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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
Optimal Power Allocation for
OFDM-Based Cognitive Radio with
New Primary Transmission Protection Criteria
Xin Kang, Student Member, IEEE, Hari Krishna Garg, Senior Member, IEEE,
Ying-Chang Liang, Senior Member, IEEE, and Rui Zhang, Member, IEEE
Abstract—This paper considers a spectrum underlay network,
where an OFDM-based cognitive radio (CR) system is allowed
to share the subcarriers of an OFDMA-based primary system
for simultaneous transmission. Instead of using the conventional
interference power constraint (IPC) to protect the primary users
(PUs) in the primary system, a new criterion referred to as rate
loss constraint (RLC), in the form of an upper bound on the
maximum rate loss of each PU due to the CR transmission,
is proposed for primary transmission protection. Assuming the
channel state information (CSI) of the PU link, the CR link,
and their mutual interference links is available to the CR, the
optimal power allocation strategy to maximize the achievable
rate of the CR system is derived under RLC together with
CR’s transmit power constraint. It is shown that the CR system
can achieve a significant rate gain under RLC as compared to
IPC. Furthermore, the relationship between RLC and IPC is
investigated, and it is shown that the rate gain is obtained by
exploiting the additional CSI of the PU link. A more general
case referred to as hybrid protection to PUs is then studied, by
taking into account that some PU links’ CSI is not available at
CR.
Index Terms—Cognitive radio, convex optimization, OFDM,
power control, spectrum sharing.
I. I NTRODUCTION
W
ITH the rapid development of wireless services and
applications, the currently deployed radio spectrum
is becoming more and more crowded. Therefore, how to
accommodate more wireless services and applications within
the limited radio spectrum becomes a big challenge faced by
modern society. A report published by Federal Communications Commission (FCC) shows that the current scarcity of
spectrum resource is mainly due to the inflexible spectrum
regulation policy rather than the physical shortage of spectrum
[1]. Most of the allocated frequency bands are under-utilized,
Manuscript received June 17, 2009; revised November 10, 2009 and February 10, 2010; accepted March 20, 2010. The associate editor coordinating the
review of this paper and approving it for publication was M. L. Merani.
This work was supported in part by research grants from National University of Singapore (project number: R-263-000-436-112, R-263-000-421-112,
and R-263-000-589-133).
X. Kang and H. K. Garg are with the Department of Electrical and
Computer Engineering, National University of Singapore (e-mail: {kangxin,
eleghk}@nus.edu.sg).
Y.-C. Liang is with the Institute for Infocomm Research, Singapore (e-mail:
[email protected]).
R. Zhang is with the Institute for Infocomm Research, Singapore (e-mail:
[email protected]), and the Department of Electrical and Computer Engineering, National University of Singapore (e-mail: [email protected]).
Digital Object Identifier 10.1109/TWC.2010.06.090912
and the utilization of the spectrum varies in time and space.
This motivates the advent of cognitive radio (CR) [2], which
makes use of spectrum flexibly, efficiently, and reliably. In
early wireless CR networks, an unlicensed user, also known as
secondary user (SU), is only allowed to opportunistically access the spectrum originally allocated to a licensed user known
as primary user (PU), when the PU is not transmitting over
the band. Recently, as proposed by many researchers, SU is
allowed to transmit with the PU over the same spectrum band
simultaneously on condition that the resultant interference at
the PU receiver is below a prescribed threshold, known as
spectrum underlay in [2] or spectrum sharing in [3].
On the other hand, with the high transmission efficiency and
the great capability in combating the inter-symbol interference
caused by frequency selective channels, orthogonal frequency
division multiplexing (OFDM) is regarded as a potential transmission technology for broadband wireless systems. Moreover,
due to its flexibility in allocating transmit resources, OFDM
is also considered as a promising candidate for the future
CR systems. In a wireless network where both the primary
system and the secondary system employ OFDM transmission
technology, the SUs can flexibly fill the spectral gaps left by
the PUs [4] or transmit over the unused subcarriers left in the
primary system [5]. Even if there are no unused subcarriers left
in the primary system, SU can flexibly share the subcarriers
with PUs on condition that PUs are sufficiently protected [6].
Due to the above reasons, OFDM-based CR systems have
attracted wide attention and the related resource allocation
problems have become hot research topics. In conventional
OFDM systems, with a total transmit power constraint, it is
proved that water-filling over the subcarriers is the optimal
power allocation strategy [7]–[9]. However, the conventional
water-filling power control policy is found to be inefficient
for OFDM-based CR systems due to the interaction with the
PUs. In [6], when SU and PU coexist in the same bands,
with individual interference power constraint imposed on each
subcarrier to protect the primary transmission, the optimal
power allocation strategy to maximize the rate of SU is
derived. While in [5], for the case that SU and PU coexist in
side-by-side bands, with a constraint in the form of an upper
bound on the cross band interference incurred to PU to protect
the primary transmission, the optimal and suboptimal power
allocation strategies to maximize the sum rate of the SUs are
derived. The case when SU explores the unused subcarriers
c 2010 IEEE
1536-1276/10$25.00 ⃝
KANG et al.: OPTIMAL POWER ALLOCATION FOR OFDM-BASED COGNITIVE RADIO WITH NEW PRIMARY TRANSMISSION PROTECTION CRITERIA
left in the primary system, the power allocation strategies to
minimize the rate loss of SU caused by the returning of the
PU to reuse the subcarriers are studied in [10]. In [11], a
best effort approach is proposed for interference mitigation by
minimizing the interference from PU to SU while guaranteeing
that PU’s own transmit rate is larger than a target rate.
The contributions of this paper are as follows. We consider
a spectrum underlay CR network where an OFDM-based
CR system coexists with an OFDMA-based primary system.
Instead of using the conventional interference power constraint
to protect PU, a new type of constraint referred to as rate loss
constraint, in the form of an upper bound on the rate loss of
PU due to the secondary transmission is used to protect PU.
Under the proposed constraints together with the SU’s transmit
power constraint, the optimal power allocation strategy for
the SU to maximize its transmission rate is derived. It is
shown that the newly obtained power allocation strategy can
achieve a rate gain over that based on the conventional
interference power constraint. The relationship between the
rate loss constraint and the interference power constraint is
also investigated. It is shown that the channel state information
(CSI) of the primary link is needed to implement the rate
loss constraint. Then, a more general and practical scenario
referred to as hybrid protection to PUs, is considered, where
we assume that only some PUs’ CSI is available at SU
transmitter, and thereby these PUs are protected by the rate
loss constraints; while the rest PUs without CSI available
at SU transmitter are protected by the interference power
constraints. The optimal power allocation strategy to maximize
the SU’s rate under such a hybrid protection constraint is then
studied. It is shown that the power allocation strategy obtained
under the hybrid protection constraints can also achieve a
rate gain as compared to that obtained under the interference
power constraint. It is worth pointing out that for the pointto-point CR network with one PU and one SU, the optimal
power allocation strategies to maximize the ergodic capacity
of the SU under the transmit power constraint together with
an ergodic capacity loss constraint or an outage capacity loss
constraint have been studied in [12] and [13], respectively.
The rest of this paper is organized as follows. Section II
presents the system model and introduces the rate loss constraint. Section III derives the optimal power allocation strategy to maximize the rate of SU under the rate loss constraint
together with a total transmit power constraint. Section IV investigates the relationship between rate loss constraint and the
interference power constraint. Section V derives the optimal
power allocation strategy to maximize the rate of SU under
the hybrid protection constraints and a total transmit power
constraint. Section VI provides numerical examples to verify
the proposed studies. Section VII concludes the paper.
II. S YSTEM M ODEL
As shown in Fig. 1, we consider an OFDMA primary
system that has a total number of 𝑁 subcarriers. The 𝑁
subcarriers are allocated to 𝑀 PUs in the primary system.
Denote the set of subcarriers allocated to PU𝑗 as 𝒦𝑗 , and
we suppose
∪𝑀be allocated to one PU,
∩ one subcarrier can only
i.e., 𝒦𝑗 𝒦𝑖 = ∅, ∀𝑖 ∕= 𝑗, then 𝑗=1 𝒦𝑗 = {1, 2, ⋅ ⋅ ⋅ , 𝑁 }.
PU 1
PU 2
PU j
2067
PU M
N
1 2
Kj
Fig. 1.
Spectrum allocation in OFDMA-based primary system.
