China Communications • June 2015 96

as low as 15% on average[1]. Cognitive radio (CR), which can solve the conflict between scarcity and underutilization of spectrum, has attracted great attention. In underlay scenarios, a fundamental challenge of spectrum sharing is to ensure the quality-of-service (QoS) of primary users (PUs) while maximizing the achievable throughput of CR users (CUs). Fortunately, the multi-antenna technique pro-vides more spatial degrees of freedom to deal with the interference.

Interference alignment (IA) is an effective user coordination technology for distributed multi-user multi-antenna networks, and is dispensable to central control and global opti-mization. Each pair of transmitter and receiver only needs to optimize the precoding and re-ceiving matrices of themselves respectively, and they can obtain half spatial degrees of freedom of the interference-free case without joint signal processing[2]. IA is suitable for spectrum sharing in CR networks owing to the CUs’ ability to sense channel state information (CSI) and their interference tolerance due to the “best-effort” QoS demand. Besides, the distributed construction and flexible reconfig-uration of CR networks prefer IA to deal with signals distributively and independently.

However, the interference from the CR system to PUs is one of the biggest chal-lenges. Due to the priority of primary (PR) system, PR terminals will not participate into

Abstract: An interference alignment (IA) spectrum sharing method based on Rayleigh quotient is proposed for distributed multi-user multi-antenna cognitive radio (CR) networks. The interference from cognitive users (CUs) to the primary (PR) system is constrained through the Rayleigh quotients of channel matrices to deal with the absence of PR users (PUs) in the IA process. As a result, the IA scheme can be applied in CR networks with-out harmful interference to PUs. Compared with existing IA based spectrum sharing meth-ods, the proposed method is more general be-cause of breaking the restriction that CUs can only transmit on the idle sub-channels of the PR system. Moreover, in comparison to other four spectrum sharing methods applicable in general scene, the proposed method leads to improved performance of achievable sum rate of the CR system as well as guarantees the transmission of PUs.Keywords: cognitive radio; spectrum sharing; Rayleigh quotient; interference alignment; MIMO

I. INTRODUCTION

In the past decade there has been explosive growth in spectrum demand due to the deploy-ment of a wide variety of wireless services. Unfortunately, the current utilization efficien-cy of the licensed radio spectrum bands can be

Rayleigh Quotient Based Interference Alignment Spectrum Sharing in MIMO Cognitive Radio NetworksROnG MeI

Department of Electronic and Information Engineering, School of Information Engineering, Chang’an University, Xi’an 710064, China

NETWORK TECHNOLOGY AND APPLICATION

China Communications • June 201597

from each CRT to PUs cannot be calculated before IA since it is iterative. On the other hand, the complexity of implementing IA in all possible CU combinations to look for the best is unaffordable. Through observation, we realize that the interference can be expressed as the Rayleigh quotients of the channel ma-trices between CRTs and PUs. Consequently, properties of Rayleigh quotient can be used to predict the maximum interference from each CRT to PUs. User access control can be imple-mented in the CR system accordingly. Based on the discussions above, in this paper, we propose a Rayleigh quotient based IA spec-trum sharing (RIASS) scheme for generalized multi-user multi-antenna CR networks without restriction on sub-channel or power allocation of PUs.

The main contributions of this paper are summarized as follows: first, we break the re-striction that CUs can only transmit on the idle sub-channels of the PR system, and introduce IA based spectrum sharing into general CR networks; Secondly, we propose an IA spec-trum sharing method for distributed multi-us-er multi-antenna CR networks. Rayleigh quotients of channel matrices are utilized to predict the maximum interference from each CRT to PUs. On the basis of predictions, CUs with less interference to PUs are permitted to access. As a result, both the number of access-ing CUs and transmit power of each active CRT are increased as well as the sum rate of CUs. Thirdly，because of the time-variant channel condition between CRTs and PUs, in-terference constraints maybe so low as no CU can access. To solve the problem, we set an adjustable transmit power baseline to achieve a better tradeoff between power gain and spa-tial degrees of freedom so as to improve the robustness of our method.

