Abstract
Cognitive radio is a novel approach to cope with spectrum scarcity, in which either a network or a wireless node changes its transmission or reception parameters to communicate efficiently. However, it is difficult to avoid the interference between licensed and unlicensed users in various scenarios. This paper analyzes the jointly optimized allocation of sensing time and power for a twouser, amplifyandforward (AF) cognitive network developed by maximizing the average aggregate throughput of its secondary network. In particular, this paper discusses diverse cooperation ratios for different scenarios and a unique cooperation ratio in spite of scenario changes. The observations of experiment results indicate that the sensing duration is within a strict interval. The results show that the optimized sensing time is 14.111 ms and the aggregate throughput equals to 1.1451 bps/Hz which are tractable by sequential optimization. This result indicates that by adopting the fixed cooperation ratios, the achievable throughput of the system is decreased. The system innovatively creates multiple independent fading channels to achieve technological diversity among partners.
Introduction
Mobile radio channels suffer from fading, which means that mobile users go through severe variations of signal attenuation within the duration of any given call. This motivates a new form of spatial diversity achieved via the cooperation of mobile users [1]. Cognitive radio (CR) [2] is a promising radio access method with the capacity to increase spectrum utilization in dynamically changing environments. It allows secondary users (SUs) to opportunistically access temporarily vacant spectrums of primary licensed networks without compromising the quality of service (QoS) of the primary users (PUs) of a network. In a CR network (CRN), spectrum sensing has a key role for the access of SUs without interference to PUs. This cooperative spectrum sensing has tradeoffs between energy consumption and system throughput [3]. Furthermore, it is important to formulate an optimization problem for the tradeoff between energy and throughput for SUs based on spectrum sensing efficiency [3]. Instead of selecting fixed sensing time for all SU frames, it seems to be encouraging to implement spectrum sensing policies to determine the sensing duration for the optimization of network performance for SUs [4]. Meanwhile, the evaluation of spectrum sensing takes energy consumption and cost into account [4]. The development of spectrum sensing also includes a multiband joint detection framework using an artificial immune system and clone selection theory to obtain the optimized solutions without any reformulations or mathematical costs [5]. To maximize SUs’ aggregate argotic throughput, researchers investigate the design of the optimized spectrum sensing time and power allocation schemes in a multibandsensing CR network [6]. This design can guarantee the QoS of PUs and satisfy power limitations of secondary transmitters. It shows that the achievable rate region of the cooperative systems will be larger than the capacity region of conventional systems (without cooperation between source nodes) [7].
However, it is difficult to avoid the interference between PUs (licensed) and SUs (unlicensed) users in various scenarios [8,9,10]. Particularly, it is insufficient in the analysis of the jointly optimized allocation of sensing time and power for a CRN with the focus of maximizing the average aggregate throughput of its secondary network [7,8,9,10]. Currently, there are mainly two different kinds of operations at the partner (or relays) level for CRN, which are distinguished as amplifyandforward and decodeandforward respectively. The characteristics of amplifyandforward method include simple power allocation strategies for AF cooperative relaying, the optimization of the sum and product of the average signaltonoise ratios (SNRs) of direct links, and an upper bound of the average SNR of relay links.
The optimization objectives of CRN usually include total throughput, joint sensing time, and power allocation under the constraints of average transmit power budget and average interference power budget [9]. For the optimization algorithms of CRN, researchers investigated greedy algorithm for optimization of energy efficiency, sensing time, and power allocation based on a nonconvex problem [10].
In 5G system [11], the benefits, principles, and potential solutions are introduced for LTEUnlicensed in chapter 20, while the cognitive radio is one of the solutions for LTEUnlicensed. And the joint energy and spectrum sharing for communication cooperation in 5G system is introduced, while the presentation of this paper is also suitable for joint energy and spectrum sharing in 5G.
Zhao and Kwak discussed a jointly optimization method for sensing time and power of a twouser cognitive network by maximizing the averaged aggregate throughput of the secondary network [12]. Nevertheless, their method does not take the power limitation of SUs into consideration, which makes the work unintegrated. Furthermore, the SUs adopt diverse cooperation ratios in different scenarios. Considering that channel conditions change frequently in the real world situations, scenarios should change frequently in the view of SUs. Thus, the method has to recompute frequently the optimized values of cooperation ratios of PUs and SUs. It could not avoid system delays in responses and is difficult to implement.
