Adaptive Power Allocation Schemes for Spectrum Sharing in Interference-Alignment-Based Cognitive Radio Networks

Interference alignment (IA) is a promising technique for interference management and can be applied to spectrum sharing in cognitive radio (CR) networks. However, the sum rate may fall short of the theoretical maximum, particularly at low signal-to-noise ratio (SNR), and the quality of service (QoS) of the primary user (PU) may not be guaranteed. In addition, power allocation (PA) in IA-based CR networks is largely ignored, which can further improve its performance. Thus, in this paper, PA in IA-based CR networks is studied. To guarantee the QoS requirement of the PU, its minimal transmitted power is derived. Then, we propose three PA algorithms to maximize the throughput of secondary users (SUs), the energy efficiency (EE) of the network, and the requirements of SUs, respectively, while guaranteeing the QoS of the PU. To reduce the complexity, the closed-form solutions of these algorithms are further studied in detail. The outage probability of the PU according to its rate threshold is also derived to analyze the performance of these algorithms. Moreover, we propose a transmission-mode adaptation scheme to further improve the PU's performance when its QoS requirement cannot be guaranteed at low SNR, and it can be easily combined with the proposed PA algorithms. Simulation results are presented to show the effectiveness of the proposed adaptive PA algorithms for IA-based CR networks.


I. INTRODUCTION
C OGNITIVE RADIO (CR) has attracted significant atten- tion as a technology to overcome the problem of spectrum scarcity [1], [2].In CR networks, spectrum sharing is a key technique allowing secondary users (SUs) to share the licensed spectrum of primary users (PUs) on the condition that the interference from SUs is not deemed harmful by the PUs [3].Generally, there are two types of spectrum sharing schemes, i.e., overlay and underlay spectrum sharing [4].In the underlay spectrum sharing, SUs can share the licensed spectrum with PUs, and the power of interference and noise at the primary receiver is constrained by interference temperature limit (ITL) [5]- [7].In [6], Clancy showed that the resulting performance of SUs from the interference temperature model is low, compared with the performance degradation of PUs due to the interference from SUs.Thus, it is still a key challenge in the underlay spectrum sharing to enhance SUs' rate while guaranteeing the quality of service (QoS) of PUs [7], [8].
Interference alignment (IA) is a promising technique for interference management [9], [10].Nevertheless, there are still some challenges when IA is utilized in practical systems, and one problem is the imperfect channel state information (CSI).Accurate global CSI should be available at all the transceivers to calculate the solutions of IA, which is difficult to achieve in practical systems, and there are several works that focus on solving this problem [11]- [16].On the other hand, the sum rate by IA can approach the capacity of the interference channel at very high SNR.However, it may decrease at moderate or low SNRs [15], since IA mainly focuses on eliminating interference, without involving the quality of desired signal [17].Some research works have focused on improving the performance of IA networks when SNR is low [15], [18].Gomadam et al. proposed a maximum signal-to-interferenceplus-noise ratio (SINR) algorithm for IA in [18] to optimize the SINR of the desired signal, and it was verified that the sum rate of interference networks can be improved obviously when SNR is low.However, its advantage tends to be lost when SNR becomes larger.An antenna-switching IA scheme was proposed to improve its sum rate in [15], and the performance degradation of IA at low SNR was also analyzed.In most of the early research works, only the equal power allocation (PA) scheme was adopted.Recently, PA and control has been adapted to IA to further improve its performance [19], [20].Farhadi et al. proposed a distributed power control algorithm for IA networks in [19], to ensure the data transmission at a fixed rate for each user.PA was introduced to IA by Shu et al., to optimize the throughput of IA networks in [20].
Due to its promising performance, IA has also been applied to CR networks [14], [17], [21]- [24].Yi et al. [21] identified the opportunity of using IA to exploit frequency-domain diversity from the available spectrum in CR networks to support transmission and improve the throughput of SUs.In [22], a MIMO CR network with relay was designed by Tang et al., and IA was used to enhance the achievable degrees of freedom (DoFs) for the network.Xu et al. [23] proposed a practical IA and cancelation algorithm for CR networks that can avoid the interference at the PU and optimize SUs' DoFs.In [14], [17], and [24], resource allocation was studied in CR networks to optimize the performance of the network.
In underlay spectrum sharing CR networks, ITL is usually adopted to guarantee the QoS of PUs [5]- [7].When IA is applied, ITL does not need to be considered because the interference among PUs and SUs can be eliminated perfectly [9].Thus, IA provides a convenient framework for spectrum sharing in CR networks free of interference.On the other hand, the received SINR of PUs in an IA-based CR network may decrease, compared with the scenario without IA and SUs.This may reduce PUs' QoS, although the residual interference is trivial [14], [17], [24].Therefore, the QoS of PUs in IA-based CR networks should be further improved and guaranteed.To this end, PA can be a potential candidate [25].
Although there exist some research works about PA in underlay spectrum sharing CR networks [26], [27], PA in IA-based CR networks is quite different due to the characteristics of IA.To the best of our knowledge, the PA problem in IA-based CR networks has not been studied systematically.The distinct features of this paper are as follows.
• PA is always an important issue in wireless communications [28].However, the PA problem in IA-based CR networks is largely ignored in the existing works.In this paper, we study the fundamental issues about PA in IA-based CR networks.• The PA problem in IA-based CR networks is quite different from the PA problem in traditional IA wireless networks because the QoS of PUs must be guaranteed.Thus, we derive the minimal transmitted power of the PU to guarantee its rate threshold in IA-based CR networks.This is an important metric for designing PA algorithms.• We propose three PA algorithms to maximize the sum rate of SUs, the energy efficiency (EE) of the network, and the satisfaction of SUs (SSU), respectively, in IA-based CR networks.To reduce the complexity of these algorithms, the closed-form solutions are studied in detail.The outage probability of a PU and SUs in the proposed algorithms is also derived according to its rate threshold with specific transmitted power.• When SNR is low, all the transmitted power may be allocated to the PU to guarantee its rate threshold.In this case, it does not make sense to still adopt the IA scheme.We propose a transmission-mode adaption scheme to further improve the PU's rate at low SNR, and it can be easily combined with the proposed PA algorithms.
The remainder of this paper is organized as follows.In Section II, the system model is presented, and the QoS requirement of the PU is analyzed.In Section III, the minimal transmitted power of the PU to guarantee its rate threshold is derived, three adaptive PA algorithms for IA-based CR networks are proposed, and the outage probability of the PU is derived according to its rate threshold.In Section IV, a transmissionmode adaptation (TMA) scheme is proposed to further improve the QoS of the PU.Simulation results are discussed in Section V, and finally, conclusions and future work are presented in Section VI.
Notation: I d represents the d × d identity matrix.A † and |A| are the Hermitian transpose and the determinant of matrix A, respectively.a is the 2 -norm of vector a. |a| is the absolute value of complex number a. C M×N is the space of complex M × N matrices.R N is the space of real N × 1 vectors.CN (a, A) is the complex Gaussian distribution with mean a and covariance matrix A. E(•) stands for expectation.

