A Mixed SINR-Balancing and SINR-Target ... - Semantic Scholar

A Mixed SINR-Balancing and SINR-Target Constraints Based Beamformer Design Technique for Spectrum Sharing Networks Yogachandran Rahulamathavan, Student Member, IEEE, Kanapathippillai Cumanan Member, IEEE, and Sangarapillai Lambotharan, Senoir Member, IEEE Abstract— We propose a transmitter beamformer design technique for a spectrum sharing network with a mixed quality of services requirement. In particular, the proposed algorithm has the ability to jointly design multiple beamformers so that a specific subset of users attains certain target signal-to-interferenceplus-noise-ratios (SINRs) while the SINRs of the remaining users are balanced subject to multiple linear constraints. The design framework considered is applicable to both the overlay cognitive radio network (CRN) and the underlay CRN, however, the algorithm is described for the example of an underlay CRN in order to include a very general mathematical framework. We maximize the worst-case users’ SINR of a particular set of users while satisfying the target SINRs of the remaining users subject to the total transmission power and the interference leakage constraints. This algorithm is solved using the uplink-downlink duality and subgradient method. The condition for convergence is derived analytically. The simulation results are provided to validate the optimality and the convergence of the proposed algorithm. Index Terms— Spectrum sharing network, beamforming, uplink-downlink duality, SINR balancing

I. INTRODUCTION Introduction of data intensive multimedia and interactive services together with exponential growth of wireless and mobile users has resulted the radio spectrum a scarce resource. There has been extensive research on various techniques for the enhancement of spectrum utilization. The spatial diversity techniques have been widely used to improve spectrum usage. For example, multiple-input and multiple-output (MIMO) systems [1], [2] have the ability to enhance the data rate by providing multiple data pipes to users. Employment of multiple antennas at the transmitter could also facilitate spatial multiplexing [3]–[8]. Various spatial multiplexing designs are known in the literature, e.g., zero-forcing block diagonalization [3], convex optimization based transmitter beamformer design [4] and the uplink-downlink duality based signal-tointerference-plus-noise-ratio (SINR) balancing beamformers [6]. The method in [3] is based on the null space of the channel matrices so that the inter-user interference is forced to zero. The method in [4] is based on the optimization of transmission Copyright (c) 2011 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. Manuscript received March 31, 2011; revised July 30, 2011; accepted August 29, 2011. Date of publication xxxxx; date of current version xxxxx. The review of this paper was coordinated by Dr. Ngoc-Dung Dao. The authors are with the Advanced Signal Processing Group, School of Electronic, Electrical and Systems Engineering, Loughborough University, UK (e-mails: {R.Yogachandran, K.Cumanan, S.Lambotharan}@lboro.ac.uk) This work has been supported by the Engineering and Physical Sciences Research Council (EPSRC) of UK under Grant EPSRC G020442.

power and beamformer patterns while constraining the SINR target of each downlink user. This was solved using convex optimization techniques. The method in [6] uses the uplinkdownlink duality to design a set of beamformers in order to maximize the SINR of the worst-case user resulting into SINR balancing beamformers, i.e., all users attain equal SINR. In this paper, we address the problem of designing transmitter beamformers to satisfy two design criteria simultaneously. Beamformers should be jointly designed so that a set of users should be satisfied with specific target SINRs while the SINRs of the remaining users should be balanced and maximized. The beamformer design techniques based on the satisfaction of target SINRs [9]–[12] and the beamformer design techniques based on the SINR balancing [13]–[17] have been treated separately in the literature. However, the beamformer design to include above mentioned both criteria simultaneously is not a trivial extension. This is because these set of beamformers can not be designed separately, as the interference introduced to each other set of beamformers is a function of all the beamformer weight vectors and power allocations. We tackle this problem by introducing some mathematical manipulations to the design framework as explained in Section III. The solution to this problem has potential applications in a network that has users with delay intolerant real time (RT) services and delay tolerant packet data (non-real time (NRT)) services [18], [19] as shown in Fig. 1(a). For the delay intolerant services, the resources should be allocated so that the users should attain a specific target SINR. The remaining resources can be allocated to the delay tolerant packet data services fairly so that users attain identical SINR ratio. The fairness criterion has been widely considered for non-real time users (NRTUs) in the work of [20]–[34]. Also, a popular proportional fair scheduling (PFS) algorithm [34], [35] is considered in 3G wireless network for delay-tolerant applications [36]. The aim of this paper is only on the development of beamformer design for a given mixed quality of services (QoS) requirement (i.e., SINR) and how these target SINRs are determined at the medium access control (MAC) layer and above is not within the immediate scope of this paper. The proposed technique has also potential applications in spectrum sharing networks such as the cognitive radio network (CRN) [37]. The cognitive radios have the ability to enhance spectrum utilization by allowing secondary users (SUs) to access the spectrum of the primary users (PUs) provided that the SU transmissions do not harmfully affect the PUs [38]. There are generally three types of CRNs namely overlay, underlay and interweaved networks [39]. In the interweave

RT-PU 1

PU 1

RT-1

RT-PU L RT-K1

PU L

RT-SU L+1

RT-SU 1

RT-SU L+K1

RT-SU K1

NRT-K1+1

Basestation

NRT- K

(a) Fig. 1.

