 Research
 Open Access
Optimal resource allocation for cognitive radio networks with primary user outage constraint
 Peng Lan^{1},
 Lizhen Chen^{1},
 Guowei Zhang^{2} and
 Fenggang Sun^{1}Email author
https://doi.org/10.1186/s1363801504654
© Lan et al. 2015
 Received: 15 September 2015
 Accepted: 16 October 2015
 Published: 31 October 2015
Abstract
In this paper, we investigate the problem of power allocation in cognitive underlay networks, where a secondary user (SU) is allowed to coexist with a primary user (PU). We consider three transmission models for the secondary link: (i) oneway transmission with relay assisted, (ii) twoway transmission with a direct link, and (iii) twoway transmission with relay assisted. In conventional interferencelimited cognitive networks, the instantaneous channel state information (CSI) of a PU is required to suppress SU’s transmit power to guarantee the quality of service (QoS) of the PU, which increases the feedback burden in practice. To tackle this issue, in this article we take primary outage probability as a new criterion to measure the QoS of the PU, where only the statistical CSI of the PU is required. Firstly, we derive the primary outage constraints for the three models, respectively. Then, with the newly obtained constraints, we formulate optimization problems to maximize the channel rate of the SU. Finally, we derive the optimal solutions for power allocation with respect to different parameters, respectively. Simulation results verify the performance improvement of the proposed schemes.
Keywords
 Cognitive radio
 Cooperative communication
 Power allocation
 Quality of service
 Twoway networks
1 Introduction
In cognitive underlay networks, a secondary user (SU) is allowed to share the spectrum with a primary user (PU) as long as the qualityofservice (QoS) requirement of the primary transmission is guaranteed [1]. The main advantage of cognitive underlay systems lies in its efficient utilization of radio spectrum, which makes it as a promising solution to tackle the spectrum scarcity problem [2]. However, the transmit power of the SU needs to be strictly controlled to satisfy PU’s QoS, which consequently degrades SU’s performance. To tackle this issue, power allocation and cooperative relaying techniques are considered as two potential ways to improve SU’s performance.
To protect the PU while optimizing the SU’s performance, various power allocation (PA) strategies have been investigated for underlay cognitive relay networks [9, 10]. Subject to average/peak interference power constraint for PU, an optimal PA strategy was proposed in [9] to achieve the ergodic capacity for SU in cognitive oneway relay networks. With power limit for SU being further considered, the authors in [11] proposed optimal PA schemes to maximize the ergodic/outage capacity of SU. In [12], PA schemes were proposed under the joint constraints of outage probability requirement for PU and average/peak transmit power limit for SU. With relay selection, the secondary transmission can be further enhanced. Joint PA and relay selection was investigated in [13] to maximize the system throughput with limited interference to PU. In [14], the transmit power limit for SU was also taken into consideration. For amplifyandforward (AF) cognitive relay networks with multiple SUs, joint relay assignment and PA was proposed in [15]. In cognitive twoway networks, closedform solutions for optimal PA were derived in [16] under the joint peak interference constraint for PU. For cognitive twoway relaying networks, optimal PA and relay selection scheme were studied in [17], where a pair of secondary transceivers communicate with each other assisted by a set of twoway AF relays. Further, the problem of relay selection and PA for the twoway relaying cognitive radio networks was investigated in [10], where the relays select between the AF and decodeandforward (DF) protocols to maximize SU’s sum rate.
In the aforementioned works [9, 10], SU’s transmit power is controlled to keep the interference of PU under a predefined limit. In such interferencelimited cognitive networks, the instantaneous channel state information (CSI) of PU is required to be known at SU to protect PU. Since a secondary network is typically not coordinated with a primary network and no dedicated feedback channel is available from PU to SU, the instantaneous CSI of PU can hardly be obtained, and unreliable CSI results in violation of the interference constraint. To deal with this critical obstacle, Zou et al. investigated the problem of relay selection to maximize the received signaltointerferencenoise ratio (SINR) at SU under a novel independent primary outage constraint for PU [18], in which only the statistical CSI of PU is required at SU. The work in [18] restrained the transmit powers of secondary transmitter and secondary relay under individual primary outage constraint for PU. In [19], the authors introduced a new cooperative transmission scheme for overlay cognitive radio in which the secondary network exploits the primary retransmissions without requiring global CSI. Further in [20], the primary outage constraint due to secondary transmitter and secondary relay was jointly considered for the first time, and a closedform solution for optimal PA and relay selection was derived in DF cognitive relay networks.