The rest of this paper is organized as fol-lows. Section II provides the system model of CR network under IA based spectrum shar-ing. Section III describes the RIASS method. Performance analysis is shown in Section IV, in terms of achievable sum rate, degrees of freedom, robustness and complexity. Section

the IA process among CUs. Consequently, this interference should be reduced by CUs independently but without joint processing with PR receivers (PRRs). The precoding matrices of CUs for interference cancellation are basically not consistent with the optimal solution of IA. Considering the conflict, re-searchers have taken helpful explorations on IA application in CR networks, and proposed some IA based spectrum sharing methods for single-user multi-antenna CR networks to make use of the unused spatial directions of PUs brought by power limitations in water-filling[3-8]. The transmitted signals from CRTs are first projected onto the idle sub-channels of the PR system and then processed with IA. The conclusions were extended to multi-user scenarios[9-14]. These methods are only applica-ble to special scenes with waterfilling power allocation and unused spatial sub-channels in PR systems. Based on these results, Lu etc. developed an IA based spatial-frequency sig-nal alignment spectrum sharing scheme in MI-MO-OFDM CR systems. Spatial waterfilling power allocation and cyclic-prefix (CP) of the PR system bring the idle spatial and frequen-cy sub-channels, respectively. Consequently, both spatial and frequency space left over by the PR system are used for IA among CUs[15,

16]. Moreover, Maso etc. realized IA in CR net-works through the cooperation between the PR and CR system[17]. These methods are valuable for application of IA spectrum sharing in CR networks. However, all the methods above are not suitable for general CR networks, where PUs can neither be aware of the coexistence of CUs nor cooperate with the CR system. Furthermore, waterfilling power allocation and idle sub-channels in the PR system cannot be guaranteed. Motivated by all these reasons, we investigate IA spectrum sharing method for general CR networks to increase the sum rate of CUs through access and power control. Considering the interference constraints, we prefer CUs with less interference to PUs ac-cessing in order to increase both the number of active CUs and the transmit power of each CR transmitter (CRT). However, the interference

I n t h i s p a p e r , a Ray le igh quot i en t based IA spectrum shar ing method i s proposed to maximize the achievable sum rate of CR systems on the premise of satis-fying the IC of PUs for general CR networks.

China Communications • June 2015 98

Wk properly. Due to the complexity, we prefer to look for the suboptimal solution through IA. From the mathematics model, we can observe that there is not any specific on the sub-channel allocation of the PR system, so the proposed method is more general than the methods in Ref. [3-16].

III. RAYLEIGH QUOTIENT BASED IA SPECTRUM SHARING

As mentioned before, although IA based spec-trum sharing is appropriate for CR networks, the challenge of interference at the PRR cannot be neglected. Since it is impossible to compel PUs to join in IA, the signal pre-pro-cessing can be carried out only by designing Fk properly. Unfortunately, IA is an iterative process of searching optimal Fk and Wk, which is always conflicting with the interference re-duction. To solve the conflict, we propose an IA based spectrum sharing scheme by utilizing the Rayleigh quotients[18] of channel matri-ces. Alternating minimization[19] is adopted, in which Ck is assumed to meet the equation Wk = IN×N − CkCH

k . The dimensions of Fk and Ck are M × rk, 1 ≤ rk < N and N × (N − rk), re-spectively.

Here we show the RIASS method. First,

V provides the simulation results and discus-sions. Finally, we conclude the paper in Sec-tion VI.

II. SYSTEM MODEL

We consider a CR network with K pairs of CUs, each including a CRT with M antennas and a CR receiver (CRR) with N antennas. For brevity, the CRTs and CRRs are collectively called CUs when it is not necessary to distin-guish them. The CR system coexists with a PR transmitter (PRT) and a PRR with Mp and Np antennas, respectively. The CR network is shown in Fig. 1.

The channels between users are all qua-si-static Rayleigh flat. Hi,k and Gk denote the channel matrices from CRTk to CRRi and PRR, respectively. Dk is the channel matrix between PRT and CRRk. The set of active pairs of CUs is A, and |A|=Ka. The transmitted signal from CRTk is xk, and the signal from PRT is s. The receiving signal at CRRk is

yk = Hk,kxk +∑

i�k,i∈A

Hk,ixi + Dks + zk (1)

where zk is the complex Gaussian white noise with power σ2

0 . Every CU is attainable to H, G and D by sensing[3, 4, 8, 9, 12].