NRT-SU L+K1+1

Basestation

NRT-SU K1+1

Basestation

NRT-SU L+K

NRT-SU K

(b)

(c)

The possible applications of the proposed scheme: (a) A conventional wireless system. (b) An overlay CRN. (c) An underlay CRN.

approach, SU network is required to sense the availability of spectrum and transmit signals only when frequency holes are available. In the overlay approach, the SU network helps the PU network to offset the interference caused by the secondary transmission by assisting and relaying PU signals. For the underlay approach, the SUs could coexist with the PU network, however they need to ensure the interference caused to the primary receivers is below a predefined threshold to ensure that the primary network is not affected harmfully. The beamformer design considered in this paper can be applied to the overlay CRN. In this scenario, as shown in Fig. 1(b), the basestation of the SUs uses the primary spectrum to transmit signals to its users while relaying the PU signals to their destination using beamforming techniques. In this case, resources should be allocated (power and beams) such that each PU receiver should attain a target SINR specified by the PU network and the remaining resources can be used to serve the SUs. The proposed technique could also be used for an underlay CRN scenario shown in Fig. 1(c). In this scenario, the SUs could transmit signals provided that the interference leakage to the PU receivers is below a specific threshold. Therefore, additional constraints for interference leakage should be included in the beamformer design. This paper proposes novel approaches for beamformer designs that consider simultaneously SINR balancing and the target SINRs for both the conventional and the spectrum sharing networks. The remaining paper is organized as follows. In Section II we describe the system model and problem formulation. The analytical framework is presented in Section III and in Section IV complexity of the proposed algorithm is analyzed. The optimality of the proposed algorithm is verified using simulation results in Section V. Finally, conclusions are drawn in Section VI. Notation: We use uppercase boldface letters for matrices and lowercase boldface letters for vectors. AT and AH denote the transpose and Hermitian transpose operations of A, respectively. ρ(A) denotes the spectral radius of square matrix A. 1n denotes n×1 vector composed of ones. I denotes an identity matrix of an appropriate size. diag(x) denotes a diagonal matrix whose elements are drawn from the vector x.

II. S YSTEM MODEL AND P ROBLEM FORMULATION The algorithm is described for an underlay CRN as the optimization framework considers both the transmission power constraint and the interference leakage constraints. The beamformer design techniques for the overlay CRN and the conventional wireless network can readily be obtained by dropping the interference constraints. There are K SUs and L PUs and the secondary network basestation (SNBS) consists of Nt transmit antennas and each of the SUs and PUs is equipped with single antenna. By defining sk (n), uk ∈ C Nt ×1 and pk as the transmitted symbol, transmit beamformer weight vector and the power allocation for the k th SU respectively, the signal transmitted by the SNBS is written as x(n) =

K X √ pk uk sk (n),

(1)

k=1

where kuk k2 = 1, ∀k. The variance of the symbol sk (n), ∀k is assumed to be unity. The received signal at the k th SU can be written as yk (n) = hH k x(n) + ηk (n),

k = 1, . . . , K,

(2)

Nt ×1

where hk ∈ C is the channel gain vector between the SNBS and the k th SU. We assume that ηk (n) is a zero-mean circularly symmetric complex additive white Gaussian noise (AWGN) with variance σk2 = 1. Let gl = h H iT H T and p = [p1 . . . pK ] , where kh˜l u1 k2 . . . kh˜l uK k2 2

2

h˜l ∈ C Nt ×1 is the channel gain vector between the SNBS and the lth PU. The interference leakage to the lth PU due to the SU transmission is εl = glT p. Similar to the work in [17], [25], [40]–[45], the SNBS is assumed to have the channel state information (CSI) of the PUs (from SNBS to PUs). As mentioned in the introduction, the proposed scheme has potential applications in conventional wireless network (non cognitive radio setting), overlay and underlay CRN. The application scenario considered for the overlay CRN is that all the PUs are delay intolerable. In this scenario, the SNBS exploits the PUs’ spectrum to transmit signals to the SUs while relaying the PUs’ signals using beamforming techniques to ensure a target SINR for the PU. In this case, resources should be allocated (power and beams) such that each PU should attain a target SINR specified by the primary network

and the remaining resources can be used to serve the SUs. Since SNBS is assisting the primary network to relay the PUs signal, we can expect a degree of cooperation between the primary and the secondary networks. In this case, it is possible for the SNBS to acquire the CSI of the PUs. In the underlay CRN, the scenario considered is that the beamformer should be designed so that a subset of SUs should attain a target SINR while the SINRs of remaining SUs are balanced subject to certain interference constraints to PUs. The required protocols to obtain CSI (between primary users and secondary basestation) are discussed in [17], [40]. For example, a primary network could sublease the spectrum to the secondary network for monitory purposes. In this case, a degree of cooperation between the primary network and the secondary network can be expected. This kind of scenario has been discussed in [25], [41]–[45]. In the following, we consider an underlay CRN as the results for the conventional network and the overlay CRN can be obtained from the results of the underlay CRN. By defining th Rk , hk hH SU in the downlink can be k , the SINR of the k written as pk uH k Rk uk . H 2 p u i6=k i i Rk ui + σk

SINRDL k = P

(3)