In this article, we investigate the problem of power allocation to maximize the achievable rate of SU in cognitive networks. Three secondary transmission models are considered: (i) oneway transmission with relay assisted (OWTRA), (ii) twoway transmission with a direct link (TWTDL), and (iii) twoway transmission with relay assisted (TWTRA). We first derive the joint primary outage constraints for the three models, respectively. With these constraints, optimization problems for power allocation are formulated and optimal solutions are derived. Simulation results are provided to verify the performance improvement of our proposed schemes compared with the equal resource allocation schemes.

We derived the joint primary outage constraints for three secondary transmission models, OWTRA, TWTDL, and TWTRA, respectively. Compared to the traditional interferencelimited constraint for PU, the key advantage is that only the statistical CSI of PU is required at SU, which is more practical.

We proposed power allocation schemes for OWTRA, TWTDL, and TWTRA to maximize the achievable rate of SU, respectively. In addition, we also proposed power allocation schemes to ensure fairness for TWTDL and TWTRA.

For comparison, we provided corresponding lowcomplexity equal resource allocation (ERA) schemes for each model. Simulation results show that our proposed schemes outperform greatly the corresponding ERA scheme. The reason is that primary outage constraint due to secondary transmission is considered jointly in our proposed schemes, while it is considered individually in the ERA schemes.
The rest of this article is organized as follows. System model and the joint primary outage constraint of the three transmit models are, respectively, described in Sections 2 and 3. Then, the proposed power allocation schemes of different models are provided in Sections 4, 5, and 6, respectively. Simulation results are given in Section 7. Section 8 concludes the article.
2 System and channel model
In this article, we consider underlay cognitive networks with three secondary transmission models as shown in Fig. 1, in which primary system and secondary system coexists simultaneously. In the primary system, a primary transmitter (PT) sends data to a primary destination (PD). The secondary system is a cooperative relay system which consists of a pair of transceivers (denoted as S1 and S2) and/or M SRs, in which SR_{ i } (for i=1,…,M) denotes the ith relay. The M relays are considered to be in a cluster, so that they are assumed to be approximately at the same position. This assumption simplifies the analysis and can represent a number of practical scenarios [19, 21, 22]. Time division multiple access (TDMA)based protocol is used for the secondary transmissions, and three different transmission models for the secondary system are considered in this article.
Model 1—oneway transmission with relay assisted (OWTRA)
In the OWTRA model as shown in Fig. 1 a, S1 is the transmit node and S2 is the receive node. The secondary transmission between S1 and S2 is assisted by SRs (i.e. S1→ SR →S2). The whole transmission process is divided into two phases equally. In the first phase, S1 broadcasts its message and SRs receive. In the second phase, the best relay is selected to amplify the received signal and forward it to S2.
Model 2—twoway transmission with a direct link (TWTDL)
In the TWTDL model, S1 and S2 transmit to each other without the assistance of SRs (i.e. S1⇔S2). The transmission procedure is shown in Fig. 1 b. The whole secondary transmission process is divided into two phases equally, where S1 transmits to S2 in the first phase and S2 transmits to S1 in the second phase.
Model 3—twoway transmission with relay assisted (TWTRA)
In TWTRA model, S1 and S2 exchange information with the help of SRs (i.e. S1⇔ SR ⇔S2) when no direct link between S1 and S2 is available. The twoway secondary transmission is completed in three equal phases, as shown in Fig. 1 c. S1 and S2 transmit to SRs in the first and second phases, respectively. In the third phase, a decodeandforward (DF) protocol is considered, the best SR decodes the signals received from S1 and S2 and jointly encodes the signal through XOR operation and forwards the signal to S1 and S2.
