# 认知无线网下的beamforming技术_图文

SINR Balancing Technique and its Comparison to Semide?nite Programming Based QoS Provision for

Cognitive Radios.

K. Cumanan, R. Krishna, Z. Xiong, and S. Lambotharan Loughborough University, Leicestershire, UK, LE11 3TU Email:{K.Cumanan, R.Krishna, Z.Xiong, S.Lambotharan}@lboro.ac.uk

Abstract— Cognitive radio networks opportunistically operate in frequency bands that have been licensed to other networks. Therefore, communication between unlicensed users should ensure the interference leaked to the licensed users is kept below an acceptable level while achieving the required quality of services. In this paper, we extend SINR balancing technique to serve multiple cognitive users in the downlink while imposing constraints on interference temperature of primary users. We show that when the set interference temperatures is ?xed, the proposed SINR balancing technique will always have a unique solution that is identical to semide?nite programming based optimal solution. The advantages and disadvantages of the SINR balancing technique and semide?nite programming based techniques are also discussed.

Index Terms— Cognitive radio, Spatial diversity, SINR balancing.

I. INTRODUCTION

Recently, opportunistic access of radio spectrum has been proposed to utilize the scarce radio spectrum more ef?ciently. The frequency bands owned by licensed users (also known as primary users) are continuously monitored and accessed by unlicensed users (also known as secondary users) when the licensed spectrum is not in use. Cognitive radios (CRs) [1], [2], identify spectrum holes and exploit them to communicate between the secondary users.

Resource allocation for secondary users with constraints on primary user interference temperature is one of the key challenges in a CR network. In wireless communications, employing multiple antennas at the basestation signi?cantly improves the spectral ef?ciency and accommodates multiple users. In the past decade, power allocation and beamforming for multiple users in wireless communication have been well studied. In [3], an optimal downlink power assignment technique has been proposed for a given set of beamforming weight vectors. SINR balancing technique has been presented in [4], where the beamformers designed in the uplink, have been employed in the downlink using the principle of uplinkdownlink duality [5]. In [6], joint beamforming and power allocation techniques have been proposed for an uplink in a CR network. A capped multi-level water ?lling algorithm and recursive decoupled power allocation algorithm have been presented to maximize the achievable sum-rate of the secondary users. In this paper, we adopt the SINR balancing

This work has been supported by the Engineering and Physical Sciences Research Council of U.K. under Grant EPSRC E041817.

technique to serve multiple secondary users in the downlink of a CR network while imposing constraints on the interference temperature of the primary users. The total power allocation is controlled by the primary user interference temperature. The beamforming weight vectors and the power allocation vector are obtained using SINR balancing technique and compared with those obtained using semide?nite programming based techniques [7], [8]. The merits and demerits are discussed in Section V.

II. PROBLEM STATEMENT

We consider a downlink CR network with L primary users and K secondary users. Each of the primary and the secondary

users consists of single antenna at the receiver. The secondary user basestation is equipped with Nt transmit antennas. The signal transmitted by the secondary user’s basestation is

x(n) = Us(n)

(1)

where s(n) = [s1(n) s2(n) · · · sK (n)]T and sk(n), k = 1, 2, · · · , K is the symbol intended for the kth secondary user, U = [u1u2 · · · uK ], uk 2 = 1 and uk ∈ CNt×1 is the transmit beamforming weight vector for the kth secondary

user. The variance of the symbol s(n) is assumed to be unity. i.e. σs2(n) = 1. The received signal at the kth secondary user can be written as

yk(n) = hHk x(n) + ηk(n)

(2)

where hk = [h(1k) h(2k) ... h(Nkt)]H is the channel responses from the secondary user basestation to the kth secondary user

and ηk(n) is assumed to be zero mean circularly symmetric AWGN component with variance σk2. Further, pk is the power allocated to the kth secondary user and the total transmit

power at the secondary user basestation is denoted as Pmax.

The power of the interference leaked to the lth primary

user is εl = E{h?Hl x(n)x(n)H h?l} =

K k=1

pk

h?Hl uk

22,

where h?l = [h(1l) h(2l) · · · h(Nl)t ]H is the channel responses

from the secondary user basestation to the lth primary user

and pk is the power allocated to the kth secondary user.

Let

gl

=

[ h?Hl u1

2 2

h?Hl u2

2 2

···

h?Hl uK

2 2

]T

and

the

interference leaked to the lth primary user is εl = glT p where

p = [p1 p2 · · · pK ]T .

978-1-4244-2517-4/09/$20.00 ?2009 IEEE

1

The SINR of the kth secondary user can be written as

SINRk =

pkuHk Rkuk i=k piuHi Rkui

+

σk2

.