We assume a general scenario where the first K1 SUs (i.e., real time users (RTUs)) out of the K SUs employ delay intolerant RT services. Hence, a target SINR should be satisfied for these users all the time. The rest of the SUs (i.e., NRTUs) employ delay tolerant packet data services, hence, a target SINR is not a priority, however, in order to maintain user fairness, their SINRs should be balanced and maximized. From these requirements, a mixed QoS problem for the beamformer design is formulated as follows: SINRDL k (U, p) , k = K1 + 1, . . . , K, k δk s.t. SINRDL k (U, p) ≥ γk , k = 1, . . . , K1,

max min U,p

(l) glT p ≤ Pint , 1TK p ≤ Pmax ,

l = 1, . . . , L,

(4) (5) (6) (7)

where U = [u1 . . . uK ], δk is the preferred target SINR of (l) k th NRTU, γk is the target SINR for the k th RTU, Pint is the interference threshold for the lth PU and Pmax is the available total transmission power at the SNBS. The first set of constraints in (5) ensures that the RTUs should achieve their target SINRs, provided the problem is feasible. The constraints in (6) and (7) account for the interference leakage to the PUs and the total transmission power respectively. To consider an overlay CRN, the first K1 users should be treated as PUs whose SINR targets should be satisfied all the time and the SINRs of the remaining users (who are SUs) can be balanced. The interference constraints in (6) will also be dropped in this case. III. ANALYTICAL FRAMEWORK The solution to the mixed QoS problem stated in (4)-(7) is nontrivial. This is because the SINR of each user is a function of the beamformer weight vectors of both the users with target

SINRs (considered as RTUs) and the users whose SINRs are to be balanced (NRTUs). Hence beamformers for RT and NRT users can not be designed separately. We solve this problem by writing the beamformer weight vectors of the RTUs as a function of that of the NRTUs and by invoking the uplinkdownlink duality. In the interest of keeping the development of the algorithm as closer as possible to the algorithms known in the literature, we start our design framework by relating our beamformer design problem with mixed-SINRs requirement to that of an ordinary SINR balancing problem of [6], however, the main differentiating factor will be seen in Section III-A. First, we put multiple linear constraints in (6) - (7) into a single linear constraint by introducing auxiliary variables as described in [15] as follows; L X

´ ´ ³ ³ (l) al glT p − Pint + aL+1 1TK p − Pmax ≤ 0,

(8)

l=1

where al , l = 1, . . . , L and aL+1 are the auxiliary variables associated with the interference constraints and the power constraint respectively. These auxiliary variables can be updated to find the optimal solution based on the subgradient method [46], as explained later in this section. Let us define vectors T a = [a1 . . . aL aL+1 ] and b = [g1 . . . gL 1K ] a and a scalar PL (l) P := l=1 al Pint + aL+1 Pmax . Using these definitions, the problem in (4)-(7) can be simplified to max min U,p

k

s.t.

SINRDL k (U, p) , k = K1 + 1, . . . , K, δk SINRDL k (U, p) ≥ γk , k = 1, . . . , K1, bT p ≤ P.

(9)

All the auxiliary variables should satisfy the nonnegative conditions all the time. This will ensure that the problem defined in (9) by introducing auxiliary variables yields an upper-bound solution of the original problem in (4)-(7). For a given set of auxiliary variables, the following dual uplink mixed QoS problem can be formulated by modifying the noise covariance and the linear constraint based on the uplinkdownlink duality [15]. SINRUL k (uk , q) , k = K1 + 1, . . . , K, (10) U,q k δk s.t. SINRUL k (uk , q) ≥ γk , k = 1, . . . , K1, (11) T σ q ≤ P, (12) £ 2 ¤ T T 2 where σ = σ1 . . . σK and q = [q1 . . . qK ] , qk is the virth tual uplink power allocated to the k SU and SINRUL k (uk , q) is the virtual uplink SINR of the k th SU which is given as max min

SINRUL k (uk , q) =

q uH R u ³P k k k k ´ , uH k i6=k qi Ri + Ω uk

(13)

PL H where Ω = l=1 al h˜l h˜l + aL+1 I is the interference-plusnoise-covariance matrix at the £ SNBS for Hthe virtual ¤T uplink problem. Note that b = uH Ωu . . . u Ωu . In the 1 K 1 K following subsections, we describe the solution to the above problem for a fixed set of auxiliary variables. The auxiliary variables will then be updated using a subgradient method.

A. Uplink power allocation for a given set of beamformers We first consider the optimal power allocation for a given set of beamformers u ˜ k , ∀k in the uplink. At the optimal setting, the constraints in (11)-(12) must be satisfied with equality. Hence, at optimal power allocation, (10) - (12) will satisfy the following set of simultaneous equations SINRUL uk , q ˜) k (˜

1 , k = K1 + 1, . . . , K, λ γk , k = 1, 2, . . . , K1,

=

δk SINRUL uk , q ˜) k (˜ T

=

σ q ˜ =

P,

(14) (15) (16)

[ΨD ]ki

£ H ¤T and bB = u ˜ K1+1 Ω˜ uK1+1 . . . u ˜H uK . Note that the K Ω˜ power allocation q ˜ is composed of the power allocation for the RTUs q ˜A and the power allocation for the NRTUs q ˜B , i.e., q ˜ = [˜ qTA q ˜TB ]T . Hence, the constraint in (16) can be written as σ TA q ˜A + σ TB q ˜B = P,

where σ A = [σ1 . . . σK1 ]T and σ B = [σK1+1 . . . σK ]T . Therefore, the equations (14) - (16) can be reformulated into the following matrix forms:

