Hybrid spectrum access with relay assisting both primary and secondary networks under imperfect spectrum sensing
 Thi My Chinh Chu^{1}Email authorView ORCID ID profile,
 HansJürgen Zepernick^{1} and
 Hoc Phan^{2}
https://doi.org/10.1186/s1363801607007
© The Author(s) 2016
Received: 2 February 2016
Accepted: 17 August 2016
Published: 12 October 2016
Abstract
This paper proposes a novel hybrid interweaveunderlay spectrum access for a cognitive amplifyandforward relay network where the relay forwards the signals of both the primary and secondary networks. In particular, the secondary network (SN) opportunistically operates in interweave spectrum access mode when the primary network (PN) is sensed to be inactive and switches to underlay spectrum access mode if the SN detects that the PN is active. A continuoustime Markov chain approach is utilized to model the state transitions of the system. This enables us to obtain the probability of each state in the Markov chain. Based on these probabilities and taking into account the impact of imperfect spectrum sensing of the SN, the probability of each operation mode of the hybrid scheme is obtained. To assess the performance of the PN and SN, we derive analytical expressions for the outage probability, outage capacity, and symbol error rate over Nakagamim fading channels. Furthermore, we present comparisons between the performance of underlay cognitive cooperative radio networks (CCRNs) and the performance of the considered hybrid interweaveunderlay CCRN in order to reveal the advantages of the proposed hybrid spectrum access scheme. Eventually, with the assistance of the secondary relay, performance improvements for the PN are illustrated by means of selected numerical results.
Keywords
Cognitive radio network Continuoustime Markov chain Hybrid spectrum access Amplifyandforward Relay networks Nakagamim fading1 Introduction
In recent years, the growing demand of mobile multimedia services has led to a serious shortage of radio frequency spectrum. However, measurement campaigns have shown that most of the allocated spectrum bands are underutilized [1]. To alleviate the severe scarcity of spectrum resources, a promising technique, called cognitive radio, has been developed. This technology allows secondary users (SUs) to access the licensed spectrum of the primary users (PUs) by utilizing interweave, underlay, or overlay spectrum access techniques [2–4]. In particular, in the interweave spectrum access, the SU periodically monitors the radio spectrum to detect occupancy in the different parts of the spectrum and then opportunistically utilizes the spectrum holes for its communication [4]. A drawback of this scheme is the limitation on access time which leads to a reduction in the effectiveness of spectrum utilization compared to other spectrum access schemes. However, an interweave cognitive radio network (CRN) may achieve high performance since the transmit power of the secondary transmitter is not bounded by the interference power constraint of the primary receiver. In the overlay spectrum access, the SU is only allowed to access frequency bands if the secondary network (SN) uses interference cancelation techniques to mitigate interference to the primary network (PN) [2]. In the underlay spectrum access, the PU and SU can concurrently operate in the same licensed spectrum, but the secondary transmitter must adjust its transmit power to meet the interference power constraint of the primary receiver [3]. Due to this interference power constraint, the SUs have to reduce their transmit powers which degrades the performance of the SN. To reveal the performance of CRNs using different spectrum access schemes, comparisons of the outage probability for CRNs using interweave, underlay, and overlay techniques were presented in [5].
Inspired by the inherent benefits of the overlay and underlay spectrum access, hybrid overlayunderlay CRNs have been proposed in [6, 7]. Specifically, Oh and Choi [6] derived the optimal switching rate between spectrum access modes in order to maximize throughput of a hybrid overlayunderlay CRN. A power allocation strategy for a hybrid overlayunderlay CRN was proposed in [7] to maximize the channel capacity of the CRN. The works of [8, 9] focused on incorporating cooperative communications into hybrid overlayunderlay CRNs to benefit from both the cognitive radio and cooperative communication techniques such as efficient spectrum utilization and improved link reliability, respectively. In [10], a novel hybrid scheme which combines the interweave and underlay schemes for a single hop CRN was introduced. Specifically, an M/M/1 queuing model with Poisson traffic arrival is utilized in [10] to analyze the average service rate of video services for a hybrid interweaveunderlay CRN. In [11], the spectrum access of a hybrid interweaveunderlay cognitive cooperative radio network (CCRN) was modeled as a continuoustime Markov chain (CTMC) to obtain the steady state probabilities of the interweave and underlay modes. Nevertheless, the relay in [11] only forwards the signals of the secondary network but does not support the primary network. Recently, in [12], an amplifyandforward (AF) relay has been employed to forward the signals of both the primary and secondary networks which provides a performance gain for the PN as well as an improved spectrum utilization. However, the SUs of the CCRN in [12] utilize the underlay spectrum access under the interference constraint of the PUs all the time regardless of the activity of the PUs. To the best of our knowledge, no work has deployed hybrid interweaveunderlay spectrum access for CCRNs where the relay forwards the signals of the PNs and SNs in which the effect of imperfect spectrum sensing is considered.
In this paper, we examine a hybrid interweaveunderlay spectrum access scheme for CCRNs. The motivation for this kind of hybrid spectrum access scheme is to inherent benefits of both the underlay and interweave spectrum access schemes. Specifically, the CCRN basically operates in the underlay spectrum access to obtain high spectrum efficiency, i.e., if the CCRN senses that it is concurrently active with the PN, the SUs must operate in the underlay spectrum access. In this operation mode, the SUs must control their transmit powers to satisfy the interference power constraint. However, if the PN is sensed to be inactive, the SUs opportunistically operate in the interweave mode without facing the interference power constraint imposed by the primary receiver to improve the system performance. In order to benefit from the advantages of relaying communication, in our proposed scheme, the secondary relay forwards the signals of the PN and CCRN. This is different from overlay CCRNs where the SUs have knowledge of the codebook of the PU and utilize this information to decode the message of primary users [2]. In our system, the secondary relay simply amplifies the signals of the PN and CCRN and broadcasts the resulting signals. It is noted that, in our study, the characteristics of traffic patterns of the PUs and SUs are taken into account when modeling the spectrum access. Further, the impact of imperfect spectrum sensing of the SUs are also considered when analyzing the system performance of the PN and SN. It is assumed that the PN and SN are subject to Nakagamim fading. The Nakagamim fading model is used here as it comprises a wide variety of fading channels as special cases by setting the fading severity parameter m to particular values. For example, m=0.5 represents onesided Gaussian fading and m=1 represents Rayleigh fading. The Nakagamim fading model also closely approximates the Nakagamiq (Hoyt) and the Nakagamin (Rice) models.