(3)

where Rk = hkhHk . Based on the work in [4], the SINR balancing for the downlink of a CR network is de?ned as,

max min

U,p 1≤k≤K

SINRk(U, p) γk

Subject to 1T p ≤ Pmax

glT p ≤ P(inl)t

l = 1, · · · , L (4)

where γk is the target SINR of the kth secondary user, P(inl)t is the interference limit on the lth primary user and 1 [1 1 · · · 1]. Downlink beamforming problem is more complicated compared to the uplink beamforming problem because SINR of each user is a function of the beamforming weight vectors of all the users in the downlink. In this problem, beamforming weight vectors and power allocation must be jointly optimized while satisfying the power constraints and interference constraints of all primary users. It should also be noticed that in the downlink, the power gain between the secondary user basestation and the primary user is a function of beamforming weight vectors uk.

III. SOLUTION TO THE SINGLE PRIMARY USER

The SINR balancing problem in a CR network with single primary user case is similar to the problem under the sum power constraint in [4]. The joint beamforming and power allocation in the downlink of a CR network is different from the SINR balancing problem considered for the uplink in [6], because the power gain between the secondary user basestation and the primary user is a function of beamforming weight vectors. In the dowlink, any changes in the beamforming weight vectors will change the power gain, but this dif?culty does not arise in the uplink problem. Here, we adopt an iterative algorithm to obtain the joint optimal power allocation and beamforming matrix for SINR balancing in a CR network. The optimum beamforming matrix and power allocation must satisfy the following conditions to achieve balanced SINR for all the secondary users.

SINRk(U, p) γk

=

C1(U, Pi(n1t)), for

k = 1, · · · , K

(5)

g1T p = P(in1)t

(6)

where C1(U, Pi(n1t)) is the balanced SINR and the interference constraint satis?es with the equality. The ?rst step of the iter-

ative algorithm is to determine the optimum power allocation

for a given beamforming matrix. The following equation is

formulated using the power gain between the secondary user

basestation and the primary user. This equation is identical

to the one presented in [4], except for the explicit control on interference Pi(n1t).

1 C1(U, Pi(n1t))

p

=

DΨ(U

)p

+

Dσ

(7)

where σ

=

[σ12, · · · , σK2 ] and

diag{(γ1/(uH1 R1u1)), · · · , γK /(uHK RK uK ))}.

Ψ(U ) is de?ned as follows,

D

=

Moreover

[Ψ(U )]ik =

uHk Riuk, k = i;

0,

k = i.

(8)

The following equation is obtained by multiplying both sides of (7) by g1 = [g1, · · · , gK ]T . Note that power gain between the secondary user basestation and the primary user is a

function of the beamforming weight vectors.

1 C1(U, Pi(n1t))

=

1 P(in1)t

g1T

DΨ(U

)p

+

1 P(in1)t

g1T

Dσ

(9)

From (7) and (9), extended coupling matrix is formed as

φ(U, P(in1)t) =

DΨ(U )

1 P(in1)t

g1T

DΨ(U

)

Dσ

1 P(in1)t

g1T

Dσ

(10)

Note (10) is similar to (12) in [4], but instead of Pmax, Pint is

directly used together with the primary user power gain vector

g. By de?ning an extended power vector pext =

p 1 , (7)

and (9) can be formulated into an eigenvector problem as,

1 C1(U,

Pi(n1t))

pext

=

DΨ(U )

1 P(in1)t

g1T

DΨ(U

)

Dσ

1 P(in1)t

g1T

Dσ

pext (11)

From

(11),

it

can

be

seen

that

1 C1 (U,Pi(n1t) )

and

pext

represent

the eigenvalue and the corresponding eigenvector, respectively,

of the extended coupling matrix. As stated in [4], from Perron-

Frobenius theory [9], it is always possible to ?nd a positive

eigenvalue and the corresponding positive eigenvector for a

non-negative matrix. Moreover, in [3], it has been proven that

only maximal eigenvalue and the corresponding eigenvector

satisfy the positivity requirement. Next step of the iterative

algorithm is to determine the optimum beamforiming matrix

for a given interference level for the primary user. From

the uplink-downlink duality [5], it is well known that the

beamforming vectors used in the uplink can be used to achieve

the same SINR values in the downlink. In a CR network, the

interference leaked to the primary user should be incorporated

in ?nding optimum beamforming vectors in the uplink. To

incorporate the primary user interference in the design of

optimum beamforming vectors in the uplink, we need the

following Lemma,

Lemma 1: For the dowlink SINR balancing, for a given set of

channels and beamforming matrix, each interference level for

the primary user has a corresponding unique total power at the

secondary user basestation. i.e. each Pint has a corresponding unique Ptot.

