SR is equal to the peak interference power constraint (I th = 2Q) at the PUs.As such, the transmit power at the SS and the kth SR in CogHRN are given by respectively.With AF relaying, the received signals at the kth SR and at the SD via the kth SR are given by respectively, where G k = g H k I Ns is the relay gain matrix for the kth SR, and . Therefore, the corresponding e2e-SINR of CogHRN at the SD is given by where the SNR from the SS to the kth SR is denoted as and the SNR from the kth SR

III. DISTRIBUTIONS OF SNR AND SINR
In this section, we first derive the CDFs and PDFs of the Y 1 and Y k based on the Definition 1 and Definition 2 in the following.We then utilize these CDFs and PDFs to facilitate the derivations of CDFs of γ s,k F , γ s,k H , and γ k,d H . Definition 1.The PDF and the CDF of a RV X distributed as a gamma distribution with shape N and scale α are given, respectively, as and where U(•) denotes the discrete unit step function.In the sequel, a RV X distributed according to a gamma distribution with shape N and scale α is denoted by X ∼ Ga(N, α).Here, shape N is positive integer.
, then the CDF and the PDF of a RV X max = max{a 1 X 1 , a 2 X 2 , . . ., a L X L } are given, respectively, as and where L,jt,{Ni},{ai} Note that the magnitudes of the four channel vectors 2 are distributed as gamma distributions with shapes N 1,k , N 2,k , N 3,l , and N 4,k,l , respectively, and scale 1.Also, |h k | 2 is distributed as a gamma distribution with shape 1 and scale 1.We have also defined the two RVs andf where j = l t=1 j t and β1 = l t=1 1 α3,n t .

A. CogFRN
From the definition of the SNR from the SS to the kth SR γ s,k F = min(Q/Y 1 , P T )X k γ, we have the following CDF of γ s,k F as where µ T = Q P T and Γ(•, •) denotes the incomplete gamma function.
Proof.See Appendix A.

B. CogHRN
In cooperative CP-SC CogHRN, we have γ s,k H = min(2Q/Y 1 , P T )X k γ.We derive the CDF of γ s,k H as IV. ASYMPTOTIC DESCRIPTION In this section, we assume To examine the effect of power scaling on the outage probability, we have also defined ρ = P R P T .When γT → ∞, we can easily observe γR → ∞ and γQ → ∞.This will benefit the secondary network without violating the transmission of the primary network [8].

A. CogFRN
To derive the asymptotic results, ( 8) is simplified to one term for high SNRs.Since the second order term is dominating compared with the linear terms i.e., E γ k,d , at high SNRs, we can obtain an approximate e2e-SINR expression as We see that the high e2e-SINR is only determined by the first hop and residual loop interference, and is independent of the second hop.By eliminating γT in ( 23 2) Asymptotic INR at the kth SR: From the definition of The derivation of ( 24) and ( 25) are similar to those provided in Appendix A.

B. CogHRN
Different from the approach used in deriving the asymptotic e2e-SINR of CogFRN, in CogHRN, we use the first order expansion for the CDFs of γ s,k H and γ k,d H to derive the asymptotic e2e-SNR of CogHRN.
1) Asymptotic SNR from the SS to the kth SR: When γT → ∞ and γQ → ∞, an asymptotic expression of F X k (γ/γ T ) is derived by applying [38, eq. (1.211.1)] and [38, eq. (3.354 The asymptotic CDF of γ s,k H is derived as 2) Asymptotic SNR from the kth SR to the SD: When γR → ∞ and γQ → ∞, the asymptotic CDF of γ k,d H is derived as Having ( 27) and ( 28) for the CDFs of γ s,k H and γ k,d H in closed-form, respectively, we derive the lower bound on the outage probability of CogHRN in Section VI.

V. OUTAGE PROBABILITY OF COGFRN
In this section, we derive the expression for the lower bound on the outage probabilities of CogFRN with various relay selection policies based on the max-min criterion, partial relay selection criterion, and maximum interference criterion.We then derive the corresponding asymptotic outage probabilities to observe the diversity gains of the three selection policies.

A. CogFRN with MM
Compared with the conventional MM policy in CogHRN, the MM policy in CogFRN takes into account the loop interference.Let k MM be the selected relay based on the maxmin criterion.The employed relay selection is mathematically given by 1) Outage Probability: The lower bound on the outage probability of CogHRN at a given threshold η F is given by Theorem 1.The lower bound on the outage probability of CogFRN with MM policy is derived as where Proof.See Appendix B.
Note that our derived outage probability with the MM policy is valid for different types of SRs and PUs having arbitrary channel lengths and path loss components.
2) Asymptotic Outage Probability: Based on (23), the asymptotic outage probability can be written as Having ( 24) and ( 25), we derive the asymptotic CDF of L,jt,{N3},{α3} where the two terms R 1 and R 2 are derived in Appendx C.
Substituting the derived closed-form expression of F ∞ γ in (36) at a given η F into (35), we obtain the asymptotic outage probability with MM policy.Since P ∞,out MM (η F ) is independent of γT , γR , and γQ (as shown in ( 24) and ( 25) which are independent of γQ , γT and γR ), the diversity gain collapes to zero regardless of the spatial diversity and multipath diversity in the high SNR regime.