Assume the background noise is additive white Gaussian noise
(AWGN), and the noise power of each subcarrier is denoted
by 𝑁0 . Let 𝑓𝑖 be the channel power gain between the PU’s
transmitter and receiver at subcarrier 𝑖 (see Fig. 2), 𝑇𝑖 be
the transmit power of PU allocated to subcarrier 𝑖, then the
transmission rate of PU𝑗 is given by
(
)
1 ∑
𝑓𝑖 𝑇𝑖
log2 1 +
𝑅𝑗𝑝 =
, ∀𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 }. (1)
𝑁
𝑁0
𝑖∈𝒦𝑗
The secondary system is supposed to be a single-user
OFDM system sharing the same 𝑁 subcarriers of the primary
system. This kind of architecture is also known as spectrum
underlay. It is assumed that SU’s transmit signals are Gaussian
distributed, and PU does not know SU’s codebook. Let 𝑃𝑖 be
the transmit power of SU allocated to subcarrier 𝑖, and 𝑔𝑖 be
the channel power gain between the SU’s transmitter and PU’s
receiver, due to SU’s transmission, for any 𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 },
the average rate of PU𝑗 then becomes
(
)
1 ∑
𝑓𝑖 𝑇𝑖
𝑠
log2 1 +
𝑅𝑗 =
.
(2)
𝑁
𝑁0 + 𝑔𝑖 𝑃𝑖
𝑖∈𝒦𝑗
Let Δ𝑅𝑗 be the maximum rate loss that PU can tolerate,
then SU’s transmission is allowed only when the following
constraint is satisfied
𝑅𝑗𝑝 − 𝑅𝑗𝑠 ≤ Δ𝑅𝑗 , ∀𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 }.
(3)
The above constraints are referred to as the PUs’ rate loss constraints. If we define 𝑅𝑗 ≜ 𝑅𝑗𝑝 − Δ𝑅𝑗 , ∀𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 },
then the rate loss constraints can be rewritten as
𝑅𝑗𝑠 ≥ 𝑅𝑗 , ∀𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 }.
(4)
Let 𝑒𝑖 be the channel power gain of the channel between
SU’s transmitter and receiver at subcarrier 𝑖, and let 𝑜𝑖 be the
channel power gain of the channel between PU’s transmitter
and SU’s receiver at subcarrier 𝑖. It is assumed that PU’s
transmit signals are Gaussian distributed, and SU does not
know PU’s codebook. Then, the interference introduced to SU
by the PU can be modeled as AWGN with power 𝐼𝑖 , where
𝐼𝑖 = 𝑜𝑖 𝑇𝑖 . The achievable rate of SU is then given by
(
)
𝑁
𝑒𝑖 𝑃𝑖
1 ∑
.
(5)
log2 1 +
𝑟𝑠 =
𝑁 𝑖=1
𝐼𝑖 + 𝑁0
Take note that CSI of the primary link (PU-TX to PU-RX),
the secondary link (SU-TX to SU-RX), and the interference
links (PU-TX to SU-RX and SU-TX to PU-RX) is required
to implement the rate loss constraint. In practice, CSI of
the secondary link can be obtained at SU-TX by the classic
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
fi
PU-TX
PU-RX
oi
gi
SU-TX
Fig. 2.
ei
SU-RX
Channel model at subcarrier 𝑖, 𝑖 ∈ {1, ⋅ ⋅ ⋅ , 𝑁 }.
channel training, estimation, and feedback mechanisms, while
CSI on the link between PU-TX and SU-RX can be obtained
by SU-RX via estimating the received signal power from PUTX. Similarly, CSI on the primary link and the interference
link between SU-TX and PU-RX can be easily obtained at PURX. Such information is readily obtained at SU-TX, if PU-RX
is aware of the existence of SU-TX and would like to feedback
the information to SU-TX. Otherwise, some dedicated means
must be employed by SU-TX to obtain those CSI, e.g., the
feedback from a cooperative sensor that is located in the
vicinity of PU-RX and is thus able to eavesdrop the CSI
feedback from PU-RX to PU-TX.
feasible solution 𝒛 such that 𝑹𝑠 (𝒛) ≥ 𝛽𝑹𝑥 + (1 − 𝛽)𝑹𝑦
and 𝑓 (𝒛) ≥ 𝛽𝑓 (𝒙) + (1 − 𝛽)𝑓 (𝒚), where 𝑓 (⋅) is the objective
function of P1. Due to the space limitation, the proof is omitted
here. Actually, the time-sharing condition implies that the
maximum transmission rate of SU is a concave function of
𝑹. Since P1 satisfies the “time-sharing” condition, the duality
gap for P1 is virtually negligible with realistic (large) number
of subcarriers, and this makes it possible to solve P1 by using
the Lagrange dual decomposition method, similarly as in [12].
The Lagrangian of P1 is
)
(
𝑁
𝑁
1 ∑
1 ∑
log2 (1 + ℎ𝑖 𝑃𝑖 ) − 𝜆
𝑃𝑖 − 𝑃𝑎
ℒ(P, 𝜆, 𝝁) =
𝑁 𝑖=1
𝑁 𝑖=1
⎞
⎛
(
)
𝑀
∑
∑
𝑇
1
𝑓
𝑖 𝑖
⎠,
−
𝜇𝑗 ⎝𝑅𝑗 −
log2 1 +
(10)
𝑁
𝑁
+
𝑔
𝑃
0
𝑖
𝑖
𝑗=1
𝑖∈𝒦𝑗
where 𝜆 is the dual variable associated with the transmit power
constraint given in (7), and 𝝁 = [𝜇1 , 𝜇2 , ⋅ ⋅ ⋅ , 𝜇𝑀 ] is a vector
of dual variables each associated with one corresponding rate
constraint given in (9).
The Lagrange dual function is then expressed as
𝑔 (𝜆, 𝝁) = max ℒ(P,𝜆, 𝝁).
The dual optimization problem becomes
min 𝑔 (𝜆, 𝝁)
s. t. 𝜆 ≥ 0, 𝝁 ર 0.
III. ACHIEVABLE R ATE OF SU U NDER THE R ATE L OSS
C ONSTRAINT
𝑒𝑖
Define ℎ𝑖 ≜ 𝐼𝑖 +𝑁
and let 𝑃𝑎 be the average transmit power
0
budget of SU, the achievable rate of SU under PUs’ rate loss
constraints can be formulated as
𝑁
1 ∑
P1 : max
log2 (1 + ℎ𝑖 𝑃𝑖 )
P 𝑁
𝑖=1
s.t.
1
𝑁
𝑁
∑
𝑃𝑖 ≤ 𝑃𝑎 ,
(6)
(7)
𝑃𝑖 ≥ 0, ∀𝑖 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑁 },
(8)
𝑅𝑗𝑠
(9)
≥ 𝑅𝑗 , ∀𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 },
where P is a vector of transmit power allocation over subcarriers for the SU, which is given by [𝑃1 , 𝑃2 , ⋅ ⋅ ⋅ , 𝑃𝑁 ]. To
avoid trivial solutions, we assume that at least one of the rate
loss constraints in (9) satisfies the equality. If none of the
constraints in (9) satisfies the equality, the problem reduces
to the conventional power allocation problem for OFDM
systems.
Unfortunately, the rate constraints given in (9) are nonconvex and thus make the problem P1 a non-convex optimization problem. Therefore, if we solve the problem by
considering its Lagrange dual problem, the duality gap between the primal problem and its dual problem will not be
zero. However, it can be verified that P1 satisfies the “timesharing” condition given in [14] when the size of 𝒦𝑗 goes
to infinity, ∀𝑗. To show that P1 satisfies the “time-sharing”
𝑠 𝑇
] and
condition, we first define 𝑹𝑠 = [𝑅1𝑠 , 𝑅2𝑠 , ⋅ ⋅ ⋅ , 𝑅𝑀
𝑇
𝑹 = [𝑅1 , 𝑅2 , ⋅ ⋅ ⋅ , 𝑅𝑀 ] . Then, we let 𝒙 and 𝒚 be the optimal
solutions to the P1 with 𝑹 = 𝑹𝑥 and 𝑹 = 𝑹𝑦 , respectively.
Finally, we show that for any 0 ≤ 𝛽 ≤ 1, there exists a
(12)
(13)
In the following, the dual decomposition method introduced
in [12] is employed to solve this problem. It is observed that
(10) can be rewritten as
ℒ(P, 𝜆, 𝝁) =
𝑀
𝑀
𝜆 ∑∑
1 ∑∑
log2 (1 + ℎ𝑖 𝑃𝑖 ) −
𝑃𝑖
𝑁 𝑗=1
𝑁 𝑗=1
𝑖∈𝒦𝑗
+
𝑀
∑
𝑗=1
𝑖=1
(11)
P
(
𝜇𝑗 ∑
𝑓𝑖 𝑇𝑖
log2 1 +
𝑁
𝑁0 + 𝑔𝑖 𝑃𝑖
𝑖∈𝒦𝑗
)
−
𝑀
∑
𝜇𝑗 𝑅𝑗 + 𝜆𝑃𝑎 .