II. SYSTEM DESCRIPTION
Here, we first introduce the model for IA-based CR networks.Then, the QoS requirement of the PU is analyzed.

A. IA-Based CR Networks
Consider a K-user interference channel in a CR network as shown in Fig. 1, including one PU and K − 1 SUs sharing the spectrum in the same frequency band simultaneously.The PU can be seen as user 1, and users 2 to K are SUs.M [k]  and N [k] antennas are equipped at the kth transmitter and receiver, respectively.Perfect CSI of the network is assumed to be available at each node, and linear IA is harnessed to avoid interference among the PU and SUs in the CR network [9], [18].The received signal with d [k] data streams at the kth receiver can be expressed as where is the channel coefficient matrix from the jth transmitter to the kth receiver in the time slot n, with each of its entities independent and identically distributed (i.i.d.) and following CN (0, 1).We assume that the channels follow block fading [29].For clarity, the time slot number n is henceforth omitted.V [k] and U [k] are the unitary M [k] × d [k] precoding matrix, and N [k] × d [k] interference suppression matrix of the kth user, respectively.x [k] consists of d [k] data streams of user k with power constraint is the additive white Gaussian noise (AWGN) vector with distribution CN (0, σ 2 I N [k] ) at the receiver k, where σ 2 is the noise power at each antenna of the receiver.
When IA is feasible [30], the interference in the CR network can be assumed to be completely eliminated if the following conditions are met: [18] Thus, the desired signals of user k can be assumed to be received through a d [k] × d [k] full rank channel matrix H [kk] [k] ; thus, (1) can be rewritten as where z [k] = U [k] † z [k] , which also follows CN (0, Since this paper mainly concentrates on the adaptive PA of CR networks among different users instead of DoFs, it is assumed that there is only one data stream for each user in the rest of this paper.In addition, symmetric networks are considered, and all the users are assumed to have the same parameters, i.e., M [k] = M , N [k] = N , and d [k] = 1 for all k.Thus, the largest number of users that can be accommodated in the IA-based CR network should be [30] K ≤ M + N − 1. ( The transmission rate of user k in the IA-based CR network when interference is perfectly eliminated can be denoted as where [k] .u [k] and v [k] are the unitary precoding and decoding vectors for the kth user, respectively.

B. QoS Requirement of the PU in the IA-Based CR Network
In the underlay spectrum sharing CR networks, SUs can coexist with the PU on the condition that the interference from SUs will not be deemed harmful by the PU.The power of interference and noise at the primary receiver is usually constrained by the ITL, which can be used to guarantee the QoS requirement of the PU.
When IA is leveraged in the CR network, the interference among the PU and SUs can be eliminated perfectly, and the ITL can be always satisfied with reasonable power of the background noise because there is no residual interference at the primary receiver.Therefore, IA can provide a convenient framework for the spectrum sharing, in which the interference among the PU and SUs need not be considered any longer.
Nevertheless, the SINR of the received signal at the primary receiver will decrease [15], [18], compared with the scenario with one MIMO PU and no SUs.Thus, the problem of QoS requirement of PU should also be considered in the IA-based CR network.The transmission rate can reflect the variability of PU's QoS directly, and it is leveraged to measure the QoS of received signal in this paper.We define a rate threshold R [1] th according to the QoS requirement of the PU, and the following constraint should be satisfied based on the principle of CR: R [1] ≥ R [1] th . ( In the IA-based CR network, the SUs should try to satisfy the QoS requirement of the PU defined in (7); otherwise, they will not be allowed to access the licensed spectrum.

