This paper introduces a full duplex single secondary user multipleinput multipleoutput (FDSSUMIMO) cognitive radio network, where secondary user (SU) opportunistically accesses the authorized spectrum unoccupied by primary user (PU) and transmits data based on FDMIMO mode. Then we study the network achievable average sumrate maximization problem under sum transmit power budget constraint at SU communication nodes. In order to solve the tradeoff problem between SU’s sensing time and data transmission time based on opportunistic spectrum access (OSA) and the power allocation problem based on FDMIMO transmit mode, we propose a simple trisection algorithm to obtain the optimal sensing time and apply an alternating optimization (AO) algorithm to tackle the FDMIMO based network achievable sumrate maximization problem. Simulation results show that our proposed sensing time optimization and AObased optimal power allocation strategies obtain a higher achievable average sumrate than sequential convex approximations for matrixvariable programming (SCAMP)based power allocation for the FD transmission mode, as well as equal power allocation for the half duplex (HD) transmission mode.
1. Introduction
W
ith the rapidly increasing spectrum requirements of emerging wireless communication service and application, cognitive radio (CR) is proposed to improve the spectrum utilization efficiency and solve the problem of congestion caused by traditional regular spectrum assignment. In cognitive radio network, opportunistic spectrum access (OSA), which is one of the most promising technologies to be implemented in dynamic spectrum access system as a replacement of static spectrum utilization rule, has a capability to access the spectrum holes according to prior primary spectrum sensing results. The basic idea of OSA is allowing the secondary user to identify the spectrum holes unoccupied by a primary user and access the authorized spectrum
[1]
. However, secondary user (SU) must vacate the spectrum holes once primary user returns back to access the channel again in order to protect the primary user from the harmful interference.
In order to satisfy the qualityofservice (QoS) requirements of SU and maximize the network achievable rate under the constraint that the primary user (PU) is sufficiently protected, a lots of previous works have studied on opportunistic spectrum sharing and power allocation strategies in cognitive radio network. The work in
[2]
proposed a joint power control and spectrum access scheme in CR network, which tackles the power allocation problem from the cooperative game perspective and solves the optimization problem of the proposed model with the differential evolution algorithm. In
[3]
, the authors proposed continuous sensingbased power allocation strategies to maximize the achievable throughput of the SU in a multiband CR network with perfect and quantized channel state information (CSI).
On the other hand, as a result of the requirement of high speed rate data transmission, multipleinput multipleoutput (MIMO) communication techniques
[4

5]
have been paid considerable attention in recent years, because of the capability of greatly improving system reliability and spectral efficiency without more additional power. In
[4]
, the authors considered the transmit optimization problem for a single secondary user MIMO and multipleinput singleoutput (MISO) channel in CR network under constraint of opportunistic spectrum sharing. In
[5]
, the authors researched the joint beamforming and power allocation problem in cognitive MIMO systems via game theory in order to maximize the total throughput of secondary users. However, these works have focused on the spectrum access mode, power allocation strategies or MIMO.
Recently, the research on improving the spectral efficiency by the FD transmission mode has increased
[6

7]
. Obviously, comparing with half duplex (HD) transmission mode, the FD mode has the capability to greatly increase the communication system capacity, if the selfinterference from the transmit antennas to the receive antennas at the same node can be efficiently eliminated
[8

10]
. Thus, the FD transmission mode has the potential to achieve more system sumrate than the conventional HD transmission mode. However, the combination of power allocation and FD–MIMO in a CR network is not wellresearched.
Motivated by these techniques, in this paper, we investigate joint opportunistic spectrum access and optimal power allocation strategies for the full duplex single secondary user MIMO (FDSSUMIMO) cognitive radio network. In our proposed network model, we pay much attention to how to solve the spectrum sensing time and data transmission time design problem and the power allocation problem of transmit antennas. In order to maximize the network achievable average sumrate, we apply to a simple trisection algorithm to search the optimal spectrum sensing time, and then propose an alternating optimization (AO) algorithm to solve the power allocation optimization problem for the FDSSUMIMO cognitive radio network.
The rest of this paper is organized as follows. In Section 2, the FDSSUMIMO cognitive radio network model is introduced, and then the achievable average sumrate maximization problem is formulated. In section 3, we study the tradeoff problem between sensing time and data transmission time to maximize the average probability of spectrum holes discovery in the secondary network. And we propose AObased optimal power allocation strategies applied to the FDSSUMIMO cognitive radio network in this section. Simulation results and discussions are presented in Section 4. Conclusions are drawn in Section 5.