This paper discusses joint sensing time and power allocation schemes in cooperative CRN under different kinds of power constraints. It has invariable cooperation ratio which is suitable for frequent scenario changes (or under varied channel conditions). The research objective is to design a CRN with optimized sensing time and power allocation strategies as well as maximized average aggregate throughput. The design restrictions are to avoid causing harmful interference to PUs and exceed the power limitation of each SU node. The implementation conditions of the system are twoSUs and amplifyandforward cooperation.
The remainder of this paper is organized as follows. “System model and algorithm” describes the system model of cooperative sensing and transmission. It also discusses the algorithm for calculating joint sensing time and power allocation, with the assumption that two nodes are with different cooperation ratios. “Numerical results and discussion” shows the system simulations and discusses the proposed joint optimization. “Conclusions and future work” concludes the paper and suggests future work.
System model and algorithm
Figure 1 shows the system under consideration, in which two SUs (Nodes 1 and 2) wish to cooperate in transmission of data to another SU (Node 0). Nodes 1 and 2 adopt orthogonal channels (e.g., in a frequency domain) to transmit and relay the data. Therefore, in the view of Node 1 or Node 2, each of them needs three channels (two channels for transmission and one channel for acceptance) to carry out the proposed cooperative transmission. In the view of the whole network, there needs to be four channels: two channels for transmission and two channels for acceptance. The channels are all assumed to be memoryless with additional white Gaussian noise.
Figure 2 shows the frame structure and power allocation of Nodes 1 and 2, where Phase 0 is for the cooperative sensing of the PUs’ statuses. Phase 1 and Phase 2 are for the cooperative transmission of SUs’ data. In particular, Nodes 1 and 2 transmit their selfdata directly to Node 0 (i.e., the terminal node) in Phase 1 with the power \( {P}_1^t \) and \( {P}_2^t \), respectively. Nodes 1 and 2 receive data from their corresponding partners afterwards. In Phase 2, Nodes 1 and 2 help their partners to forward the data received in Phase 1 to Node 0 with the power \( {P}_1^c \) and \( {P}_2^c \), respectively.
Cooperative sensing
In this research, the cognitive network can opportunistically access a wideband spectrum licensed to a primary network. The wideband channel contains K nonoverlapping narrow subbands. As explained in Fig. 2, the spectrum is sensed by SUs in Phase 0. The SUs need to exploit the spectrum opportunities over the subbands, which requires to sense the presence of the PUs’ signals on all subbands simultaneously through wideband spectrum sensing. Equations 1a and 1b are binary hypotheses to decide whether the k ^{th} subband is occupied or not:
where z_{ k }(m) is the m ^{th} received sample from subband k; h_{ k } represents the channel detected on the k ^{th} subband. We assume that h_{ k } undergoes flat fading and is constant during the sensing period. s_{ k }(m) is the m ^{th} symbol transmitted on the subband k from the k^{th} PU. u_{ k }(m) is the noise of the process. For each subband, we compute the sum of received signal energy over an interval of M samples using the following formula \( {Z}_k\triangleq {\sum}_{m=1}^M{\left{z}_k(m)\right}^2 \), m ∈ {1, 2, …, M}.
We assume that all PUs’ signals are complexvalued phaseshift keying (PSK) signals and the noise has the circularly symmetric complex Gaussian (CSCG) values with the mean of zero and the variance of \( {\sigma}_{u,k}^2 \). For M samples, using central limit theorem, the statistic tests of the energy detector can be approximated by normal distributions. Eq. 2a and Eq. 2b [2] are the probabilities of detections and false alarms in the k^{th} subband:
where \( Q(x)=\frac{1}{\sqrt{2\pi }}{\int}_x^{+\infty }{e}^{\frac{y^2}{2}} dy \); ζ_{ k } is the received SNR. The PUs’ signal is measured at a secondary receiver under the hypothesis H_{1, k}. ε_{ k } is the threshold of the energy detector on subband k. f_{ s } is the sampling rate. For a target probability of detection \( {P}_{d_k} \), Eq. 3 calculates the probability of its false alarm [2]:
In the opportunistic spectrum access model, there are two scenarios in which a SU can operate at the PU’s frequency band [2]. These scenarios include (1) hypothesis H_{0, k}, in which the PU is not present and no false alarm is generated by the SU; and (2) hypothesis H_{1, k}, in which the PU is active but it is not correctly detected by the SU. Equation 4 shows the probability that subband k is accessible to the SU:
where Pr(H_{0, k}) is the probability that the PU is absent on subband k, and Pr(H_{1, k}) is the probability that the PU is present on subband k.