We develop a CTMC to model the spectrum access of a hybrid interweaveunderlay CCRN wherein the secondary AF relay assists primary and secondary communications.

On this basis, the equilibrium probability of each operation mode of the hybrid interweaveunderlay CCRN over Nakagamim fading is derived.

The effect of imperfect spectrum sensing in terms of false alarm and missed detection of the CCRN on the system performance is also investigated.

We further develop an analytical framework to assess the performance of the secondary and primary networks in terms of outage probability, outage capacity, and symbol error rate.

We also make performance comparisons between the proposed hybrid CCRN and the conventional underlay CCRN to reveal the superior performance of the hybrid CCRN.

Finally, through the provided numerical results, essential insights into the impact of network parameters on system performance are revealed.
The rest of this paper is organized as follows. Section 2 describes the system model, constructs the CTMC, and calculates the probabilities of operation modes in the PN and SN. Sections 3, 4, and 5 derive the expressions for outage probability, outage capacity, and symbol error rate (SER) for the PN and SN, respectively. Selected numerical results are provided in Section 6. Finally, conclusions are given in Section 7.
2 System and channel model
Let the arrival traffics of PU_{TX} and SU_{TX} be modeled as Poisson processes with arrival rates λ _{ p } and λ _{ s }, respectively. Further, the departure traffics of PU_{TX} and SU_{TX} are also Poisson processes with departure rates μ _{ p } and μ _{ s }, respectively. Since relaying transmission is deployed, departure traffics of PU_{TX} and SU_{TX} become arrival traffics of SU_{R} with rates μ _{ p } and μ _{ s }, respectively. Finally, the departure traffics at SU_{R} resulting from the arrival traffics of PU_{TX} and SU_{TX} are also Poisson processes with rates \({\mu ^{p}_{r}}\) and \({\mu ^{s}_{r}}\), respectively. As a result, when the PN and SN are concurrently active, the arrival of the system is λ _{p}+λ _{s}. Further, the arrival and departure rates of SU_{R} are μ _{p}+μ _{s} and μrs+μrp, respectively.
When considering the impact of imperfect spectrum sensing at the SUs, the missed detection and false alarm probabilities must be taken into account. The missed detection probability, p _{ m }, is the probability that the SUs consider the licensed spectrum as being vacant although it is occupied by the PU. False alarm probability, p _{ f }, is the probability that the SUs consider the licensed spectrum as occupied by the PU even though the PU is inactive. Then, the detection probability and no false alarm probability can be calculated from the missed detection and false alarm probabilities as (1−p _{ m }) and (1−p _{ f }), respectively. Taking into account imperfect spectrum sensing, the following scenarios of the PN and CCRN can occur depending on the states of the terminals in the networks.
2.1 Scenario 1: The secondary network operates in underlay mode