𝑗=1
𝑖∈𝒦𝑗
(14)
Then, the Lagrange dual function can be rewritten as
𝑔 (𝜆, 𝝁) =
𝑀
∑
𝑔𝑗′ (𝜆, 𝝁) + 𝜆𝑃𝑎 ,
(15)
𝑗=1
where
1 ∑
𝜆 ∑
log2 (1 + ℎ𝑖 𝑃𝑖 ) −
𝑃𝑖
𝑃𝑖 ∈ℱ𝑗 𝑁
𝑁
𝑖∈𝒦𝑗
𝑖∈𝒦𝑗
⎛
⎞
(
)
∑
1
𝑇
𝑓
𝑖
𝑖
(16)
+ 𝜇𝑗 ⎝
log2 1 +
− 𝑅𝑗 ⎠ ,
𝑁
𝑁0 + 𝑔𝑖 𝑃𝑖
𝑔𝑗′ (𝜆, 𝝁) = max
𝑖∈𝒦𝑗
with ℱ𝑗 ≜ {𝑃𝑖 : 𝑃𝑖 ≥ 0, ∀𝑖 ∈ 𝒦𝑗 }.
For a given 𝜆, it is clear that (15) can be decomposed into
𝑀 independent subproblems, each for one PU with the same
structure given by
P2: max 𝑓 (P𝑗 )
(17)
P𝑗
(
∑
log2 1 +
s.t.
𝑖∈𝒦𝑗
𝑓𝑖 𝑇𝑖
𝑁0 + 𝑔𝑖 𝑃𝑖
)
≥ 𝑁 𝑅𝑗 ,
(18)
KANG et al.: OPTIMAL POWER ALLOCATION FOR OFDM-BASED COGNITIVE RADIO WITH NEW PRIMARY TRANSMISSION PROTECTION CRITERIA
where P𝑗 is the power allocation vector for the subcarriers
sharing the spectrum with PU𝑗 , and 𝑓 (P𝑗 ) is defined as
∑
∑
𝑓 (P𝑗 ) ≜
log2 (1 + ℎ𝑖 𝑃𝑖 ) − 𝜆
𝑃𝑖 .
(19)
𝑖∈𝒦𝑗
𝑖∈𝒦𝑗
The Lagrangian of P2 is
𝐿˜𝑗 (P𝑗 , 𝜇𝑗 ) = 𝑓 (P𝑗 )
⎛
(
∑
⎝
+ 𝜇𝑗
log2 1 +
𝑖∈𝒦𝑗
𝑓𝑖 𝑇𝑖
𝑁0 + 𝑔𝑖 𝑃𝑖
)
⎞
− 𝑁 𝑅𝑗 ⎠ ,
(20)
where 𝜇𝑗 is the non-negative dual variable associated with the
constraint (18).
The dual function of P2 is given by
𝑔˜𝑗 (𝜇𝑗 ) = max 𝐿˜𝑗 (P𝑗 , 𝜇𝑗 ) .
P𝑗
(21)
The dual problem is then expressed as
min 𝑔˜𝑗 (𝜇𝑗 )
(22)
s.t. 𝜇𝑗 ≥ 0.
(23)
𝜇𝑗
Thus, the Karush-Kuhn-Tucker (KKT) conditions [15] of
P2 can be written as
⎛
⎞
(
)
∑
𝑓𝑖 𝑇𝑖
(24)
log2 1 +
𝜇𝑗 ⎝
− 𝑁 𝑅𝑗 ⎠ = 0,
𝑁0 + 𝑔𝑖 𝑃𝑖
𝑖∈𝒦𝑗
∂ 𝐿˜𝑗
= 0.
(25)
∂𝑃𝑖
From the KKT conditions listed above, it is not difficult to
obtain the following theorem for determining 𝑃𝑖 for P2:
Theorem 1: The optimal power allocation 𝑃𝑖∗ for P2 is
∀𝑖 ∈ 𝒦𝑗 ,
𝑃𝑖∗ = max {𝜂0 , 0} ,
(26)
where 𝜂0 is the positive root (if no positive root is found, set
𝜂0 = −∞) of the following equation
𝜂=
1
1
− ,
𝜆 ln 2 + 𝜇𝑗 𝑔𝑖 𝜈𝑖 (𝜂) ℎ𝑖
(27)
and 𝜈𝑖 (𝜂) is a function of 𝜂, which can be expressed as
𝜈𝑖 (𝜂) =
𝑓𝑖 𝑇𝑖
,
(𝑁0 + 𝑔𝑖 𝜂) (𝑁0 + 𝑔𝑖 𝜂 + 𝑓𝑖 𝑇𝑖 )
(28)
where 𝜇𝑗 is equal to zero or determined by solving (18) with
equality.
Proof: Please see Part A of the Appendix.
From Theorem 1, it is observed that the optimal power
allocation given in (27) is similar to the conventional waterfilling solution given in [16]. The major difference is that
the water level for the conventional water-filling strategy is
determined by only one parameter, 𝜆, which is the same for
all the subcarriers. However, the water level for the solution
given in (27) not only depends on 𝜆, but also depends on 𝜇𝑗 ,
𝑔𝑖 and 𝜈𝑖 (𝜂). Water level is very important since it directly
relates to the power allocation strategy. For the same channel
condition ℎ𝑖 , a higher water level indicates a higher transmit
power and thus a higher transmission rate. Therefore, it is
important to have a clear understanding of the parameters
that impact the water level. Firstly, 𝜆 is the dual variable
2069
associated with the transmit power constraint, and it reflects
the influence of the transmit power budget on the water level.
A larger power budget results in a smaller 𝜆, and thus results
in a higher water level, and vice versa. Secondly, 𝜇𝑗 is the
dual variable associated with the rate loss constraint, and it
reflects the influence of PU𝑗 ’s rate loss on the water level. If
PU𝑗 can accommodate a larger rate loss, 𝜇𝑗 will be smaller,
and thus result in a higher water level, and vice versa. In the
extreme case that PU𝑗 cannot accommodate any rate loss, 𝜇𝑗
will be infinity, and thus the water level will be zero, which
indicates that the secondary transmission is not permitted over
PU𝑗 ’s band. Thirdly, 𝑔𝑖 is the power gain of the channel
from SU-TX to PU-RX over subcarrier 𝑖. It is clear that a
smaller 𝑔𝑖 will result in a higher water level. This is intuitively
correct because the secondary transmission will not cause too
much rate loss when 𝑔𝑖 is small. Finally, 𝜈𝑖 (𝜂) is a parameter
related to the primary transmission, and it indirectly reflects
the influence of the primary transmission on the water level.
For instance, in the case of 𝑓𝑖 𝑇𝑖 = 0, which indicates that there
is no primary transmission, 𝜈𝑖 (𝜂) will be equal to zero, and
thus the power allocation reduces to the conventional waterfilling strategy. This is true as SU will not cause any rate
loss to PU no matter how large its transmit power is, when
PU is not transmitting. Furthermore, it is observed that 𝜆 is
the same for all the subcarriers, 𝜇𝑗 is the same only for the
subcarriers belonging to PU𝑗 , and 𝑔𝑖 , 𝜈𝑖 (𝜂) are different for
each subcarrier. This suggests that a hierarchical algorithm can
be developed to tackle the problem.
For fixed 𝜆 and fixed 𝜇𝑗 , 𝜂0 can be found by the bisection
search [15]. Let 𝑄(𝜂) ≜ 𝜆 ln 2+𝜇1𝑗 𝑔𝑖 𝜈𝑖 (𝜂) − ℎ1𝑖 . It is easy to
observe that 𝑄(𝜂) is a monotonically increasing function of
𝜂 for 𝜂 ≥ 0. It is clear that 𝜂0 is intersection between the
straight line 𝑦 = 𝜂 with the curve 𝑄(𝜂) for 𝜂 ≥ 0. Suppose
𝜂 is within the range [𝜂𝑚𝑖𝑛 , 𝜂𝑚𝑎𝑥 ]. For the first iteration, we
𝑚𝑎𝑥
compute 𝜂𝑐 = 𝜂𝑚𝑖𝑛 +𝜂
and 𝑄(𝜂𝑐 ), then compare 𝑄(𝜂𝑐 )
2
with 𝜂𝑐 . If 𝑄(𝜂𝑐 ) > 𝜂𝑐 , it is clear that 𝜂0 is within the range
(𝜂𝑐 , 𝜂𝑚𝑎𝑥 ], and we remove the left half interval by setting
𝜂𝑚𝑖𝑛 = 𝜂𝑐 . Otherwise, if 𝑄(𝜂𝑐 ) ≤ 𝜂𝑐 , 𝜂0 must be within
the range [𝜂𝑚𝑖𝑛 , 𝜂𝑐 ), and we remove the right half interval by
setting 𝜂𝑚𝑎𝑥 = 𝜂𝑐 . We repeat the above process until 𝜂0 is
found with the required accuracy. Then, the nonnegative dual
variable 𝜇𝑗 can be updated by its subgradient, which is given
by Proposition 1.