where 1/λ is a balanced SINR of the NRTUs and q ˜ is the ˜ optimal power allocation for the given set of beamformers U, ˜ where U = [˜ u1 , . . . , u ˜ K ]. From (15) we could write q˜ u ˜H R u ˜ ³P k k k k ´ = γk , k = 1, . . . , K1 H u ˜k ˜i Ri + Ω u ˜k i6=k, q ³P ´ u ˜H ˜i Ri + Ω u ˜k k i6=k q ⇒ q˜k = γk = u ˜H ˜k k Rk u ´ ³P PK K1 q ˜ R ˜k q ˜ R + Ω + u ˜H k i=K1+1 i i u j6=k,j=1 j j γk , u ˜H ˜k k Rk u k = 1, . . . , K1. (17)

  u ˜H ˜ k , i 6= k, k = K1 + 1 . . . K, k Ri u = i = K1 + 1 . . . K,  0, i = k,

λ˜ qB q ˜A

= =

DB ΨD q ˜ B + D B b B + D B ΨC q ˜A , DA ΨA q ˜A + DA bA + DA ΨB q ˜B ,

P

=

σ TA q ˜A + σ TB q ˜B .

(19) (20) (21) −1

Let us assume (I−DA ΨA ) is invertible and (I−DA ΨA ) is a nonnegative matrix. The conditions required to satisfy these assumptions will be provided in the subsequent sections. Using these assumptions and (20), q ˜A can be written in terms of q ˜B as q ˜A = (I − DA ΨA )−1 DA ΨB q ˜B + (I − DA ΨA )−1 DA bA . (22)

By substituting (22) into (19), the following is obtained, λ˜ qB

=

(23)

D˜ qB + d,

By rearranging (17), the optimum power allocation vector T q ˜A = [˜ q1 . . . q˜K1 ] for the first K1 SUs (i.e., RTUs) can be written as,

where −1 D = DB ΨD + DB ΨC (I − DA ΨA ) DA ΨB , (24)

q ˜A = DA ΨA q ˜A + DA bA + DA ΨB q ˜B ,

The constraint in (21) can also be written in terms of q ˜B by substituting (22) into (21) as follows:

T

where q ˜B = [˜ qK1+1 . . . q˜K ] is the optimum power allocation ¤T £ H u1 . . . u ˜H uK1 , vector for the ˜i1 Ω˜ K1 Ω˜ h NRTUs and bA = u γ1 γK1 DA = diag u˜H R . . . u˜H R and ˜1 ˜ K1 1u K1 u 1

½ [ΨA ]ki = [ΨB ]ki

K1

u ˜H ˜k , i k Ri u

6= k, k = 1 . . . K1, i = 1 . . . K1,

0, i = k, © H = u ˜ k Ri u ˜k ,

i = K1 + 1..K, k = 1 . . . K1.

q˜ u ˜H R u ˜ δ ³P k k k k ´ = k , k = K1 + 1, .., K, H u ˜k ˜i Ri + Ω u ˜k λ i6=k q

³P

´ q ˜ R + Ω u ˜k i i i6=k

⇒ λ˜ qk = δk = u ˜H ˜k k Rk u ³P ´ PK1 K u ˜H q ˜ R + Ω + q ˜ R u ˜k i i j j k i6=k,i=K1+1 j=1 , δk H u ˜ k Rk u ˜k k = K1 + 1, . . . , K. (18) By rearranging (18), the following equation is obtained λ˜ qB

DB ΨD q ˜ B + D B b B + D B ΨC q ˜A , h i δK1+1 δK where DB = diag u˜H R , . . . H ˜ K1+1 u ˜ RK u ˜K K1+1 u [ΨC ]ki =

=

©

K1+1

u ˜H ˜k , k Ri u

K

k = K1 + 1 . . . K, i = 1 . . . K1,

−1

DA bA .

cT q ˜B = P − c,

(25)

(26)

where cT

−1

DA ΨB + σ TB ,

(27)

−1

DA bA .

(28)

σ TA (I − DA ΨA )

=

σ TA

c =

Similarly, from (14) we could write the following

u ˜H k

d = DB bB + DB ΨC (I − DA ΨA )

(I − DA ΨA )

Multiplying both sides of (23) by cT and using (26) we obtain 1 1 cT D˜ qB + cT d. P −c P −c Therefore, the uplink problem in (19)-(21) for the power allocation can be converted into the determination of power allocation for only the NRTUs, subject to (I − DA ΨA ) is invertible and (I − DA ΨA )−1 is a nonnegative matrix, as follows: λ

λ˜ qB

=

=

D˜ qB + d, 1 1 cT D˜ qB + cT d. (29) λ = P −c P −c These equations can be formulated into the following eigensystem [6]: ˜ qext . λ˜ qext = Υ(U)˜

(30)

where q ˜ext = [˜ qTB 1]T and ·

˜ = Υ(U)

D 1 cT D P −c

d 1 cT d P −c

¸ .

(31)

From the Perron-Frobenious theory, the eigenvector corresponding to the largest eigenvalue of a nonnegative matrix is always nonnegative and unique, which is called the Perron vector [6], [47]. There is no nonnegative eigenvector except positive multipliers of Perron vector for a given nonnegative matrix. To this end, the following Lemma is necessary to ˜ satisfy the nonnegativity of Υ(U). Lemma 1: A sufficient condition to enable (I−DA ΨA ) nonsingular and (I − DA ΨA )−1 nonnegative is ρ(DA ΨA ) ≤ 1. Also the conditions ≤ c ≤

ρ(DA ΨA )

1, P,

(32)

B. Beamformer design for a given power allocation For a given power allocation, the optimal beamformers for all the SUs in the virtual uplink can be obtained by maximizing the SINR of each SU independently [6]. Hence, the optimal beamformers for all the SUs in the uplink can be determined by solving the following optimization problem: = argmax ui

s.t.

uH R u ´ , ³P i i i uH ˜k Rk + Ω ui i k6=i q

kui k = 1, ∀i.