Case 1: The PN and SN are active and the SN correctly senses the active state of the PN. The probability of this event is$$\begin{array}{*{20}l} p_{p}^{(1,1)}=p_{s}^{(1,1)}=(1p_{m}) P_{PS}P_{R} \end{array} $$(9)The communication in Case 1 is described as follows. In the first time slot, PU_{TX} and SU_{TX} broadcast their signals to the relay and to their respective receivers, PU_{RX} and SU_{RX}. Since the PN and SN concurrently share the same spectrum, SU_{TX} must control its transmit power subject to the interference power threshold Q. As a result, the received signal \(y_{1p}^{(1,1)}\) at PU_{RX}, the received signal \(y_{1s}^{(1,1)}\) at SU_{RX}, and the received signal \(y_{r}^{(1,1)}\) at SU_{R} in the first time slot are, respectively, given by$$\begin{array}{*{20}l} y_{1p}^{(1,1)}= h_{3} x_{p} + h_{7} x_{s}^{(1)} + n_{p} \end{array} $$(10)$$\begin{array}{*{20}l} y_{1s}^{(1,1)}=h_{6} x_{s}^{(1)} + h_{8} x_{p} + n_{s} \end{array} $$(11)$$\begin{array}{*{20}l} y_{r}^{(1,1)}=h_{1} x_{p} + h_{4} x_{s}^{(1)}+n_{r} \end{array} $$(12)
where x _{ p } is the transmit signal at PU_{TX} with average power P _{ p }. Further, n _{ p },n _{ s }, and n _{ r } denote the additive white Gaussian noise (AWGN) with zero mean and variance N _{0} at PU_{RX},SU_{RX}, and SU_{R}, respectively. As for the received signal at the relay, the term \(h_{4} x_{s}^{(1)}\) in (12) can be considered as interference from the SN to the PN and the term h _{1} x _{ p } can be considered as the interference from the PN to the SN. Usually, these terms are much larger than the noise term n _{ r }, such that n _{ r } in (12) can be neglected as suggested in [15].
In the second time slot, SU_{R} amplifies the received signal \(y_{r}^{(1,1)}\) with the gain G _{1} and then forwards the primary and secondary signals. Consequently, the received signal \(y_{2p}^{(1,1)}\) at PU_{RX} and the received signal \(y_{2s}^{(1,1)}\) at SU_{RX} in the second time slot are, respectively, given by$$\begin{array}{*{20}l} y_{2p}^{(1,1)}=G_{1} h_{1} h_{2} x_{p} +G_{1} h_{2} h_{4} x_{s}^{(1)}+n_{p} \end{array} $$(13)$$\begin{array}{*{20}l} y_{2s}^{(1,1)}=G_{1} h_{1} h_{5} x_{p} + G_{1} h_{4} h_{5} x_{s}^{(1)}+n_{s} \end{array} $$(14)Assume that PU_{RX} and SU_{RX} deploy selection combining (SC) to select the best signal among the direct link and relaying link. From (10) and (13), the instantaneous signaltointerferenceplusnoise ratio (SINR) of the PN is obtained as$$\begin{array}{*{20}l} \gamma_{p}^{(1,1)}=\max \left[\gamma_{1p}^{(1,1)}, \gamma_{2p}^{(1,1)}\right] \end{array} $$(15)where \(\gamma _{1p}^{(1,1)}\) and \(\gamma _{2p}^{(1,1)}\) are, respectively, the instantaneous SINRs of the direct link and relaying link of the PN, i.e.,$$\begin{array}{*{20}l} \gamma_{1p}^{(1,1)}&=X_{3}/(\beta_{1}+\beta_{2}) \end{array} $$(16)$$\begin{array}{*{20}l} \gamma_{2p}^{(1,1)}&=X_{1} X_{7}/[(\beta_{1}+\beta_{2})X_{4}] \end{array} $$(17)Here, β _{1} = Q/P _{ p }, β _{2} = N _{0}/P _{ p }, and X _{ l } = h _{ l }^{2}, l=1,2,…,8. From (11) and (14), the instantaneous SINR of the direct link \(\gamma _{1s}^{(1,1)}\) and relaying link \(\gamma _{2s}^{(1,1)}\) of the SN are, respectively, given by$$\begin{array}{*{20}l} \gamma_{1s}^{(1,1)}&=\beta_{1}X_{6}/[X_{7}X_{8}+\beta_{2}X_{7}] \end{array} $$(18)$$\begin{array}{*{20}l} \gamma_{2s}^{(1,1)}&=\beta_{1} X_{4}X_{5}/[X_{1}X_{5}X_{7}+\beta_{2}X_{2}X_{4}] \end{array} $$(19)Thus, the instantaneous SINR of the SN is obtained as$$\begin{array}{*{20}l} \gamma_{s}^{(1,1)}&=\max \left[\gamma_{1s}^{(1,1)},\gamma_{2s}^{(1,1)}\right] \end{array} $$(20) 
Case 2: The PN is inactive but the SN senses that the PN is active. The probability of this event is$$\begin{array}{*{20}l} p_{s}^{(1,2)}=p_{f} P_{S} P_{R^{S}} \end{array} $$(21)Though the PN is inactive, due to the false alarm, SU_{TX} senses the PN as active and still controls its transmit power \(P_{s}^{(1)}\) subject to Q. As a result, the received signal \(y_{1s}^{(1,2)}\) at SU_{RX}, and the received signal \(y_{r}^{(1,2)}\) at SU_{R} in the first time slot are, respectively, given by$$\begin{array}{*{20}l} y_{r}^{(1,2)}= h_{4}x_{s}^{(1)}+n_{r} \end{array} $$(22)$$\begin{array}{*{20}l} y_{1s}^{(1,2)}=h_{6}x_{s}^{(1)}+n_{s} \end{array} $$(23)In the second time slot, SU_{R} amplifies the received signal with the gain G _{1} and then forwards the resulting signal. Hence, the received signal at SU_{RX} in the second time slot is given by$$\begin{array}{*{20}l} y_{2s}^{(1,2)}=G_{1} h_{5}h_{4}x_{s}^{(1)}+G_{1}h_{5}\text{ }n_{r}+n_{s} \end{array} $$(24)Considering the instantaneous signaltonoise ratio as a particular case of SINR where no interference exists, the SINR of the CCRN is obtained from (23) and (24) as$$\begin{array}{*{20}l} \gamma_{s}^{(1,2)}&=\max \left[\gamma_{1s}^{(1,2)},\gamma_{2s}^{(1,2)}\right] \end{array} $$(25)where \(\gamma _{1s}^{(1,2)}\) and \(\gamma _{2s}^{(1,2)}\) are, respectively, the instantaneous SINRs of the direct link and the relaying link of the SN, i.e.,$$\begin{array}{*{20}l} &\gamma_{1s}^{(1,2)}=\beta_{1}X_{6}/(\beta_{2}X_{7}) \end{array} $$(26)$$\begin{array}{*{20}l} &\gamma_{2s}^{(1,2)}=\beta_{1}X_{4}X_{5}/[\beta_{2}(X_{5}X_{7}+X_{2}X_{4})] \end{array} $$(27)
2.2 Scenario 2: The secondary network operates in interweave mode

