This paper studies the power control problem in cognitive radio networks where a primary user and multiple secondary users (SUs) coexist. Imperfect channel state information is considered. The objective is to maximize the SUs' sum rate while guaranteeing the proportional rate fairness among SUs. The problem under consideration is nonconvex. By doing a transformation, it is equivalently changed to a secondorder cone programming problem, which can be effi ciently solved by existing standard methods. Simulations have been done to verify the network performance under different channel uncertainty conditions.
1. Introduction
As the rapid development of advanced technologies on wireless communications, a lot of high transmission rate services and applications have emerged, which increases the demand for spectrum. On the other hand, experimental results have shown that traditional fi xed spectrum allocation schemes yield inefficient spectrum utilization
[1]
. To improve the spectrum utilization and provide high quality of services (QoS), cognitive radio networks (CRNs) that allow the unlicensed secondary users (SUs) share the licensed spectrum with the licensed primary users (PUs) have been proposed.
Spectrum allocation problem in CRNs has drawn large attention in recent years
[2

7]
. In most of these works, it is assumed that perfect channel state information (CSI) is known
[2

4]
. However, in practice perfect CSI, especially the channel gain from the SUs to PUs, cannot be obtained due to the lack of cooperation among PUs and SUs. Therefore, this motivates the research on resource allocation problem in CRNs with imperfect CSI
[4