𝜇𝑗 ) is given by
Proposition
of 𝑔˜𝑗 (ˆ
( 1: The subgradient
)
∑
𝑓𝑖 𝑇𝑖
ˆ
𝑖∈𝒦𝑗 log2 1 + 𝑁0 +𝑔𝑖 𝑃ˆ𝑖 − 𝑁 𝑅𝑗 , where P𝑗 is the optimal
solution obtained at 𝜇
ˆ𝑗 .
Proof: Please see Part B of the Appendix.
When 𝜇𝑗 , ∀𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑀 are obtained, 𝜆 is updated by
its subgradient, which is given by Proposition 2.
(
)
ˆ 𝝁
Proposition 2: For given 𝝁, the subgradient of 𝑔 𝜆,
∑𝑁
given by (11) is 𝑁1 𝑖=1 𝑃𝑖 − 𝑃𝑎 , where P̂ is the optimal
ˆ under the given 𝝁.
solution obtained at 𝜆
Proposition 2 can be proved using the same method as that
has been used for proving Proposition 1. Details are omitted
here for brevity.
The algorithm to solve P1 can be summarized as follows.
2070
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
Algorithm 1: Power allocation under the rate loss constraints
1) Initialization: 𝜆1 , 𝑘 = 1,
2) Repeat
a) Initialization: 𝜇𝑗,1 , 𝑘 ′ = 1, ∀𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 }
b) For all 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑀 , repeat
i) Find 𝑃𝑖∗ ,∀𝑖 ∈ 𝒦𝑗 by the bisection search
ii) Update 𝜇𝑗,𝑘′ by
(
(
))
∑
𝑖 𝑇𝑖
𝜇𝑗,𝑘′ +1 = 𝜇𝑗,𝑘′ + 𝛽 𝑁 𝑅𝑗 − 𝑖∈𝒦𝑗 log2 1+ 𝑁 𝑓+𝑔
0
𝑖 𝑃𝑖
iii) If 𝜇𝑗,𝑘′ +1 < 0, set 𝜇𝑗,𝑘′ +1 = 0 and stop;
Otherwise, stop when ∣𝜇𝑗,𝑘′ +1 − 𝜇𝑗,𝑘′ ∣ ≤ 𝜖.
c) Update 𝜆𝑘+1 by (
)
1 ∑𝑁
𝜆𝑘+1 = 𝜆𝑘 + 𝛼 𝑁
𝑖=1 𝑃𝑖 − 𝑃𝑎
3) If 𝜆𝑘+1 < 0, set 𝜆𝑘+1 = 0 and stop;
Otherwise, stop when ∣𝜆𝑘+1 − 𝜆𝑘 ∣ ≤ 𝜖.
Where 𝛼 and 𝛽 are the step size, and 𝜖 > 0 is a given small constant.
IV. R ELATIONSHIP B ETWEEN THE R ATE L OSS
C ONSTRAINT AND THE I NTERFERENCE P OWER
C ONSTRAINT
In the previous section, the optimal power allocation strategy to maximize the rate of SU under the rate loss constraint
together with the transmit power constraint is derived. The
novelty and difficulty of P1 result from the rate loss constraint.
In this section, we investigate the relationship between this
newly proposed constraint with two types of widely used
interference power constraints in the literature. It is proved that
the interference power constraint can serve as an upper bound
on the maximum rate loss of PU, and thus the power allocation
strategies obtained under the interference power constraint can
serve as the sub-optimal power allocation strategies for P1.
A. The per User Based Interference Power Constraint
Let Γ𝑗 be the maximum total interference power that PU𝑗
can tolerate. The per user based interference power constraint
now can be written as
∑
𝑔𝑖 𝑃𝑖 ≤ Γ𝑗 , ∀𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 }.
(29)
𝑖∈𝒦𝑗
Γ𝑗
≥ 𝑅𝑗𝑝 −
,
𝑁 𝑁0 ln 2
where the notation (⋅)+ is defined as (⋅)+ ≜ max {⋅, 0}, and
𝜆 and 𝜇𝑗 are the non-negative dual variables associated with
∑
∑𝑁
the constraints 𝑁1
𝑖∈𝒦𝑗 𝑔𝑖 𝑃𝑖 ≤ Γ𝑗 , ∀𝑗 ∈
𝑖=1 𝑃𝑖 ≤ 𝑃𝑎 and
{1, 2, ⋅ ⋅ ⋅ , 𝑀 }, respectively.
Comparing the above results with the optimal power allocation under the rate loss constraint given in Theorem 1, it is
observed that (31) does not contain the parameter 𝜈𝑖 (𝜂) given
in (28). This indicates that (31) lacks one degree of freedom
as compared to the optimal one in Theorem 1, and this results
in its suboptimality. It is also noted from (28) that the one
additional degree of freedom for the optimal power allocation
is obtained by exploiting the additional information of 𝑓𝑖 𝑇𝑖
from PU. This reveals the fact that with more information
on PUs’ CSI, SU can regulate its power in a more efficient
way, and thus achieves a higher rate over the conventional
interference power constraint.
B. The per Subcarrier Based Interference Power Constraint
Let Γ̃𝑗 be the maximum interference power that each
subcarrier of PU𝑗 can tolerate, then the per subcarrier based
interference power constraint can be written as
𝑔𝑖 𝑃𝑖 ≤ Γ̃𝑗 , ∀𝑖 ∈ 𝒦𝑗 .
The relationship between the per user based interference
power constraint and the rate loss constraint is given by the
following proposition.
Proposition
3: If there exists a threshold Γ𝑗 for PU𝑗 such
∑
that 𝑖∈𝒦𝑗 𝑔𝑖 𝑃𝑖 ≤ Γ𝑗 , then the maximum rate loss of PU𝑗 is
upper-bounded by Γ𝑗 / (𝑁 𝑁0 ln 2).
Proof:
(
)
1 ∑
𝑓𝑖 𝑇𝑖
𝑅𝑗𝑠 =
log2 1 +
𝑁
𝑁0 + 𝑔𝑖 𝑃𝑖
𝑖∈𝒦𝑗
⎞
⎛
(
) ∑
(
)
1 ⎝∑
𝑓𝑖 𝑇𝑖
𝑔𝑖 𝑃𝑖 ⎠
≥
log2 1 +
log2 1 +
−
𝑁
𝑁0
𝑁0
𝑖∈𝒦𝑗
𝑖∈𝒦𝑗
(
)
𝑎 1 ∑
𝑓𝑖 𝑇𝑖
1 ∑ 𝑔𝑖 𝑃𝑖
≥
log2 1 +
−
𝑁
𝑁0
𝑁
𝑁0 ln 2
𝑖∈𝒦𝑗
Proposition 3 reveals the fact that the interference power
constraint is related to PU’s rate loss in an indirect way. This
indicates that by properly choosing Γ𝑗 , ∀𝑗 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑀 },
the rate loss of PU𝑗 can also be regulated to be less than the
prescribed threshold under the interference power constraint.
Therefore, the power allocation obtained under the interference power can be regarded as a sub-optimal power allocation
for the rate loss constraint case, similarly as observed in [12].
Replacing the rate loss constraint in P1 with the interference
power constraint, the resulting new problem becomes a convex
optimization problem. Using the method similar to that given
in [17], it can be shown that the power allocation is
(
)+
1
1
−
𝑃𝑖 =
,
(31)
ln 2 (𝜆 + 𝜇𝑗 𝑔𝑖 /𝑁0 ) ℎ𝑖
𝑖∈𝒦𝑗
(30)
where the inequality “𝑎” results from the fact that 𝑥 log2 (𝑒) ≥
log2 (1 + 𝑥), ∀𝑥 ≥ 0.
(32)
The relationship between the per subcarrier based interference power constraint and the rate loss constraint is given by
the following proposition.
