1 Introduction

Recently, significant attention has been given to cognitive radio as a new technology that uses radio spectrum opportunistically and efficiently and which allows new applications to share currently allocated spectrum while the quality of service (QoS) at primary side is guaranteed.

There are three approaches of cognitive radio used to access licensed or unlicensed band [1]: spectrum interweave, spectrum underlay, and spectrum overlay.

a- Spectrum interweave (interference avoiding): the secondary user (SU) uses the first part of time slot (i T) (first phase) for ‘spectrum sensing’ to detect spectrum hole (spectrum not currently used by primary user) and make a secondary user decision H s (i) at time slot (i T), and beginning transmission in the second phase, only if the result of sensing in the first phase is H s (i)=H 0 (hole is detected, i.e., primary user is absent). In this approach, the SU does not interfere with primary user.

b- Spectrum underlay (interference controlling): the SU can transmit simultaneously with primary user over the same spectrum and through all time slots (i T) only if the interference generated by secondary transmitter (cognitive transmitter) at primary receiver is tolerable and controlled by an acceptable level, i.e., below a threshold or an outage probability limit that guarantees certain QoS for primary user’s communication.

c- Spectrum overlay (interference mitigating): the SU (cognitive transmitter) can transmit simultaneously with primary user over the same spectrum. In this approach, cognitive transmitter can transmit with any power, part of its power and time slot (i T) used for relaying primary user’s transmission, and the remainder of its power and time slot (i T) for own transmission.

Also, cooperative communications with amplify-and-forward (AF) or with decode-and-forward (DF) strategy have gained a great interest in the academic research circles. This is due to benefits over a fading channel, like combating fading and shadowing, and increasing the system capacity. In [2], the authors shown that the best relay selection scheme can achieve the same diversity-multiplexing tradeoff performance as the traditional protocols. As researches go deep and broad, some works introduce cooperative communication techniques into cognitive radio, to enhance the secondary user’s data transmission and the reliability of primary user’s detection.

One of the QoS parameters for secondary system is the continuity of service. Continuous transmission for secondary users is weakly addressed. Indeed, in [3,4], the SU transmits through underlay method either by controlling interference level [3,4] or outage probability limit. Nevertheless, underlay method underutilizes the spectrum, and SU does not transmit with high power even when primary user is absent. In [5,6], the SU transmits with the interweave method in the second phase only if the result of sensing in the first phase is H s (i)=H 0. In [7], mixed strategies introduced for cognitive transmission in the second phase based on the sensing results in the first phase. Many studies in the literature on such mixed strategies have been conducted [8-13], in their scenarios, the secondary user transmits through interweave method in the second phase if the result of sensing in the first phase is H s (i)=H 0 or transmits with underlay method in the second phase if the result of sensing in the first phase is H s (i)=H 1 (hole is not detected, i.e., primary user is active). However, none of these papers have considered the transmission in two phases, SU does not transmit over all time slots, it transmits only in the second phase, which decreases the spectrum efficiency as some part of the time slot is used for sensing instead of data transmission.

In most of the above presented strategies, sensing is a key functionality that affects the spectral efficiency of SU due to the time dedicated to sensing. However, two different architectures could be adopted to operate spectrum sensing and communication by using single radio or dual radio [14,15]. Nevertheless, in the single radio architecture, no communication is allowed during sensing phase. Our work aims at exploiting sensing phase for transmission also optimizes the SU spectrum efficiency and thus adopts double radios architecture. To be able to do so without creating harmful interference to the primary user, we propose an underlay scheme with limited interference threshold.

In our previous works [16], where we have introduced our new strategy of transmission ‘Always Transmit (AT): to enhance the cognitive sources of data transmission and to provide continuous transmit solution by using mixed strategies in cognitive radio context, which it performed via the combination of the underlay and interweave approaches depending on the activity of the primary system’ and have shown that it outperforms other conventional cognitive radio strategies for power transmission and interference management, in [17], where we have derived the closed form and analyzed the performance of outage probability over Rayleigh fading, and in [18], where we have studied the trade-off between spectrum sensing duration and the QoS at cognitive network while guaranteeing certain QoS at primary network controlled by the outage probability limit, over independent but not necessarily identically distributed Nakagami-m channels. In (M Ridouani, A Hayar, A Haqiq, Continuous-transmit in cognitive radio systems: perform sensing and transmission in parallel anytime. submitted to IEEE TWC), we have shown that the use of (average transmit P av, instantaneous interference I) as power constraint is recommended for transmission, because it allows preserving the QoS of primary system at each time while guaranteing an outage performance limit for the secondary of about ≃20% of maximum degradation, and due to fading and by using average transmit power constraint, the energy efficiency is improved.

Motivated by cooperative communications and our results in [17,18], in this paper, we extend our strategy ‘AT’ (the cognitive source (CS) will be able to transmit over all time slots and in any time slot, then, it will be able to operate continuously whenever either the primary user is active or absent) into cognitive relay selection. We derive the closed form of outage probability of relay selection in dual-hop transmission over independent but not necessarily identically distributed (i.n.i.d.) Nakagami-m channels, with integer values of fading severity parameter m.