6]
. Mitliagkas et al. investigated the joint power control and admission control problem in
[5]
. Kim et al. in
[6]
studied the sum rate maximization problem under the total power and interference power constraints. Parsaeefard et al. in
[7]
worked on the social utility of SUs while satisfying each SU's signal to noise ratio requirement and interference power constraint. However, all those works do not explicitly consider SUs' different transmission rate requirements and fairness issue, thus they are not suitable for a situation where different SUs have different transmission rate requirements. To fl exibly allocate transmission rates to each SU and guarantee fairness among SUs, we will investigate the resource allocation problem with proportional rate fairness requirements in CRNs under imperfect CSI.
In this paper, we will investigate the power control problem in CRNs, where imperfect CSI from secondary BS to the primary user is considered. The objective is to maximize the SUs' sum rate subject to the proportional rate fairness constraint among SUs, the total power constraint at secondary BS, and the interference power constraint to the PU. The problem is formulated as a nonconvex optimization problem. By doing a transformation, the problem is changed to an equivalent secondorder cone programming (SOCP) problem, which can be efficiently solved by existing standard methods. Simulations have been done to demonstrate the network performance under different channel uncertainty conditions.
2. System Model and Problem Formulation
Consider a network setting where a PU and
K
SUs coexist. Downlink transmission from the secondary base station (BS) to SUs is considered. The SUs can adopt the available channels that are licensed to the PU for its own data transmission. It is assumed that the total available bandwidth is divided into multiple nonoverlapping channels. And each SU is allocated one such channel for its own data transmission.
The channel gain from the secondary BS to SU
k
, ∀
k
∈ {1,2,⋯,
K
} is denoted by
h_{k}
.
σ_{k}
is the variance of the additive white Gaussian noise in that channel. For notational brevity, let
H_{k}
=
h_{k}
/
σ_{k}
. The data rate for SU
k
isdenoted by
where
P_{k}
is the transmission power for SU
k
at BS.
To protect the PU’s QoS, the interference to the PU should not be greater than the given threshold
T_{th}
, which can be expressed by
Where
d_{k}
is the channel gain for SU
k
from the secondary BS to the primary user. In practice, imperfect channel information cannot be obtained, especially the channel gain from the secondary users to the primary users. Because generally there is a lack of cooperation between primary user and SUs, and thus the primary user will not feedback the CSI to the SUs. Ellipsoidal uncertainty will be adopted to model the uncertainty of channel gain
d_{k}
. Let us define vector
d
= [
d
_{1}
d
_{2}
⋯
d_{K}
]
^{T}
. Adopting the ellipsoidal uncertainty
[5]
, the uncertainty region of
d
can be expressed by
where
is the nominal value of
d
,
D
is a
K
×
K
matrix, and
u
is a
K
dimensional vector. To facilitate the following analysis, let us define a vector
P
_{s}
= [
P
_{1}
P
_{2}
⋯
P_{K}
]
^{T}
, and then (2) can be rewritten as
Since
d
satisfies (3), to guarantee (4) hold, it is equivalent to make sure the following inequality (5) holds,
From (5), by invoking the CauchySchwarz inequality, one gets that
We desire to study the power control problem to maximize the sum rate of SUs under several constraints. The problem under consideration can be formulated as follows,
Where
C
1 represents the BS total power constraint, and
P_{th}
is the power threshold at the BS.
C
2 indicates that the consumed power for each SU at the BS should be nonnegative.
C
3 is the interference power constraint to the primary user.
C
4 is the proportional rate fairness constraint;
γ
_{1}
,
γ
_{2}
,⋯,
γ_{K}
are given constants, and they indicate the proportional rate requirements of SUs.
C
5 represents the SU’s transmission rate constraint.
3. Optimal Solution
Problem (7) is a nonconvex optimization problem since the nonlinear equality constraint
C
5. To make the problem easy to solve, we will transform problem (7) into its equivalent form.
By replacing the equality constraint in
C
5 by an inequality constraint
problem (7) becomes
Problem (9) is an SOCP problem, since its objective function is a linear function, its constraint set is a convex set, and
C
3 is a secondorder cone constraint. A proposition will be given in the following to show that the optimal solution of problem (9) satisfies
R_{k}
= 0.5log
_{2}
(1 +
H_{k}P_{k}
), and thus problem (9) is equivalent to problem (7). Hence, we can solve Problem (9) instead of Problem (7).
Proposition 1.
The rates that optimize problem (9) satisfy that
R_{k}
= 0.5log
_{2}
(1 +
H_{k}P_{k}
), ∀
k
∈ {1,2,⋯,
K
}.
Proof.
Because the objective function of problem (9) is an increasing function with respect to
R_{k}
, and
R_{k}
satisfies constraint
C
5’. It is easy to see that when problem (9) admits its optimal solution
R_{k}
satisfies that
R_{k}
= 0.5log
_{2}
(1 +
H_{k}P_{k}
), ∀
k
∈ {1,2,⋯,
K
}.
Problem (9) is an SOCP problem, existing standard methods such as interiorpoint methods can solve it efficiently. In Section 3, CVX toolbox
[8]
will be used to find the optimal solution of problem (9).
4. Numerical Results
In this Section, simulation results are presented to illustrate the network performance under different channel uncertainty conditions.
Consider a simulation model shown in
Fig.1
, where the secondary BS is located at (0, 0) and the primary user is located at (294meter, 500meter). There are eight SUs, which are randomly generated around the BS. The channel gain in any transmission pair contains a largescale Rayleigh fading component and a large scale path loss component with path loss factor four. The uncertainty of channel gain in (6) is given by
[3]
Simulation model
Where
D
(
i, j
) indicates the element on the
i
th row and
j
th column of
D
, and
α
,
θ_{i}
∈ (0,1].
Fig. 2
shows the sum rate of the SUs versus the interference power threshold. The parameters are
P_{th}
= 4
W
and
γ
_{1}
:
γ
_{2}
: ⋯ :
γ
_{8}
= 1:1:2:2:3:3:4:4. The curves can be parted into two parts. In the first part, i.e.,
T_{th}
< 0.4 , the SU’s sum rate increases as the interference power threshold
T_{th}
increases, that is because during this period the interference power constraint is the dominating constraint. As the increase of
T_{th}
, more transmission power can be used for SUs' data transmission, and thus the sum rate increases. During this period, the sum rate obtained with a smaller
α
is always much higher than that achieved with a greater
α
. That is because the uncertainty region of the channel gain with a greater
α
is much larger than that with a smaller
α
. The algorithm needs to sacrifice much more sum rate to guarantee all the constraints are satisfied when the uncertainty region of the channel gain is much larger. In the second part, as the increase of
T_{th}
, the total power constraint gradually becomes the dominating constraint at some different points of
T_{th}
for the three cases with different values of
α
. After these points they will keep at their highest sum rate no matter how large
T_{th}
becomes. Although the three cases achieve their highest sum rates at different values of
T_{th}
, the final sum rates are the same.
Sum rate versus T_{th}
Fig. 3
shows each SU’s transmission rate distribution with different values of
α
when
T_{th}
= 0.25 . The other simulation parameters are the same as those in
Fig. 2
. It is evident from
Fig. 3
that the SU’s transmission rate satisfies the proportional fairness constraint. And each SU’s transmission rate with a smaller value of
α
is much higher than that with a greater value of
α
.
SU′s transmission rate distribution
Fig. 4
shows the sum rate changes with total power
P_{th}
. The parameters are
γ
_{1}
:
γ
_{2}
: ⋯ :
γ
_{8}
= 1:1:2:2:3:3:4:4 and
T_{th}
= 0.3
W
. When
P_{th}
≤ 2.6 , the figure shows that the problems with different values of
α
obtain the same sum rate, that is because when
P_{th}
≤ 2.6 , the total power constraint is the dominating constraint for the three cases, thus all the curves achieve the same sum rate. As the increase of
P_{th}
, the interference power constraint gradually becomes the dominating constraint. The greater the value of
α
, the earlier the interference constraint becomes a dominating constraint. When the interference constraint becomes a dominating constraint, the sum rate will keep constant even if
P_{th}
changes.
Sum rate versus total power
5. Conclusion
In this work, we have considered the power control problem in CRNs with channel uncertainty. Our objective is to maximize the sum rate of SUs while guaranteeing the proportional rate fairness among SUs. The problem is formulated as a nonconvex optimization problem. By doing a problem transformation, it becomes an SOCP problem. And it can be efficiently solved by existing methods. In the future, we will propose distributed algorithms to solve this problem.
Acknowledgements
This research was supported by the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the ITRC (Information Technology Research Center) support program (NIPA2014H0301141042) supervised by the NIPA (National IT Industry Promotion Agency).
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