10) 11)

until λ(n−1) − λ(n) ≤ ² ˜∗ = U ˜ (n−1) λ∗ = λ(n) and U TABLE I B EAMFORMER A LLOCATION (BA) ALGORITHM

(33)

˜ is a nonnegative matrix. will imply that Υ(U) Proof. See Appendix A. ¥ The following Corollary is an immediate consequence of Lemma 1. Corollary 1: If a given set of beamformers satisfy the conditions in (32) and (33), then from (30), q ˜ext is equal to ˜ Hence, q the Perron vector of Υ(U). ˜B can be obtained by scaling q ˜ext such that the last element of it is equal to one. Once q ˜B is determined, q ˜A can be obtained from q ˜B using (22). In the next subsection, we will study how to obtain the beamformers in the uplink for a given power allocation.

u ˜i

1) Initialize q(0) with an uplink power (as will be explained in Subsection D) 2) n = 0 3) repeat 4) n←n+1 ˜ (n−1) 5) Solve (34) using q(n−1) to obtain U (n−1) (n−1) (n−1) 6) Generate DA , DB , ΨA , (n−1) (n−1) (n−1) (n−1) ΨB , ΨC , ΨD , bA (n−1) ˜ (n−1) and bB using U (n) 7) Solve (30) and obtain λ(n) and q ˜B (n) (n) 8) Obtain q ˜A from q ˜ and (22) iT h BT (n) (n)T (n) 9) Define q = q q ˜B ˜A

(34)

The solution can be obtained by finding the dominant h ³P generalized ´ieigenvector of the matrix pairs Ri , ˜k Rk + Ω , ∀i. From Corollary 1, these k6=i q beamformers are required to satisfy the conditions in (32) and ˜ It is apparent that, (33) in order to obtain nonnegative Υ(U). the conditions (32) and (33) are not necessarily satisfied for any arbitrary power allocation. However, in Subsection D, we will propose a method for an appropriate power initialization so that we will be able to satisfy (32) and (33). In the next subsection, we propose an iterative algorithm to obtain the optimal beamformers and the power allocation for a given set of auxiliary variables with an appropriate power initialization. C. Iterative solution We initialize the virtual uplink power vector with q(0) and ˜ (0) using (34) obtain the corresponding beamformer matrix U

˜ and q(0) . Let us denote the matrices DA , ΨA and Υ(U) (0) (0) (0) (0) ˜ ˜ that are obtained using U as DA , ΨA and Υ(U ) re˜ (0) obtained spectively. Let us assume that the beamformers U (0) from this power initialization q will satisfy the required ˜ (0) ) in (31) is nonnegative conditions in (32) and (33) i.e., Υ(U matrix (Subsection D will elaborate how to obtain this initial power allocation q(0) ). In the first iteration, equation (30) is given by (1) (1) ˜ (0) )˜ λ(1) q ˜ext = Υ(U qext . (35) The superscript associated with each quantity denotes the itera(1) tion number. A feasible power allocation q ˜B is obtained for a ˜ (0) by finding the Perron vector of given set of beamformers U (0) ˜ Υ(U ) and scaling it such that the last element of the Perron (1) (0) (0) vector q ˜ext is one. From Lemma 1 (i.e., (I − DA ΨA )−1 (1) is a nonnegative matrix) and q ˜B , we obtain a valid power (1) allocation q ˜A using (22). Power vector q ˜(1) can be obtained (1) (1) (1) T (1) (1) using q ˜A and q ˜B as q ˜ = [(˜ qA ) (˜ qB )T ]T . Using q ˜(1) (1) ˜ and (34) we can obtain U for the second iteration. Similar to (35), in the second iteration, we will solve the following equation: (2) (2) ˜ (1) )˜ λ(2) q ˜ext = Υ(U qext . (36) Now we wish to invoke the following Lemma. ˜ (1) ) in (36) is a nonnegative matrix Lemma 2: Matrix Υ(U (2) (1) and λ ≤ λ . ¥ Proof: See Appendix B. We now present an iterative algorithm in Table I, namely the Beamformer Allocation (BA) algorithm to obtain the beamformers for a given auxiliary vector a. The quantities associated with the nth iteration are denoted by the superscript (n). Using Lemma 2 and mathematical induction, we can prove ˜ (n) ) is a nonnegative matrix and λ(n) ≤ λ(n−1) that Υ(U at the nth iteration (i.e., n > 2) for a feasible initial power allocation q(0) . Hence, λ is monotonically decreasing with the iteration number, where 1/λ is the balanced SINR of the NRTUs. Hence, SINRs of the NRTUs are increasing monotonically with the iteration number. Since, the system

is limited by the transmission power constraint, SINR cannot increase beyond a certain value. Hence, λ must converge to ˜ as a value denoted as λ∗ . Let us denote the corresponding U ∗ ˜ U . From the uplink-downlink duality, identical SINR values can be achieved in both the uplink and the downlink with the same set of beamformers but with a different power allocation. ˜ ∗ obtained from BA From this, the uplink beamformers U algorithm can be used to achieve the same SINR values in the downlink. £ T T ¤TLet us denote the downlink power allocation p = pA pB , where pA and pB are the downlink power allocation vectors for the RT and the NRT users respectively. Similar to the uplink equations in (23)-(25), we can write the following equations for the power allocation of the NRTUs in the downlink: λ∗ p ˜ ∗B