Proposition 4: If there exists a threshold Γ̃𝑗 for PU𝑗 such
that 𝑔𝑖 𝑃𝑖 ≤ Γ̃𝑗 , ∀𝑖 ∈ 𝒦𝑗 , the maximum
(
)rate loss of PU𝑗 then
Γ̃𝑗
∣𝒦𝑗 ∣
is upper-bounded by 𝑁 log2 1 + 𝑁0 , where ∣𝒦𝑗 ∣ denotes
the cardinality of the set 𝒦𝑗 .
Proof:
(
)
1 ∑
𝑓𝑖 𝑇𝑖
𝑅𝑗𝑠 =
log2 1 +
𝑁
𝑁0 + 𝑔𝑖 𝑃𝑖
𝑖∈𝒦𝑗
(
)
𝑎 1 ∑
𝑓𝑖 𝑇𝑖
≥
log2 1 +
𝑁
𝑁0 + Γ̃𝑗
𝑖∈𝒦𝑗
⎛
(
)⎞
(
) ∑
Γ̃𝑗 ⎠
1 ⎝∑
𝑓𝑖 𝑇𝑖
≥
log2 1 +
log2 1 +
−
𝑁
𝑁0
𝑁0
𝑖∈𝒦𝑗
𝑖∈𝒦𝑗
(
)
Γ̃𝑗
∣𝒦𝑗 ∣
log2 1 +
,
(33)
≥ 𝑅𝑗𝑝 −
𝑁
𝑁0
KANG et al.: OPTIMAL POWER ALLOCATION FOR OFDM-BASED COGNITIVE RADIO WITH NEW PRIMARY TRANSMISSION PROTECTION CRITERIA
where the inequality “𝑎” results from the fact that 𝑔𝑖 𝑃𝑖 ≤
Γ̃𝑗 , ∀𝑖 ∈ 𝒦𝑗 .
It is seen from Proposition 4 that if the transmit power of SU
satisfies the constraint 𝑔𝑖 𝑃𝑖 ≤ Γ̃𝑗 , ∀𝑖 ∈ 𝒦𝑗(, the maximum
rate
)
Γ̃
∣𝒦 ∣
loss of PU is upper-bounded by 𝑁𝑗 log2 1 + 𝑁𝑗0 . If we let
(
)
Γ̃
∣𝒦 ∣
this bound satisfy 𝑁𝑗 log2 1 + 𝑁𝑗0 ≤ Δ𝑅𝑗 , then it is clear
that the rate
( loss constraint is
) satisfied. Choosing the threshold
Γ̃𝑗 as 𝑁0 2𝑁 Δ𝑅𝑗 /∣𝒦𝑗 ∣ − 1 , the rate loss is regulated to be
less than Δ𝑅𝑗 . Under the constraint 𝑔𝑖 𝑃𝑖 ≤ Γ̃𝑗 , ∀𝑖 ∈ 𝒦𝑗 ,
P1 becomes a convex optimization problem. Then, it can be
shown that the power allocation is
{(
}
)+
1
Γ̃𝑗
1
−
,
,
(34)
𝑃𝑖 = min
𝜆 ln 2 ℎ𝑖
𝑔𝑖
where 𝜆 is the
dual variable associated with the
∑non-negative
𝑁
𝑃
≤
𝑃
.
constraint 𝑁1
𝑎
𝑖=1 𝑖
V. ACHIEVABLE R ATE OF SU W ITH H YBRID P ROTECTION
TO PU S
In the previous section, the relationship between the rate
loss constraint and the interference power constraint is investigated. It is shown that additional information (𝑓𝑖 𝑇𝑖 ) of the
primary links is needed at the SU to implement the rate loss
constraint. Such information can be obtained at the SU via
the feedback from the PUs. However, in practice, some PUs
may be not able to feedback such information to SU. For such
a scenario, it is more reasonable to protect these PUs by the
interference power constraint. Consequently, in this section,
we propose that different types of constraints should be used to
protect different types of PUs instead of using a homogeneous
criterion to protect all the PUs. We study the case when some
PUs are protected by the rate loss constraints, and some PUs
are protected by the interference power constraints. We refer
to this kind of protection to the primary system as hybrid
protection.
Denote the set of PUs protected by the rate loss constraints
by 𝒮𝑟 , the set of PUs protected by the interference power
constraints by 𝒮Γ , the problem can be formulated as
P3 : max
P
𝑁
1 ∑
log2 (1 + ℎ𝑖 𝑃𝑖 )
𝑁 𝑖=1
(35)
𝒮𝑟
∪
𝒮Γ = {1, 2, ⋅ ⋅ ⋅ , 𝑀 }, 𝒮𝑟
(
)
𝑁
𝑁
1 ∑
1 ∑
log2 (1 + ℎ𝑖 𝑃𝑖 ) − 𝜆
𝑃𝑖 − 𝑃𝑎
ℒ(P, 𝜆, 𝝁) =
𝑁 𝑖=1
𝑁 𝑖=1
⎛
⎞
∑
∑
∑
(
)
(41)
−
𝜇𝑗 𝑅𝑗 − 𝑅𝑗𝑠 −
𝛾𝑗 ⎝
𝑔𝑖 𝑃𝑖 − Γ𝑗 ⎠ ,
𝑗∈𝒮𝑟
𝑗∈𝒮Γ
(37)
(38)
∩
𝒮Γ = ∅,
𝑔 (𝜆, 𝝁, 𝜸) = max ℒ(P,𝜆, 𝝁, 𝜸).
(42)
P
The dual optimization problem becomes
min 𝑔 (𝜆, 𝝁, 𝜸) ,
s. t. 𝜆 ≥ 0, 𝝁 ર 0, 𝜸 ર 0.
(43)
(44)
Then, it is not difficult to show that the Lagrange dual
function of P3 can be rewritten as
∑
∑
𝑔 (𝜆, 𝝁, 𝜸) =
𝑔𝑗′ (𝜆, 𝝁) +
𝑔𝑗′′ (𝜆, 𝜸) + 𝜆𝑃𝑎 , (45)
𝑗∈𝒮𝑟
𝑗∈𝒮Γ
where
𝜆 ∑
1 ∑
log2 (1 + ℎ𝑖 𝑃𝑖 ) −
𝑃𝑖
𝑃𝑖 ∈ℱ𝑗 𝑁
𝑁
𝑖∈𝒦𝑗
𝑖∈𝒦𝑗
⎛
⎞
(
)
∑
1
𝑓
𝑇
𝑖
𝑖
(46)
+ 𝜇𝑗 ⎝
log2 1 +
− 𝑅𝑗 ⎠ ,
𝑁
𝑁0 + 𝑔𝑖 𝑃𝑖
𝑔𝑗′ (𝜆, 𝝁) = max
𝑖∈𝒦𝑗
and
𝑔𝑗′′ (𝜆, 𝝁) = max
⎛
+ 𝛾𝑗 ⎝
𝑃𝑖 ∈ℱ𝑗
∑
1 ∑
𝜆 ∑
log2 (1 + ℎ𝑖 𝑃𝑖 ) −
𝑃𝑖
𝑁
𝑁
𝑖∈𝒦𝑗
𝑖∈𝒦𝑗
⎞
𝑔𝑖 𝑃𝑖 − Γ𝑗 ⎠ ,
(47)
𝑖∈𝒦𝑗
with ℱ𝑗 ≜ {𝑃𝑖 : 𝑃𝑖 ≥ 0, ∀𝑖 ∈ 𝒦𝑗 }.
Thus, for a given 𝜆, it is clear that (45) can be decomposed
into 𝑀 independent subproblems, one for each PU and in one
of the following two kinds of structures:
∑
∑
log2 (1 + ℎ𝑖 𝑃𝑖 ) − 𝜆
𝑃𝑖 ,
(48)
P4: max
s.t.
𝑖∈𝒦𝑗
(
log2 1 +
∑
𝑖∈𝒦𝑗
and
(39)
P5: max
(40)
s.t.
where P is a vector of transmit power for SU given by
[𝑃1 , 𝑃2 , ⋅ ⋅ ⋅ , 𝑃𝑁 ], 𝑃𝑎 is the average transmit power budget of
SU, and Γ𝑗 is the maximum interference that PU𝑗 can tolerate.
𝑖∈𝒦𝑗
where 𝜆 is the dual variable associated with the transmit power
constraint given in (36), and 𝝁 and 𝜸 are two vectors of
the dual variables associated with the rate constraints given
in (38) and the interference power constraint given in (39),
respectively.