III. ADAPTIVE POWER ALLOCATION ALGORITHMS IN INTERFERENCE ALIGNMENT-BASED COGNITIVE RADIO NETWORKS
In most previous works of IA, equal transmitted power P t is allocated to each user as usually assumed.However, this may hinder the improvement of IA's performance.Here, PA among users in IA-based CR networks is studied, under the condition that the sum transmitted power of all the users is constrained to be lower than a constant, i.e., K k=1 P [k] t ≤ P max t .The minimal transmitted power of the PU to guarantee its rate threshold is first presented.Then, three adaptive PA algorithms with different objectives are proposed for IA-based CR networks.Finally, the outage probability of PU and SUs is analyzed.

A. Minimal Power of PU to Guarantee its QoS Requirement
In the IA-based CR network, when the PA among users is considered, the threshold of the PU's transmission rate should be satisfied.Proposition 1 is presented to define the minimal transmitted power of the PU that can guarantee its transmission threshold R [1] th .Proposition 1: To satisfy the threshold of the PU's rate in the IA-based CR network R [1] th , the transmitted power of the PU should be Proof: See Appendix A. When IA is adopted in CR networks, the residual interference at the primary receiver is trivial and can be assumed to be perfectly eliminated.Thus, we can deem P [1] t−min as the minimal transmitted power of PU to satisfy R [1] th requirement.Remark 1: To guarantee the QoS requirement of the PU in the IA-based CR network, P [1]  Thus, we can discuss the PA problem in the IA-based CR network as follows.
• P [1] t−min > P max t : This means that when the constraint of the sum transmitted power of the network P max t is all allocated to the PU (corresponding to its rate of R [1] max ), its rate threshold R [1] th still cannot be satisfied, as shown in Fig. 2. Thus, we should assign all the power P max t to the PU to maximize its rate, and the SUs cannot access the licensed spectrum.
• P [1] t−min ≤ P max t : If we want to optimize the performance of the network, P [1] t ∈ [P [1] t−min , P max t ] can be determined by the specific optimization problem.For example, when we want to optimize the performance of SUs, only P [1] t−min should be assigned to the PU to guarantee its rate threshold while maximizing SUs' performance.

B. PA Algorithm for Maximizing the Sum Rate of SUs
In the spectrum trading based CR network [31], the income of PUs is proportional to the sum rate of SUs they provided.Moreover, when there are multiple PUs selling spectrum to multiple SUs [31], the SUs can adapt their behavior by observing the variations in price and quality of spectrum offered by these PUs.Thus, the sum rate of SUs should be maximized by means of PU with its R [1] th constraint to maximize its utility and maintain trading with SUs, and a PA algorithm for maximizing the rate of SUs (PAMRSU) is proposed here.SUs' sum rate can also be called the spectrum efficiency if we consider unit bandwidth.
In the PAMRSU algorithm, when R [1] th constraint to the PU can be satisfied, i.e., P [1] t−min < P max t , allocate minimal power that can satisfy the R [1] th constraint to the PU, i.e., P [1] t = P [1] t−min .All the remaining power P max t − P [1] t−min is allocated to SUs to maximize their throughput.In this case, the PA optimization problem of the K − 1 SUs can be represented as t ,P [3] t ,...,P From (P1), we can see that it is similar to the PA problem in multiple parallel channels.Thus, the famous water-filling PA strategy [32] can be exploited to solve (P1) when P [1] t−min < P max t , and its closed-form solution can be represented as where x + max(x, 0), and ν should satisfy The closed-form solution of (P1) expressed in ( 11) and ( 12) is easy to obtain; thus, its computational complexity can be significantly reduced.The PAMRSU algorithm in each time slot can be expressed in Algorithm 1.

2: P
t−min is allocated to SUs by ( 11) and ( 12).6: else 7: Allocate P max t to the PU.8: SUs are switched into sleep mode.9: end if 10: Transmission for duration T with the power allocated.11: The time slot ends.