Cooperative transmission
In Phase 2, Node 1 and Node 2 amplify and forward the data received in Phase 1 by the following factors respectively.
where h_{ ij } captures the fading effect between nodes i ∈ {1, 2} and j ∈ {0, 1, 2}; \( {\sigma}_n^2 \) is the variance of zeromean Gaussian noise at node n ∈ {0, 1, 2}. We assume that h_{ ij } follows an exponential distribution with a mean of h_{ ij }; therefore, the amplitude gain \( \sqrt{h_{ij}} \) follows a Rayleigh distribution.
For simplicity, the total available power is fixed for each user. Hence, \( {P}_i^t+{P}_i^c=P \), where P is the total available power for each user. A cooperation ratio implies the ratio of the power used for selfinformation transmission over the total power. For example, β_{1} and β_{2}, are adopted by Node 1 and Node 2 respectively. Therefore, the corresponding power allocations are \( {P}_i^t={\beta}_iP \) and \( {P}_i^c=\left(1{\beta}_i\right)P \).
By applying the maximum combining ratio, the SNR achieved at the terminal node (Node 0) is the sum of the SNRs at each phase. Therefore, Eq.6a and Eq. 6b calculate the achievable SNR regions of Node 1 and Node 2 transmitted to Node 0 [13, 14]:
where \( {\gamma}_1=P{h}_{10}/{\sigma}_0^2 \), \( {\gamma}_2=P{h}_{20}/{\sigma}_0^2 \), \( {\gamma}_3=P{h}_{12}/{\sigma}_2^2 \), and \( {\gamma}_4=P{h}_{21}/{\sigma}_1^2 \). We further assume the following conditions: (1) a flat fading channel in every cooperation interval, (2) the fading gain does not change, and (3) the channel is independent with each other and is independent for each cooperation interval. Thus, h_{ ij } is replaceable by \( \overline{h_{i,j}} \). When the full channel state information is available, the optimized power allocation in a cooperative transmission is to the desirable value of β_{ i }, i ∈ {1, 2}.
Joint sensing time and power allocation
Based on the previous discussion, Node 1 or Node 2 needs three available channels to carry out the proposed cooperative transmission. Four channels are needed in the view of the whole network. To make sure that there is no collision at Node 0, we assume that Nodes 1 and 2 have the same vision of four available channels. The four channels are allocated to these two nodes through the higher layer protocol, which is out of the scope of this paper. Therefore, at least four common channels are available for Node 1 and Node 2. When K channels or subbands are available to the cognitive network, there are L possible scenarios for the SUs to use the channels.
Equation 8 calculates the occurrence probability of scenario l:
where Ψ_{ l } is the set of active channels in scenario l. \( {P}_{S_k,1} \) and \( {P}_{S_k,2} \) are calculated in Eq. 4 and reflect the accessible probabilities of subband k to Nodes 1 and 2 respectively.
One objective of this research is to maximize the averaged aggregate throughput of the cooperative SUs for a particular PUs’ detection probability and the available power of each SU node. Therefore, the cooperative sensing and transmission problem can be formulated as in Eq. 9a–9e.
where β = {{β_{i, l}, }, iϵ{1, 2}, lϵ{1, …, L}}. β_{i, l} reflects the cooperation ratio of Node i in the scenario l. γ_{c, l} = {γ_{1, l}, γ_{2, l}, γ_{3, l}, γ_{4, l}} is the channel fading index. γ_{ c , l } = {γ_{c, l}, ∀l}; Ψ = {Ψ_{ l }, ∀l}; \( \overline{P_{d_k}} \) is the target probability of detection on subband k. R_{ l } is the achievable throughput in scenario l. In a specific scenario, four available channels are allocated for the cooperative transmission. Thus, R_{ l } is achieved based on Eq. 9b. μ ∈ [0, 1] is the weighting factor to make the two SUs to achieve the desirable rates. The corresponding fading gains should be substituted into the expressions of R_{1}(β_{1, l}, β_{2, l}) and R_{2}(β_{1, l}, β_{2, l}) in Eq. 6a and Eq. 6b respectively.