The Lagrange dual function of P3 is expressed as
(36)
𝑃𝑖 ≥ 0, ∀𝑖 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝑁 },
𝑅𝑗𝑠 ≥ 𝑅𝑗 , ∀𝑗 ∈ 𝒮𝑟 ,
∑
𝑔𝑖 𝑃𝑖 ≤ Γ𝑗 , ∀𝑗 ∈ 𝒮Γ ,
𝑖∈𝒦𝑗
The Lagrangian of P3 is
𝑃𝑖 ∈ℱ𝑗
𝑁
1 ∑
s.t.
𝑃𝑖 ≤ 𝑃𝑎 ,
𝑁 𝑖=1
2071
𝑃𝑖 ∈ℱ𝑗
∑
𝑓𝑖 𝑇𝑖
𝑁0 + 𝑔𝑖 𝑃𝑖
𝑖∈𝒦𝑗
log2 (1 + ℎ𝑖 𝑃𝑖 ) − 𝜆
𝑖∈𝒦𝑗
∑
)
𝑔𝑖 𝑃𝑖 ≤ Γ𝑗 .
≥ 𝑁 𝑅𝑗 .
∑
𝑃𝑖 ,
(49)
(50)
𝑖∈𝒦𝑗
(51)
𝑖∈𝒦𝑗
It is observed that P4 has the same structure as P2 studied
in the previous section. Therefore, the solution of P4 is the
same as that for P2 given by Theorem 1. For P5, it is
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
Algorithm 2: Power allocation under the hybrid protection constraints
1) Initialization: 𝜆1 , 𝑘 = 1,
2) Repeat
a) Initialization: 𝜇𝑗,1 , ∀𝑗 ∈ 𝒮𝑟 , 𝑘 ′ = 1
b) ∀𝑗 ∈ 𝒮𝑟 , repeat
i) Find 𝑃𝑖∗ ,∀𝑖 ∈ 𝒦𝑗 by the bisection search
ii) Update 𝜇𝑗,𝑘′ +1 by (
(
))
∑
𝑖 𝑇𝑖
𝜇𝑗,𝑘′ +1 = 𝜇𝑗,𝑘′ + 𝛽 𝑁 𝑅𝑗 − 𝑖∈𝒦𝑗 log2 1 + 𝑁 𝑓+𝑔
0
𝑖 𝑃𝑖
iii) If 𝜇𝑗,𝑘′ +1 < 0, set 𝜇𝑗,𝑘′ +1 = 0 and stop;
otherwise, stop when ∣𝜇𝑗,𝑘′ +1 − 𝜇𝑗,𝑘′ ∣ ≤ 𝜖.
c) Initialization: 𝛾𝑗𝑚𝑖𝑛 ,𝛾𝑗𝑚𝑎𝑥 , ∀𝑗 ∈ 𝒮Γ ,
∑
d) ∀𝑗 ∈ 𝒮Γ , repeat until ∣ 𝑖∈𝒦𝑗 𝑔𝑖 𝑃𝑖 − Γ𝑗 ∣ ≤ 𝜖
)
(
i) 𝛾𝑗 = 𝛾𝑗𝑚𝑖𝑛 + 𝛾𝑗𝑚𝑎𝑥 /2
)+
(
1
ii) Calculate 𝑃𝑖∗ ,∀𝑖 ∈ 𝒦𝑗 by
− ℎ1
ln 2(𝜆+𝛾𝑗 𝑔𝑖 /𝑁0 )
𝑖
∑
iii) If 𝑖∈𝒦𝑗 𝑔𝑖 𝑃𝑖 < Γ𝑗 , set 𝛾𝑗𝑚𝑎𝑥 = 𝛾𝑗 ;
otherwise, set 𝛾𝑗𝑚𝑖𝑛 = 𝛾𝑗 .
e) Update 𝜆𝑘+1 by (
)
1 ∑𝑁
𝜆𝑘+1 = 𝜆𝑘 + 𝛼 𝑁
𝑖=1 𝑃𝑖 − 𝑃𝑎
3) If 𝜆𝑘+1 < 0, set 𝜆𝑘+1 = 0 and stop;
Otherwise, stop when ∣𝜆𝑘+1 − 𝜆𝑘 ∣ ≤ 𝜖.
Where 𝛼 and 𝛽 are the step size, and 𝜖 > 0 is a given small constant.
not difficult
to show that the
)+ optimal solution is given by
(
1
1
, where 𝛾𝑗 is a nonnegative
𝑃𝑖 = ln 2(𝜆+𝛾𝑗 𝑔𝑖 /𝑁0 ) − ℎ𝑖
dual variable associated with the constraint (51). It is either
equal to zero or determined by solving (51) with equality.
Numerically, 𝛾𝑗 can be found by the bisection search.
When all the 𝑀 subproblems are solved, 𝜆 can be found by
the subgradient method. Thus, the entire problem P3 can be
solved by the following iterative power allocation algorithm.
Remark: From the above decomposition-based solutions, it
can be observed that P3 includes the SU’s rate maximization
problem under only the rate loss constraint or under only the
interference power constraint as two special cases. If we set
𝒮Γ = ∅, P3 reduces to SU’s rate maximization problem under
the rate loss constraint. Similarly, if we set 𝒮𝑟 = ∅, P3 reduces
to SU’s rate maximization problem under the interference
power constraint.
VI. N UMERICAL R ESULTS
In this section, several numerical examples are presented
to verify the effectiveness of the proposed power allocation
strategies. In these numerical examples, we assume that all
the involved channels (i.e., the primary link, the secondary
link and the interference links) are Rayleigh distributed. Consequently, the channel power gains for these channels are
exponentially distributed. Since the channel power gains can
be different for different channel realizations, all the numerical
results presented in this part are obtained by averaging over
10, 000 independent simulation runs. The average channel
power gains for the primary link 𝑓𝑖 and the secondary link
𝑒𝑖 are assumed to be 1, i.e. 𝔼{𝑓𝑖 } = 1, 𝔼{𝑒𝑖 } = 1, ∀𝑖.
The average channel power gain for the interference links
are assumed to be 0.1, i.e. 𝔼{𝑔𝑖 } = 0.1, 𝔼{𝑜𝑖 } = 0.1, ∀𝑖.
Moreover, we assume that the number of subcarriers 𝑁 of the
primary system is 128, and the transmit power budget of the
primary system is 10𝑑𝐵. It is also assumed that the primary
system adopts equal power allocation over its subcarriers. The
4.5
Transmission rate of SU (bits/complex dim.)
2072
4
3.5
PU‘s rate loss 20%
PU‘s rate loss 10%
PU‘s rate loss 5%
3
2.5
2
1.5
1
0.5
0
−10
−5
0
5
10
Transmit power constraint, Pa(dB)
15
20
Fig. 3. Transmission rate of SU vs. the transmit power constraint under
different PU’s rate loss constraints.
noise power on each subcarrier is assumed to be identical, and
equal to 1, i.e. 𝑁0 = 1.
A. Example 1: Effects of rate loss constraints on SU’s transmission rate
In this example, for clarity of exposition, we assume that
the 128 subcarriers of the primary system are all allocated to
one PU, and it is protected by the rate loss constraint. Then,
the rate of SU under different rate loss constraints are plotted
in Fig. 3. It is observed that the rate increases with 𝑃𝑎 and
PU’s rate loss constraint. It is also observed that when 𝑃𝑎
is small, the difference of the SU’s rate under different PU’s
rate loss constraints is almost the same. This is due to the
fact that the transmit power constraint will be the dominant
constraint when 𝑃𝑎 is small. With the increase of 𝑃𝑎 , the rate
loss constraint gradually becomes the dominant constraint, and
thus the difference of the SU’s rate under different PU’s rate
loss constraints becomes large.
B. Example 2: Comparison of the rate loss constraint and per
subcarrier based interference power constraint
In this example, we compare the rate of the SU under the
rate loss constraint with that under the per subcarrier based
interference power constraint. It is assumed that there are two
PUs in the primary system, and each of them occupies 64
subcarriers. We assume that one of them can tolerate 10%
rate loss and the other one can tolerate 20% rate loss. For the
per subcarrier based interference power constraint case, the
interference thresholds of the two PUs are chosen as Γ̃1 =
20.2𝑅1 −1 and Γ̃2 = 20.4𝑅2 −1, respectively, which guarantees
that the two PUs’ rate losses upper bounds are the same as
the PUs’ rate loss constraint case. It is observed from Fig. 4
that SU can achieve a rate gain under the rate loss constraint
over the interference power constraint. It is also observed that
the rate gain is very small when 𝑃𝑎 is small. However, with
the increase of 𝑃𝑎 , the rate gain gradually becomes large. This
suggests that the proposed constraint is more effective for large
values of 𝑃𝑎 .