Case 1: Only the SN is active and the inactive state of the PN is correctly sensed. The probability of this event is$$\begin{array}{*{20}l} p_{s}^{(2,1)}=(1p_{f})P_{S}P_{R^{S}} \end{array} $$(29)Since the SUs sense that the PN is inactive, the SN operates in interweave mode without suffering from the interference power constraint of PU_{RX}. Furthermore, there is no interference incurred by the PN to the SN. As a result, the received signal \(y_{r}^{(2,1)}\) at SU_{R} and the received signal \(y_{1s}^{(2,1)}\) at SU_{RX} in the first time slot are, respectively, given by$$\begin{array}{*{20}l} &y_{r}^{(2,1)}=h_{4} x_{s}^{(2)}+n_{r} \end{array} $$(30)$$\begin{array}{*{20}l} &y_{1s}^{(2,1)}=h_{6} x_{s}^{(2)}+n_{s} \end{array} $$(31)In the second time slot, SU_{R} amplifies the received signal \(y_{r}^{(2,1)}\) with the gain G _{2} and then forwards the resulting signal to SU_{RX}. Thus, the received signal \(y_{2s}^{(2,1)}\) at SU_{RX} in the second time slot can be expressed as$$\begin{array}{*{20}l} y_{2s}^{(2,1)}=G_{2} h_{5}h_{4}x_{s}^{(2)}+G_{2}h_{5}\text{ }n_{r}+n_{s} \end{array} $$(32)Defining β _{3}=P _{ s }/N _{0}, the instantaneous SINRs of the direct link \(\gamma _{1s}^{(2,1)}\) and relaying link \(\gamma _{2s}^{(2,1)}\) of the SN are, respectively, obtained from (31) and (32) as$$\begin{array}{*{20}l} &\gamma_{1s}^{(2,1)}=\beta_{3} X_{6} \end{array} $$(33)$$\begin{array}{*{20}l} &\gamma_{2s}^{(2,1)}=\beta_{3}X_{4}X_{5}/(X_{4}+X_{5}) \end{array} $$(34)Thus, the instantaneous SINR of the SN is obtained as$$\begin{array}{*{20}l} \gamma_{s}^{(2,1)} = \max\left[\gamma_{1s}^{(2,1)},\gamma_{2s}^{(2,1)}\right] \end{array} $$(35)