In the primary transmission, assume that PT transmits signal x _{ P }(E(x _{ P }^{2})=1) to PD with fixed power P _{ PT } and a data rate R _{ P }; in the meantime, S1 and/or S2 intends to reuse this time resource to transmit their signals x _{ S1} and x _{ S2}(E(x _{ S1}^{2})=1 and/or E(x _{ S2}^{2})=1) to each other with powers P _{ S1} and P _{ S2}, respectively. The channels are invariant during the transmission phases. The channel gain between any transmitter i∈{S1,S2,SR,PT} and any receiver j∈{S1,S2,SR,PD} is denoted as h _{ i−j }. Assuming all links are independent and identically distributed (i.i.d.), zeromean Rayleigh flat fading channels with variance \(1 \big / {\sigma ^{2}_{i  j}}\), where \(\sigma _{i  j}^{2}\) is defined as \(\sigma _{i  j}^{2}=d_{ij}^{\gamma }\), d _{ i−j } is the distance between transmitter i and receiver j, and γ is the pathloss exponent [19]. It is assumed that SU has the instantaneous CSI of the secondary transmission links and the links from PT to SU, which can be obtained by a pilotaided channel estimation or CSI feedback [14, 23]. Moreover, the two SUs are assumed to know the average CSI between themselves and PD and also the average CSI between PT and PD. The thermal noises of receivers n _{ j }(j∈{S1,S2, SR, PD}) are modeled as additive white Gaussian noises (AWGN) with mean zero and variance N _{0}.
3 The primary outage constraint
In cognitive underlay networks, SU is allowed to share PU’s spectrum on a condition that the QoS of the primary transmission is not affected. Therefore, the transmit power of SU must be allocated appropriately to satisfy the primary QoS requirement. Traditionally, SU’s transmit power is limited by the peak/average interference constraint for PU [9, 10]. Specifically, SU can access PU’s licensed spectrum as long as the induced interference from SU to PU is below the threshold. In such an interferencelimited network, the instantaneous CSI of the link from SU to PU is required. However, this CSI can hardly be obtained, since the secondary network is typically not coordinated with the primary network and no dedicated feedback channel is available from PU to SU.
To tackle this issue in this article, we use the primary outage probability as a metric to quantify the QoS of the primary transmission. Specifically, we consider that the primary outage probability should be kept below a predefined threshold Pout _{ P r i,T h r }. The main advantage is that only the statistical CSI of the link from SU to PU is required, which is more practical. In the following, we derive the primary outage constraints for the three transmission models, OWTRA, TWTDL, and TWTRA, respectively.
which should satisfy the constraint Pout _{ Pri }≤Pout _{ P r i,T h r }.
3.1 The primary outage constraint for the OWTRA model
where \(\rho = \left ({1  P\mathrm {out_{\Pr i,Thr}}} \right)\exp (\frac {{{2^{{R_{P}}}}  1}}{{{\widehat P_{\text {PT}}}\sigma _{\mathrm {PT  PD}}^{2}}})\), \(g=\widehat P_{\text {PT}}\sigma _{\mathrm {PT  PD}}^{2}\), and \({\lambda _{n}} = \sigma _{n \text {PD}}^{2}\left ({{2^{{R_{P}}}}  1} \right), n \in \{ S1,\text {SR}\}\). \(\widehat P_{\text {SR}_{i}}\) is the equivalent transmit power of the ith relay, and \({\widehat P_{S1_{(i)}}}\) is the equivalent transmit power of S1 corresponding to the ith relay.
For the underlay cognitive networks, the QoS of PU should be conservatively guaranteed, that is, Pout _{ P r i,T h r } should take a small value or at least no larger than 0.5 [12]. Therefore, ρ>0.5 always holds. Moreover, since the upper bound of \(\frac {g}{{{\widehat P_{S1_{(i)}}}\lambda _{S{1 }}+ g}} + \frac {g}{{{{\widehat P_{\text {RS}_{i}}}\lambda _{\text {SR}_{i} }} + g}}\) is 2, the secondary transmission is enabled only when ρ<1. Otherwise, the powers \({\widehat P_{S1_{(i)}}}\) and \({\widehat P_{\text {SR}_{i}}}\) should be set to zero, and the secondary transmission is not available. Based on the above analysis, the constraint 0.5<ρ<1 will always be satisfied for SU power allocation in this model.
3.2 The primary outage constraint for the TWTDL model
where g, ρ, and λ _{ n },n∈{S1,S2} follow the similar definition as in the OWTRA model.
3.3 The primary outage constraint for the TWTRA model
where \(\widehat {P}_{S1_{(i)}}\) and \( \widehat {P}_{S2_{(i)}}\) denote the equivalent transmit powers of S1 and S2 corresponding to the ith relay.