Our main contributions are summarized as follows:

1) The continuity of service: we propose a scenario that enhances secondary user QoS in terms of outage probability by allowing it to transmit over all time slots and in any time slots. Therefore, the secondary can continue its transmission whether the primary user is active or absent.

2) We derive the closed-form expressions of the secondary user outage probability for transmission scenario over both phases and over independent but not necessarily identically distributed Nakagami-m fading channels. And we analyze for this considered scenario the impact of spectrum sensing duration and spectrum hole detection, the transmit and the interference power constraints, and the number of relays and fading on the secondary outage probability (QoS).

3) We show that the proposed strategy ‘Always transmit + Relay’ outperforms other conventional cognitive radio strategies.

This paper is organized as follows. In Section 3, we outline the system model (scenario, spectrum sensing, and the optimal power allocation) of the proposed scheme of cognitive transmission. The closed form of outage probability of cognitive relay selection over a Nakagami-m fading channel is given in Section 3. In Section 3, we present numerical results. Section 3 draws the conclusion.

2 System model

In this section, we describe the cognitive relay network of our scenario. As shown in Figure 1b, the primary destination (PD) coexists with the cognitive nodes (CS, cognitive destination (CD), and N potential cognitive relay (CR k ) k=1,..,N). Each cognitive node has a dual-radio architecture, one radio chain is dedicated for spectrum sensing (see Section 3) while the other radio chain operates on a half-duplex mode, and it is dedicated for data transmission and reception.

Figure 1
figure 1

The mode of transmission and the allocation of time durations (a) and fading coefficient of the channels (b).

Full size image

Notice here that, if we look only at the second radio chain, we can see that it operates on a half-duplex mode. If we look at the cognitive node (which consists of a two radio chains), one radio is dedicated for spectrum sensing (reception), the second chain for communication (for example, transmitting data: transmission), we can see that it operates, globally, on a full-duplex mode.

In this model, we consider a half-duplex mode and a dual-hop communication system with regenerative mode for data transmitting. Also, it is assumed that there is no direct transmission between CS and CD. So the data transmission from CS to CD is subdivided into two steps. In step one, the CS transmits its data (one packet) to (CR k ) which is selected prior to a new transmission. In step two, the CR k decodes the received data and then re-encodes and forwards it to a CD.

Also, we consider that the transmission of CS and CR k process can move into two phases through a time slot (i T) as shown in Figure 1a. In the first phase (α T), the CS (resp. CR k ) transmits through underlay approach, in the meantime, each cognitive node senses the spectrum hole existence (activity of primary source (PS)). In the second phase ((1−α)T), the CS (resp. CR k ) transmits through interweave or underlay approach whether or not a hole is detected in the first phase, respectively.

The data (frame i) structure of the proposed system is shown in Figure 1a. The time duration of the transmission of each frame is T. Each frame consists of many packet transmission periods, and we assume that each packet transmission period is very small subject to the time slot T, which we allow to suppose that each packet transmitted from CS to the best cognitive relay CR k and from CR k to the CD is via the same transmission mode (underlay or interweave), and the best cognitive relay CR k is selected to transmit all packets of a frame i during the time slot iT.

We assume that the channel changes slowly, so the channel remains constant during a one time slot. Also, we assume that the channels between any of the two nodes are independent but not necessarily identically distributed (i.n.i.d.) Nakagami-m channels, with integer values of fading severity parameter m, so that the probability distribution function (PDF) of the channel gains (|g sp |2, |h SK |2, |g kp |2, and |h KD |2) is statistically characterized by the Nakagami distribution [19], where g sp , h SK , g kp , and h KD are the fading coefficients of the channel from CS to PD, from CS to CR k , from CR k to PD, and from CR k to CD, respectively, as shown in Figure 1. Let X k =|h SK |2, Y k =|h KD |2, Z 0=|g sp |2, and Z k =|g kp |2.

Note that X k , Y k , and Z k (k=0,1,..,N) follow the probability distribution p k (x), p k (y), and p k (z), respectively, and are independent of each other. Thus, the channel gain, γ (γ = X k , Y k , Z k ), is distributed according to a gamma distribution given by (1) with different fading parameters γ xk , γ yk , γ zk and fading severity parameters m xk , m yk , m zk :

$$ p_{\gamma}(\gamma) = \left(\frac{m}{\overline{\gamma}}\right)^{m}\frac{\gamma^{m-1}}{\Gamma(m)}\exp\left(-m\frac{\gamma}{\overline{\gamma}}\right), \gamma\geq0,m\geq1/2 $$
((1))

where \(\overline {\gamma }\) is the average channel gain (\(\overline {\gamma } = E[\gamma ]\)), m is the Nakagami fading severity parameter, and Γ(.) is the gamma function [20].

The Nakagami fading model has proven useful for modeling multipath faded envelope in wireless channels and is widely accepted as best fit to model multipath propagation from severe to light fading. Indeed, it is known to accurately characterize a range of multipath and to model different fading conditions ranging from Rayleigh to strong Rician channels. As the fading parameter m tends to infinity, the Nakagami multipath fading channel converges to an AWGN channel.