The following notations are used in this paper. Bold upper case letter denotes matrix, bold low letter denotes vector, and nonbold letter denotes scalar.
G
^{H}
represents the Hermitian transpose of matrix
G
, ｜
G
｜ denotes the determinant of matrix
G
, and Tr{
G
} is the trace of matrix
G
. E[⋅] denotes the mathematical expectation operation.
I
_{m}
represents the m×m unit matrix.
Q
⪰0 indicates that
Q
is a positive semidefinite matrix.
2. Network Model and Problem Formulation
 2.1 Network Model
We consider a FDSSUMIMO cognitive radio network, which is comprised of a pair of primary user transmitter (PUTx) and primary user receiver (PURx), and two SU communication nodes as depicted in
Fig. 1
. Either of SU nodes is equipped with
N_{t}
transmit antennas and
M_{r}
receive antennas, which transmit and receive data respectively at the same time on the same frequency. SU can opportunistically access the primary channel when PU is detected to be absent. Once PU reoccupies the primary channel, SU must vacate the current channel and search a new available channel.
FDSSUMIMO cognitive radio network model
In
Fig. 1
, we show the FDSSUMIMO cognitive radio network model, where
G
_{j}
(
j
= 1,2) denote the
M_{r}
×
N_{t}
channel power gain matrix from the one node’s transmit antennas to the other node’s receive antennas, and
H
_{i}
(
i
= 1,2) denote the
M_{r}
×
N_{t}
channel selfinterference matrix from the
i
th node’s transmit antennas to the
i
th node’s receive antennas.
s
_{i}
(
i
= 1,2) is regarded as the
N_{t}
× 1 transmitted signals vector of the
i
th node. Let
P
_{i}
be the
N_{t}
×
N_{t}
transmitted power matrix for the transmit antennas of the
i
th node. Therefore, the expression for the received signal at the node 1 and node 2 are written as, respectively
where
w
_{i}
(
i
= 1,2) is the
M_{r}
× 1 background noise at the
i
th node which is assumed to be zeromean complex Gaussian vector. The first part of (1) or (2) represents the received signals, and the second part represents the selfinterference signals caused by the transmit antennas at the same node, which is treated as the background noise. Here, according to
[7]
, we assume that E[
s
_{i}
s
_{i}
^{H}
]=
I
_{Nt}
and E[
s
_{i}
s
_{i}
^{H}
]=0(
i
≠
j
). On the other hand, we suppose that the channel power gain matrix
G
_{j}
(
j
= 1,2) is known, and the selfinterference channel matrix
H
_{i}
(
i
= 1,2) need to be estimated.
Let △
_{i}
be the estimated error matrix, and
is the estimated channel matrix. Then, the actual selfinterference channel matrix is given by
Let
Σ
_{i}
denote
. From (1), we have
The achievable rate at the node 1 and node 2 are:
where
represents the transmit power covariance matrix at the
i
th node. Then, the FD sumrate is
 2.2 Spectrum Sensing and Data Transmission Design
In this work, we assume that the CR network operates on frame structure of fixed duration. The duration of each frame consists of two slots: sensing slot
τ
and data transmission slot
T

τ
, as shown in
Fig. 2
. Then the SU carries out periodic spectrum sensing to decide whether the PU is absent or not. In OSA mode, the SU must frequently sense the spectrum before accessing the licensed spectrum. The spectrum holes appear only when the PU are detected to be not busy, for the sake of protecting the PU from the harmful interference.
Frame structure design for the CR network
Let
H
_{0}
and
H
_{1}
be two hypotheses that the PU is absent and the PU is present, respectively. In the single threshold based energy detection method, the final decision result depend on the predefined threshold
λ_{th}
, shown as
where
E_{s}
denotes the energy of the received sample signal. CR makes a final decision whether PU is absent or not in accordance with the sample signal energy
E_{s}
and the predefined threshold
λ_{th}
. Usually, two metrics are used to evaluate the detection performance: the false alarm probability
P_{f}
and the detection probability
P_{d}
.
According to the central limit theorem, the sample signal statistic can be approximated by a Gaussian distribution when the sample number is large enough. Let
f_{s}
stand for the sample frequency.
denotes the variance of Gaussian noise and
γ
represents the received PU signal to noise ratio. Thus, in the energy detection method, the
P_{f}
and
P_{d}
are derived as
[11]
where
Q
(.) denotes the
Q
function defined as
.