Case 2: Both the PN and SN are active but the SN senses that the PN is inactive. With the incorrect sensing outcome, the SN operates in interweave mode though the PN is active. The probability of this event is$$\begin{array}{*{20}l} p_{p}^{(2,2)}=p_{s}^{(2,2)}=p_{m} P_{PS} P_{R} \end{array} $$(36)Then, the PU_{TX} and SU_{TX} transmit the signals x _{ p } and \(x_{s}^{(2)}\) to the relay and to their respective receivers. As a result, the received signal \(y_{1p}^{(2,2)}\) at PU_{RX}, the received signal \(y_{1s}^{(2,2)}\) at SU_{RX}, and the received signal \(y_{r}^{(2,2)}\) at SU_{R} in the first time slot are, respectively, given by$$\begin{array}{*{20}l} y_{1p}^{(2,2)}=h_{3}x_{p}+h_{7}x_{s}^{(2)}+n_{s} \end{array} $$(37)$$\begin{array}{*{20}l} y_{1s}^{(2,2)}=h_{6}x_{s}^{(2)}+h_{8}x_{p}+n_{s} \end{array} $$(38)$$\begin{array}{*{20}l} y_{r}^{(2,2)}=h_{1}x_{p}+ h_{4}x_{s}^{(2)}+n_{r} \end{array} $$(39)Again, the term \(h_{4} x_{s}^{(2)}\) in (39) represents interference from the SN to the PN while the term h _{1} x _{ p } can be considered as the interference from the PN to the SN. Usually, these terms are much larger than the noise term n _{ r }, such that n _{ r } in (39) can be neglected [15]. As a result, the received signals, \(y_{2p}^{(2,2)}\) at PU_{RX} and \(y_{2s}^{(2,2)}\) at SU_{RX}, in the second time slot are, respectively, given by$$\begin{array}{*{20}l} y_{2p}^{(2,2)}=G_{2}h_{1}h_{2}x_{p}+ G_{2}h_{2}h_{4}x_{s}^{(2)}+n_{s} \end{array} $$(40)$$\begin{array}{*{20}l} y_{2s}^{(2,2)}=G_{2}h_{1}h_{5}x_{p}+ G_{2}h_{4}h_{5}x_{s}^{(2)}+n_{s} \end{array} $$(41)From (37) and (40), the instantaneous SINR of the PN is obtained as$$\begin{array}{*{20}l} \gamma_{p}^{(2,2)}=\max \left[\gamma_{1p}^{(2,2)},\gamma_{2p}^{(2,2)}\right] \end{array} $$(42)where \(\gamma _{1p}^{(2,2)}\) and \(\gamma _{2p}^{(2,2)}\) are, respectively, the instantaneous SINRs of the direct and relaying links of the PN:$$\begin{array}{*{20}l} \gamma_{1p}^{(2,2)}&=X_{3}/(\beta_{2}\beta_{3}X_{7}+\beta_{2}) \end{array} $$(43)$$\begin{array}{*{20}l} \gamma_{2p}^{(2,2)}&=X_{1}X_{2}/(\beta_{2}\beta_{3}X_{2}X_{4}+\beta_{2}X_{4}) \end{array} $$(44)From (38) and (41), the instantaneous SINR of the SN is obtained as$$\begin{array}{*{20}l} \gamma_{s}^{(2,2)}&=\max \left[\gamma_{1s}^{(2,2)},\gamma_{2s}^{(2,2)}\right] \end{array} $$(45)where \(\gamma _{1s}^{(2,2)}\) and \(\gamma _{2s}^{(2,2)}\) are, respectively, the instantaneous SINRs of the direct link and the relaying link of the SN, i.e.,$$\begin{array}{*{20}l} \gamma_{1s}^{(2,2)}&=\beta_{2}\beta_{3}X_{6}/(X_{8}+\beta_{2}) \end{array} $$(46)$$\begin{array}{*{20}l} \gamma_{2s}^{(2,2)}&=\beta_{2}\beta_{3}X_{4}X_{5}/(X_{1}X_{5}+\beta_{2}X_{4}) \end{array} $$(47)
2.3 Scenario 3: Only the PN is active
3 Outage probability
where \(\mathcal {S}\) is the state space containing all operation modes, S _{ i } is the ith operation mode, and p _{ i } is the probability that the system operates in the ith mode. Equation (55) will be utilized in the sequel to analyze the performance of the PN and SN in terms of outage probability, outage capacity, and symbol error rate.
3.1 Outage probability of the PN
where \(P_{p,\text {out}}^{(i,j)}\) is the outage probability of the PN in Scenario iCase j, i.e., \(P_{p,\text {out}}^{(i,j)}=F_{\gamma _{p}^{(i,j)}}(\gamma _{\text {th}})\). Here, \(F_{\gamma _{p}^{(i,j)}}(\gamma)\) is the CDF of the instantaneous SINR of the PN in Scenario iCase j. Specifically, \(F_{\gamma _{p}^{(1,1)}}(\gamma)\), \(F_{\gamma _{p}^{(2,2)}}(\gamma)\), and \(F_{\gamma _{p}^{(3,1)}}(\gamma)\) are, respectively, given as follows:
Theorem 1
Proof
See Appendix 1. □
Theorem 2
Proof
See Appendix 2. □
Theorem 3
Proof
See Appendix 3. □
Substituting (9), (36), (48), (57), (58), and (61) into (56) and using γ=γ _{th} as an argument, the outage probability of the PN can be straightforwardly derived.
3.2 Outage probability of the CCRN
Here, \(P_{s,\text {out}}^{(i,j)}\) is the outage probability of the SN in Scenario iCase j, i.e., \(P_{s,\text {out}}^{(i,j)}\,=\,F_{\gamma _{s}^{(i,j)}}(\!\gamma _{\text {th}}\!)\) where \(F_{\gamma _{s}^{(i,j)}}(\!\gamma \!)\) is the CDF of the instantaneous SINR \(\gamma _{s}^{(i,j)}\).
Finally, substituting (9), (21), (29), (36), (70), (73), (74), and (78) into (62) and using γ=γ _{ th } as an argument, the outage probability for the CCRN is obtained.
4 Outage capacity analysis
4.1 Outage capacity of the PN
where \(T(a,b,c,d,e,f,g,h,i,j,k,l)=\int _{0}^{\infty }\frac {e^{\gamma a}\gamma ^{b}}{(\gamma +c)^{d}}\times \ U\left (e,f,\frac {g \gamma +h}{i \gamma + j}\right)\frac {(\ln (1+\gamma))^{k}}{(1+\gamma)^{l}}d\gamma \).