For notation convenience, we use the notations as follows:

  1. ᅟ

    k: the k-th cognitive node (k=0 refer to CS and k=1,…,N refer to CR k ).

  2. ᅟ

    P SK and P KD are the transmit power of CS and CR k , respectively.

  3. ᅟ

    H p (i): denote whether or not the licensed band is occupied by PS in the first phase of time slot iT.

  4. ᅟ

    H 0: the band is unoccupied by PS (and H 1: otherwise).

  5. ᅟ

    CS and CR k will make a decision H s (i) to transmit through interweave or underlay method in the second phase

  6. ᅟ

    H s (i)=H 0: considers the band is available, then, CS and CR k transmit through interweave method

  7. ᅟ

    H s (i)=H 1: considers the band is unavailable, then, CS and CR k transmit through underlay method.

  8. ᅟ

    P a =P r(H p (i)=H 0): the probability that there is a hole (and P r(H p (i)=H 1)=1−P a otherwise).

  9. ᅟ

    \({P_{f}^{k}} =Pr(H_{s}(i)=H_{1} /H_{p}(i)=H_{0})\): the probability that the k-th cognitive node has a false alarm.

  10. ᅟ

    \({P_{d}^{k}} = Pr(H_{s}(i) = H_{1} /H_{p}(i) = H_{1})\): the probability that the k-th cognitive node detects the PS presence.

  11. ᅟ

    ε: represent the fraction of a time slot utilized for data transmission in the first phase (ε=α) and second phase (ε=β=1−α)

  12. ᅟ

    P α : the probability to be in the first phase.

  13. ᅟ

    P β =1−P α : the probability to be in the second phase.

2.1 Spectrum sensing

Cooperative spectrum sensing can improve the performance of spectrum sensing by combating fading and shadowing and decreasing the detection time required. In this technique, each of N+1 cognitive nodes (CS or CR k ) contributes to sense spectrum holes. To protect primary’s communication when PS is active (H p (i)=H 1), we use the OR rule [21] (spectrum is not available (H s (i)=H 1) if any of N+1 cognitive nodes sense the presence of PS) which minimizes the probability of interfering at PD.

Each cognitive node periodically senses the channel like in [5,6,22]. We assume that it first estimates the noise floor and distinguishes the rest of the information bearing samples for wireless signals [23]. After, it identifies the CS’s signal (the signal transmitted through underlay approach in the first phase (α T)) [23]. And finally, if it detects a collision (i.e., the CS’s signal is combined with other signal), then, it decides that the PS is active (a hole is not detected), else, it decides that PS is absent (a hole is detected).

We assume that the N+1 cognitive nodes use the same spectrum sensing detector and sense the spectrum hole existence through the same duration (α T), then, the cooperative probabilities of detection P d and false alarm P f are as follows:

$$ P_{d} = 1 -\prod\limits_{k = 0}^{N}\left(1 - {P_{d}^{k}}\right) $$
((2))
$$ P_{f} = 1 - \prod\limits_{k = 0}^{N}\left(1 - {P_{f}^{k}}\right) $$
((3))

2.2 Optimal power allocation

In the underlay approach, the CS’s (resp. CR k ) transmission power is constrained as P SK Z 0≤I (resp. P KD Z k ≤I), where I is the interference threshold that the interference generates by CS (resp. CR k ) on the PD remains below it, and is constrained as P SK ≤P (resp. P KD ≤P), where P is the maximum transmission power.

In the interweave approach, there is just the maximum transmission power constraint, then, P SK ≤P and P KD ≤P.

For each time slot iT, in the first phase (first fraction of time α T), the CS’s (resp. CR k ) transmits through underlay approach, then, it transmits with a low power \(P_{\textit {US}}^{k}\) (resp. \(P_{\textit {UR}}^{k}\)). During the second phase ((1−α)T), the CS’s (resp. CR k ) transmits through underlay approach with a low power \(P_{\textit {US}}^{k}\) (resp. \(P_{\textit {UR}}^{k}\)) or switch to interweave approach with a high power \(P_{\textit {IS}}^{k}\) (resp. \(P_{\textit {IR}}^{k}\)) based on the sensing result.

Subject to P SK ≤P (resp. P KD ≤P) and P SK Z 0≤I (resp. P KD Z k ≤I), then, the optimal power \(P_{\textit {IS}}^{k}\) (resp. \(P_{\textit {IR}}^{k}\)) and \(P_{\textit {US}}^{k}\) (resp. \(P_{\textit {UR}}^{k}\)) used by CS (resp. CR k ) in interweave and underlay respectively are as follow:

$$ P_{IS}^{k} \;= P_{IR}^{k} = P $$
((4))
$$ P_{US}^{k} = \min\left(P,\frac{I}{Z_{0}}\right) = \left\{ \begin{array}{ll} P,& Z_{0}\leq\frac{I}{P}\\ \frac{I}{Z_{0}},& Z_{0}\geq\frac{I}{P} \end{array}\right. $$
((5))
$$ \setlength{\nulldelimiterspace}{0pt} P_{UR}^{k} = \min\left(P,\frac{I}{Z_{k}}\right) = \left\{ \begin{array}{ll} P,& Z_{k}\leq\frac{I}{P}\\ \frac{I}{Z_{k}},& Z_{k}\geq\frac{I}{P} \end{array}\right. $$
((6))

According to the activity of PS and the sensing result, the synthesis of our scenario of transmission and the optimal power allocation in both phases are listed in Table 1.