Equation 9a calculates the throughput with consideration of the statistics of PUs’ activities and the imperfection of spectrum sensing (e.g., false alarm and miss detection). Basically, it presents the cooperation throughput that two SUs can achieve on average. The sensing time is correlated with the frame structure and difficult to change when implemented in a secondary network. The optimized sensing duration τ is independent of scenario varieties. This research focuses on identifying the optimized sensing duration τ. The research also targets on designing the optimized power allocation strategies, β_{i, l}, for each transmission node in a scenario l.
Optimization
The following discussion includes two optimizations, namely optimization in a settled channel condition and optimization in a changeable channel condition respectively. Generally, to develop an algorithm to identify the optimized design strategy needs to consider complex forms of the objective function and constraint set. The problem is tractable since for any given τ, we can find the optimized β_{i, l} by solving the optimization problem shown in the sequel. Furthermore, the optimized sensing time τ falls within the interval (0, T), which enables the optimized resolution using onedimensional exhaustive search.
Optimization in settled channel condition
For any given sensing time τ, the optimization in Eq. 9 is changed to the following problem:
After the determination of scenario l, the channel condition is stable. The optimized power allocation strategy (β_{1, l} and β_{2, l}) for the given scenario is independent of the activities of other scenarios. Hence, the optimization problem is equivalent to the following problem:
After the transformation of Eq. 11, the optimization becomes twodimensional, which simplifies the verification of R(β_{1, l}, β_{2, l}) as a concave function in Eq. 6. Further explanation of the verification is in Appendix 1. When the first derivatives of R(β_{1, l}, β_{2, l}) with respect to β_{1, l} and β_{2, l} are set to zero, the optimized values of β_{1, l} and β_{2, l} can be achieved in a scenario l. Therefore, the optimized power allocation vector β can be achieved by solving the optimization problem shown in Eq. 11 for various scenarios. In addition, the optimized sensing time can be found by the exhaustive search over (0, T). The transmission power in each scenario relates to the instantaneous transmission power of cooperative nodes. Equation 12 shows the additional transmission power constraint for the optimization problem formulated in Eq. 11.
By limiting β_{i, l} in a stricter interval β_{i, l}∈[0, \( {\overline{\beta}}_{i,l} \)], we can use the previously described method to calculate β_{1, l} and β_{2, l} for various scenarios.
In previous discussion, the SUs adopt diverse cooperation ratios in different scenarios. The channel conditions frequently change in real applications. Hence, scenarios change frequently in the view of SUs. Thus, the SUs have to frequently recompute the optimized values of cooperation ratios. This increases the difficulty of the strategy implementation. In this research, we develop joint sensing time and power allocation schemes in cooperatively cognitive networks with an invariable cooperation ratio at each cooperative node, in spite of scenario changes. Furthermore, we use experiment results to confirm the effectiveness of the ratio.
Optimization in changeable channel conditions
To simplify the system implantation, we adopt the invariable cooperation ratios to the cooperative nodes. Each cooperative node selects the fixed cooperation ratio in spite of scenario changes or the changes of channel conditions. The optimization problem transforms to maximizing the averaged throughput. In particular, the following discussion of joint optimization strategies includes the setting of no power constraint and average power constraint respectively.
To apply the invariable cooperation ratio in the system at each node, we need to modify the optimization problem shown in Eq. 9. We achieve the optimized β_{ i } by solving the optimization problem shown in the sequel. The optimized sensing time τ lies within the interval (0, T), which enables the optimized solution by a onedimensional exhaustive search.
In Eq. 10, the cooperation ratio changes in diverse scenarios. Considering that the research goal is to achieve the optimized value in each single scenario, the channel condition is invariable in each scenario. In comparison, in Eq. 13, the goal is to achieve a unique optimized cooperation ratio in diverse scenarios. The channel condition changes with the scenario changes. Therefore, the goal of the optimization is to find the values of β_{1} and β_{2} which maximize the system throughput with the constraint shown in Eq. 13c.
The second partial derivatives of the optimization goal are smaller than zero. They are ∂^{2}R/∂^{2}β_{1}<0 and ∂^{2}R/∂^{2}β_{2}<0 when β_{1},β_{2}∈[0,1]. Appendix 1 shows the details of the calculation process. Thus, the function has a concave surface and there must exist a single maximized value of the optimization goal R_{1}(τ, γ_{ c,l }, Ψ) when β_{1}, β_{2}∈[0,1]. Equation 14 helps to locate the optimized position. The optimized values of β_{1} and β_{2} are thus achieved.