C. PA Algorithm for Maximizing the EE of the Network
EE becomes an important design criterion recently in wireless communications due to rapidly rising energy consumption in information and communication technology [33]- [36].The EE of IA-based CR networks can be defined as the transmitted information per unit frequency per Joule energy consumption (bits/Hz/Joule).The PA problem aiming at maximizing the EE of the whole CR network with R [1] th constraint of the PU can be formulated as where P [k] is the total power consumption of user k, which comprises the transmitter-circuit power consumption cr , and transmitted power P [k] t [35], [37].The objective function of (P2) has a concave numerator and an affine denominator with linear constraints; thus, (P2) is a concave-convex fractional programming [38].When P [1] t−min ≤ P max t , (P2) has optimal solution.To obtain the closed-form solution of (P2), Lemmas 1 and 2 are first provided.
Lemma 1: , the closed-form solution of (P2) can be calculated as where ν should satisfy | h [1] | can be denoted as h [1] = h [1] 2 Proof: See Appendix B. The solution of (P2) when K k=1 P [k] t = P max t as in Lemma 1 is different from that of (P1).This is because in (P1) it is required that P [1] t = P [1] t−min , whereas in (P2), the constraint is changed into P [1] t ≥ P [1] t−min .In addition, when SNR is low, can be satisfied after optimization of (P2), and Lemma 1 can be leveraged to obtain the solution.However, when SNR becomes higher, K k=1 P [k] t will become smaller than P max t to maximize the EE of the network, and the waterfilling strategy is no longer suitable.We will obtain the optimal solution of (P2) through fractional programming as in Lemma 2 and Theorem 1.
Lemma 2: We have an equation with variable λ as The solution of (17) λ * can be expressed as where Ψ(•) denotes the Lambert W function.
Proof: See Appendix C. Therefore, based on Lemmas 1 and 2, we can obtain the closed-form solution of (P2) as in Theorem 1 when P [1] t−min < P max t .Theorem 1: We define By substituting 20) by ( 19) and ( 20), λ * can be obtained.Thus, we can also define (P2) can be solved by the fractional programming, and its closed-form solution can be discussed as follows.

1)
The closed-form solution of (P2) can be defined as in (21). 2) The closedform solution of (P2) can be defined as in (14).
Proof: See Appendix D. Remark 2: When SNR becomes lower, (P2) may have no solutions.This happens when That is, when the lower bound of the PU's transmitted power P [1] t−min is larger than the constraint of P max t as in Fig. 2, the three constraints in (P2) cannot be satisfied simultaneously; thus, (P2) has no solutions.
Thus, we propose a PA algorithm for Maximizing the EE of the Network (PAMEEN) based on (P2).The PAMEEN algorithm can be represented in Algorithm 2.

2: P [1]
t−min is calculated according to (8).In Step 7 of Algorithm 2, SUs are turned into sleep mode, and the power consumption of SUs mainly arise from the leaking current of the switching transistors when circuits are properly designed [37].The power consumption of leaking current is usually much lower than the circuit power consumption in the active mode; thus, it can be neglected in the proposed PAMEEN algorithm in this paper, i.e., P

D. PA Algorithm for Maximizing the Satisfaction of SUs
In the proposed PAMRSU and PAMEEN algorithms, the rate constraint is imposed only on the PU, and there is no requirement for the rate of SUs.If some rate constraints on SUs are also involved, they should also be met on condition that the PU's threshold is satisfied.
Assume that the rate requirements of the K users are R [1] th , R [2] th ,. . ., R [K] th , and Proposition 1 is also suitable for SUs.Thus, the minimal value of transmitted power of user k to meet its rate requirement R [k] th can be expressed as The rate threshold of the PU R [1] th should be satisfied primarily in the IA-based CR network.If R [1] th can be met, we can allocate the remaining power to SUs to satisfy their requirements.We define a parameter to qualify the SSU as The largest value of Ω is K − 1 when rate requirements of all the SUs can be met, whereas its smallest value is 0 when no power is allocated to SUs; thus, R [k] = 0, k = 2, . . ., K.
From (24), we can also know that still increasing th will decrease the value of Ω.This is because the QoS requirement of user k is already met, and increasing P [k] t will result in the decrease in the power allocated to other SUs.
According to the definition of Ω in (24), we can define a PA optimization problem to maximize the SSU of SUs in the IAbased CR network as (P3), i.e., (P3) max t ,P [3] t ,...,P The solution of (P3) is difficult to obtain.To reduce the computational complexity in solving (P3), we propose a PA algorithm for maximizing SSU (PAMSSU), and it is discussed in different cases as follows.
1) P max t ≤ P [1] t−min : The threshold of the PU R [1] th cannot be met; thus, all the power P max t is allocated to the PU. 2) t−min : The rate requirements of K users can all be satisfied.Thus, P [1] t−min is allocated to the PU, and the remaining power P max t − P [1] t−min is allocated to SUs to maximize their sum rate with their rate requirements met.The PA problem can be expressed as t ,P [3] t ,...,P 3) P t−min : R [1] th can be met, and P [1] t−min is allocated to the PU.The remaining power P max t − P [1] t−min should be allocated to SUs to maximize the value of Ω, and the transmitted power of user k (k = 2, . . ., K) cannot exceed P [k] t−min to facilitate the maximizing of SSU.The PA problem can be expressed as t ,P [3] t ,...,P where Ω is not denoted as the expression in (24) because the transmitted power of each SU is already constrained to be lower than its minimal transmitted power to meet its rate requirement.According to Lemma 1, (P4) in ( 26) can be rewritten into a new format, and it can be solved by water-filling strategy as where ν should satisfy can be denoted as 27) is a convex optimization problem, and it is easy to be solved by Karush-Kuhn-Tucker (KKT) conditions as where ν should satisfy t−min .
(32) The PAMSSU algorithm in each time slot can be expressed in Algorithm 3.

