Optimal Power Allocation for OFDM-Based Cognitive Radio with New ...

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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

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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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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)


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


)

⎞ − 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

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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 ˆ is the optimal given by (11) is 𝑁1 𝑖=1 𝑃𝑖 − 𝑃𝑎 , where P 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.

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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) 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 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 ˜ 𝑗 be the maximum interference power that each Let Γ subcarrier of PU𝑗 can tolerate, then the per subcarrier based interference power constraint can be written as ˜ 𝑗 , ∀𝑖 ∈ 𝒦𝑗 . 𝑔𝑖 𝑃𝑖 ≤ Γ

𝑖∈𝒦𝑗

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. ˜ 𝑗 for PU𝑗 such Proposition 4: If there exists a threshold Γ ˜ that 𝑔𝑖 𝑃𝑖 ≤ Γ𝑗 , ∀𝑖 ∈ 𝒦𝑗 , the maximum ( )rate loss of PU𝑗 then ˜𝑗 Γ ∣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 + ˜𝑗 𝑁 𝑁0 + Γ 𝑖∈𝒦𝑗 ⎛ ( )⎞ ( ) ∑ ˜𝑗 Γ 1 ⎝∑ 𝑓𝑖 𝑇𝑖 ⎠ ≥ log2 1 + log2 1 + − 𝑁 𝑁0 𝑁0 𝑖∈𝒦𝑗 𝑖∈𝒦𝑗 ( ) ˜𝑗 Γ ∣𝒦𝑗 ∣ log2 1 + , (33) ≥ 𝑅𝑗𝑝 − 𝑁 𝑁0


where the inequality “𝑎” results from the fact that 𝑔𝑖 𝑃𝑖 ≤ ˜ 𝑗 , ∀𝑖 ∈ 𝒦𝑗 . Γ It is seen from Proposition 4 that if the transmit power of SU ˜ 𝑗 , ∀𝑖 ∈ 𝒦𝑗 , the maximum rate satisfies the constraint 𝑔𝑖 𝑃𝑖 ≤ Γ ( ) ˜ Γ ∣𝒦 ∣ loss of PU is upper-bounded by 𝑁𝑗 log2 1 + 𝑁𝑗0 . If we let ( ) ˜ Γ ∣𝒦 ∣ this bound satisfy 𝑁𝑗 log2 1 + 𝑁𝑗0 ≤ Δ𝑅𝑗 , then it is clear that the rate ( loss constraint is ) satisfied. Choosing the threshold ˜ 𝑗 as 𝑁0 2𝑁 Δ𝑅𝑗 /∣𝒦𝑗 ∣ − 1 , the rate loss is regulated to be Γ ˜ 𝑗 , ∀𝑖 ∈ 𝒦𝑗 , less than Δ𝑅𝑗 . Under the constraint 𝑔𝑖 𝑃𝑖 ≤ Γ P1 becomes a convex optimization problem. Then, it can be shown that the power allocation is {( )+ ˜ } 1 Γ𝑗 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

∑


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


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.)

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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 ˜1 = interference thresholds of the two PUs are chosen as Γ 0.2𝑅1 0.4𝑅2 ˜ 2 −1 and Γ2 = 2 −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 .


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

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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

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) ( 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 ˆ 𝑗 + 𝜇′ ⎝ ≥𝑓 P log2 1 + − 𝑁 𝑅𝑗 ⎠ 𝑗 ˆ 𝑁 0 + 𝑔𝑖 𝑃𝑖 𝑖∈𝒦𝑗 ⎛ ⎞ ( ) ( ) ∑ 𝑇 𝑓 𝑖 𝑖 ˆ𝑗 + 𝜇 ˆ𝑗 ⎝ =𝑓 P 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 , ˆ 𝑗 is the optimal solution associated with 𝜇𝑗 = 𝜇 and P ˆ𝑗 . The inequality 𝑎 results from the fact that P′𝑗 is the optimal solution for 𝜇𝑗 = 𝜇′𝑗 . R EFERENCES [1] “Spectrum policy task force,” Federal Communications Commission, ET Docket No. 02-135, Tech. Rep., Nov. 2002. [2] J. Mitola and G. Q. Maguire, “Cognitive radio: making software radios more personal,” IEEE Personal Commun., vol. 6, no. 6, pp. 13–18, Aug. 1999. [3] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005.


[4] T. Weiss and F. Jondal, “Spectrum pooling: an innovative strategy for the enhancement of spectrum efficiency,” IEEE Commun. Mag., vol. 42, no. 3, pp. S8–S14, Mar. 2004. [5] G. Bansal, J. Hossain, and V. K. Bhargava, “Optimal and suboptimal power allocation schemes for OFDM-based cognitive radio systems,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4710–4718, Nov. 2008. [6] P. Wang, M. Zhao, L. Xiao, S. Zhou, and J. Wang, “Power allocation in OFDM-based cognitive radio systems,” in Proc. IEEE Global Telecommun. Conf. (Globecom), 2007, pp. 4061–4065. [7] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. New York: Cambridge University Press, 2005. [8] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999. [9] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp. 171–178, Feb. 2003. [10] Z. Hasan, E. Hossain, C. Despins, and V. K. Bhargava, “Power allocation for cognitive radios based on primary user activity in an OFDM system,” in Proc. IEEE Global Telecommun. Conf. (Globecom), 2008. [11] S. Geirhofer, L. Tong, and B. M. Sadler, “A cognitive framework for improving coexistence among heterogeneous wireless networks,” in Proc. IEEE Global Telecommun. Conf. (Globecom), 2008. [12] R. Zhang, “Optimal power control over fading cognitive radio channels by exploiting primary user CSI,” in Proc. IEEE Global Telecommun. Conf. (Globecom), 2008. Available: arXiv: 0804.1617 [13] X. Kang, R. Zhang, Y.-C. Liang, and H. K. Garg, “Optimal power allocation for cognitive radio under primary user’s outage loss constraint,” in Proc. IEEE Int. Conf. Commun. (ICC), 2009. [14] W. Yu and R. Lui, “Dual methods for non-convex spectrum optimization of multi-carrier systems,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1310–1322, July 2006. [15] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge University Press, 2004. [16] T. Cover and J. Thomas, Elements of Information Theory. New York: Wiley, 1991. [17] X. Kang, Y.-C. Liang, A. Nallanathan, H. K. Garg, and R. Zhang, “Optimal power allocation for fading channels in cognitive radio networks: ergodic capacity and outage capacity,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 940–950, Feb. 2009. [18] S. Boyd, L. Xiao, and A. Mutapcic, “Subgradient methods.” [Online]. Available: http://www.stanford.edu/class/ee392o/subgrad_method.pdf 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.

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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.