See Appendix for the proof.

Next we consider determining the optimum beamforming

weight vectors. In [4], an iterative algorithm has been proposed

to obtain the optimum beamforming weight vectors for sum

power constraint. The uplink coupling matrix is de?ned as

follows,

φ2(U, Ptot) =

DΨ(U )T

1 Ptot

1T

DΨ(U

)T

Dσ

1 Ptot

1T

Dσ

(12)

2

where the last row of the matrix controls the available total power and ?nding eigenvector corresponding the maximum eigenvalue yields the power allocation of each user in the uplink. Similarly, in a CR network, power allocation can be controlled by primary user interference. In order to ?nd the beamformers, the uplink coupling matrix is modi?ed by incorporating the interference to the primary user as follows.

λ(U, P(in1)t) =

DΨ(U )T

1 P(in1)t

g1T

DΨ(U

)T

Dσ

1 P(in1)t

g1T

Dσ

(13)

where last row of the modi?ed uplink coupling matrix ensures

that the interference leaked to the primary user does not

exceed the threshold. Moreover, it de?nes the total power

that can be allocated in the downlink through Pint and power gain between the secondary user basestation and the primary

user. The beamformer will be formulated by maximizing each

uplink SINR separately. In [4], it has been proven that for a

given amount of total power, the beamforming weight vectors

obtained using iterative algorithm is unique. Further, the last

row of both the coupling matrices de?nes the same total power.

Hence the beamforming weight vectors obtained using both

the coupling matrices will be the same. Hence every Ptot has a unique Pint and interestingly Ptot and Pint forms a one

to one mapping. In [4], it was shown that the beamforming

matrix and power allocation obtained using iterative algorithm

are optimal. Since, the adopted algorithm generates the same

beamforming matrix and power allocation, they are optimal.

The steps used to ?nd optimum beamforming weight vectors

are given in Table 1. In the original problem in (4), two

constraints are de?ned: primary user interference and available

total power. In the design of beamforming weight vectors,

the modi?ed uplink coupling matrix incorporates only the

primary user interference. After ?nding the power allocation

in the downlink according to the beamforming weight vectors

obtained in the uplink, the required power could turn out to

be greater than the maximum available power Pmax. Hence,

the available total power should be considered as a constraint

in the algorithm. In order to introduce total power constraint,

we need the following Lemma,

Lemma primary

2u:seSruipnpteorsfeerpen(to1ct)eisPit(nh1te).

total Any

power power

allocated using the allocation p(to2t) less

than p(to1t) will cause an interference Pi(n2t) less than Pi(n1t).i.e. if

p(to1t) > p(to2t), then Pi(n1t) > Pi(n2t).

See Appendix for the proof.

Hence, if the power utilization obtained using primary user

interference control is more than the total power constraint,

then the beamforming weight vectors can be formulated using

uplink coupling matrix instead of modi?ed coupling matrix.

IV. SOLUTION TO MULTIPLE PRIMARY USERS

For simplicity, however without loss of generality, ?rst we consider only two primary users. The problem can be expressed as,

max min

U,p 1≤k≤K

Subject to

SINRk(U, p) γk

1T p ≤ Pmax,

g1T p ≤ P(in1)t,

g2T p ≤ P(in2)t

The above problem based on two interference constraints can be decoupled into the following two sub-problems with single interference constraint. Sub-problem 1:

max min

U,p 1≤k≤K

SINRk(U, p) γk

Subject to

1T p ≤ Pmax, g1T p ≤ P(in1)t

(14)

Sub-problem 2:

max min

U,p 1≤k≤K

SINRk(U, p) γk

Subject to

1T p ≤ Pmax, g2T p ≤ P(in2)t

(15)

Next, we show only one of the solutions of the sub-problems in (14) or (15) will be optimal and will satisfy the constraints in both the sub-problems. Suppose (U1, p1) and (U2, p2) are the optimal solutions for the sub-problem 1 and sub-problem 2 respectively. In order to prove the decoupling property, we need the following two Lemmas. Lemma 3: The solutions of both the sub-problems can not satisfy simultaneously. i.e. g1T p2 ≤ P(in1)t and g2T p1 ≤ P(in2)t. Proof: Let the optimal solution of sub-problem 1 satis?es the sub-problem 2. At the optimal solution of sub-problem 1, the interference constraint holds equality. i.e. g1T p1 = P(in1)t. Since it satis?es the sub-problem 2, the interference leaked to the second primary user should be less than the threshold. i.e. P(in2,tnew) = g2T p1 ≤ P(in2)t. For the single primary user case, we have already proven that primary user interference and total power allocation at the secondary user basestation form a one to one mapping. The amount of power that can be allocated using interference P(in2)t is higher than the power allocated using interference P(in1)t. Then, it is obvious that the inequality g1T p2 ≤ P(in1)t can not be satis?ed. Hence, it is infeasible that both solutions of the sub-problems satisfy simultaneously. Lemma 4: The inequalities g1T p2 ≥ P(in1)t and g2T p1 ≥ P(in2)t can not hold simultaneously. Proof: Similar to proof of Lemma 3.