4 Power allocation for the OWTRA model
In this section, the problem of power allocation is studied to maximize the achievable rate of SU for the OWTRA model.
4.1 Equal resource allocation
As noted in Section 3, the equivalent powers \({\widehat {P}_{S1_{(i)}}}\) and \({\widehat {P}_{\text {SR}_{i}}}\) must satisfy the primary outage constraint (5). A simple but not optimal way to meet the constraint without coordination between S1 and SR_{ i } would be \(\frac {g}{{{\widehat {P}_{S1_{(i)}}}\lambda _{S{1 }}+ g}} \ge \rho \) and \(\frac {g}{{{{\widehat {P}_{\text {SR}_{i}}}\lambda _{\text {SR}_{i} }} + g}} \ge \rho \) [18], and the equivalent transmit powers \({\widehat {P}_{S1_{(i)}}}\) and \({\widehat {P}_{\text {SR}_{i}}}\) should satisfy
We denote this scheme as the equal resource allocation (ERA) scheme. From (15), the powers of S1 and SR_{ i } are dominantly determined by the QoS requirement of PU and the average channel gain of the links from PT to PD and from itself to PD. The static property allows the ERA scheme to allocate powers individually with low complexity. However, the only concern of the ERA scheme is to guarantee the primary transmission while the secondary transmission is ignored; therefore, it can not reach an optimal performance. Next, we will jointly allocate the transmit power between S1 and SR by taking into consideration the primary outage constraint (5).
4.2 Optimal power allocation
subject to
From the derived solution (20), the optimal powers of S1 and SR can be allocated. Different from the ERA scheme in (15), the powers of S1 and SR_{ i } are decided not only by the QoS requirement of PU and the average channel gain of the links from PT to PD and from itself to PD but also by the instantaneous channel gain of the secondary link. Moreover, the average channel gains of the interference links from S1 and SR to PD are jointly considered in our schemes. To be more specific, if the average channel gain of one interference link is dominantly stronger that the another, without loss of generality, we assume λ _{ S1} is larger than \({\phantom {\dot {i}\!}\lambda _{\text {SR}_{i}}}\). According to (20), the proposed scheme will suppress the transmit power of S1 but allocate more power to SR_{ i } without violating the primary outage constraint. However, the ERA scheme only suppresses the power of S1 when λ _{ S1} is larger than \({\phantom {\dot {i}\!}\lambda _{\text {SR}_{i}}}\). Therefore, the proposed scheme can reach a better tradeoff between S1 and SR than the ERA scheme, which can enhance the secondary transmission obviously.
5 Power allocation for the TWTDL model
in which the two intermediate parameters of Eq. (22) are given as \(\alpha = \frac {{{{ {{h_{S1  S2}}} }^{2}}}}{{{\widehat {P}_{\text {PT}}}{{ {{h_{\text {PT}  S2}}} }^{2}} + {1}}}\) and \(\beta =\frac {{{{ {{h_{S1  S2}}} }^{2}}}}{{{\widehat {P}_{\text {PT}}}{{ {{h_{\text {PT}  S1}}} }^{2}} + {1}}}\).
In the TWTDL model, we enhance the secondary transmission by power allocation, and the two different goals are as follows: (1) to maximize the sum achievable rate and (2) to maximize the achievable rate of the weaker link which is referred as to guarantee fairness.
5.1 Equal resource allocation
respectively. The ERA scheme in the TWTDL also allocates powers individually. However, when considering the primary outage constraint, it is more meaningful to improve the SU’s performance based on the joint primary outage constraint for S1 and S2. The SU performance desires further improvement through jointly allocating the transmission powers of S1 and S2.