KANG et al.: OPTIMAL POWER ALLOCATION FOR OFDM-BASED COGNITIVE RADIO WITH NEW PRIMARY TRANSMISSION PROTECTION CRITERIA
35%
PU‘s rate loss constraint
Interference power constraint
30%
3.5
3
Rate loss of PU
Transmission rate of SU (bits/complex dim.)
4.5
4
2073
2.5
2
25%
20%
1.5
Δν=0, Rs=2.1845
1
Δν=0.1, Rs=1.2648
15%
Δν=0.5, R =0.5983
s
0.5
0
−10
Δν=+∞, Rs=0
−5
0
5
10
Transmit power constraint, Pa(dB)
15
20
Fig. 4.
Comparison of the SU’s transmission rate under the rate loss
constraint vs. per subcarrier based interference power constraint.
C. Example 3: Effects of imperfect CSI on PU’s rate loss
In this subsection, we investigate the impact of imperfect
channel estimations on the performance of the proposed power
allocation strategies. To study the effects of imperfect CSI
on the proposed SU power control policy, we only consider
imperfect estimations of the PU channel and the channel
between SU-TX and PU-RX , while the SU channel and the
channel between PU-TX and SU-RX are both assumed to be
perfect in the sequel.
Let 𝑓𝑖 and 𝑓ˆ𝑖 be the true and the estimated fading channel
coefficients for the primary link, respectively. Similarly, let 𝑔𝑖
and 𝑔ˆ𝑖 be the true and the estimated coefficients of the fading
channel between SU-TX and PU-RX, respectively. Then the
relationship between the true and estimated fading coefficients
is given by
√
√
(52)
𝑓𝑖 = 1 − 𝜎 2 𝑓ˆ𝑖 + 𝜎 2 𝑛1
√
√
2
2
𝑔𝑖 = 1 − 𝜎 𝑔ˆ𝑖 + 𝜎 𝑛2
(53)
where 𝑛1 and 𝑛2 are independent CSCG random variables
each having zero mean and unit variance, and 𝜎 2 is the
variance for the effective channel estimation errors, 𝜎 2 ≤ 1.
Under the above assumptions, it is observed that the proposed SU power control strategy will cause additional rate loss
of PU due to the imperfect channel estimation. To alleviate
this, we modify the SU power control strategy to improve
its robustness against channel estimation errors. First, we
compute the power allocation strategy according to Theorem
1 based on the estimated channels power gains. Then, we
modify the obtained power allocation strategy by introducing
a protection gap, denoted by Δ𝜈, where Δ𝜈 ≥ 0. The
modified (
power allocation )strategy can then be expressed
+
1
1
, where 𝜈𝑖′ = 𝑔𝑖 𝜈𝑖 + Δ𝜈, ∀𝑖.
as 𝑃𝑖 = 𝜆 ln 2+𝜇
′ − ℎ
𝑗 𝜈𝑖
𝑖
The values of 𝜆 and 𝜇𝑗 remain the same as those obtained
from the unmodified power allocation strategy. It is observed
that, by introducing the protection gap, we actually lower the
water level of SU, reduce the interference caused to PU, and
thus decrease the rate loss of PU. As such, it is expected
that this modified SU power control strategy will not cause
10%
0
0.2
0.4
0.6
0.8
1
2
Variance of channel estimation errors, σ
Fig. 5.
Effects of imperfect channel estimation on the PU rate loss.
too much additional rate loss of PU provided that Δ𝜈 is
chosen sufficiently large to incorporate the channel estimation
errors measured by 𝜎 2 . On the other hand, the introduction
of the protection gap decrease the transmit power of SU,
and thus decrease the transmission rate of SU. The larger the
protection gap is, the lower the SU’s transmission rate is. This
indicates that the improvement of the power control strategy’s
robustness is achieved by sacrificing SU’s transmission rate.
In Fig. 5, we show PU’s rate loss due to imperfect CSI
versus 𝜎 2 for SU’s power control strategy given in (27) (i.e.,
Δ𝜈 = 0) and the modified strategy proposed above with
different values of Δ𝜈. It is assumed that PU’s target rate
loss is 10%. It is observed that the PU’s rate loss increases
with 𝜎 2 for a given protection gap Δ𝜈, while it decreases
with increasing Δ𝜈 for a given 𝜎 2 . In the extreme case of
Δ𝜈 = +∞, the PU’s rate loss is reduced to 10% for all
values of 𝜎 2 , since in this case SU in fact switches off
its transmission. Furthermore, it is noted that the improved
robustness of the SU power control against imperfect CSI
to protect the PU transmission is achieved at the cost of
SU’s transmission rate, which is shown as 𝑅𝑠 in the legend
field of Fig. 5. Therefore, the SU needs to choose a proper
protection gap Δ𝜈 to effectively balance the tradeoff between
SU’s transmission rate and the transmission protection of PU.
D. Example 4: Comparison of the hybrid protection constraint
and per user based interference power constraint
In this example, we assume that there are two PUs in the
primary system, and each of them occupies 64 subcarriers. For
the hybrid protection case, one of PUs is protected by the rate
loss constraint with 10% tolerable rate loss, and the other one
is protected by the per user based interference power constraint
where Γ is chosen such that the resultant rate loss of this PU
is also 10%. For the interference protection case, both of PUs
are assumed to be protected by the per user based interference
power constraints, and the two interference thresholds Γ1 and
Γ2 are chosen such that the resultant rate losses of the two
PUs are both 10%. It can be observed from Fig. 6 that SU
can achieve a rate gain under the hybrid protection constraints
2074
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
)
(
𝑔𝑖
∂𝑓 (P𝑗 )
𝜇𝑗
𝑔𝑖
−
−
∂𝑃𝑖
ln 2 𝑁0 + 𝑔𝑖 𝑃𝑖 + 𝑓𝑖 𝑇𝑖 𝑁0 + 𝑔𝑖 𝑃𝑖
𝑓𝑖 𝑇𝑖
∂𝑓 (P𝑗 ) 𝜇𝑗 𝑔𝑖
⋅
,
−
=
∂𝑃𝑖
ln 2 (𝑁0 + 𝑔𝑖 𝑃𝑖 + 𝑓𝑖 𝑇𝑖 ) (𝑁0 + 𝑔𝑖 𝑃𝑖 )
(54)
=
Transmission rate of SU (bits/complex dim.)
4
3.5
Hybrid protection constraint
Interference power constraint
3
2.5
∂𝑓 (P )
ℎ𝑖
− 𝜆 obtained by taking derivative
where ∂𝑃𝑖𝑗 = ln 2(1+ℎ
𝑖 𝑃𝑖 )
of (19). Then, define 𝜈𝑖 (𝑃𝑖 ) ≜ (𝑁0 +𝑔𝑖 𝑃𝑖 +𝑓𝑓𝑖𝑖 𝑇𝑇𝑖𝑖 )(𝑁0 +𝑔𝑖 𝑃𝑖 ) and
set (54) to zero, we have
2
1.5
𝑃𝑖 =
1
0.5
0
−10
−5
0
5
10
Transmit power constraint, P (dB)
15
20
a
Fig. 6. Comparison of the SU’s transmission rate under the hybrid protection
constraint vs. per user based interference power constraint.
over the interference power constraints, and this gain increases
with 𝑃𝑎 .
VII. C ONCLUSION
The achievable rate of an OFDM-based cognitive radio
system sharing the spectrum with an OFMDA-based primary
system is studied in this paper. A new criterion referred to
as rate loss constraint for primary transmission protection
is proposed. This newly proposed constraint protects PU
by regulating the maximum rate loss of PU due to the
SU’s transmission to be below a prescribed threshold. The
relationship between the rate loss constraint and the interference power constraint is then investigated. Then, hybrid
protection to the primary system is proposed by protecting
some PUs by the rate loss constraint and some PUs by the
interference power constraint. The optimal power allocation
strategy to maximize the rate of SU subject to the rate loss
constraint/hybrid protection constraint together with the total
transmit power constraint of the SU is derived. It is shown
that the proposed power allocation scheme obtained under the
rate loss constraint/hybrid protection constraint can achieve
substantial rate gains over the conventional power allocation
scheme obtained under the interference power constraint.
ACKNOWLEDGMENT
The authors would like to thank the associate editor and
the anonymous reviewers for their time and effort spent in
reviewing this manuscript. This has resulted in a significantly
improved manuscript.
A PPENDIX
A. Proof of Theorem 1
From KKT conditions given in (25), it is observed that the
∂ 𝐿˜
optimal solution satisfies ∂𝑃𝑗𝑖 = 0. Then, from (20), it follows
)]
[ (
(
)
∑
𝑓𝑖 𝑇𝑖
∂ 𝜇𝑗
log2 1+ 𝑁0+𝑔𝑖 𝑃𝑖 −𝑁 𝑅𝑗
𝑖∈𝒦𝑗
∂𝑓 (P𝑗 )
∂ 𝐿˜𝑗
=
+
∂𝑃𝑖
∂𝑃𝑖
∂𝑃𝑖
1
1
− .