2: P [1]
t−min is calculated according to (8) t−min is allocated to SUs according to (P5).12: end if 13: Transmission for duration T with the power allocated.14: The time slot ends.

E. Outage Probability Analysis of the PU and SUs
Outage probability simply means the probability that a given rate threshold cannot be satisfied because of channel variations [39], and it can reflect the variability of the transmission rate instantaneously.Thus, it is suitable to be used in analyzing the rate performance of the PU in the IA-based CR networks.If the threshold of the PU's transmission rate is R [1] th (bits/s/Hz), the outage probability of the PU in the IA-based CR network can be defined as Pr [1] {outage} = Pr log 2 1 + SINR [1] < R [1] th where SINR [1] is the SINR of the desired signal at the primary receiver.
Here, the outage probability of the PU in the IA-based CR Network is analyzed.
Lemma 3: In a K-user IA-based CR network with one data stream each user, if the interference is eliminated perfectly, Based on the results in Lemma 3, we can derive the expression of the outage probability of the PU in the IA-based CR network.
Proposition 2: The outage probability of the PU in the IAbased CR network can be expressed as Pr [1] Proof: See Appendix F. From Proposition 2, we can know that the outage probability of the PU in the IA-based CR network is determined by the transmit SNR, i.e., (P [1] t /σ 2 ) (the ratio between the transmitted power and the noise power at the receiver).Thus, through increasing the transmitted power P [1] t of the PU, its outage probability performance can be improved.
The outage probability of SUs can also be similarly defined according to Proposition 2 as follows, when the rate requirement of user k, i.e., R [k] th , is deemed as its threshold k = 2, 3, . . ., K.
In CR networks, we should try to satisfy the rate requirements of SUs with the QoS of the PU guaranteed.Nevertheless, the spectrum sharing is performed among the PU and SUs even when the rate requirements of SUs cannot be met, i.e., the outage probability of the PU is more important to achieve than that of SUs.Thus.only the PU's outage probability is analyzed through simulation in Section V.

IV. TRANSMISSION-MODE ADAPTATION BASED ON POWER ALLOCATION IN THE INTERFERENCE ALIGNMENT-BASED COGNITIVE RADIO NETWORK
When IA is performed in the CR network, we can know that it is equal to a single-input and single-output (SISO) channel for each user if one data stream is sent at each transmitter; thus, the rate of each user in IA (equal to SISO) is lower than that of the MIMO single-user channel [15].On the other hand, in the proposed PAMRSU, PAMEEN, and PAMSSU algorithms, when P [1] t−min ≥ P max t , the transmitted power is all allocated to the PU, and SUs are switched into sleep mode.
Thus, we propose a TMA scheme when P [1] t−min ≥ P max t to change the transmission mode from IA to a single-user MIMO system with SUs sleeping to further improve the rate of the PU to approach its constraint R [1] th .In the proposed TMA scheme, when P [1] t−min ≥ P max t , SUs are switched into sleep mode, and the PU adopts MIMO to communicate in the time slot solely.The transmission rate of the PU using MIMO can be expressed as [32] R [11] Q [1] H [11] † . ( The CSI at transmitters (CSIT) of the network is available due to the calculation of IA; thus, in (36), the transmitted power at each antenna can be optimized through using water-filling strategy.The optimal signal covariance Q [1] = V [1] S [1] V [1] † , and V [1] can be obtained by singular value decomposition of the channel matrix as U [1] D [1] V [1] † = H [11] .The optimal diagonal PA matrix S [1] = diag(s 1 , . . ., s min(M [1] ,N [1] ) , 0, . . ., 0).The optimal PA among antennas of user k can be achieved through using the water-filling strategy as . ., min M [1] , N [1]   (37) where x + max(x, 0).δ [1] 1 , . . ., δ [1] min(M [1] ,N [1] ) are the diagonal elements of D [i] , and μ should satisfy min(M [1] ,N [1] ) We can easily obtain that [11] Q [1] H [11] † > log 2 1 + h [1] 2 Thus, all the proposed PAMRSU, PAMEEN, and PAMSSU algorithms can be further combined with a TMA scheme to improve the PU's rate when R [1] th constraint cannot be guaranteed in the IA-based CR network, and we call them PAMRSU-TMA, PAMEEN-TMA, and PAMSSU-TMA algorithms, respectively.Only Step 7 of PAMRSU, Step 6 of PAMEEN algorithms, and Step 7 of PAMSSU algorithm should be changed accordingly when TMA is involved.
The TMA scheme can improve the transmission rate of the PU in the proposed algorithms when P [1] t−min ≥ P max t , and the outage probability of the PU will be significantly reduced according to (33) and (39) when P [1] t−min ≥ P max t .