Table 1 The synthesis of the transmission scenario and the optimal power allocation in both phases
Full size table

3 Outage analysis of cognitive relay selection

3.1 Cognitive relay selection

If we select the k-th cognitive relay CR k for data transmission in dual-hop, the instantaneous capacity in underlay approach \({C_{k}^{U}}\) (resp. interweave approach \({C_{k}^{I}}\)) between cognitive source and cognitive destination is given by [24]:

$${} {C_{k}^{U}} = \min \left\{ \varepsilon\log_{2}\left(1+\frac{X_{k} P_{US}^{k}}{N_{0}}\right),\varepsilon\log_{2}\left(1+\frac{Y_{k} P_{UR}^{k}}{N_{0}}\right)\right\} $$
((7))
$${} {C_{k}^{I}} = \min \left\{\beta \log_{2}\left(1+\frac{X_{k} P_{IS}^{k}}{N_{0}}\right),\beta \log_{2}\left(1+\frac{Y_{k} P_{IR}^{k}}{N_{0}}\right)\right\} $$
((8))

In (7) and (8), we have assumed that any interference from the primary source is neglected (this can be possible if the primary transmitter is located far away from the secondary users, or the interference is represented by the noise term under an assumption that the primary transmitter’s signal is generated by random Gaussian codebooks [4]). The coefficient ε (resp. β) is due to the fact that only ε (resp. β) fraction of a time slot is utilized for the CS’s and CR k ’s data transmission phase. And N 0 is the noise variance.

Substituting (5) and (6) in (7) and (4) in (8), we obtain:

$${} {\fontsize{8.8pt}{9.6pt}\selectfont{\begin{aligned} {C_{k}^{U}} = \varepsilon \log_{2}\!\left(\!1+\frac{P}{N_{0}}\min \!\left\{\!X_{k} \min\left(\!1,\frac{I}{{PZ}_{0}}\!\right),Y_{k} \min\!\left(\!1,\frac{I}{{PZ}_{k}}\!\right)\!\right\}\right) \end{aligned}}} $$
((9))
$$ {C_{k}^{I}} = \beta \log_{2}\left(1+\frac{P}{N_{0}}\min \left\{ X_{k},Y_{k}\right\}\right) $$
((10))

We introduce two variables \(\left ({W_{k}^{I}}, {W_{k}^{U}}\right)\) for briefly. The short-hand notation is as follow:

$$ {W_{k}^{U}} = \min \left\{X_{k} \min\left(1,\frac{I}{{PZ}_{0}}\right),Y_{k} \min\left(1,\frac{I}{{PZ}_{k}}\right)\right\} $$
((11))
$$ {W_{k}^{I}} = \min \left\{ X_{k},Y_{k}\right\} $$
((12))

For the best relay selection scheme, and by using the max-min criterion [4,25] which maximizes the minimum of signal-to-noise ratios (SNRs) of the source-relay link and relay-destination link, we select the relay for which \({W_{k}^{U}}\) (resp. \({W_{k}^{I}}\)) is maximal in underlay (resp. interweave) approach.Let:

$$ W^{U} = \max_{1,..,N}{W_{k}^{U}} $$
((13))
$$ W^{I} = \max_{1,..,N}{W_{k}^{I}} $$
((14))

then, consequently, we obtain for the instantaneous capacity in underlay approach C U (resp. interweave approach C I)

$$ C^{U} = \max_{1,..,N}{C_{k}^{U}} = \varepsilon \log_{2}\left(1+\frac{P}{N_{0}}W^{U}\right) $$
((15))
$$ C^{I} = \max_{1,..,N}{C_{k}^{I}} = \beta \log_{2}\left(1+\frac{P}{N_{0}}W^{I}\right) $$
((16))

3.2 Outage probability

Given a predetermined transmission rate of the source C T , the outage probability mathematically defined as P out=P r(C<C T ), where C is the capacity of channel.