where \(\Upsilon (a,\!b,\!c,\!d,\!e,\!f) =\! \int _{0}^{\infty }\!\frac {\gamma ^{a} \!\ln ^{b}(1\,+\,\gamma)\exp (c \gamma)}{1\,+\,\gamma } \mathcal {K}_{d}{~}^{e}(\,f \gamma \!)d\gamma \).
Substituting (82), (83), and (84) into (79), we obtain the outage capacity \(C_{p,\epsilon }^{(i,j)}\) of the PN in Scenario iCase j. Then, substituting these outcomes together with (9), (36), and (48) into (81) allows us to obtain the outage capacity for the PN.
4.2 Outage capacity of the CCRN
By substituting (??), (87), (88), and (89) into (79), we obtain the outage capacity \(C_{s,\epsilon }^{(i,j)}\). Then, substituting these outcomes together with (9), (21), (29), and (36) into (85), the outage capacity of the SN is obtained.
5 Symbol error rate analysis
5.1 Symbol error rate of the PN
By substituting (9), (36), (48), (92), (93), and (94) into (91), we obtain the SER of the PN.
5.2 Symbol error rate of the CCRN
Finally, substituting (9), (21), (29), (36), (96), (97), (98), and (99) into (95), the SER of the CCRN can be found.
6 Numerical results
In this section, we present numerical results to evaluate the performance for the PN and the CCRN for various network parameters. In particular, the numerical results are provided to quantify the considered metrics and to illustrate the performance improvement obtained when applying the proposed relayingassisted hybrid spectrum access scheme.
Let us recall that d _{1},d _{2},d _{3},d _{4},d _{5},d _{6},d _{7}, and d _{8} denote the normalized distances of the links PU_{TX}→SU_{R}, SU_{R}→PU_{RX},PU_{TX}→PU_{RX},SU_{TX}→SU_{R},SU_{R}→SU_{RX},SU_{TX}→SU_{RX},SU_{TX}→PU_{RX}, and PU_{TX}→SU_{RX}, respectively. Assume that the considered system operates in a suburban environment such that all channel mean powers are attenuated with the link distances according to the exponential decaying path loss model with path loss exponent of 4. The SINR threshold used for calculating the outage probability is selected as γ _{th}=3 dB. Further, the expected outage threshold is chosen as ε=0.1 % when calculating the outage capacity.
First, we will illustrate the impact of imperfect spectrum sensing at the SUs, the fading severity parameters, and the communication link distances d _{1},d _{2}, and d _{3} of the PN on the system performance in terms of outage probability, outage capacity, and SER of the PN. To reveal the superior performance of the proposed scheme, we then provide a comparison of the performance of the PN with and without the use of the secondary relay. Assume that the arrival processes of the PN and CCRN follow a Poisson distribution with arrival rates λ _{ p }=λ _{ s }=85 packets/sec. Furthermore, the departure processes of the PN and secondary network at SU_{TX} and PU_{TX} are also modeled as Poisson distribution with rate μ _{ s }=μ _{ p }=100 packets/sec. Moreover, departure rates at SU_{R} for the traffics from the primary transmitter and secondary transmitter are set as \({\mu _{r}^{s}}={\mu _{r}^{p}}=100\) packets/s. In addition, the socalled transmit signaltonoise ratio (SNR) of the CCRN in the interweave mode and the interferencepowertonoise ratio in the underlay mode are selected as P _{ s }/N _{0}=10 dB and Q/N _{0}=5 dB, respectively. Finally, the distances, d _{4}, d _{5}, d _{6}, associated with the communication links and the distances, d _{7}, d _{8}, associated with the interference links of the CCRN are fixed as d _{4}=d _{5}=0.6, d _{6}=1.2, and d _{7}=d _{8}=0.9.
 ∙
Case 1: (m _{1},m _{2},m _{3},m _{4},m _{5},m _{6},m _{7},m _{8})=(3,3,3,3,3,3,3,3)
 ∙
Case 2: (m _{1},m _{2},m _{3},m _{4},m _{5},m _{6},m _{7},m _{8})=(2,2,2,2,2,2,2,2)
 ∙
Case 3: (m _{1},m _{2},m _{3},m _{4},m _{5},m _{6},m _{7},m _{8})=(2,2,2,2,2,2,1,1)
As can be seen from Fig. 3 a–c, Case 1 provides the lowest outage probability and SER, and the highest outage capacity as compared to Cases 2 and 3 because Case 1 has more favorable fading conditions. Furthermore, Case 3 considers different fading severity parameters on different channels. Specifically, Rayleigh fading is present on the interference channels from the secondary transmitter to the primary receiver and from the primary transmitter to the secondary receiver, i.e., m _{7}=m _{8}=1, while the communication channels inside the PN and the CCRN have the some lineofsight links with m _{ i }=2, i=1,2,…,6. As Case 3 induces more severe fading on the interference links compared to Case 2, while communication links in both cases have the same fading severity parameter, a performance improvement in the PN is observed for Case 3. Finally, imperfect spectrum sensing of the SUs degrades the performance of the PN, i.e, the outage probability and SER increase and the outage capacity decreases as the false alarm and missed detection probabilities of the SUs increase.