Next, we prove that the solution to one of the sub-problems will be globally optimum. For the single primary user case, it has been proven that the primary user interference and the total power at the secondary user basestation form a one to one mapping. In ?nding beamforming matrix, interference power can be replaced by the corresponding total power. Hence, any amount of power which is more than the corresponding total power would cause more interference to the primary user. In [4], it has been proven that the beamforming matrix and power allocation vector are unique and globally optimal for a given total power. From Lemma 3 and Lemma 4, both solutions of the sub-problems can not satisfy simultaneously. Hence, one of the solutions of the sub-problems which satis?es the constraints of both sub-problems is globally optimum.

From the above Lemmas and global optimality, it has been proven that two primary users SINR balancing problem can be decoupled into two sub-problems. Moreover, using mathematical induction method, it can be shown that the two

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User 1 Power (W) 0.6894 0.2175 0.2499 0.4338 0.3822

User 2 Power (W) 0.5538 0.3169 0.1602 0.3881 0.3984

User 3 Power (W) 0.3989 0.2848 0.0985 0.1669 0.3976

Total Power (W) 1.6421 0.8192 0.5086 0.9888 1.1782

Balanced SINR (dB) 15.7672 11.2668 8.4838 12.0242 12.3058

TABLE III POWER ALLOCATION AND BALANCED SINR BY USING SINR BALANCING FOR A CR NETWORK WITH THREE SECONDARY USERS AND ONE PRIMARY

USER.

Target SINR (dB) 15.7672 11.2668 8.4838 12.0242 12.3058

Required Total Power (W) 1.6421 0.8192 0.5086 0.9888 1.1782

User 1 Power (W) 0.6894 0.2175 0.2499 0.4338 0.3822

User 2 Power (W) 0.5538 0.3169 0.1602 0.3881 0.3984

User 3 Power (W) 0.3989 0.2848 0.0985 0.1669 0.3976

TABLE IV TARGET SINR AND REQUIRED TOTAL POWER USING SEMIDEFINITE PROGRAMMING.

primary user SINR balancing problem can be extended to multiple primary users. The SINR balancing algorithm for multiple primary users is given in Table 2.

1. initialize: q(0) = [0, · · · , 0]T 2. R? k = Rk/σk2, k = 1, · · · , K 3. σk2 = 1, k = 1, · · · , K. 4. repeat 此处进入循环 5. n = n+1

6. Q(kn) =

K m=1,m=k

qm(n?1)R? k

+

I,

k = 1, · · · , K

7. u(kn) = generalized eigenvector of R? k and Q(kn)

8. u(kn) = u(kn)/ u(kn) 2 k = 1, · · · , K

9. solve λ(U, P(in1)t)

q(n) 1

= λmax(n)

q(n) 1

10. until λmax(n) ? λmax(n ? 1) < ε

11. solve φ(U, P(in1)t)

p(opt) 1

= λmax(n)

p(opt) 1

12. if 1T p(opt) < Pmax

13.

Popt = p(opt)

14. else

15. Replace step 9 by

16.

solve φ2(U, Pmax)

q(n) 1

= λmax(n)

q(n) 1

17. repeat steps 1 to 10.

18. end

TABLE I ALGORITHM FOR SINR BALANCING FOR SINGLE PRIMARY USER

V. SIMULATION RESULTS

In order to validate the performance of SINR balancing in a CR network, we compute the balanced SINR and the allocated total power for a given interference level of the primary user. We consider a network with three secondary users and a primary user. The basestation of the secondary user consists

1. initialize: n = 0 2. repeat 3. n = n+1 4. Find optimal beamforming matrix U(on) and power

allocation p(on) for sub-problem n. 5. Check whether U(on) and p(on) satisfy

other sub-problems. 6. if yes 7. exit 8. else 9. continue 10. until n = L

TABLE II ALGORITHM FOR SINR BALANCING FOR MULTIPLE PRIMARY USERS.

of ?ve transmitting antennas while both the primary user and the secondary users have only single antenna. The channels between the basestation and the secondary users as well as the primary user are assumed to be known to the secondary user basestation, and they have been generated using zero mean, unity variance, circularly symmetric additive white Gaussian noise. The primary user interference level and the noise power at the secondary user receivers have been set to Pi(n1t) = 1 and σk2 = 0.05 for k = 1, 2, 3 respectively. Target SINR of each user has been set to 20 dB and the available total power at the secondary user basestation is restricted to 2W. Table 3 provides the power allocation of each secondary user and balanced SINR for ?ve different set of random channels. We then set the achieved SINR balanced values as a set of target SINRs for the semide?nite optimization based problem [7], [8], [10].