The outage probability in underlay approach \(P_{\text {out}}^{U,\varepsilon }\) (resp. interweave approach \(P_{\text {out}}^{I}\)) is given:

$$ P_{\text{out}}^{U,\varepsilon} = Pr\left(C^{U}<C_{T}\right) $$
((17))
$$ P_{\text{out}}^{I} = Pr\left(C^{I}<C_{T}\right) $$
((18))

Substituting (15) in (17) and (16) in (18), we obtain:

$$ P_{\text{out}}^{U,\varepsilon} = Pr\left(W^{U}<\frac{U_{\varepsilon}}{P}\right) $$
((19))
$$ P_{\text{out}}^{I} = Pr\left(W^{I}<\frac{U_{\beta}}{P}\right) $$
((20))

where \(U_{\varepsilon } = N_{0}(2^{\frac {C_{T}}{\varepsilon }}-1)\)Hence, by using (11) and (12), we get:

$$ P_{\text{out}}^{U,\varepsilon} = \prod_{k = 1}^{N} Pr\left({W_{k}^{U}}<\frac{U_{\varepsilon}}{P}\right) $$
((21))
$$ P_{\text{out}}^{I} = \prod_{k = 1}^{N} Pr\left({W_{k}^{I}}<\frac{U_{\beta}}{P}\right) $$
((22))

Using Appendix (Equations 32 and 39) and replacing A by \(\frac {U_{\varepsilon }}{P}\) and B by \(\frac {I}{P}\), the closed form of (21) yields:

$$ \begin{aligned} P_{\text{out}}^{U,\varepsilon} &= \prod_{k = 1}^{N} \left(1-\Phi\left(m_{xk},\gamma_{xk},m_{z0},\gamma_{z0},\frac{I}{P},\frac{U_{\varepsilon}}{P}\right)\right. \\ &\quad \left.\Phi\left(m_{yk},\gamma_{yk},m_{zk},\gamma_{zk},\frac{I}{P},\frac{U_{\varepsilon}}{P}\right)\right) \end{aligned} $$
((23))

Replace A by \(\frac {U_{\beta }}{P}\) in Appendix (Equation 33), the closed form of (22) yields:

$$ P_{\text{out}}^{I} = \prod_{k = 1}^{N} \left(1-\frac{\Gamma\left(m_{xk},\frac{m_{xk}}{\gamma_{xk}}\frac{U_{\beta}}{P}\right)}{\Gamma\left(m_{xk}\right)} \frac{\Gamma\left(m_{yk},\frac{m_{yk}}{\gamma_{yk}}\frac{U_{\beta}}{P}\right)}{\Gamma\left(m_{yk}\right)}\right) $$
((24))

Finally, the outage probability of our system is:

$$ P_{\text{out}} = P_{\alpha}P_{\text{out}}^{\text{first-phase}}+(1- P_{\alpha})P_{\text{out}}^{\text{second-phase}} $$
((25))

where:

$$P_{\text{out}}^{\text{first-phase}} = P_{\text{out}}^{U,\alpha} $$

and:

$$\begin{array}{@{}rcl@{}} P_{\text{out}}^{\text{second-phase}}&=&\! \left[Pr(H_{s}(i) = H_{0},H_{p}(i) = H_{0}) + Pr(H_{s}(i)\right.\\ &=&\! \left. H_{0},H_{p}(i) = H_{1})\right]P_{\text{out}}^{I}\notag\\ &&+ \!\left[Pr(H_{s}(i) = \!H_{1},H_{p}(i) = H_{0}) + Pr(H_{s}(i)\right.\notag\\ &=&\! \left.H_{1},H_{p}(i) = H_{1})\right]P_{\text{out}}^{U,\beta} \end{array} $$
((26))

which can also be written as:

$$\begin{array}{@{}rcl@{}}{} P_{\text{out}}^{\text{second-phase}} &=& \left[P_{a}(1-P_{f})+(1-P_{a})(1-P_{d})\right]P_{\text{out}}^{I} \notag\\&& +\left[P_{a}P_{f} + (1-P_{a})P_{d}\right]P_{\text{out}}^{U,\beta} \end{array} $$
((27))

Then, the closed form of outage probability of our strategy (Always transmit + Relay) is derived in (28):

$${} {\fontsize{8.89pt}{9.6pt}\selectfont{\begin{aligned} P_{{\text{out}}-{\text{AT}}-R}& = P_{\alpha} \prod_{k = 1}^{N} \left(1-\Phi\left(m_{xk},\gamma_{xk},m_{z0},\gamma_{z0},\frac{I}{P},\frac{U_{\alpha}}{P}\right) \right. \\&\quad\times \left.\Phi\left(m_{yk},\gamma_{yk},m_{zk},\gamma_{zk},\frac{I}{P},\frac{U_{\alpha}}{P}\right)\right)\\ &\quad+ \left(1- P_{\alpha}\right)\left(\!\vphantom{\left.\qquad\qquad\quad\! \Phi\left(m_{yk},\gamma_{yk},m_{zk},\gamma_{zk},\frac{I}{P},\frac{U_{\beta}}{P}\right)\right)}\left[P_{a}\left(1-P_{f}\right)+\left(1-P_{a}\right)\left(1-P_{d}\right)\right] \right. \\ & \quad\times\left. \prod_{k = 1}^{N} \left(\!1-\frac{\Gamma\left(m_{xk},\frac{m_{xk}}{\gamma_{xk}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{xk})} \frac{\Gamma\left(m_{yk},\frac{m_{yk}}{\gamma_{yk}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{yk})}\!\right)\right.\\ &\quad+\left.\left[P_{a}P_{f} + (1-P_{a})P_{d}\right]\right. \\ & \left.\quad\times \prod_{k = 1}^{N} \left(1-\Phi\left(m_{xk},\gamma_{xk},m_{z0},\gamma_{z0},\frac{I}{P},\frac{U_{\beta}}{P}\right)\right.\right. \\ &\left.\left.\quad\times \Phi\left(m_{yk},\gamma_{yk},m_{zk},\gamma_{zk},\frac{I}{P},\frac{U_{\beta}}{P}\right)\right)\right) \end{aligned}}} $$
((28))