𝜆 ln 2 + 𝜇𝑗 𝑔𝑖 𝜈𝑖 (𝑃𝑖 ) ℎ𝑖
Since the transmit power cannot be negative, it is easy to
show{that the optimal power} allocation strategy is 𝑃𝑖∗ =
max 𝜆 ln 2+𝜇1𝑗 𝑔𝑖 𝜈𝑖 (𝑃𝑖 ) − ℎ1𝑖 , 0 . Theorem 1 is thus proved.
B. Proof of Proposition 1
Let 𝜇′𝑗 (be a) feasible value of
( 𝑔˜𝑗 (𝜇𝑗 ).) From [18], it is known
𝜇𝑗 ) + 𝑠 𝜇′𝑗 − 𝜇
ˆ𝑗 holds for any feasible
that if 𝑔˜𝑗 𝜇′𝑗 ≥ 𝑔˜𝑗 (ˆ
𝜇′𝑗 , then 𝑠 must be the subgradient of 𝑔˜𝑗 (ˆ
𝜇𝑗 ) at 𝜇
ˆ𝑗 . Now,
( ′)
𝑔˜𝑗 𝜇𝑗
⎛
⎛
⎞⎞
(
)
∑
𝑓
𝑇
𝑖 𝑖
log2 1+
= max⎝𝑓 (P𝑗 )+𝜇′𝑗⎝
−𝑁 𝑅𝑗 ⎠⎠
P𝑗
𝑁0 +𝑔𝑖 𝑃𝑖
𝑖∈𝒦𝑗
⎛
⎞
(
)
∑
( ′)
𝑇
𝑓
𝑖
𝑖
= 𝑓 P𝑗 + 𝜇′𝑗 ⎝
log2 1 +
− 𝑁 𝑅𝑗 ⎠
𝑁0 + 𝑔𝑖 𝑃𝑖′
𝑖∈𝒦𝑗
⎛
⎞
(
)
( )
∑
𝑎
𝑓𝑖 𝑇𝑖
≥ 𝑓 P̂𝑗 + 𝜇′𝑗 ⎝
log2 1 +
− 𝑁 𝑅𝑗 ⎠
ˆ
𝑁
0 + 𝑔𝑖 𝑃𝑖
𝑖∈𝒦𝑗
⎛
⎞
(
)
( )
∑
𝑇
𝑓
𝑖 𝑖
ˆ𝑗 ⎝
= 𝑓 P̂𝑗 + 𝜇
log2 1 +
− 𝑁 𝑅𝑗 ⎠
𝑁
+
𝑔𝑖 𝑃ˆ𝑖
0
𝑖∈𝒦𝑗
⎛
⎞
(
)
∑
𝑇
𝑓
𝑖
𝑖
+ 𝜇′𝑗 ⎝
log2 1 +
− 𝑁 𝑅𝑗 ⎠
ˆ𝑖
𝑃
𝑁
+
𝑔
0
𝑖
𝑖∈𝒦𝑗
⎛
⎞
(
)
∑
𝑓
𝑇
𝑖
𝑖
−𝜇
ˆ𝑗 ⎝
− 𝑁 𝑅𝑗 ⎠
log2 1 +
𝑁0 + 𝑔𝑖 𝑃ˆ𝑖
𝑖∈𝒦𝑗
⎞
⎛
(
)
) ∑
( ′
𝑓𝑖 𝑇𝑖
= 𝑔˜𝑗 (ˆ
𝜇𝑗 )+ 𝜇𝑗 − 𝜇
ˆ𝑗 ⎝
log2 1+
−𝑁 𝑅𝑗⎠ ,
𝑁0 +𝑔𝑖 𝑃ˆ𝑖
𝑖∈𝒦𝑗
where P′𝑗 is the optimal solution associated with 𝜇𝑗 = 𝜇′𝑗 ,
and P̂𝑗 is the optimal solution associated with 𝜇𝑗 = 𝜇
ˆ𝑗 . The
inequality 𝑎 results from the fact that P′𝑗 is the optimal solution
for 𝜇𝑗 = 𝜇′𝑗 .
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Xin Kang (S’08) received his B.Sc. degree in electrical engineering from Xi’an Jiao Tong University,
China, in 2005. He is currently working toward
his Ph.D. degree in the Electrical and Computer
Engineering Department at the National University
of Singapore. His research interests include convex
optimization, centralized and decentralized power
allocation strategies, game theory, information theory, cognitive radio networks, and multiuser multicarrier communications systems.
Hari Krishna Garg received the B.Tech. degree
from the Indian Institute of Technology (IIT), Delhi,
the M.Eng. and Ph.D. degrees from Concordia University, Montreal, PQ, Canada, and the MBA degree
from Syracuse University, Syracuse, NY, USA.
He has been a faculty member of the Electrical
and Computer Engineering Department at Syracuse
University. Currently, he is with the Electrical and
Computer Engineering Department at the National
University of Singapore. His research area of interest
is mobile communications from the physical layer
to the applications on both technology as well as applications’ front. More
recently, he has been active as an entrepreneur having founded or co-founded
four companies. In his leisure time, he enjoys spending time with his children
and listening to Bollywood music.
2075
Ying-Chang Liang (SM’00) is now a Senior Scientist in the Institute for Infocomm Research (I2R),
Singapore, where he has been leading the research
activities in the area of cognitive radio and cooperative communications. He also has held an adjunct
associate professorship position at Nanyang Technological University (NTU) since 2004. From Dec.
2002 to Dec. 2003, he was a visiting scholar with
the Department of Electrical Engineering, Stanford
University. His research interests include cognitive
radio, dynamic spectrum access, reconfigurable signal processing for broadband communications, space-time wireless communications, wireless networking, information theory, and statistical signal
processing.
Dr. Liang now an Associate Editor of IEEE T RANSACTIONS ON V EHIC ULAR T ECHNOLOGY . He served as an Associate Editor of IEEE T RANSAC TIONS ON W IRELESS C OMMUNICATIONS from 2002 to 2005, Lead GuestEditor of IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS,
Special Issue on Cognitive Radio: Theory and Applications, and Special
Issue on Advances in Cognitive Radio Networking and Communications,
and Guest-Editor of the C OMPUTER N ETWORKS J OURNAL (Elsevier) Special
Issue on Cognitive Wireless Networks. He received the Best Paper Awards
from IEEE VTC-Fall 1999 and IEEE PIMRC 2005, and the 2007 Institute of
Engineers Singapore (IES) Prestigious Engineering Achievement Award. He
has served for various IEEE conferences as a technical program committee
(TPC) member. He was Publication Chair of the 2001 IEEE Workshop on
Statistical Signal Processing, TPC Co-Chair of the 2006 IEEE International
Conference on Communication Systems (ICCS 2006), Panel Co-Chair of the
2008 IEEE Vehicular Technology Conference Spring (VTC 2008-Spring),
TPC Co-chair of the 3rd International Conference on Cognitive Radio
Oriented Wireless Networks and Communications (CrownCom 2008), TPC
Co-Chair of 2010 IEEE Symposium on New Frontiers in Dynamic Spectrum
Access Networks (DySPAN 2010), and Co-chair of the Thematic Program
on random matrix theory and its applications in statistics and wireless
communications, the Institute for Mathematical Sciences, National University
of Singapore, 2006. Dr. Liang is a Senior Member of IEEE.
Rui Zhang (S’00-M’07) received the B.Eng and
M.Eng degrees in electrical and computer engineering from the National University of Singapore
(NUS) in 2000 and 2001, respectively, and the
Ph.D. degree in electrical engineering from Stanford University, California, USA, in 2007. He is
now a Senior Research Fellow with the Institute
for Infocomm Research (I2R), Singapore. He also
holds an Assistant Professorship position with the
Department of Electrical and Computer Engineering,
NUS. He has authored/co-authored more than 100
refereed international journal and conference papers. He was the co-recipient
of the Best Paper Award from IEEE PIMRC 2005. He was a Guest-Editor
of the EURASIP J OURNAL ON A PPLIED S IGNAL P ROCESSING, special
issue on Advanced Signal Processing for Cognitive Radio Networks. He has
served for various IEEE conferences as a technical program committee (TPC)
member and organizing committee member. His current research interests
include cognitive radio, cooperative communication, and multiuser MIMO
systems.