=

D∗D p ˜ ∗B + d∗D ,

(37)

where T T T T D∗D = D∗B Ψ∗D + D∗B Ψ∗B (I − D∗A Ψ∗A )−1 D∗A Ψ∗C ,

1) Initialize a(0) , t, ², m = 0 . 2) repeat 3) m←m+1 ˜ ∗ and λ∗ using BA 4) Obtain U 5) Obtain p ˜ A and p ˜ B using (40) and (41) 6) p(m) = [˜ pTA p ˜ TB ]T 7) Update auxiliary variables using (42) and (43) 8) until (44) and (45) are satisfied. TABLE II C OMPLETE ALGORITHM

following stopping criteria are used to terminate the algorithm: ¯ ³ ´¯ ¯ ¯ (m+1) T (m) g p − Pint ¯ ≤ ², l = 1 . . . L, (44) ¯al ¯ ³ ´¯ ¯ (m+1) T (m) ¯ 1 p − Pmax ¯ ≤ ², (45) ¯aL+1 where ² is a very small value. The complete algorithm is summarized in Table II.

(38) d∗D

=

D∗B σ B

+

T D∗B Ψ∗B (I

−

T D∗A Ψ∗A )−1 D∗A σ A .

D. Power initialization (39)

The matrices D∗A , D∗B , Ψ∗A , Ψ∗B , Ψ∗C and Ψ∗D are generated ˜ ∗ . To this end, the following Lemma is necessary to using U determine a feasible p ˜ ∗B .

Lemma 3: (λ∗ I − D∗D ) is nonsingular and (λ∗ I − D∗D )−1 is a nonnegative matrix. Proof: See Appendix C. ¥ Using Lemma 3, the power allocation for the NRTUs in the downlink can be obtained from (37) as follows: p ˜ ∗B

=

(λ∗ I − D∗D )−1 d∗D .

(40)

Similar to the uplink power allocation for the RTUs in (22), we can obtain the downlink power allocation for RTUs as follows: p ˜ ∗A

T

T

˜ ∗B + = (I − D∗A Ψ∗A )−1 D∗A Ψ∗C p T

(I − D∗A Ψ∗A )−1 D∗A σ A .

In the previous subsections, we have assumed that there ˜ (0) ) exists a particular initial power allocation q(0) so that Υ(U is nonnegative and the proposed algorithm converges. In this subsection, we discuss how to obtain this power initialization q(0) . To proceed, let us first consider only RTUs (i.e., SNBS allocates power only to the RTUs). The virtual uplink SINR balancing problem with only the RTUs can be written as follows: max min

UA ,qA

k

s.t.

(46) (47)

where UA = [u1 . . . uK1 ] is the matrix containing the beamformers for the RTUs. At the global optimal point, for a ˜ A = [˜ set of fixed beamformers, U u1 . . . u ˜ K1 ], the following equations can be obtained similar to (19), (29), (31) and (30):

(41)

λA qA

=

So far we have shown how to obtain the beamformers and the power allocations for a given set of auxiliary variables. For a given auxiliary variable vector a, we can obtain the beamformers using the BA algorithm in Table I and the corresponding downlink power allocation using (40) and (41). Also we have shown that the BA algorithm converges to a feasible point for a given set of auxiliary variables. Now we explain how to update the auxiliary vector using a subgradient method. Elements of the auxiliary vector al , l = 1 . . . L + 1 are updated via a subgradient algorithm [46], [48] according to the downlink power allocation as follows: ³ ´ (m+1) (m) al = al + t glT p(m) − Pint , l = 1 . . . L, (42) ³ ´ (m+1) (m) aL+1 = aL+1 + t 1T p(m) − Pmax , (43)

λA

=

ΥA

=

λA q ˜Aext

=

where t denotes the step-size of the subgradient algorithm. The

SINRUL k (uk , qA ) , k = 1, . . . , K1, γk σ TA qA ≤ P,

DA ΨA q ˜ A + DA bA , 1 T 1 σ A DA ΨA qA + σ TA DA bA , P P · ¸ D A ΨA DA bA , 1 T 1 T P σ A DA ΨA P σ A DA bA ΥA q ˜Aext ,

(48) (49) (50) (51)

where 1/λA is the ratio between the achieved SINR and the target SINR for the RTUs and q ˜Aext = [˜ qTA 1]T is the corresponding extended power vector. The optimal beamformers UA and the power allocation qA can be determined using the iterative algorithm proposed in [6]. For any initial power allocation, the beamformers can be obtained using (34) for the RTUs (i.e., k = 1, . . . , K1). For this set of beamformers, a power allocation will be performed again using (51). This will result into a higher SINR value [6]. This iteration will be continued until the desired accuracy (i.e., there is no significant difference in λA ). After the convergence of the algorithm, the

Step No. 5 7 8

Matrix Inversion KO(Nt3 ) O{(NK

Eigenvalue £ decomposition ¤ KO Nt3 + (Nt log2 Nt )logb O{(NK + 1)3 +¤ £ (NK + 1)log2 (NK + 1) logb)} + 1)3 } -