If CS and CR k transmit just through interweave approach, (28) can reduced to (29):

$$ {}{\fontsize{8.9pt}{9.6pt}\selectfont{\begin{aligned} P_{{\text{out}}-IN-R}& = P_{\alpha}+\left(1- P_{\alpha}\right)\left(\!\vphantom{\left[P_{a}P_{f} + (1-P_{a})P_{d}\right]{\vphantom{\left(1-\frac{\Gamma\left(m_{xk},\frac{m_{xk}}{\gamma_{xk}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{xk})} \frac{\Gamma\left(m_{yk},\frac{m_{yk}}{\gamma_{yk}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{yk})}\right)}}}\left[P_{a}\left(1-P_{f}\right)+\left(1-P_{a}\right)\left(1-P_{d}\right)\right]\right.\\ &\quad\times\left.\prod_{k = 1}^{N} \left(1-\frac{\Gamma\left(m_{xk},\frac{m_{xk}}{\gamma_{xk}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{xk})} \frac{\Gamma\left(m_{yk},\frac{m_{yk}}{\gamma_{yk}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{yk})}\right)\right.\\ &\quad+\left.\left[P_{a}P_{f} + (1-P_{a})P_{d}\right]{\vphantom{\left(1-\frac{\Gamma\left(m_{xk},\frac{m_{xk}}{\gamma_{xk}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{xk})} \frac{\Gamma\left(m_{yk},\frac{m_{yk}}{\gamma_{yk}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{yk})}\right)}}\right) \end{aligned}}} $$
((29))

If CS and CR k transmit just through underlay approach, (28) can reduced to (30):

$${} \begin{aligned} P_{{\text{out}}-UN-R} &= P_{\alpha} \prod_{k = 1}^{N} \left(1-\Phi\left(m_{xk},\gamma_{xk},m_{z0},\gamma_{z0},\frac{I}{P},\frac{U_{\alpha}}{P}\right)\right. \\&\left.\quad\times \Phi\left(m_{yk},\gamma_{yk},m_{zk},\gamma_{zk},\frac{I}{P},\frac{U_{\alpha}}{P}\right)\right)\\ &\quad +\left(1- P_{\alpha}\right)\left(\left[P_{a} P_{f} + (1-P_{a})P_{d}\right] \prod_{k = 1}^{N} \right. \\ &\quad\times \left.\left(1-\Phi\left(m_{xk},\gamma_{xk},m_{z0},\gamma_{z0},\frac{I}{P},\frac{U_{\beta}}{P}\right) \right.\right.\\ & \quad \times \left.\left.\Phi\left(m_{yk},\gamma_{yk},m_{zk},\gamma_{zk},\frac{I}{P},\frac{U_{\beta}}{P}\right)\right)\right) \end{aligned} $$
((30))

If there is no rely (k=0), CS transmits data directly to PD without help of relays, (28) can reduced to (31):

$${} {\fontsize{8.8pt}{9.6pt}\selectfont{\begin{aligned} P_{{\text{out}}-{\text{AT}}-NR} =&\; P_{\alpha}\left(1-\Phi\left(m_{x1},\gamma_{x1},m_{z0},\gamma_{z0},\frac{I}{P},\frac{U_{\alpha}}{P}\right) \right)\\ & + \left(1- P_{\alpha}\right)\left(\!\vphantom{\frac{1}{2}}\left[P_{a}\left(1-{P_{f}^{1}}\right)+(1-P_{a})\left(1-{P_{d}^{1}}\right)\right]\right.\\ &\times \left. \!\left(1\,-\,\frac{\Gamma\left(m_{x1},\frac{m_{x1}}{\gamma_{x1}}\frac{U_{\beta}}{P}\right)}{\Gamma(m_{x1})}\right)\right.\\ &+ \left[P_{a} {P_{f}^{1}} \,+\, (1-P_{a}){P_{d}^{1}}\right] \\ &\times \left. \left(1-\Phi\left(m_{x1},\gamma_{x1},m_{z0},\gamma_{z0},\frac{I}{P},\frac{U_{\beta}}{P}\right) \right)\right) \end{aligned}}} $$
((31))

4 Numerical results

Computer simulations were carried out in order to validate the performance of the proposed scenario (‘Always transmit + Relay’) based on the outage probability. We analyze the impact of the fraction α of time slot used for spectrum sensing and other parameters (number of relay N, fading severity parameters (m xk , m yk , m zk , m z0), the transmit (resp. interference) power constraint P (resp. I)) over data transmission. Throughout simulations, we have assumed a perfect spectrum hole detection, thus, \({P_{f}^{k}} = 0.1 (k = 1,\dots,N)\) to use the cognitive channel with a higher chance with the interweave method, and \({P_{d}^{k}} = 0.9 (k = 1,..,N)\) to protect the primary source transmission. We consider that P α follows a uniform distribution, then, P α =α and P β =1−α. Also, in all simulations, we fix: γ xk =γ yk =γ zk =γ z0=1 dB (k=1,..,N), N 0=1, P a =0.8, C T =0.5, and we vary α, m, N, I, and P.