5.2 Optimal power allocation for data rate maximization (DRM)
Take the limit \({\widetilde {P}_{S1}}\! \ge 1\) into account, we can see that when \(\frac {{{2\widetilde {\beta } \rho + 1} }}{{{2\widetilde {\alpha } \rho + 1} \! }} < {\left ({2\rho  1} \right)^{2}}\), the partial derivative \(\frac {{dg\left ({{{\widetilde {P}}_{S1}}} \right)}}{{d{{\widetilde {P}}_{S1}}}}\! > \!0\) and Eq. (30) is a monotonously increase function with \({\widetilde {P}_{S1}}\). Therefore, \(g\left ({{{\widetilde {P}}_{S1}}} \right)\) can be maximized by maximizing \({\widetilde {P}_{S1}}=\frac {1}{2\rho 1}\) and \({\widetilde {P}_{S2}}=1\), i.e., \({ \widehat {P}_{S1}=\frac {2(1\rho)g}{(2\rho 1)\lambda _{S1}}}\) and \(\widehat {P}_{S2}=0\). In this case, the twoway transmission is retrograded as a oneway transmission, i.e., S1→S2. Similarly, if \(\frac {{ {2\widetilde {\alpha } \rho + 1}}}{{ {2\widetilde {\beta } \rho + 1}}} \le {\left ({2\rho  1} \right)^{2}}\), the twoway transmission is retrograded as a oneway transmission from S2 to S1 with powers \(\widehat {P}_{S1}=0\) and \({\widehat {P}_{S2}=\frac {2(1\rho)g}{(2\rho 1)\lambda _{S2}}}\). With the above analysis, if \(\min \left [{\frac {{{2\widetilde {\beta } \rho + 1} }}{{{2\widetilde {\alpha } \rho + 1} }},\;\frac {{ {2\widetilde {\alpha } \rho + 1}}}{{ {2\widetilde {\beta } \rho + 1}}}} \right ] \le {\left ({2 \rho  1} \right)^{2}}\), the sum achievable rate can be maximized by \(\text {max} \left [\alpha \frac {2(1\rho)g}{(2\rho 1)\lambda _{S1}},\beta \frac {2(1\rho)g}{(2\rho 1)\lambda _{S2}}\right ]\). Specifically, if \(\frac {\alpha }{\beta } > \frac {{{\lambda _{S1}}}}{{{\lambda _{S2}}}}\), the link of S1→S2 is enabled with powers \(\left ({\widehat {P}_{S1}^{\text {opt}},\widehat {P}_{S2}^{\text {opt}}} \right) = \left [ {\frac {{2(1  \rho)g}}{{(2\rho  1){\lambda _{S1}}}},0} \right ]\). Otherwise, if \(\frac {\alpha }{\beta } \le \frac {{{\lambda _{S1}}}}{{{\lambda _{S2}}}}\), the link of S2→S1 is enabled with powers \(\left ({\widehat {P}_{S1}^{\text {opt}},\widehat {P}_{S2}^{\text {opt}}} \right) = \left [0, {\frac {{2(1  \rho)g}}{{(2\rho  1){\lambda _{S2}}}}} \right ]\).
5.3 Optimal power allocation for data rate fairness (DRF)
with constraints (27a) and (27b).
and \(\widehat {P}_{S1}^{\text {opt}}\) can be achieved as shown in (38).
6 Power allocation for the TWTRA model
In the cognitive TWTRA network, assume that the signal is severely attenuated between the two transceivers; thus, the direct link is not considered for transmission. The twoway transmission is completed with the help of secondary relays. The TDMA protocol is used, and the secondary transmission process is divided into three phases equally.
6.1 Equal resource allocation
With the similar analysis as the ERA schemes in the OWTRA and TWTDL networks, the ERA scheme in the TWTRA network allocates powers of \(\widehat {P}_{S1}\), \(\widehat {P}_{S2}\), and \(\widehat {P}_{\textit {SR}}\) individually, which should satisfy the outage constraint (9). A simple way is to let \(\frac {g}{{{\widehat {P}_{S1_{(i)}}}\lambda _{S1} + g}}=\rho \), \(\frac {g}{{{\widehat {P}_{S2_{(i)}}}\lambda _{S2} + g}}=\rho \), and \(\frac {g}{{{\widehat {P}_{\text {SR}_{i}}}\lambda _{\text {SR}_{i}} + g}}=\rho \), and the equivalent transmit powers of the ERA scheme are \(\widehat {P}_{S1_{(i)}}^{\text {ERA}}=\frac {g(1\rho)}{\rho \lambda _{S1}}\), \(\widehat {P}_{S2_{(i)}}^{\text {ERA}}=\frac {g(1\rho)}{\rho \lambda _{S2}}\), and \(\widehat {P}_{\text {SR}_{i}}^{\text {ERA}}=\frac {g(1\rho)}{\rho \lambda _{\text {SR}_{i}}}\) with \(\widehat {P}_{SR_{i, (1)}}^{\text {ERA}}=\widehat {P}_{SR_{i, (2)}}^{\text {ERA}}=\frac {1}{2}\widehat {P}_{\text {SR}_{i}}^{\text {ERA}}\).