Case 4: d _{1}=d _{2}=0.4 and d _{3}=0.6

Case 5: d _{1}=d _{2}=0.5 and d _{3}=0.7
As can be seen from Fig. 4 a–c, with the communication link distances d _{1}=d _{2}=0.4 and d _{3}=0.6, Case 4 provides lower outage probability and SER, and higher outage capacity as compared to Case 5. This is because all channel mean powers are attenuated with the link distances according to an exponentially decaying function. Thus, the average channel power gains of the communication channels of the PN in Case 4 are higher than those of Case 5. It can also be observed from these figures that the proposed relaying scheme where the relay assists the communication of both the PN and SN can improve the performance of the PN. In particular, with the help of the secondary relay, the outage probability, outage capacity, and SER of the PN always outperform those results obtained for the PN without the assistance of the secondary relay in both Case 4 and Case 5.
Next, we investigate the effect of the arrival rate of the PN and imperfect spectrum sensing of the SUs on the outage probability, outage capacity, and SER of the CCRN. Then, we demonstrate the influence of the transmit SNR P _{ s }/N _{0} in interweave mode and the interferencepowertonoise ratio Q/N _{0} of PU_{RX} in underlay mode on the performance of the CCRN. In addition, a comparison of the performance of the hybrid interweaveunderlay CCRN with that of the conventional underlay CCRN is presented. In these examples, we select the arrival rate of SU_{TX} as λ _{ s } = 85 packets/s, and the departure rates of PU_{TX}, SU_{TX}, and SU_{R} as \(\mu _{p}\,=\,\mu _{s}\,=\,{\mu _{r}^{p}}\,=\,{\mu _{r}^{s}}\,=\,100\) packets/s. The transmit SNR of the PN is selected as P _{ p }/N _{0} = 10 dB. Furthermore, for these examples, the communication link distances, d _{1}, d _{2}, d _{3}, of the PN and the interference link distances, d _{7}, d _{8}, are fixed as d _{1} = d _{2} = 0.8, d _{3}=1.1, and d _{7} = d _{8} = 1.1. Finally, the fading severity parameters are chosen to be the same for all links as m _{ i } = 2, i=1,2,…,8.