From the figures (Figures 2, 3, 4, 5, and 6), it has been observed that outage performances are degraded when α increases; in contrast, we see a higher reduction of outage probability, thus improving the system performance, when α is near 0. Indeed, when α is near 0, which is referred to as perfect sensing, the CS and CR k transmit with underlay method just in short time α T in the first phase and can have more chance to transmit with the interweave method in a large amount of time (1−α)T in the second phase. But when α increases, the CS and CR k transmit in large time with the underlay method, and thus, the system is more probable to have an outage.These figures also show that the outage probability decreases (which is the indication of performance improvement) as P, I, or N increase. Indeed, a) if the transmit power P at (CS and CR k ) increases, then, the SNRs increase which denote the increase of their capacity and finally the decrease of their outage probability, b) when the interference threshold I is increasing, the constraints are relaxed at the cognitive system side, so the system performance increases as I increases, c) more relays in fading channel network can combat fading and consequently, enhance the performance.

Figure 2
figure 2

Outage probability versus spectrum sensing overheadα for different values of the transmission power P at (CS and CR k ).

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Figure 3
figure 3

Outage probability versus spectrum sensing overheadα for different values of the interference power constraintI.

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Figure 4
figure 4

Outage probability versus spectrum sensing overheadα for different values of the number of relay N.

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Figure 5
figure 5

Outage probability versus spectrum sensing overhead α for a fixed/random values of the probabilities of false alarm and of detection.

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Figure 6
figure 6

Outage probability versus spectrum sensing overhead α for different values of the fading severity parameters (m xk,m yk,m zk,m z0,k=1,2,3).

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From Figure 5, we show that the outage probability decreases as probability of false alarm P f decreases. Indeed, when P f is small, which means that the system of detection is more reliable, which thus leads the CS to transmit with interweave method instead of underlay method when primary system is absent. Also, while looking at Figure 5, we can observe that the outage probability of secondary system converges to the same point when α go to 1, under a different value of P f and P d . In fact, when α go to 1, the CS expires the whole of the time slot (iT) for sensing and transmits through the whole of the time slot (iT) via underlay mode in the first phase and can have no time in the second phase for transmitting through interweave mode whatever the probability value of detection P d and/or of a false alarm P f , and even if primary user is not detected or there is no false alarm.

From Figure 6, we can see the advantage of the use relays in deep fading channel. In contrast, in poor fading channel, the use relays decrease the performance.

An interesting result found from Figure 7 (resp. Figure 8) is that our strategy (Always transmit + Relay) outperforms other conventional cognitive radio strategies like interweave or underlay approach in deep (resp. poor) fading channel.

Figure 7
figure 7

The outage probability of different strategies of transmission under a deep fading and a random average channel gain.

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Figure 8
figure 8

The outage probability of different strategies of transmission under a poor fading and a random average channel gain.

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Also, Figure 7 (resp. Figure 8) presents the comparison of the outage probability between our strategies (Always transmit + Relay) and (Always transmit + No-Relay) in deep (resp. poor) fading channel. As we have shown in [16] that the strategy ‘AT’ enhances the performance, the performance of a cognitive system can be more improved by using relays in deep fading channel, and it recommended not to use the relays in poor fading channel.

5 Conclusion

We have presented ‘Always transmit + Relay’ strategy for dual-hop transmission with decode-and-forward and relay selection. The closed-form expressions of the outage probability have been derived over independent but not identically distributed Nakagami-m fading channels on all nodes. Simulation results have shown that the performance increases as either the transmit power P, the interference power constraint I, and the number of relays N increases and when spectrum sensing duration α is small (near to 0). Finally, we have shown that the strategy ‘Always transmit + Relay’ outperforms other conventional cognitive radio strategies and the use of relays enhances the performance in deep fading channel. A future work is dedicated to explore some new ideas on how to perform sensing and detecting the presence or the absence of the primary system while transmitting with underlay approach.