The spectrum sharing problem aims at maximizing the sum rate of CR system on the premise of satisfying the interference con-straint (IC) of PUs. Based on IA, we design the precoding matrices for xk and post-process-ing matrices for yk as Fk and Wk, respectively. The signal model isxk =WkHk,kFkxk +Wk

∑i�k,i∈A

Hk,iFixi +WkDks +Wkzk

(2)The maximum transmit power of each CRT and PRT are Pc and Pp with equally power on each antenna. The interference at PRR is

Ip =∑k∈A

Prk‖GkFk‖2 (3)

where P is the transmit power of each CRT, and rk is the rank of Fk. Accordingly, the math-ematics model of spectrum sharing problem is shown at top of the next page, where Icon is the IC of the PRR. The problem can be solved by searching the optimal A and designing Fk and Fig.1 Cognitive radio network

1

M

1

M

1

N

1

N

1

Np

1

Mp

s r

1x

Kx

1x

Ky

1y

Kx

1F 1W

KF KW

1,1H

,1KH

1G

KG

,K KH

1DKD

1,KH

s

CR system

PR system

……

…

……

…

China Communications • June 201599

Ip ≤ Ip,max = P0

∑k∈Aλk ≤ Icon (10)

So the transmit power of each CRT can be ad-justed as

P = min(P0

Icon

Ip, Pc

) (11)

to increase the sum rate of CR system and sat-isfy IC.

IV. PERFORMANCE ANALYSIS

4.1 Achievable sum rate of the CR system

With IA based spectrum sharing, the achiev-able sum rate of CR system can be represented as

R =∑k∈A

log2 det

I +

Prk

(WkHk,kFk

) (WkHk,kFk

)H

Wk

( Pp

MpDkDH

k + σ20

)WH

k

(12)Since Fk and Wk are calculated through iter-ative IA process, we substitute the optimized results in each random channel realization into (12) to obtain the average sum rate from a large amount of simulations as the ergodic sum rate of CUs.

In this subsection, we compare the RIASS method with IA based power control spectrum sharing (IAPSS) method and one user access (OUA) method qualitatively. IAPSS permits all the K pairs of CUs accessing and guaran-tees IC by reducing the transmit power of all the CRTs, while OUA allows only one user to access at a time. Both of them are intuitive re-sults of introducing IA based spectrum sharing to CR networks without any restriction on the resource allocation of the PR system.

In IA based spectrum sharing, sum of the transmit power of all the CRTs is limited by IC. We can reduce either the transmit power of every CRT or the number of accessing CUs to

select proper CUs to access. We set the base-line of transmit power from CRT as P0=BPc

[20], where B ∈ (0, 1) is adjustable. We suppose rk = 1, 1 ≤ k ≤ K to enlarge the spaces for de-sired signals. The interference at PRR is

Ip =∑k∈A

P0‖GkFk‖2 =∑k∈A

P0FHk GH

k GkFk (5)

where GHk Gk is Hermit ian. We assume

FHk Fk = 1 [19], so

FHk GH

k GkFk =FH

k

(GH

k Gk

)Fk

FHk Fk

(6)

is the Rayleigh quotient[18] of GHk Gk . Due

to the property of Rayleigh quotient[18], Ip is maximized when Fk is equal to the eigenvec-tor corresponding to the largest eigenvalue of GH

k Gk , and the largest eigenvalue can be rep-resented as λk. So

Ip,max = max{Ip} = P0

∑k∈Aλk (7)

Sort λk of GHk Gk, 1 ≤ k ≤ K , and permit the

first Ka pairs of CUs with smallest λk accessing to guarantee that Ip,max ≤ Icon .

Secondly, alternating minimization[19] is adopted to realize IA of accessing CUs. Initialize the Fk as an arbitrary M×1 vector, and then calculate the columns of Ck as the eigenvectors corresponding to the first N-1

eigenvalues of ∑

i�k,i∈AHk,iFiFH

i HHk,i . Accordingly,

acquire Fk, k ∈ A as the last eigenvector of ∑

i�k,i∈AHH

i,k

(IN − CiCH

i

)Hi,k . Repeat the above cal-

culations until convergence, i.e.,

∑k∈A

∑i�k,i∈A

∥∥∥Hk,iFi − CkCHk Hk,iFi

∥∥∥2< δ (8)

where δ=10-4 represents the convergence accu-racy.