B. CogFRN with PS
In this policy, partial CSI is required, the SR which has the maximum SNR from the SS to the kth SR is selected.Thus, the index of the selected relay is denoted as To see the diversity gain of the outage probability, in the rest of this section we have assumed that As such, we have the same distribution for each SR to the SD link, that is,

1) Outage Probability:
The lower bound on the outage probability is evaluated as where .
Proof.See Appendix D.
2) Asymptotic Outage Probability: The asymptotic outage probability with PS policy is given as Having ( 24) and ( 25), we derive the asymptotic outage probability.The asymptotic diversity gain with PS policy is zero.

C. CogFRN with MI
In the MI policy, the SR resulting in the maximum interference on the PU is selected in order to achieve the minimum loop interference, thus the index of the selected relay is given as (41)

1) Outage Probability:
Theorem 3. The lower bound on the outage probability of CogFRN with MI policy is derived as (42) at the top of next page.
Proof.See Appendix E.
2) Asymptotic Outage Probability: In the high SNR regime, the e2e-SINR expression of CogFRN with the MI policy becomes where With the derived CDF of γ s,k p in ( 24) and the PDF of γ k M I ,I p as and we substitute them into we derive the asymptotic outage probability with MI policy.In CogFRN, the diversity gain of the MI policy is identical to those of the MM and PS policies.

VI. OUTAGE PROBABILITY OF COGHRN
In this section, we present the lower bound on the exact and asymptotic outage probabilities of CogHRN with the MM policy and the PS policy.

A. CogHRN with MM
In this policy, a relay with the maximum e2e-SNR is selected based on the CSI from the SS to the kth SR link and from the kth SR to the SD link .Thus, the index of the selected relay is denoted as Based on (46), the lower bound on the outage probability at a given η H is written as Substituting ( 21) and ( 22) into (47), we can easily derive the lower bound on the outage probability of CogHRN with the MM policy, which is applicable to different types of SRs and PUs having arbitrary channel lengths and pass loss components.
Lemma 1.For the proportional interference case, the asymptotic diversity gain of CogHRN with the MM policy is K min(N 1 , N 2 ).
Proof: As γQ → ∞, it can be seen that In (48), , where Therefore, this policy provides K min(N 1 , K 2 ) diversity gain.

B. CogHRN with PS
In this policy, the relay with the maximum SNR from the SS to the kth SR is selected.The corresponding relay index is given by Here, we have assumed The lower bound on the outage probability is evaluated as Substituting ( 21) and ( 22) into (51), we can easily derive the lower bound on the outage probability of CogHRN with the PS policy.
Lemma 2. The diversity gain with the PS policy is min(KN 1 , N 2 ) as γQ → ∞.
Proof: Based on (27) and (28), we can easily see that Thus, the diversity gain is min(KN 1 , N 2 ).
We can readily see that the number of PUs has no effect on the diversity gain with the MM and the PS policies.
Table I highlights the required CSI for the three relay selection strategies of CogFDR and CogHDR.