Let
P
(
H
_{0}
) and
P
(
H
_{1}
) be the probability that PU is absent and the probability that PU is present, respectively. Then, the probability of spectrum holes discovery in OSA mode is shown as
where (1 
P_{f}
)
P
(
H
_{0}
) indicates the probability that the PU is idle and SU make a right decision, and (1 
P_{d}
)
P
(
H
_{1}
) represents the probability that the PU is busy but SU do not detect accurately.
 2.3 Problem Formulation
In this paper, we are interested in maximizing the achievable average FD sumrate under the sum transmit power budget constraint of SU node. The achievable average FD sumrate in the OSA mode can be given by
Therefore, this problem can be formulated as
where the positive semidefinite constraint conditions guarantee that the transmit power covariance matrices are feasible.
P_{max}
stands for the total transmit power budget of SU node.
P_{d,tar}
and
P_{f,tar}
are the target detection probability and the target false alarm probability on condition that the PU is sufficiently protected, respectively. Usually, in order to improve the unoccupied spectrum utilization and reduce the interference to PU, they satisfy
P_{d,tar}
≥ 0.9 and
P_{f,tar}
≤ 0.1 . It is pointed out that if the primary user requires 100% protection in its authorized spectrum, the secondary user is not allowed to access the authorized spectrum in OSA mode because it is not guaranteed that the detection probability
P_{d}
is equal to 1. However, since the target detection probability
P_{d,tar}
is more than 0.9 and
P_{d}
≥
P_{d,tar}
, the probability (1 
P_{d}
)
P
(
H
_{1}
) of producing the harmful interference to PU is very small and acceptable.
According to (7) and (8), it is obvious that (9) is related with the variable
τ
and independent of
Q
_{i}
. However,
R_{i}
(
i
= 1,2) is independent of
τ
and is related with
Q
_{i}
. Thus, the maximization problem (11) can be divided into two subproblems. Then (9) is equivalent to (12) and (13):
In the next section, we will solve the above optimization problem (12) and (13), respectively.
3. Optimal Sensing Time Design and Optimal Power Allocation Strategies
 3.1 Optimal Sensing Time Design
In the previous section, the relationship between sensing time and the achievable average FD sumrate in the OSA mode has been derived. In this section, we will design the sensing time and data transmission time to maximize the achievable average FD sumrate of the cognitive radio network. In OSA mode, SU need to perform spectrum sensing so that it could find spectrum holes and access the unused licensed spectrum without the harmful interference to PU. For a fixed frame duration
T
, the longer the spectrum sensing time
τ
, the shorter the data transmission time
T

τ
. The longer spectrum sensing time causes much overhead and mitigates data transmission time of SU, while short sensing time makes it difficulty to guarantee the acceptable detection probability and false alarm probability requirement. Therefore, it is necessary to consider the tradeoff between the spectrum sensing time and data transmission time to find the optimal sensing time
τ
in order to achieve the maximal sumrate while PU is sufficiently protected.
Next, we will demonstrate the existence of the optimal sensing time to obtain the object function maximal value of (12). Let
F
(
τ
) represent the average probability of spectrum holes discovery.
Thus, (12) is equivalent to
From (7) and (8), for a given target detection probability
P_{d,tar}
and a given target false alarm probability
P_{f,tar}
, we have
P_{f}
=
Q
(
α
) and
P_{d}
=
Q
(
β
) , where
and
. According to literature
[12]
,
P_{d}
is an increasing and concave function of
τ
under
P_{d}
> 0.5 and
P_{f}
is a decreasing and convex function of
τ
under
P_{f}
< 0.5 . Thus, we have
α
> 0 and
β
< 0. By using
Q
(
x
) = 1 
Q
(
x
),
x
> 0 , the
Q
(
x
) is approximately equal to
[13]
where
C
_{1}
= 1.98 and
C
_{2}
= 1.135.
Furthermore, by using (16), we have
Then, we will prove that there indeed exists a maximum value
F
(
τ
) about
τ
within the interval (0,
T
) .
Proof: Differentiating (14) with respect to
τ
, we have
Obviously, as a result of fact that the lower bound of
Q
(
x
) is 0 and the upper bound is 1, we have
Proof of (21): See Appendix A.
According to the zero theorem, there exists a value
τ
_{0}
within (0,
T
) at least to satisfy
, because of
is a continuous differential function of variable
τ
. It means that
F
(
τ
) is a increasing function for the smaller
τ
, and it becomes a decreasing function when
τ
approach to
T
. Thus, there exists a maximal value of
F
(
τ
) within (0,
T
) .
As a result of not obtaining the optimal sensing time
τ
in a closed form expression from (15), we will adopt a simple trisection Algorithm to search the optimal
τ
that make
F
(
τ
) acquire the maximal value, as shown in the following Algorithm 1.