Given that the transmission power is the same for all scenarios, we can calculate the average transmission power of the cooperative nodes using Eq. 15. The average transmission power constraint leads to an additional constraint in Eq. 13.
Numerical results and discussion
To test the system, we use a fixed frame of T = 100 ms for the secondary network. The target probability of detection is set to \( \overline{P_{d_k}} \) = 0.9 for all subbands. The PU is assumed to be a quadrature phaseshift keying (QPSK) modulated signal with the bandwidth of 6 MHz. The sampling frequency is the same as the bandwidth of the PU. The weighting factor μ is 0.6. We consider a total of K = 10 subbands in the experiment. The probability of a PU using each channel is Pr(H_{1, k}) = 0.2. Correspondingly, the received SNR of the PUs’ signal measured at Node 1 and Node 2 is set as [−20, −19, ..., −11] and [−11, −12, ..., −20] respectively. Without loss of generality, we make the following assumptions: the SUs have receivers with SNRs only decided by the corresponding transmission power and the SNRs are independent of the channels selected in the scenario. Thus, we set γ_{1}=6 dB, γ_{2} = 12 dB, γ_{3} = 20 dB, and γ_{4} = 24 dB (This value set demonstrates the invariability of β_{ l } over various scenarios.)
According to Eq. 10, we simulate the theoretical values of β_{1} and β_{2} in a specified scenario, and achieve the throughput of the cooperative CR network. Figure 3 shows the simulation results. In the figure, the optimized values of β_{1} and β_{2} are indicated by the green line and the blue line respectively. For example, if β_{2} is 0.2, the optimized value of β_{1} is 0.68, which is shown as a yellow dot in the figure. The value set achieves the maximum capacity of 3.2725. The joint optimized value of the capacity is achieved when β_{1}=1 and β_{2}=0.523, which makes the achievable maximum capacity of 3.4332.
Using Fig. 3, we can confirm the conclusion expressed in the research of Mesbah and Davidson [14]: During the cooperative transmission in a situation with an unequal cooperative ratio, at most one node will act as the relay for the boundary points of the achievable rate region. For instance, Nodes 1 and 2 decide to cooperatively transmit and not depend on the channel conditions of γ_{1}, γ_{2}, γ_{3}, and γ_{4}. The rate ratio of μ influences the cooperation degree. If γ_{2} is much larger than γ_{1}, Node 1 will wish to transmit its information through the relay from Node 2. Thus, the value of β_{2} will be smaller than 1 while the value of β_{1} will be equal to 1. At the same time, the value of μ influences the value of β_{2} which is the degree of cooperation.
In the following simulation, we set each node to be selfish, which means that they want to save their own power. Figure 4 shows the simulation results. When the power constraints of Eq. 10 is set to be β_{1}, β_{2} ≤ {1, 0.75, 0.5}, respectively, the optimized sensing time τ and the average aggregate throughput of the secondary network will be achieved. Furthermore, the optimized sensing time is τ = 14.111 ms. Meanwhile, the experiment results indicate that the aggregate throughput in this case equals to 1.1474 bps/Hz. Compared to the results in the research of Zhao and Kwak [12], this value is larger than the corresponding cooperative ratio (1.1189 bps/Hz in [12]). This result indicates that the action of adopting different cooperation ratios in two transmission/relaying nodes endues the cooperative system with more flexible cooperation and finally achieves a larger capacity.
In above simulations, we set the SUs to adopt diverse cooperation ratios in different scenarios. In the following part, we simulate the proposed optimization with certain fixed cooperation ratios in two cooperative nodes in spite of scenario changes. Most of the simulation parameters are set to be the same as those in the previous simulations. As explained in “Cooperative transmission”, the amplitude gain \( \sqrt{h_{ij}} \) follows a Rayleigh distribution. Thus, h_{i, j} follows an exponential distribution with the mean of \( \overline{h_{i,j}} \). To reflect the diversity of channels in various scenarios, we set γ_{1}, γ_{2}, γ_{3}, and γ_{4} to follow the exponential distributions with the means of 6, 12, 18, and 24 dB respectively.
Figure 5 shows the simulation results. The joint optimized value of the network capacity is achieved when β_{1}= 0.946 and β_{2}= 0.550, while the achievable maximum capacity is 2.9732. The maximum throughput is achieved when at most one node acts as a relay in a single scenario. However, to reduce the system complexity, the fixed cooperation ratios for two nodes despite scenario changes require that they act as the relays for each other simultaneously.