Thirdly, adjust the transmit power of each CRT according to the actual interference at PRR:

Ip =∑k∈A

P0‖GkFk‖2 (9)

Due to the property of Rayleigh quotient[18],

maxA,F

k,W

k

∑

k∈A

log2 det

I +Prk

(WkHk,kFk

) (WkHk,kFk

)H

Wk

∑i�k,i∈A

Prk

(Hk,iFi

) (Hk,iFi

)H+

Pp

MpDkDH

k + σ20

WHk

s.t. Ip � Icon

(4)

China Communications • June 2015 100

(14)where γi,k represents the ith singular value of Ak. In RIASS, IAPSS and OUA, |A|=Ka,K, and 1, respectively. Based on IA theory[2],

the degrees of freedom are DR = 12 KaN ,

DI = 12 KN and DO = N , which character-

ize the slopes of the sum rate curves when P→∞, i.e., the performance at high-SNR re-gime[21]. Normally, K≥Ka>2, so DI ≥ DR > DO

, especially in large CR systems. In contrast, transmit power is more effective at low-SNR

regime[21]. Since P =Icon∑

k∈A‖GkFk‖2 and K≥Ka,

PR ≥ PI .In summary, RIASS can usually obtain

larger transmit power than IAPSS at low-SNR regime, as well as larger degrees of freedom than OUA at high-SNR regime. Consequently, RIASS can achieve a better tradeoff between degrees of freedom and transmit power than the other two methods.

4.3 Robustness and flexibility

In this subsection, we discuss the robustness and flexibility in different cases of Icon incor-porating the selection of B. On the basis of (12), the problem of maximizing the sum rate of CUs can be converted into a power allo-cation issue if only Pk and Ka are regarded as variables, such as

maxA,P

k

∑k∈A

log2 det[I + Pk

(WkHk,kFk

) (WkHk,kFk

)HϕkWk

(PpDkDH

k /Mp + σ20

)WH

k

]

s.t.∑k∈A

Pk ≤ Icon

(15)where ϕk = FH

k GHk GkFk and Pk = ϕkP . The

equivalent channel gain of the kth pair of CUs is assumed to be

hk = tr

(WkHk,kFk

) (WkHk,kFk

)H

ϕkWk

( Pp

MpDkDH

k + σ20

)WH

k

(16)

so the optimal P∗k to the power allocation problem can be obtain through waterfilling al-gorithm[21]. However, Wk and Fk are unattain-able before A and Ka are selected. To solve the conflict, we choose the suboptimal solution on

meet the IC condition. Hence, as the number of accessing CUs (Ka) increases for more de-grees of freedom, the transmit power of each CRT decreases. To the limit of Ka→∞, P→0, the sum rate of CUs is R→0. On the contrary, we can reduce Ka to increase the transmit power of each CRT, as well as decrease the interference between CUs. In the extreme case of Ka→0, the sum rate is R→0. The above analysis indicates that there should be a best tradeoff between the number of accessing CUs (Ka) and the transmit power of each CRT (P), which can maximize the sum rate of the CR system (R) in each channel realization. Ba-sically, neither IAPSS nor OUA can achieve any tradeoff since each of them focuses on one aspect. Fortunately, RIASS gives con-sideration to both Ka and P, so it can obtain a better tradeoff between them. Besides, RIASS method is attainable to multi-user diversity gain through sorting the largest ei-genvalues of the interference channel matrices Gk, k = 1, · · · ,K and permitting less-inter-fering CUs to access, so it can achieve larger sum rate of the CR system than the other two methods.

4.2 Tradeoff between degrees of freedom and transmit power

In the preceding subsection, RIASS, IAPSS and OUA are compared in a qualita-tive way. Here, some further analysis on the tradeoff between degrees of freedom and transmit power is provided. The degrees of freedom brought by each method are denoted by DR, DI and DO, respectively. In the mean-time, PR, PI and PO are assumed to be the transmit power of every active CRT in the three methods.