Algorithm 1
 3.2 Optimal Power Allocation Strategies
Despite of the many previous literates on power allocation strategies in wireless communication network, the power allocation problem of transmit antennas about the FDSSUMIMO cognitive radio network under total transmit power constraints is not wellstudied. Therefore, in this paper, we consider the power allocation strategies applied to the FDSSUMIMO cognitive radio network in order to maximize the achievable average FD sumrate. As described in previous section, in order to reduce the effect of the selfinterference for FD transmission mode, it is necessary to optimize the transmit antennas power allocation at each node under the node total transmit power constraints.
From the maximization problem (13), we can obtain the follow equivalent problem
where
f_{i}
and
g_{i}
are represented by
Obviously,
f_{i}
is concave and
g_{i}
is nonconcave. Thus, the maximization optimization problem (23) is nonconcave and difficult to solve directly. In this paper, we will apply an alternating optimization (AO) algorithm
[14]
to solve the nonconcave optimization problem (22). Before describing the AO algorithm, we need to introduce a lemma
[15]
.
Lemma 1: Let
E
be any
m
×
m
matrix such that
E
≻0 and ｜
E
｜≤1 . Consider the function
h
(
S
)=−Tr{
SE
}+log
_{2}
｜
S
｜+
m
, where
S
is the
m
×
m
matrix. Then,
with the optimum value
S
_{opt}
=
E
^{1}
.
Let
, where
E
is the
m
×
m
matrix, by applying lemma 1 to (24), we have
where (26) is a concave function. Thus, equivalent formulation of problem (22) is given by
The objective function of (28) is concave and equivalent to the original objective function of (22). The AObased Algorithm solves the approximate concave programming problem by iteratively updating the objective function of (28) until convergence by using CVX package in Matlab, as described in the following Algorithm 2.
Algorithm 2
According to Algorithm 1 and Algorithm 2, the achievable average sumrate in FDSSUMIMO network in OSA mode is obtained by
4. Simulation Results and Discussion
In this section, we provide simulation examples to evaluate the network performance of the proposed sensing time optimization and optimal power allocation strategies. In spectrum sensing simulation process, we assume that the target detection probability
P_{d,tar}
= 0.95 , the target false alarm probability
P_{f,tar}
= 0.05 , the fixed time duration
T
= 100ms, the sample frequency
f_{s}
= 10KHz
Fig. 3
illustrates the relationship between the average probability of spectrum holes discovery
F
(
τ
) and sensing time
τ
under different
P
(
H
_{0}
). Obviously, from the
Fig. 3
, the average probability of spectrum holes discovery
F
(
τ
) indeed exist a maximum value. For example, the maximal value of
F
(
τ
) is about 0.71 under
P
(
H
_{0}
) = 0.7 .
F(τ) vs. sensing time τ with T = 100ms
Then in data transmission simulation process, the noise covariance matrix
Σ
_{i}
is normalized to unit matrix. In
[7]
, the authors point out that no standard reference selfinterference channel model has been reported, and selfinterference channel matrix simply is generated as a zeromean complex Gaussian random variable. In
[16]
, it is assumed that the true selfinterference channel matrix consists of the estimated channel matrix and the channel estimation error matrix, which are generated as a zeromean complex Gaussian random variable. In this paper, we assume that the channel power gain matrix
G
_{j}
, and the channel selfinterference gain matrix
are zeromean complex Gaussian random variables, and variance are equal to the signal to noise ratio (
SNR
) and the selfinterference to noise ratio (
INR
), respectively. Furthermore, we suppose that the estimated error matrix △
_{i}
is also zeromean complex Gaussian random variable with variance equal to
. Let
SIR
represent the ratio of
SNR
and
INR
. The transmit power budget
P_{max}
of two nodes are identical. The number of transmit antennas
N_{t}
and receive antennas
M_{r}
are identical.