6.2 Optimal power allocation scheme for data rate maximization (DRM)
In this section, we consider the power allocation to maximize the sum achievable rate for the cognitive TWTRA networks.
The optimal power allocation is to maximize the sum achievable rate C _{ S i,Sum}. It is obvious from Eqs. (44) and (45) that the achievable rate of S m(m=1,2) is determined by the minimal value of the twohop links: S m→SR_{ i } and SR_{ i }→S n (m,n=1,2 and m≠n). Therefore, the two links should achieve the same SINR, i.e., \({\alpha _{Sm  \text {SR}_{i}}}{\widehat {P}_{Sm_{(i)}}} = {\alpha _{\text {SR}_{i}  Sn}}{\widehat {P}_{S{R_{i,(n)}}}}\). We have \({\widehat {P}_{\text {SR}_{i,(1)}}} = \beta _{1i}{\widehat {P}_{S1_{(i)}}}\), \({\widehat {P}_{\text {SR}_{i,(2)}}} = \beta _{2i}{\widehat {P}_{S2_{(i)}}}\), and \({\widehat {P}_{\text {SR}_{i}}} = {\widehat {P}_{S{R_{i,(1)}}}} + {\widehat {P}_{\text {SR}_{i,(2)}}} = \beta _{1i}{\widehat {P}_{S1_{(i)}}} + \beta _{2i}{\widehat {P}_{S2_{(i)}}}\), where \(\beta _{1i}=\frac {\alpha _{S1  \text {SR}_{i}}}{\alpha _{\text {SR}_{i}  S2}}\) and \(\beta _{2i}=\frac {\alpha _{S2  \text {SR}_{i}}}{\alpha _{\text {SR}_{i}  S1}}\).
where λ _{1} and λ _{2} represent nonnegative dual variables.
It is noteworthy that solving a dual problem is not always equivalent to solving the primal problem. It has been proven from duality theory that the optimal duality gap d=D ^{∗}−f ^{∗}≥0 always holds [25], where D ^{∗} and f ^{∗} denote the primal and dual optimal values, respectively. The optimal duality gap d=0 when the primal problem is convex. For the problem of our interest, the objective function (47) is concave, and constraints (48a) and (48b) are convex; therefore, the primal and dual problems have the same optimal solutions.
According to [26], the dual problem in (51) can be further decomposed into the following sequentially iterative subproblems:
where (·)^{+}= max(·,0), \({u_{m}} = \frac {1}{{{\alpha _{Sm  \text {SR}_{i}}}}} + 2\frac {{{g}}}{{{\lambda _{\textit {Sm}}}}}  \frac {1}{{\lambda _{m}}{\beta _{\textit {mi}}}}\), \({v_{m}} = {\left ({\frac {{{g}}}{{{\lambda _{\textit {Sm}}}}}} \right)^{2}} + 2\frac {{{g}}}{{{\alpha _{Sm  \text {SR}_{i}}}{\lambda _{\textit {Sm}}}}} + \frac {{({\lambda _{\overline m}}  2){g}}}{{{\lambda _{m}}{\beta _{\textit {mi}}}{\lambda _{\textit {Sm}}}}}\), \({w_{m}} = \left (\frac {1}{{{\alpha _{Sm  \text {SR}_{i}}}}}  \frac {1}{{{\lambda _{m}}{\beta _{\textit {mi}}}}}\right){\left (\frac {{{g}}}{{{\lambda _{\textit {Sm}}}}}\right)^{2}} + \frac {{{\lambda _{\overline m}}}}{{{\lambda _{m}}}}\frac {1}{{{\alpha _{Sm  \text {SR}_{i}}}{\beta _{\textit {mi}}}}}\frac {{{g}}}{{{\lambda _{\textit {Sm}}}}}\), \({p_{m}} = {v_{m}}  \frac {{{u_{m}^{2}}}}{3}\), \({q_{m}} = {w_{m}} + \frac {{2{u_{m}^{3}}}}{{27}}  \frac {{{u_{m}}{v_{m}}}}{3}\), and \(\overline m (\ne m) \in \{ 1,2\}\).
Through the subgradient method, the powers for all relays can be allocated.