Figure 6 shows the simulation results achieved when the optimized sensing time τ = 14.111 ms. At the same time, the aggregate throughput in this case equals to 1.1451 bps/Hz, which is smaller than the previous one. This result indicates that by adopting the fixed cooperation ratios in two nodes in spite of scenario changes, the achievable throughput of the system is decreased. Afterwards, we set each node to be selfish and assume that they want to save their own power. We set the power constraints of Eq. 13 to be β_{1}, β_{2} ≤ {1, 0.75, 0.5}, respectively. The optimized sensing time τ and the averaged aggregate throughput of the secondary network are thus achieved.
To find out the optimized values, we simulate the optimizations under different power constraints, namely without power constraint and under average power constraint. The results are presented in Figs. 4 and 6, respectively. Considering that all the nodes are selfish in practice, partial power used for cooperation is more reasonable for the deployment. Hence, the situation shown in Fig. 6 would be more common in real world operations.
Conclusions and future work
In this paper, we addressed the research problem of joint sensing time and power allocation for cognitive networks with twouser amplifyandforward cooperative schemes. By specifying two cases, we discussed the joint optimization problem for different cases sequentially. Considering that the sensing duration of a network lies within a strict interval, the sensing time and the power are tractable by sequential optimization. The simulation results indicate that the sequential optimization is able to seize the globally optimized value satisfactorily. The research analysis also includes the optimized sensing time which is achieved by exhaustive search and is relatively independent of the power allocation procedure. Most importantly, by comparing with the outcomes from various power allocation schemes adopted in related literatures, the results in this research show that the proposed joint optimization is compatible with those schemes and can be introduced into practice with minor and appropriate modifications.
The limitation of this research is that only the AF cooperation is studied. In future work, we plan to study more flexible cooperation between two or more users. We also notice that the power budgets of the users are both fixed values, which is not flexible enough to satisfy the requirements of real systems. One potential research topic could be the schemes suitable for the systems with variable power budgets.
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Acknowledgements
This research was supported by the Education Department of Shaanxi Province (17JZ047, 17JK0425); the Special Foundation for Young Scientists of Xi’an University of Architecture and Technology (6040516148); and the Talent Technology Foundation of Xi’an University of Architecture and Technology (6040300613).
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Appendix 1: Proof of R(β1, l, β2, l) as a concave function
Appendix 1: Proof of R(β_{1, l}, β_{2, l}) as a concave function
The proof starts with the characteristics of a concave function. If the function R(β_{1, l}, β_{2, l}) is a second order derivative in a certain interval, the necessary and sufficient condition for this function to be a concave function is that \( \frac{\partial^2R\left({\beta}_i\right)}{\partial^2{\beta}_1}<0 \) and \( \frac{\partial^2R\left({\beta}_i\right)}{\partial^2{\beta}_2}<0 \).Therefore, it is necessary to prove that the second derivative of \( \frac{\partial^2\mathrm{R}\left({\upbeta}_{\mathrm{i}}\right)}{\partial^2{\upbeta}_1}<0 \) and \( \frac{\partial^2\mathrm{R}\left({\upbeta}_{\mathrm{i}}\right)}{\partial^2{\upbeta}_2}<0 \). The following steps are to prove \( \frac{\partial^2R\left({\beta}_i\right)}{\partial^2{\beta}_1}<0 \). The same explanation is applicable to the proof of \( \frac{\partial^2\mathrm{R}\left({\upbeta}_{\mathrm{i}}\right)}{\partial^2{\upbeta}_2}<0 \).

(1)
Calculate the first derivative:
where:

(2)
Calculate the second derivative:
where:
So we know that,
And we can say that,
From the process we know that \( \frac{\partial^2R\left({\beta}_i\right)}{\partial^2{\beta}_1} \)<0. Similarly, it is easy to prove that \( \frac{\partial^2R\left({\beta}_i\right)}{\partial^2{\beta}_2} \)<0. In conclusion, we knew that R(β_{1, l}, β_{2, l}) is a concave function.
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Ruan, S., Zhao, C., Jiang, S. et al. Joint sensing time and power allocation in cognitive networks with amplifyandforward cooperation. Ann. Telecommun. 73, 391–399 (2018). https://doi.org/10.1007/s1224301806258
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Keywords
 Cognitive radio
 Spectrum sensing
 Power allocation
 Amplifyandforward relaying
 Cooperative communications
 Optimization