V. SIMULATION RESULTS AND DISCUSSIONS
In our simulations, we consider a five-user IA-based CR network with M [k] = N [k] = 3 antennas equipped at each transceiver, and each transmitter sends one data stream to its corresponding receiver.The minimizing interference leakage algorithm is adopted to calculate the solutions of IA [18].Rayleigh block fading [29] is adopted, and perfect CSI is assumed to be available at each node.According to [35] and [37], the transmitter-circuit power consumption P [k] ct and the receiver-circuit power consumption P [k] cr of all the users are set to 98 and 112 mW, respectively.P max t /K is set to 20 dbm; thus, the constrained total transmitted power of the network (also the maximum transmitted power of each user) is equal to 500 mW.The iterative algorithm is adopted to obtained the solutions of IA [18].The performance of the proposed three PA algorithms is compared jointly to demonstrate that they are suitable to be applied in different scenarios.Fig. 4. Minimal transmitted power of the PU P [1] t−min to guarantee the PU's threshold R [1] th = 5 bits/s/Hz and its achieved rate of the PU over 200 time slots.
The analytical values of the outage probability of the PU according to (34), and its simulated values in the IA-based CR network are shown in Fig. 3, with R [1] th equal to 7.5, 5, and 2.5 bits/s/Hz, respectively.From the results, we can see that the outage probability of the PU increases when its rate threshold R [1] th becomes larger, which means the QoS requirement of the PU is becoming more rigorous.In addition, the simulated values of outage probability are quite consistent with its analytical values in (34), which proves the conclusions in Proposition 2.
In Proposition 1, we have derived the minimal transmitted power of the PU P [1] t−min to guarantee its rate threshold R [1] th .Thus, the values of P [1] t−min and its achieved R [1] when R [1] th = 5 bits/s/Hz are shown in Fig. 4 over 200 time slots with block fading is adopted.From the results, we can observe that with P [1] t−min assigned to the PU, the rate of the PU R [1] is unchanged and equal to 5 bits/s/Hz, which proves the results in Proposition 1.In addition, the minimal transmitted power of the PU P [1] t−min to guarantee its threshold varies dramatically over the time slots, and the largest value of P [1] t−min is more than 1000 times of its smallest value.Thus, the transmitted power of the users in the IA-based CR network should be carefully allocated to guarantee the QoS of the PU while improving the performance of SUs.
First the average SUs' sum rate of the IA-based CR network with these algorithms is compared in Fig. 5.It is shown that SUs' sum rate of the algorithms with TMA is the same as that of the algorithms without TMA.This is because only the rate of the PU is changed when TMA is performed, and TMA has no effect on SUs.SUs' sum rate of PAMRSU (PAMRSU-TMA) algorithm is much larger than that of both PAMSSU (PAMSSU-TMA) and PAMEEN(-TMA) algorithms.Besides, SUs' sum rate of PAMSSU(PAMSSU-TMA) algorithm is larger than that of PAMEEN (PAMEEN-TMA) algorithm when the SNR is higher, due to smaller transmitted power of PAMEEN (PAMEEN-TMA) to enhance the EE of the network.SUs' sum rate of PAMSSU (PAMSSU-TMA) algorithm is becoming smaller than that of PAMEEN(PAMEEN-TMA) algorithm when the SNR becomes lower because PAMEEN (PAMEEN-TMA) algorithm tends to be the same as PAMRSU (PAMRSU-TMA) algorithm when the SNR becomes lower.
Then, the average EE of the IA-based CR network with different algorithms is compared in Fig. 6.From the results, we can see that when the SNR is larger (P max t /K/σ 2 > 35 dB), the EE of PAMRSU, PAMSSU, PAMRSU-TMA, and PAMSSU-TMA algorithms are almost the same and much lower than that of PAMEEN and PAMEEN-TMA algorithms.This is because almost all of the rate requirements of the users can be met with IA, and the PAMEEN (PAMEEN-TMA) algorithm is designed particularly to optimize the EE of the network.When the SNR becomes smaller (15 dB < P max t /K/σ 2 < 35 dB), EE of PAMRSU (PAMRSU-TMA) is becoming higher than that of PAMSSU (PAMSSU-TMA), and EE of PAMEEN (PAMEEN-TMA) is getting close to that of PAMRSU (PAMRSU-TMA).This is because TMA is performed sometimes to guarantee the threshold of the PU, PAMEEN is losing its advantage in improving the EE of the network, and PAMSSU focuses on the optimizing the parameter of SSU instead of sum rate or EE.When the SNR is extremely low (P max t /K/σ 2 < 15 dB), the EE of the algorithms with TMA is getting close to each other, which is much higher than that of the algorithms without TMA.It is because the probability of TMA is becoming higher with SNR becoming lower.
The minimal transmitted power of the PU P [1] t−min to guarantee its rate threshold R [1] th is derived in Proposition 1, and it is adopted when P [1] t−min < P max t in all the proposed algorithms.When P [1] t−min > P max t , TMA of the PU is performed, and all the SUs are switched into sleep mode.Thus, the average outage probability of the PU of these algorithms is compared in the IAbased CR network in Fig. 7. From the results, we can observe that the outage probability of the algorithms with TMA is the same, which is much lower than that of the algorithms without TMA.Thus, the TMA scheme can significantly improve the performance of the PU in the IA-based CR network, which is consistent with the discussion in Section IV.