The degrees of freedom of the CR system can be calculated according to (12). Assuming that

Ak =

(WkHk,kFk

) (WkHk,kFk

)H

rkWk

( Pp

MpDkDH

k + σ20

)WH

k

(13)

we can rewrite (12) as

R =∑k∈A

log2 det [I + PAk] =∑k∈A

N∑i=1

log2

(1 + Pγi,k

)

China Communications • June 2015101

three methods (Power control, denoted by “P”; one user access, denoted by “O”; use one spa-tial sub-channel, denoted by “S”) have lower complexity than the two IA based methods, which means the latter increase the degrees of freedom and sum rate of CUs at the price of higher complexity. However, the complexity of IA is affordable and worthy.

V. SIMULATION RESULTS AND DISCUSSIONS

We consider the PR system as a single-cell scenario, and the cell is a round region with radius as l1=1000m. The base station is con-sidered as the PRT, located in the center of the cell. There are K=15 pairs of CUs in a round region with radius of l2=30m inside the cell. For simplicity, we suppose all the CRRs are at the same distance, d3, to their corresponding CRTs, and the same distance, d4, to the PRT, as the assumptions in Ref. [22]. Since l1 � l2, the distance between each CRT and the PRR can be assumed to be d1. The CR system shares the spectrum with the PRR whose distance is greater than 500m from the CRTs to reduce the interference, i.e. d1≥500. The schematic of the simulation scenario is shown in Fig. 2. Both PRT and PRR have 4 antennas, and each CU has 16 antennas, i.e. Mp=Np=4 and M=N=16. It is worth emphasizing that there is no idle sub-channel in the PR system in this general scenario, so all the methods in Ref. [3-16] are all ineffective.

In the above analyses, we only consider small scale fading for IA convergence as in [2-10, 12-16, 19]. However, large scale fad-ing is necessary for calculating Icon and the signal-to-interference and noise-ratio (SINR) at each receiver. For fair comparisons, we establish typical scenes, and then calculate Icon and receiving SINR with large scale fad-ing. IA is implemented only with small scale fading. In other words, the equivalent IC and SNR at each CRR are used in the IA process. Both channel matrices Hk and Gk represent small scale fading channels, and they are ran-dom matrices whose elements are complex

the basis of the water-level[21], which decided by Icon. That is to say, all CRTs transmit signals with equal power if water-level (Icon) is high; in contrast, only the best pair of CUs can ac-cess if water-level (Icon) is low[21].

The discussions above indicate that only RIASS method with the adaptive parameter B can increase Ka if Icon is large or decrease Ka if Icon is small, which leads to a better tradeoff between P and Ka and a larger achievable sum rate of the CR system than IAPSS and OUA. In a word, RIASS is more flexible and robust than the other two methods.

4.4 Complexity analysis

We analyze the complexity of spectrum sharing methods in terms of an addition or a multiplication. Since IA is iterative, it needs much more time than the Rayleigh quotient calculation and power adjustment. The time complexity is shown in Tab. 1, where T de-notes the average number of iterations for each IA process, usually in 101~102 order[19]. From the comparisons, it is concluded that our method (denoted by “R”) has the same order of time complexity with IAPSS(denoted by “I”). Besides, Table 1 shows that the other

Table I The complexity of five methodsCU Selection IA Power Adjustment In all

R O(KM3) O(KaM3T) O(KaM

2) O(KaM3T)

I — O(KM3T) O(KM2) O(KM3T)

P — — O(KM2) O(KM2)

O O(KM3) — O(M2) O(KM3)

S O(KM3) — O(M) O(KM3)

Fig.2 Cognitive radio network

PRTPRR

CRR1

CRTK

CRRK

CRT2

CRR2

CRT1

China Communications • June 2015 102

sum rate than the other four methods. IAPSS restrains the interference to PRR only by power control, which leads to a reduction of sum rate and is not obtainable to the multi-user diversity gain. Nevertheless, IAPSS is superior to the other three methods due to the application of IA, especially in high SNR cases. The power control (PC) method considers all the inter-ference as noise, and reduces it only through decreasing the transmit power of each CRT. Without consideration of the interactions be-tween CUs, the CR system gets less degrees of freedom, and the curve has floor effect. Since the CUs are all multi-antenna terminals, the MIMO channel between each pair of them can be decomposed by singular value decomposi-tion (SVD) and parallel spatial sub-channels can be constructed by precoding at the CRT and post-processing at the CRR. When we choose the minimal-interfering pair of CUs (as “OUA” method) or their best spatial sub-channel (as “use One Spatial Sub-channel, OSS” method) to transmit, the sum rate is low due to underuti-lization of spectrum.