Case 6: λ _{ p }=10 packets/s

Case 7: λ _{ p }=100 packets/s
For each case, we investigate various levels of imperfect spectrum sensing of the SUs on the performance of the hybrid CCRN, i.e., (p _{ f }=0,p _{ m }=0) for the case of perfect spectrum sensing, (p _{ f }=0,p _{ m }=0.2) for the case of 20 % of missed detection, and (p _{ f }=0.2,p _{ m }=0) for the case of 20 % of false alarm. In these examples, we select the communication link distances of the CCRN as d _{4} = d _{5} = 0.5 and d _{6} = 0.7. Furthermore, we fix the transmit SNR in the interweave mode as P _{ s }/N _{0} = 10 dB. As expected, when the arrival rate of the PN decreases from Case 7 to Case 6, the performance of the CCRN improves significantly, i.e., the outage probability and SER decrease and the outage capacity increases. This is attributed to the fact that as λ _{ p } decreases, the idle periods of the PN increases. Thus, the probability that the CCRN operates in the interweave mode increases. It can also be observed that false alarm degrades the performance of the SN since it reduces the opportunity for the SN to operate in the interweave mode. Finally, missed detection increases the chance for the SN to operate in the interweave mode which increases the performance of the SN. However, when missed detection occurs, the SUs assume that the licensed spectrum is vacant although it is occupied by the PUs which degrades the performance of the PN as can be seen in Fig. 3.

Case 8: d _{4}=d _{6}=0.4 and d _{6}=0.7

Case 9: d _{4}=d _{5}=0.5 and d _{6}=0.8
As expected, when d _{4}, d _{5}, and d _{6} increase from Case 8 to Case 9, the outage probability and SER of the hybrid CCRN and conventional CCRN increase. However, the outage capacity decreases according to the increase of d _{4}, d _{5}, and d _{6} since the transmit signals are attenuated with distance according to the exponential decaying path loss model. Finally, for both of the examined cases in Fig. 6, the performance of the proposed hybrid CCRN always outperforms that of the respective underlay CCRN.
7 Conclusions
In this paper, we have utilized a continuoustime Markov chain to model the spectrum access of the PU and SUs of a hybrid interweaveunderlay CCRN wherein an AF relay assists both the PN and CCRN. Based on this Markov model, we have obtained the steady state probability of each state which then is utilized to calculate the probability of each scenario of the PN and CCRN. Considering mutual interference between the PN and CCRN, we have derived expressions for the outage probability, outage capacity, and SER for both networks under imperfect spectrum sensing. Numerical results have been provided to illustrate the effect of the primary arrival rates, the fading severity parameters, the communication and interference link distances, and the interference power threshold of PU_{RX} on the performance of the PN and CCRN. Through the selected examples, it can be observed that the PN can achieve a performance improvement with the assistance of the secondary relay to forward its signals. Finally, for the selected numerical results, the performance of the CCRN applying the propose hybrid interweaveunderlay spectrum access always outperforms that of the conventional underlay CCRN.
7.1 Appendix 1: Proof of Theorem 1
Finally, substituting (A.2) and (A.5) into (A.1), we obtain \(F_{\gamma _{p}^{(1,1)}}(\gamma)\) as in (57).
7.2 Appendix 2: Proof of Theorem 2
Finally, substituting (B.2) and (B.4) into (B.1), after some algebraic manipulations, the CDF of \(\gamma _{p}^{(2,2)}\) can be derived as in (58).
7.3 Appendix 3: Proof of Theorem 3
By substituting (C.2) and (C.4) into (C.1), we finally obtain the CDF of \(\gamma _{p}^{(2,2)}\) as in (61).
Declarations
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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