No. of RTUs No. of NRTUs Ave. No. of Iteration

2 2 2 2 2 2 2 2 3 4 5 6 7 8 11.6 14.3 17.1 15.6 14.7 14.1 13.9

TABLE IV T HE AVERAGE NUMBER OF ITERATIONS REQUIRED FOR THE ALGORITHM IN TABLE I TO CONVERGE

TABLE III R EQUIRED ARITHMETIC OPERATION FOR THE ALGORITHM IN TABLE I

achieved SINR of the RTU can be written as follows: SINRUL k =

γk , k = 1, . . . , K1, λR A

(52)

R where λR A is the optimum value after the convergence. If λA ≤ 1, then the problem defined in (4)-(7) is feasible i.e., the target SINR for all the RTUs can be achieved. After the convergence, (48) can be written as R λR A qA

=

R R R R DR A ΨA qA + DA bA ,

(53)

where the superscript R means that the corresponding matrices and vectors are obtained after the convergence. qR A provides the power allocation for the RTUs in the absence of NRTUs. If the problem is feasible (i.e., λR A ≤ 1), the initial power allocation for the overall problem (i.e., RT and NRT users) is T provided as q(0) = [qR 0K−K1 T ]T (i.e., we set the initial A power allocation of the RTUs as obtained in (53) and set the initial power allocation for the NRTUs as zeros). The algorithm in Table I with this initial power allocation will converge according to the Lemma 4 provided below. R Lemma 4: For a feasible problem ρ(DR A ΨA ) ≤ 1. If R R (0) ˜ ρ(DA ΨA ) ≤ 1, then the beamformers U obtained using q(0) will satisfy the conditions (32) and (33), and the algorithm in Table I will converge. Proof: See Appendix D. ¥ IV. C OMPLEXITY ANALYSIS For a given set of auxiliary variables, the complexity of the proposed algorithm in Table I mainly depends on the complexity of a matrix inversion and eigenvalue decomposition. For a given n × n matrix, the required arithmetic operations to determine its inverse and eigenvectors are given by O(n3 ) and O(n3 +(nlog2 n)logb) respectively, where b is the relative error bound [49]. Based on this, the number of arithmetic operations required per iteration for the proposed algorithm summarized in Table I is provided in Table III, where Nt and NK are the number of antennas at the SNBS and the number of NRTUs in the network respectively. Only steps 5, 7 and 8 require the matrix inversion and the eigenvalue decomposition in the algorithm in Table I. Hence the total arithmetic operations required at each iteration is the summation of arithmetic operations needed for matrix inversion and the eigenvalue decomposition. The average number of iterations required was observed using simulations as provided in the Table IV. For this simulation, the number of antennas at the transmitter is five. The number of PUs is two. Table IV was generated using

different numbers of NRTUs (i.e., from two to eight) while keeping the number of RTUs at two. The results in Table IV have been averaged over 2000 channel realizations. As we observe the number of iterations needed for convergence does not heavily depend on the number of SUs. Hence, the complexity of the algorithm is that of the addition of what is shown in Table III for each iteration and approximately 14-17 iterations are required for convergence. V. S IMULATION RESULTS To validate the optimality of the proposed algorithm, we consider a CRN with four SUs and two PUs. The first two SUs are considered as RTUs and they need to achieve their target SINRs all the time whilst the SINRs of the remaining two SUs should be balanced. The SNBS consists of five antennas. The interference leakage threshold to PUs and the total available transmission power are set to 0.1 and 2 respectively. The channel coefficients between the SNBS and the SUs as well as those between the SNBS and the PUs are assumed to be known to the SNBS. The channel gains are generated using independent and identically distributed zero-mean circularly symmetric complex Gaussian random variables. The noise power at each SU receiver is set to 0.05. The stopping criterion ² has been set to 0.001. The auxiliary variables al , l = 1 . . . L + 1 have been initialized to 0.1 and the step-size t has been set to 0.01. The target SINRs for the first two SUs have been set to 10 and 5 respectively while preferred target SINRs of NRTUs have been set to 1 (i.e., δk = 1, k = K1 + 1, . . . , K). The power allocations for each SU and the balanced SINR values obtained using the proposed algorithm are depicted for five different set of random channels in Table V. The first two SUs achieve their target SINRs whilst the other two users achieve a balanced SINRs. Note that, the interference and the total power constraints are satisfied with equality. To validate the optimality of the proposed algorithm, we compare the solution shown in Table V with a result obtained using a semidefinite program (SDP) approach [11]. The SDP based design will provide optimum results; however, we have to set the SINR targets for all four users. For the same random channels used in Table V, we have set the SINR targets of all four users the same as that obtained in Table V. For example, according to Table V, for random channel 1, the SINR targets for all four SUs are set as [10.0000 5.0000 4.4039 4.4039], whilst the PUs interference threshold has been set to 0.1. The results obtained using the SDP approach of [11] are shown in Table VI. Comparing Table V and Table VI, the power allocation obtained using the SDP approach of [11] is the

Channels Channel 1 Channel 2 Channel 3 Channel 4 Channel 5

SU1 1.0286 0.5760 0.2372 0.5308 0.7963

Power Allocation SU2 SU3 0.4969 0.2066 0.2640 0.8025 0.2850 0.7834 0.4665 0.2612 0.4475 0.2917