Fig. 4
compares the network achievable average sumrate
R_{OSA}
versus
SNR
under conventional equal power allocation for HD, SCAMPbased power allocation for FD proposed in
[16]
, and our proposed AObased power allocation for FD with the fixed
SIR
=10db,
P
(
H
_{0}
)=0.7,
=1 and
N_{t}
=
M_{r}
= 4 . From
Fig. 4
, as the
SNR
is increased, our proposed AObased power allocation for FD obtains more increment than conventional equal power allocation for HD and SCAMPbased power allocation for FD in the network achievable average sumrate. It is obvious that conventional equal power allocation for HD is not optimal scheme under the constraint of the total transmit power because it fails to optimize the transmit antenna power and use the FD transmission mode. For example, our proposed AObased power allocation for FD obtains more 1.5 (bit/s/Hz) increment than conventional equal power allocation for HD when the
SNR
is equal to 0 (dB), and more 2.5 (bit/s/Hz) increment than conventional equal power allocation for HD when the
SNR
is equal to 10 (dB). On the other hand, for a fixed
SNR
, our proposed AObased power allocation for FD obtains about 1(bit/s/Hz) increment than SCAMPbased power allocation for FD. The results indicate that our proposed AObased power allocation for FD obtain higher performance improvement than conventional equal power allocation for HD and SCAMPbased power allocation for FD.
R_{OSA} vs. SNR with N_{t} = M_{r} = 4, SIR=10dB , P_{max}=1w,
Fig. 5
shows the network achievable average sumrate
R_{OSA}
versus the total transmit power budget
P_{max}
of SU node under conventional equal power allocation for HD, SCAMPbased power allocation for FD, and our proposed AObased power allocation for FD with the fixed
SIR
=10dB ,
P
(
H
_{0}
)=0.7,
=1,
R_{OSA} vs. P_{max} with N_{t} = M_{r} = 4, SIR=10dB , SNR=10dB,
SNR
=10dB and
N_{t}
=
M_{r}
= 4 . Obviously, the results indicate that our proposed AObased power allocation for FD obtain about 3050% performance improvement compared with conventional equal power allocation for HD, as well as about 520% performance improvement compared with SCAMPbased power allocation for FD. Therefore, our proposed AObased power allocation provides the best average sumrate.
Fig. 6
shows the comparisons of the network achievable average sumrate
R_{OSA}
with the number of different antennas under our proposed AObased power allocation for FD, SCAMPbased power allocation for FD and conventional equal power allocation for HD. As seen from
Fig. 6
, when the number of antennas increases, the network achievable average sumrate increases. Clearly, the results show that our proposed AObased power allocation for FD obtain about 2030% performance improvement compared with conventional equal power allocation for HD, as well as about 520% performance improvement compared with SCAMPbased power allocation for FD. Thus, the results indicate our proposed power allocation algorithm achieves the better performance improvement for the different antennas number.
R_{OSA} vs. N with SIR=10dB , SNR=10dB, P_{max} = 1w ,
5. Conclusions
In this paper, we have introduced a FDSSUMIMO cognitive radio network, where SU can transmit and receive data respectively at the same time on the same primary frequency when PU is detected to be absent. We research the CR network frame structure design of spectrum sensing time and data transmission time in OSA to find the maximal average probability of spectrum holes discovery. And then we propose optimal power allocation strategies for the FDSSUMIMO cognitive radio network in order to maximizing the average achievable sumrate of secondary user network.
Simulation results demonstrate that our proposed joint sensing time optimization and optimal power allocation strategies can achieve a higher average achievable sumrate than conventional equal power allocation for HD transmission mode, as well as SCAMPbased power allocation for FD transmission mode.
BIO
Wenjing Yue received her B.Sc. and M.Sc. degrees both from Taiyuan University of Technology in 2003 and 2006 respectively, and received her Ph.D. degree from Shanghai Jiaotong University in 2010. Now she is an associate professor in Nanjing University of Posts and Telecommunications. Her research interest includes wireless communications and networks, sensor networks.
Yapeng Ren received the B.S. degree from Luoyang Institute of Science and Technology in 2013. He is currently pursuing his M.S. degree in Signal and information processing in College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications. His research interests include cognitive radio network and cooperative communications.
Zhen Yang received his B.Sc. and M.Sc. degrees both from Nanjing University of Posts and Telecommunications in 1983 and 1988 respectively, and received his Ph.D. degree from Shanghai Jiaotong University in 1999. Now he is a professor in Nanjing University of Posts and Telecommunications. His research interest includes wireless communications and networks, speech processing and modern speech communications.
Zhi Chen received his B.S. and Ph.D. degrees both from Nanjing University of Posts and Telecommunications in 2000 and 2007 respectively. He currently works in Nanjing University of Posts and Telecommunications as an associate professor. His research interest includes wireless communication and networks, sensor networks, software engineering.
Qingmin Meng received his B.S. and Ph.D. degrees both from Nanjing University of Posts and Telecommunications in 2000 and 2007 respectively. He currently works in Nanjing University of Posts and Telecommunications as an associate professor. His research interest includes wireless communication and networks, sensor networks, software engineering.
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