6.3 Optimal power allocation scheme for data rate fairness (DRF)
To guarantee the fairness between S1 and S2, the sum achievable rate with respect to the ith relay can be denoted as
We will now determine the optimal equivalent power values \({\widehat {P}_{S1_{(i)}}}\), \({\widehat {P}_{S2_{(i)}}}\), \({\widehat {P}_{\text {SR}_{i,(1)}}}\), and \({\phantom {\dot {i}\!}P_{\text {SR}_{i,(2)}}}\) that lead to the fairness between S1 and S2, i.e., \({C_{S1_{(i)}}} = {C_{S2_{(i)}}}\) and then maximize the system achievable rate \({C_{S_{i}, \text {Fair}}}\).
Equation (59) is the univariate cubic equation about \(\widehat {P}_{S1_{(i)}}\) which can be solved by Cardano’s formula. Once the best power allocation for S1 is solved, the best power allocation for SR and S2 can also be achieved.
7 Simulation results
In this section, we evaluate the performance of the proposed power allocation schemes through MonteCarlo simulations and compare them with the traditional ERA schemes [18]. We assume throughout that the channel coefficients are i.i.d. and follow Rayleigh distribution. The following parameters are used throughout this section: γ=4, \({d_{\text {PT}  \text {PD}}}={d_{S1  \text {SR}_{i}}} = {d_{S2  \text {SR}_{i}}} = \frac {1}{2}{d_{S1  S2}}=1\), d _{ S1−PD}=d _{PT−S2}=4, d _{PT−S1}=d _{PT−SR}=d _{SR−PD}=d _{ S2−PD}=3, N _{0}=−50 dBW, and R _{ P }=1.5 bit/s/Hz.
8 Conclusions
In this article, we studied the problem of power allocation for underlay cognitive networks involving three different secondary transmission models: (i) OWTRA, (ii) TWTDL, and (iii) TWTRA, respectively. Since secondary network is typically not coordinated with primary network, the instantaneous CSI of PU can hardly be obtained. To tackle this issue, we adopted primary outage constraint to measure the QoS of primary transmission, where only the statistical CSI of PU required. We first derived the joint primary outage constraints due to the secondary transmission for the three transmission models. We then proposed power allocation schemes to maximize the received SINR for the cognitive oneway relay network, while in cognitive twoway scenarios, two power allocation criterions are considered: (i) fairness based and (ii) sum achievable rate maximization based, respectively. In order to further improve the SU’s performance, relay selection was also considered in the relayassisted transmission networks (e.g., OWTRA and TWTRA networks). The performance of the proposed schemes was illustrated for different operating conditions and shown to yield great enhancement compared to that of the corresponding ERA schemes.
Declarations
Acknowledgements
This work was supported in part by the Shandong Province High School Science & Technology Fund Planning Project (J12LN02) and the Research Fund for the Doctoral Program of Higher Education of China (20123702120016).
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
References
 Q Zhao, BM Sadler, A survey of dynamic spectrum access. IEEE Signal Process Mag.24(3), 79–89 (2007).View ArticleGoogle Scholar
 S Haykin, Cognitive radio: brainempowered wireless communications. IEEE J. Select. Areas. Commun.23(2), 201–220 (2005).View ArticleGoogle Scholar
 M Gastpar, M Vetterli, in On the capacity of wireless networks: The relay case. INFOCOM 2002. TwentyFirst Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE, vol. 3 (IEEENew York, 2002), pp. 1577–1586.Google Scholar
 G Kramer, M Gastpar, P Gupta, Cooperative strategies and capacity theorems for relay networks. IEEE Trans. Inf. Theory.51(9), 3037–3063 (2005).MATHMathSciNetView ArticleGoogle Scholar
 Q Zhang, J Jia, J Zhang, Cooperative relay to improve diversity in cognitive radio networks. IEEE Commun. Mag.47(2), 111–117 (2009).View ArticleGoogle Scholar
 B Rankov, A Wittneben, Spectral efficient protocols for halfduplex fading relay channels. IEEE J. Select. Areas. Commun.25(2), 379–389 (2007).View ArticleGoogle Scholar