The PAMSSU algorithm can maximize the requirements of SUs, measured by SSU Ω in Section III-D, while trying to  th = 0.5 bits/s/Hz, R [3] th = 1 bits/s/Hz, R [4] th = 5 bits/s/Hz, and R [4] th = 7.5 bits/s/Hz.satisfy the PU's threshold.Thus, the average value of Ω is compared in the IA-based CR network in Fig. 8. From the results, we can see that Ω of the PAMSSU algorithm is much larger than that of the PAMRSU algorithm, and Ω of the PAMRSU algorithm is larger than that of PAMEEN algorithm.This is because the PAMSSU algorithm is designed to optimize Ω, and the PAMEEN algorithm mainly focuses the EE of the network instead of SUs' rate.
To further quantify the fairness of these algorithms, Jain's index is utilized to compare their fairness [40].We define a length-(K − 1) vector R of nonnegative real entries {R [k] } K k=2 , where R [k] is the transmission rate of SU k in the IA-based CR network.The Jain's fairness index J of vector R can be expressed as From (40), we can know that (1/K) ≤ J(R) ≤ 1, and larger values of J(R) means better fairness.Thus, (40) can be leveraged as a metric to measure the fairness of the proposed algorithms.The average Jain's index of the proposed three algorithms is compared when K = 5 and K = 7 in Fig. 9, respectively.The rate requirements of all the users are set to 5 bits/s/Hz.From the results, we can know that the fairness of the PAMSSU algorithm is much better than the other two algorithms, and when the number of SUs increases, the fairness of the algorithm will decrease slowly.In addition, the Jain's index of the PAMSSU algorithm will not reach 1 when SNR is high, this is because when the rate requirements of all the SUs can be satisfied, it will not focus on the fairness any longer.Instead, the sum rate of SUs can be optimized with their rate requirements all satisfied.
In Figs. 3 and 5-9, the average performance of these algorithms is compared.Nevertheless, only the average performance cannot show their differences clearly, and we should compare the instantaneous performance of these algorithms to demonstrate their specific requirements.Therefore, the minimal transmitted power to guarantee rate requirement of each SU, and transmitted power of each SU of these three algorithms in a certain time slot are compared in Fig. 10 when The rate requirement of each SU and achieved transmission rate of each SU of the three algorithms in this time slot are compared in Fig. 11 when P max t /K/σ 2 is equal to 20 dB.From the results, we can see that, for users 2, 3, and 5, the minimal transmitted power to guarantee their rate requirements, i.e., P [2] t−min , P [3] t−min , and P [5] t−min , is relatively small; however, Fig. 10.Transmitted power comparison of SUs with different algorithms of the IA-based CR network in a certain time slot when P max t /K/σ 2 is equal to 20 dB.R [2] th = 0.5 bits/s/Hz, R [3] th = 1 bits/s/Hz, R [4] th = 5 bits/s/Hz, and R  R [2] th = 0.5 bits/s/Hz, R [3] th = 1 bits/s/Hz, R [4] th = 5 bits/s/Hz, and R [5] th = 7.5 bits/s/Hz.the transmitted power of these three users in the PAMEEN and PAMRSU algorithms is much higher than their requirements to maximize the EE of network and SUs' sum rate, respectively.On the other hand, in the PAMSSU algorithm, only the minimal required power P [2] t−min , P [3] t−min , and P [5] t−min is assigned to these three users to satisfy their rate requirements.Thus, transmitted power is saved, and more power can be assigned to user 4 to achieve R [4] th in PAMSSU algorithm than PAMEEN and PAMRSU algorithms.User 4 is a specific case to be further demonstrated.For the PAMEEN and PAMRSU algorithms, the allocated power to user 4 is much lower than that of its minimal required power P [4] t−min , results in the lower rate of user 4 than R [4] th .This is because the effective channel of user 4 |h [4] | 2 is the in this time slot due to the channel variation; thus, more power should be allocated to other users to achieve much higher sum rate or EE.On the contrary, in the PAMSSU algorithm, the objective is to satisfy the requirements of all the SUs, and only the minimal required power is assigned to users 2, 3, and 5 to satisfy their rate requirements.The transmitted power is thus saved, and more power can be assigned to user 4 to achieve R [4] th in the PAMSSU algorithm due to its poor effective channel |h [4] | 2 .According to the definition of SSU in (24), the value Ω of the PAMSSU algorithm in this time slot is close to its largest value of 4, which is larger than that in PAMEEN and PAMRSU algorithms.In addition, although R [5] th is larger than R [4] th , P [5] t−min is much smaller than P [4] t−min .This is because the effective channel |h [5] | 2 of user 5 is much higher than |h [4] | 2 of user 4 in this time slot due to the channel variation.