VII. SIMULATION RESULTS
In this section, we present numerical results to verify our new analytical results for three different relay selection policies in cooperative CP-SC spectrum sharing systems with the link level simulation.We assume the symbol block size as N s = 512 and CP length as N CP = 16.For the purpose of comparison, we set the target data rate as R T = 1 bit/s/Hz, thus the fixed SNR threshold for CogFRN is denoted as η F = 2 R T − 1.However, in CogHRN, two different channels are needed for CP-SC transmission.We assume that both the SS and the SRs use half of the resource, therefore a fixed Fig. 2 shows the outage probability of CogFRN for various numbers of relays and different relay selection policies.The exact plots with MM, PS, and MI relay selection policies are numerically evaluated using (31), (39), and (42).The asymptotic outage probabilities are plotted from (35), (40), and (45).First, we observe error floors in the high SNR with zero outage diversiy gain, which is due to the dominant effects of the residual loop interference.Second, for the same number of relays, for example K = 6, relay selection policy MM outperforms PS, and PS outperforms MI over all SNR values.The outage probabilities with MM policy and PS policy improve with increasing the number of SRs, while the outage probability with MI policy is not significantly improved by deploying more SRs.Interestingly, the performance gaps between each selection policy increase as the number of SRs increases.
In Fig. 3, we examine the outage probability of CogFRN for various numbers of PUs and different relay selection policies.It is easy to note that increasing the number of PUs deteriorates the outage performance of CogFRN since the secondary network has less chance to share the spectrum of the primary network when the number of PUs is large.
In Fig. 4, we compare the outage probability of CogFRN and CogHRN at the same target data rate under different relay selection policies.Interestingly, we notice that: 1) Compared with CogHRN, CogFRN sacrifice the outage probability to achieve the potential higher spectral efficiency; and 2)  CogHRN overcomes the outage floors of CogFRN in the high SNRs.This is due to the fact that the dominating effect of residual loop interference is removed in CogHRN.In Fig. 5, we examine the impact of the ratio between the peak interference power constraint at the PU and the maximum transmit power constraint at the SS (Q/P T ) on the outage performance of CogFRN with the MM relay selection policy.We see that the outage probability for the same relay selection policy improves with a more relaxed peak interference power constraint at the PU.The higher ratio between the peak interference power constraint at the PU and the maximum transmit power constraint at the SS, the lower error floors and the bigger gaps among these three policies can be achieved.It is readily observed that the diversity gain is zero regardless of µ T in the high SNR regime.Fig. 6 shows the outage probability with FDR and HDR as a function of ρ, which is the ratio between γR and γT .For the same relay transmission mode and the same relay selection policy, the parallel slopes illustrate that the diversity gain is unrelated to ρ.Interestingly, we observe that as ρ increases, a better outage performance is achieved in CogHRN, while a worse outage performance in CogFRN, and the crossover point between full-duplex and half-duplex moves to the left.This is due to the fact that with ρ increases, γR increases, which results in the enhancement of the second hop transmission in CogHRN.However, due to increased residual loop interference with increasing ρ, the adverse effect of the residual loop interference grows with increasing the transmit power of SR.
In Fig. 7, we examine the outage probability with FDR with various relay selection policies and ρ.Similar phenomenon in CogFRN is observed as Fig. 6.As ρ decreases, the outage probability with the PS policy and the MI policy degrade.This is because the residual loop interference is a detrimental characteristic of FDR, which is shown in ( 29), (37), and (41).We define γT < 12 dB as the SNR dominant region, and γT > 25 dB as the residual loop interference dominant region.
In the diversity achievable SNR dominant region, we observe that the outage proability decreases as increasing γT .In the residual loop interference dominant region, we observe the zero diversity gain, which restricted the decreasing trend of outage probability.

VIII. CONCLUSIONS
We have examined the effects of residual loop interference in cooperative CP-SC spectrum sharing with FDR.The lower bound on the outage probabilities and asymptotic outage probabilities for the MM policy requiring global CSI, as well as the PS and the MI policies requiring partial CSI have been derived and quantitatively compared.Interestingly, we observe that the diversity gain results from spatial diversity and multipath diversity can be achieved in the SNR dominant region, whereas the diversity gain lost in the residual loop interference dominant region.For comparison purposes, the lower bound on the outage probabilities and the corresponding asymptotic outage probabilities of cooperative CP-SC spectrum sharing with HDR have been derived for each of the relay selection policies.Our results show that CogFDR is a good solution to achieve the spectral efficiency and bearable outage probability for the systems that operate at low to medium SNRs, while CogHDR is more favorable to those operate in the high SNRs.APPENDIX A: DETAILED DERIVATION OF (20) We start from the definition of the CDF of γ s,k F , which is given by We use the integration by parts to solve I 1 of (A.1), which is given by Then using [38, Eq. 3.351.2]and the PDF of X k , the closedform expression for the CDF of γ s,k F can be derived as (20).
APPENDIX B: DETAILED DERIVATION OF (31) Based on (30), the outage probability with MM policy is given as APPENDIX C: DETAILED DERIVATION OF (36) Similar as the analysis in Appendix B, the first term R 1 is evaluated as

N3 2 } 2 n.
. For notational purposes, in the sequel, we have defined the normalized powers γQ = Qγ, γT = P T γ, and γR = P R γ, with γ = 1 σ According to the distribution of f s,l N3 2 , the CDF and the PDF of Y 1 are given by ), we derive the new expressions γ s,k p = min µ T Y1 , 1 X k , and γ k,I p = min µ T Y k , ρ R k .To derive the closed-form expression for γ k F e2ep , we first derive the closed-form expressions for γ s,k p and γ k,I p . 1) Asymptotic SNR from the SS to the kth SR: From the definition of γ s,k p = min µ T Y1 , 1 X k , we have the following asymptotic CDF of γ s,k p as

TABLE I REQUIRED
CSI FOR THE RELAY SELECTION IN COGFDR AND COGHDR