 R Zhang, YC Liang, CC Chai, S Cui, Optimal beamforming for twoway multiantenna relay channel with analogue network coding. IEEE J. Select. Areas. Commun.27(5), 699–712 (2009).View ArticleGoogle Scholar
 C Sun, C Yang, Energy efficiency analysis of oneway and twoway relay systems. EURASIP J. Wirel. Commun. Netw.2012(1), 1–18 (2012).View ArticleGoogle Scholar
 A Ghasemi, ES Sousa, Fundamental limits of spectrumsharing in fading environments. IEEE Trans. Wirel. Commun.6(2), 649–658 (2007).View ArticleGoogle Scholar
 A Alsharoa, F Bader, MS Alouini, Relay selection and resource allocation for twoway dfaf cognitive radio networks. IEEE Wirel. Commun. Lett.2(4), 427–430 (2013).View ArticleGoogle Scholar
 X Kang, YC Liang, A Nallanathan, HK Garg, R Zhang, Optimal power allocation for fading channels in cognitive radio networks: Ergodic capacity and outage capacity. IEEE Trans. Wirel. Commun.8(2), 940–950 (2009).View ArticleGoogle Scholar
 X Kang, R Zhang, YC Liang, HK Garg, Optimal power allocation strategies for fading cognitive radio channels with primary user outage constraint. IEEE J. Select. Areas Commun.29(2), 374–383 (2011).View ArticleGoogle Scholar
 L Li, X Zhou, H Xu, GY Li, D Wang, A Soong, Simplified relay selection and power allocation in cooperative cognitive radio systems. IEEE Trans. Wirel. Commun.10(1), 33–36 (2011).View ArticleGoogle Scholar
 L Lu, Y Li, G Wu, Optimal power allocation for cr networks with direct and relayaided transmissions. IEEE Trans. Wirel. Commun.12(4), 1–11 (2013).View ArticleGoogle Scholar
 M Naeem, U Pareek, DC Lee, A Anpalagan, Power allocation and relay assignment for sharedband nonregenerative relaying in cognitive radio systems. IEEE Trans. Veh. Technol.62(6), 2853–2859 (2013).View ArticleGoogle Scholar
 X Kang, Optimal power allocation for bidirectional cognitive radio networks with fading channels. IEEE Wirel. Commun. Lett.2(5), 567–570 (2013).View ArticleGoogle Scholar
 P Ubaidulla, S Aissa, Optimal relay selection and power allocation for cognitive twoway relaying networks. IEEE Wirel. Commun. Lett.1(3), 225–228 (2012).View ArticleGoogle Scholar
 Y Zou, J Zhu, B Zheng, YD Yao, An adaptive cooperation diversity scheme with bestrelay selection in cognitive radio networks. IEEE Trans. Signal Process.58(10), 5438–5445 (2010).MathSciNetView ArticleGoogle Scholar
 SB Mafra, RD Souza, JL Rebelatto, EM Fernandez, H Alves, Cooperative overlay secondary transmissions exploiting primary retransmissions. EURASIP J. Wirel. Commun. Netw.2013(1), 1–12 (2013).View ArticleGoogle Scholar
 P Lan, F Sun, L Chen, P Xue, J Hou, Power allocation and relay selection for cognitive relay networks with primary qos constraint. IEEE Wirel. Commun. Lett.2(6), 583–586 (2013).View ArticleGoogle Scholar
 I Krikidis, J Thompson, JS McLaughlin, N Goertz, Amplifyandforward with partial relay selection. IEEE Commun. Letters. 12(4), 235–237 (2008).View ArticleGoogle Scholar
 DB Costa, S Aissa, Performance analysis of relay selection techniques with clustered fixedgain relays. IEEE Signal Proc. Lett.17(2), 201–204 (2010).View ArticleGoogle Scholar
 Jp Hong, B Hong, TW Ban, W Choi, On the cooperative diversity gain in underlay cognitive radio systems. IEEE Trans. Commun.60(1), 209–219 (2012).View ArticleGoogle Scholar
 S Boyd, L Vandenberghe, Convex Optimization (Cambridge university press, Cambridge, 2009).Google Scholar
 DT Ngo, C Tellambura, HH Nguyen, Resource allocation for ofdmabased cognitive radio multicast networks with primary user activity consideration. IEEE Trans. Veh. Technol.59(4), 1668–1679 (2010).View ArticleGoogle Scholar
 AG Marques, X Wang, GB Giannakis, Dynamic resource management for cognitive radios using limitedrate feedback. IEEE Trans. Signal Process.57(9), 3651–3666 (2009).MathSciNetView ArticleGoogle Scholar