VI. CONCLUSION
In this paper, we have developed several PA algorithms for IA-based CR networks.The minimal transmitted power of the PU to guarantee its rate threshold was derived.Then, three PA algorithms, i.e., PAMRSU, PAMEEN and PAMSSU, were proposed for IA-based CR networks to maximize the SUs' rate, the EE of the network, and the SSU, respectively.To evaluate the rate performance, and the outage probability of PU and SUs with different value of its transmitted power was also derived.To further guarantee the rate constraint of the PU, we proposed a TMA scheme to adapt the transmission mode to improve the performance of the PU, and it can be combined with the proposed PA algorithms easily.Simulation results were presented to show the effectiveness of the proposed adaptive PA algorithms for IA-based CR networks.

APPENDIX A PROOF OF PROPOSITION 1
When the PA is considered in the IA-based CR network, the transmission rate of the PU can be calculated as R [1] = log 2 ⎛ ⎜ ⎜ ⎜ ⎝ 1 + u [1] † H [11] v [1] 2 P The QoS requirement of the PU should be guaranteed; thus, from ( 41) and (7), we can obtain u [1] † H [11] v [1] 2 . (42) In the feasible IA-based CR networks, the interference is constrained in certain subspaces at the unintended receivers, and the interference leakage at the receivers is trivial.Moreover, P [k] t is larger than 0. Thus, we have u [1] † H [11] v [1] 2 t−min .(43) Thus, P [1] t−min is the minimal value of the PU's transmitted power P [1] t to guarantee its threshold R [1] th .
The optimization problem in (45) is similar to the PA problem in multiple parallel channels, and the water-filling strategy can be leveraged to obtain the optimal solution.Thus, the closed-form solution of (P2) when K k=1 P [k] t = P max t can be denoted as (14), where ν should satisfy (15).APPENDIX C PROOF OF LEMMA 2 Equation ( 17) can be rewritten as ln 2 eK × λe Therefore, according to the definition of the Lambert W function, (46) can be changed into (18), which is the solution of (17).

APPENDIX D PROOF OF THEOREM 1
As (P2) is a concave-convex fractional programming, we can optimize the following problem in (47) to obtain the optimal solution of (P2) [41] F (λ) = max t ,P [2] t ,...,P The optimization in (47) is a convex optimization problem, and it is easy to solve by applying KKT conditions.The optimal solution of (47) can be expressed as (19) with the constraint Let F (λ) = 0, we can obtain (20).Then, P [k] t in (20) is substituted by the optimal solution in (19).
Through applying λ * to (19), we can obtain the optimal solution of (P2) as (21).However, the constraint K k=1 P [k] t ≤ P max t has not been considered.Thus, we should discuss the validity of the optimal solution in (21) as follows.

1)
The closed-form solution of (P2) can be defined as in (21).t−min ≤ P max t : The closedform solution of (P2) can be defined as in (14).

APPENDIX E PROOF OF LEMMA 3
In the design of u [k] and v [k] in the K-user IA-based CR network, k = 1, 2, . . ., K, it only concentrates on the condition in (2) without involving H [kk] in (3), i.e., only the channel matrices between different users H [kj] ∀ j = k, are utilized to achieve IA.The condition (3) will be satisfied naturally when the condition (2) is met.Thus u [k] and v [k] are i.i.d., and independent of H [kk] , and we can obtain that As mentioned in Section II, (H [kk] ) ij is i.i.d.CN (0, 1), and u [k] and v [k] are unitary vectors; thus we can achieve Thus, h [k] is a complex Gaussian random variable with zero mean and unit variance, and |h [k] | 2 follows exponential distribution with unit mean and variance.
APPENDIX F PROOF OF PROPOSITION 2 From (33), we can obtain that Pr [1] {outage} = Pr log 2 1 + h [1] 2 In Lemma 1, we see that |h [k] | 2 follows exponential distribution with unit mean and variance.Thus, the cumulative distribution function (cdf) of |h [k] | 2 can be expressed as Pr h [1] 2 ≤ x = 1 − e −x , x ≥ 0 0, x<0. (51) The rate threshold of the PU R [1] th should be set positive, and we have th − 1 From ( 50) and (51), we can obtain the expression of the outage probability of the PU as (34).

Fig. 1 .
Fig. 1.K-user IA-based CR network with one PU and K − 1 SUs sharing the spectrum in the same frequency band simultaneously.

Fig. 2 .
Fig. 2. Demonstration of the case when the PU's QoS requirement still cannot be met with P max t all allocated to it in the IA-based CR network.

t , then 4 :
Solve the energy-efficient PA problem in (P2) through the fractional programming according to Theorem 1. 5: else 6: Allocate P max t to the PU.7: SUs are switched into sleep mode.8: end if 9: Transmission for duration T with the power allocated.10: The time slot ends.

Fig. 3 .
Fig. 3. Analytical and simulated values comparison of outage probability of the PU in the IA-based CR network with different values of R [1] th .

Fig. 5 .
Fig. 5. Average SUs' sum rate comparison of different algorithms in a fiveuser IA-based CR network.

Fig. 6 .
Fig. 6.Average EE comparison of different algorithms in a five-user IA-based CR network.

Fig. 7 .
Fig. 7. PU's average outage probability comparison of different algorithms in a five-user IA-based CR network.

Fig. 8 .
Fig. 8. Average value of SSU comparison of different algorithms in a fiveuser IA-based CR network.R [2]

Fig. 9 .
Fig.9.Average Jain's index comparison of different algorithms in the IAbased CR network when there are five users and seven users, respectively.The rate requirements of all the users are set to 5 bits/s/Hz.

Fig. 11 .
Fig. 11.Rate comparison of SUs with different algorithms of the IA-based CR network in a certain time slot when P max t /K/σ 2 is equal to 20 dB.