For further investigation on the effect of Pp and Icon, we show the achievable sum rate of CR system when Pp or Icon is a constant. In Fig. 4, we assume that SNRp=19dB. The curves tell us that the sum rate increases monotonously with SNR in all methods. The proposed RI-ASS method acquires the largest sum rate for the multi-user diversity gain and greater de-

Gaussian random variables with mean 0 and unit variance[2-10, 12-16, 19]. The large scale fading model of PR system is Urban Macro NLoS model from ITU-R M.2135 report[23] :L(dB) = 161.04 − 7.1log10 (W) + 7.5log10 (h)

−(24.37 − 3.7(h/hBS )2

)log10 (hBS )

+(43.42 − 3.1log10 (hBS )

)(log10(d) − 3)

+20log10( fc) − (3.2(log10 (11.75hUT )

)2 − 4.97)

(17)where W=20m, h=20m, hBS=50m, hUT=1.6m, fc=2GHz. We assume the distance from CRTs and PRT to PRR as d1=500m and d2=500m, re-spectively, so L1=L2=115.4725dB. For the CR system, we adopt the Indoor HotSpot NLoS model[23]:L(dB) = 43.3log10 (d)+11.5 + 20log10 ( fc) (18)When fc=2GHz, the distance between each pair of CUs is assumed to be d3=30m, thus L3=81.48dB. Pc=23(dBm)=0.2w. We a s s u m e Pc = Pc/L3 a n d Pp = Pp/L1 , s o

SNR = Pc/σ20 a n d SNRp = Pp/σ

20 , w h i c h

are equivalent to the SNR in [2-5, 9, 12, 19]. The noise power is supposed to be σ2

0 = −174dBm/Hz × 20MHz = − 101dBm[23]. Because the anti-interference ability of the PR system increases with SNRp, the interference threshold at PRR is sup-p o s e d t o b e Icon = 0.01SNRpσ

20 . T h e r e -

fore, if SNRp ≤ 20dB , as the cases in our simulations, Icon ≤ σ2

0 . Accordingly, the equivalent IC at CRR can be represented asIc = IconL1/L3 ≈ 25SNRpσ

20 . If SNR synchro-

nously changes with SNRp, Ic ≈ 25SNRσ20 .

The IC condition can be turn to an equivalent

equation as ∑k∈A

PFHk GH

k GkFk ≤ Ic , or approxi-

matively as

∑k∈A

FHk GH

k GkFk � 25 (19)

The achievable sum rates of the CR system brought by five spectrum sharing methods are shown in Fig. 3. From the curves, we can note that the accessing selection of minor interfering CUs is an effective way not only to ease the IA from IC, but also to obtain multi-user diversity. In the meantime, IA brings extra degrees of freedom for the multi-user MIMO CR system. Consequently, the RIASS can acquire larger Fig.3 Achievable sum rate of CUs

-10 -5 0 5 10 15 200

20

40

60

80

100

120

140

160

180

SNR/SNRp/dB

Ach

ieva

ble

Sum

rate

of C

R Sy

stem

/bit/

s/H

z

RIASSIAPSSPCOUAOSS

China Communications • June 2015103

grees of freedom brought by CU selection and IA, respectively. IAPSS scheme obtains more degrees of freedom from IA, so the slope of the curve is bigger than any other method at high SNRs. However, at low SNRs, since the sum rate is mainly affected by transmit power of each CRT, it is smaller than other methods. The power control method has not gained multi-user diversity for the CR system, and its spatial degree of freedom is 1, so the curve has smallest slope. The OUA method can get multi-user diversity and spatial degrees of freedom, but both the transmit power at low SNRs and the degrees of freedom at high SNRs are smaller than those in RIASS, so the sum rate is less than the proposed method. Moreover, when the selected pair of CUs only chooses the best spatial sub-channel to trans-mit, they lose the spatial degrees of freedom seriously, so the sum rate is lower.