6 Appendix

Let X k , Y k , and Z k (k=0,1,..,N) follow the probability distribution p k (x), p k (y), and p k (z), respectively, and are independent random variables distributed according to gamma distribution given by the Equation (1) with different fading parameters γ xk , γ yk , γ zk and fading severity parameters m xk , m yk , m zk :

$$ Pr\left(\min\left(X_{k},Y_{k}\right)<A\right)=1-Pr\left(X_{k}>A\right)Pr\left(Y_{k}>A\right) $$
((32))

then:

$$ {} Pr\left(\min\left(X_{k},Y_{k}\right)<A\right)=1-\frac{\Gamma\left(m_{xk},\frac{m_{xk}}{\gamma_{xk}}A\right)}{\Gamma\left(m_{xk}\right)} \frac{\Gamma\left(\!m_{yk},\frac{m_{yk}}{\gamma_{yk}}A\!\right)}{\Gamma(m_{yk})} $$
((33))

Let \(t_{k}=X_{k}\min \left (1,\frac {B}{Z_{k}}\right)\), the cumulative distribution function of t k

$$ F_{t_{k}}(A)=Pr\left(t_{k}<A\right)=Pr\left(X_{k}\min \left(1,\frac{B}{Z_{k}}\right)<A\right) $$
((34))
$${} {\fontsize{8.8pt}{9.6pt}\selectfont{\begin{aligned} F_{t_{k}}(A)=\int_{x=0}^{A} \int_{z=0}^{B} p_{k}(x)p_{k}(z)dxdz+\int_{x=0}^{\frac{AZ}{B}} \int_{z=B}^{\infty} p_{k}(x)p_{k}(z)dxdz \end{aligned}}} $$
((35))
$$ {}\begin{aligned} F_{t_{k}}(A)&=\left(1-\frac{\Gamma\left(m_{xk},\frac{m_{xk}A}{\gamma_{xk}}\right)}{\Gamma\left(m_{xk}\right)}\right) \left(1-\frac{\Gamma\left(m_{zk},\frac{m_{zk}B}{\gamma_{zk}}\right)}{\Gamma\left(m_{zk}\right)}\right)\\\quad &\quad+ \frac{\Gamma\left(m_{zk},\frac{m_{zk}B}{\gamma_{zk}}\right)}{\Gamma\left(m_{zk}\right)} -\! \int_{z=B}^{\infty}\! \frac {\Gamma\left(m_{xk},\frac{m_{xk}AZ}{\gamma_{xk}B}\right)}{\Gamma\left(m_{xk}\right)}p_{k}(z)dz \end{aligned} $$
((36))

Assuming m x is integer and a≥0. And using the fact that \(\Gamma \left (m_{x},x\right)=\left (m_{x}-1\right)!e^{-x}\sum _{i=0}^{m_{x}-1}\frac {x^{i}}{i!}\), we have:

$$ \frac{\Gamma(m_{x},az)}{\Gamma(m_{x})}=\frac{\Gamma(m_{x},az)}{(m_{x}-1)!}=e^{-az}\sum_{i=0}^{m_{x}-1}\frac{a^{i}}{i!}z^{i} $$
((37))

With the help of the equation \(\int _{u}^{\infty }x^{n}e^{-\mu x}dx=\mu ^{-n-1}\Gamma (n+1,\mu u)\) ([19], equation (3.351.2)), we can get:

$$ {}{\fontsize{8.8pt}{9.6pt}\selectfont{\begin{aligned} \int_{z=B}^{\infty} \frac{\Gamma(m_{x},az)}{\Gamma(m_{x})}p_{k}(z)dz&\,=\, \left(\frac{\frac{m_{z}}{\gamma_{z}}}{\frac{m_{z}}{\gamma_{z}}+a}\right)^{m_{z}} \frac{1}{\Gamma(m_{z})} \sum_{i=0}^{m_{x}-1} \left(\frac{a }{\frac{m_{z}}{\gamma_{z}}+a}\right)^{i}\\ &\quad\times\!\frac{\Gamma\left(m_{z}+i,\left(\frac{m_{z}}{\gamma_{z}}+a\right)B\right)}{i!} \end{aligned}}} $$
((38))

Let \(a=\frac {m_{\textit {xk}}A}{\gamma _{\textit {xk}}B}\) and using (38), we can rewrite (36) as:

$$ F_{t_{k}}(A)=1-\Phi\left(m_{xk},\gamma_{xk},m_{zk},\gamma_{zk},B,A\right) $$
((39))

where:

$${} {\fontsize{8.4pt}{9.6pt}\selectfont{\begin{aligned} &\Phi(m_{xk},\gamma_{xk},m_{zk},\gamma_{zk},B,A)=\frac{\Gamma\left(m_{xk},\frac{m_{xk}A}{\gamma_{xk}}\right)}{\Gamma(m_{xk})} \left(1-\frac{\Gamma\left(m_{zk},\frac{m_{zk}B}{\gamma_{zk}}\right)}{\Gamma(m_{zk})}\right)\\ &-\left(\frac{\frac{m_{zk}}{\gamma_{z}}}{\frac{m_{zk}}{\gamma_{zk}}+\frac{m_{xk}A}{\gamma_{xk}B}}\right)^{m_{zk}} \frac{1}{\Gamma(m_{zk})} \sum_{i=0}^{m_{xk}-1} \left(\frac{\frac{m_{xk}A}{\gamma_{xk}B} }{\frac{m_{zk}}{\gamma_{zk}}+\frac{m_{xk}A}{\gamma_{xk}B}}\right)^{i} \\&\quad \times \frac{\Gamma\left(m_{zk}+i,\frac{m_{zk}}{\gamma_{zk}}B+\frac{m_{xk}}{\gamma_{xk}}A\right)}{i!} \end{aligned}}} $$
((40))