When the balanced SINR values obtained in Table III

4

have been set as the targets and the interference temperature has been set to 1, the semide?nite programming based solution provided the same power allocation and beamforming weight vectors as provided by the SINR balancing method. This con?rms that the SINR balancing technique performs as good as the semide?nite programming based solution. However, there are advantages and disadvantages over both methods. First, if SINR balancing could not attain a speci?c SINR, it will always provide a solution with a balanced SINR less than the speci?c SINR value. However, for the semide?nite programming based method, if the target can not be achieved with a given maximum available transmitter power, the problem will be classi?ed as infeasible, and one needs to try a different set of target SINRs less than the original target SINRs until a feasible solution is found. This is the disadvantage of semide?nite programming based solution for this cognitive radio problem. However, the disadvantage associated with the SINR balancing method is as follows. In the SINR balancing method, we set a maximum interference temperature, and allowed the SINR balancing method to obtain its natural solution in terms of the transmitted power and balanced SINR values. For example, for the ?rst simulation result in Table III, an SINR value of 15.7672 dB has been achieved for all three users with a power allocation of 1.6421. Suppose, if we increase the allocated power above 1.6421 W, the achieved SINR values will also increase, as well as the interference leaked to the primary users. Hence, an SINR value above 1.6421 W can not be obtained without increasing the interference temperature for the SINR balancing method. However, for the semide?nite programming based problem, it is possible to set a higher SINR target (for example 18 dB) with the same interference temperature, and if the solution is feasible, the semide?nite programme will ?nd this solution with a power consumption above 1.6421 W. In this case, a different set of beamformer weight vectors will be obtained, possibly by steering nulls along the directions of the primary users. However, the proposed SINR balancing technique does not have this capability, i.e. steering appropriate nulls along the direction of primary users and increase the transmission power to achieve a higher target SINR values. This requires further study and development of new SINR balancing algorithms for CR.

VI. CONCLUSION

In this paper, we extended the SINR balancing technique to accommodate multiple secondary users in the downlink while imposing constraints on the interference temperature of primary users. We showed that the proposed SINR balancing always has a unique solution which is identical to the semide?nte programming based results. Furthermore the merits and the demerits of the proposed SINR balancing technique have been discussed and future directions have been identi?ed.

VII. APPENDIX

A. Proof of Lemma 1

For a given beamforming matrix, let us assume that

p1 =

p0 1

satis?es

the

matrix

equation

p 1

C1(U,Pi(n1t)) 1

=

φ(U, Pint)p1, where

φ(U, Pint) =

DΨ(U )

1 P(in1)t

g1T

DΨ(U

)

Dσ

1 P(in1)t

g1T

Dσ

(16)

Hence p0 satis?es the following equation.

C1

1 (U,

Pi(n1t)

)

p0

=

DΨ(U )p0

+

Dσ

(17)

By multiplying both sides of (17) by 1T , the following

equation is obtained.

1 C1(U, Pi(n1t))

=

1 Ptot

1T

DΨ(U

)p0

+

1 1T Dσ Ptot

(18)

where Ptot = 1T p0. From the equations (17) and (18), the downlink coupling matrix for sum power constraint can be

de?ned as follows.

φ1(U, Ptot) =

DΨ(U )

Dσ

1 Ptot

1T

DΨ(U

)

1 Ptot

1T

Dσ

(19)

The same power allocation p1 satis?es the following matrix equation as well.

1 C1(U, Pi(n1t)

)

p1

=

φ1(U, Ptot)p1

(20)

In [3], it has been already proven that for a non-negative ma-

trix, only maximal eigenvalue and corresponding eigenvector

satisfy positivity. Hence p1 satis?es the (17) and (20), power allocations obtained using both φ1(U, Ptot) and φ(U, Pint) will be the same.

B. Proof of Lemma 2

It has been already proven that Ptot and Pint are one to one mapping. Then PtotvsPint should be either increasing function or decreasing function. It is obvious that when Ptot increases, Pint also will increase.

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