The simulations in Fig. 5 show the case of Icon = 0.014σ2

0 without adaptation of B. Ob-viously, since the interference from PRT to CRRs increases with SNRp and the transmit power of CRTs are limited by Icon, the sum rate decreases with SNR (SNRp). However, the proposed RIASS method can still obtain larger sum rate than other methods through a better tradeoff between Ka and P brought by adjusting B. Fig. 6 reflects the robustness and flexibility of every method. The results also illuminate that transmitting only on the best spatial sub-channel is inefficient. Due to the ineffective Icon, only the OSS method curve always increases with SNR.

Moreover, for the further investigations on the performance in practice, simulations are implemented in a more practical scene. The polar coordinates of the PRR distribute uni-formly in the cell, i.e. the radial coordinate is ρ0 ∼ U (0, 1000] and the angular coordinate is θ0 ∼ U [0, 2π] in each simulation independent-ly. The CUs distributes in a round region inside the cell, whose center has the radial coordinate γ0 ∼ U (0, 1000] and the angular coordinate φ0 ∼ U [0, 2π] . The polar coordinates of CRTk and CRRk are αk, βk ∼ U (0, 20] and

-10 -5 0 5 10 15 200

20

40

60

80

100

120

140

160

180

200

SNR/dB

Ach

ieva

ble

Sum

rate

of C

R Sy

stem

/bit/

s/H

z

RIASSIAPSSPCOUAOSS

Fig.4 Achievable sum rate of CUs w/ constant Pp

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80

90

SNR/SNRp/dB

Ach

ieva

ble

Sum

rate

of C

R Sy

stem

/bit/

s/H

z

RIASSIAPSSPCOUAOSS

Fig.5 Achievable sum rate of CUs in case of constant Icon w/o adaptation of B

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80

90

SNR/SNRp/dB

Ach

ieva

ble

Sum

rate

of C

R Sy

stem

/bit/

s/H

z

RIASSIAPSSPCOUAOSS

Fig.6 Achievable sum rate of CUs in case of constant Icon w/ adaptation of B

China Communications • June 2015 104

ϕk, ψkU [0, 2π] , respectively. In each simula-tion, we calculate the average distance between all CRTs and PRR, so as to acquire the large scale fading as L1 from (17). Similarly, the large scale fading is computed according to the average distance of all pairs of CUs as L3 by (18), so Ic = IconL1/L3 . The average results of 5×104 simulations are shown in Fig. 7, showing that in these scenes, RIASS can still achieve better performance than the other methods.

To describe the relationship between perfor-mance of methods and the numbers of termi-nal antennas, we show the case that M=N=8, Mp=Np=4 (Mp=Np=2) and K=7 in Fig. 8 (Fig. 9) . It is assumed that Icon = 0.005SNRpσ

20

. The curves show that the decrease of K and the numbers of antennas causes reduction of achievable sum rates of all the five methods. Nevertheless, RIASS method is still superior to the others.

VI. CONCLUSIONS

In this paper, a Rayleigh quotient based IA spectrum sharing method is proposed to max-imize the achievable sum rate of CR systems on the premise of satisfying the IC of PUs for general CR networks. Rayleigh quotients of channel matrices are used to select CUs for interference reduction, which solves the prob-lem of the absence of PUs in IA, and eases the restriction that CUs can only transmit on the idle sub-channels in the existing IA based spectrum sharing schemes as well as increas-ing the sum rate of the CR system.

ACKNOWLEDGEMENTS

This work was supported by National Natu-ral Science Foundation of China under Grant 61201233; 61271262 and Fundamental Re-search Funds for the Central Universities (2013G1241114).

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Fig.8 Achievable sum rate of CUs w/ Mp=Np=4

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BiographiesRong Mei, received her B.E. and Ph.D. degree in information and communications engineering from Xi’an Jiaotong University, Xi’an, Shaanxi, China, in 2003 and 2011 respectively. She is currently a lectur-er in the Department of Electronic and Information Engineering, Chang’an University. Her research in-terests include cognitive networks, MIMO and Radio Resource Management. *The corresponding author. Email: rongmei@chd.edu.cn

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