SU4 0.2678 0.3575 0.6944 0.7415 0.4645

Total Power 2 2 2 2 2

SU1 10.0000 10.0000 10.0000 10.0000 10.0000

Achieved SU2 5.0000 5.0000 5.0000 5.0000 5.0000

SINR SU3 4.4039 5.4491 9.7150 4.1820 9.6285

SU4 4.4039 5.4491 9.7150 4.1820 9.6285

Inter. Leakage PU1 PU2 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

TABLE V P OWER ALLOCATIONS AND THE ACHIEVED SINR S USING THE PROPOSED METHOD

Channels Channel 1 Channel 2 Channel 3 Channel 4 Channel 5

SU1 10.0000 10.0000 10.0000 10.0000 10.0000

Target SU2 5.0000 5.0000 5.0000 5.0000 5.0000

SINR SU3 4.4039 5.4491 9.7150 4.1820 9.6285

SU4 4.4039 5.4491 9.7150 4.1820 9.6285

Total Power 2 2 2 2 2

SU1 1.0286 0.5760 0.2372 0.5308 0.7963

Power Allocation SU2 SU3 0.4969 0.2066 0.2640 0.8025 0.2850 0.7834 0.4665 0.2612 0.4475 0.2917

SU4 0.2678 0.3575 0.6944 0.7415 0.4645

Inter. Leakage PU1 PU2 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

TABLE VI TARGET SINRS AND THE USER POWER CONSUMPTIONS USING THE SDP-BASED METHOD OF [11].

Step size t Average No. of Iteration

0.001 1063.61

0.005 210.62

0.01 105.69

(a) 3.5

TABLE VII AVERAGE NUMBER OF ITERATIONS REQUIRED FOR CONVERGENCE FOR DIFFERENT VALUES OF t.

3 2.5 SINR Target for SU1 & SU2 =1 SINR Target for SU1 & SU2 =3 SINR Target for SU1 & SU2 =5 SINR Target for SU1 & SU2 =7

λ

2 1.5 1 0.5 0

1

2

3

4 Iteration Number

5

6

7

4 5 SINR of SU1 and SU2

6

7

(b) Balanced SINR of SU3 and SU4

same as that we obtained using the proposed method. The interference leakage value for the PUs is equal 0.1 in both schemes. Also, we have observed that both our method and the optimum SDP approach provide the same set of beamformers. Therefore, the proposed algorithm yields an optimum solution for the mixed QoS problem considered in this paper. Note that the SDP-based method of [11] has been used just to demonstrate the optimality of the proposed scheme. However, it should be stressed that the approach of [11] can not be directly applied to the considered scenario as the maximum achievable balanced SINR values for the NRTUs are not known a priori. In order to verify convergence of λ, we have plotted the evolution of λ against the iteration number of the BA algorithm in Fig. 2 (a) for a given set of auxiliary variables. The results shown for various target SINRs of SU1 and SU2 demonstrate monotonically decreasing λ. The inverse of the λ value should provide the balanced SINR value for the SU3 and SU4. Hence the balanced SINR is monotonically increasing against the iteration number. In Fig. 2 (b), we have plotted the balanced SINR value of SU3 and SU4 against the target SINRs of SU1 and SU2. The target SINRs for SU1 and SU2 are set to identical value. As expected the balanced SINR value for SU3 and SU4 is decreasing as the target SINRs of SU1 and SU2 are increased. Fig. 3 reveals the convergence of the complete algorithm in Table II for three different step sizes t = 0.01, 0.005 and 0.001. The sub-figures (a) and (b) show the evolution of interference leakage to the PUs while the sub-figure (c)

8

6

4

2

0

1

2

3

Fig. 2. The sub-figure (a) depicts the convergence of λ against the iteration number of the BA algorithm for different target SINRs for SU1 and SU2. The sub-figure (b) depicts the balanced SINR of the SU3 and SU4 against varying target SINRs for SU1 and SU2.

shows the evolution of the total transmission power against the iteration number. The figures confirm that the interference leakages to PU1 and PU2 and the transmission power converge to the value of 0.1 and 2 respectively as set in the constraints. Table VII and Table VIII provide the number of iterations required for convergence and the steady state balanced SINR value achieved by NRTUs for five different random channel realizations, respectively. The same system model as that was used to generate results in Table V, has been used for this simulation. The results in Table VII have been averaged over 100 different random channel realizations, where we observe

(25) are nonnegative. If c ≤ P together with nonnegative D ˜ in (31) becomes a nonnegative matrix. ¥ and d then Υ(U)

Interferenc Leakage to PU2

(a) 0.6

t=0.01 t=0.005 t=0.001

0.4

B. Proof of Lemma 2

0.2 0

0

20

40

60

80

100

120

140

160

180

200

Interferenc Leakage to PU1

(b) 0.5 t=0.01 t=0.005 t=0.001

0.4 0.3 0.2 0.1 0

20

40

60

80

100

120

140

160

180

200

Transmission Power

(c) 2 t=0.01 t=0.005 t=0.001

1.5

1

0

20

40

60

80 100 120 Iteration Number

140

160

180

200

Fig. 3. The convergence of the complete algorithm in Table II against the iteration of the auxiliary variables.

Channel Channel Channel Channel Channel BALANCED

1 2 3 4 5

t = 0.001 3.1672 1.3024 0.3988 1.9191 4.8956

t = 0.005 3.1391 1.2992 0.3968 1.8997 4.8767

t = 0.01 3.1056 1.1512 0.3775 1.8887 4.8653

TABLE VIII SINR VALUES ACHIEVED BY NRTU S (i.e., SU3 AND SU4).

˜ (0) (i.e., obtained using q(0) ) We have assumed that U satisfies the conditions in (32) and (33). Hence, a feasible power allocation, q(1) , can be obtained at the first iteration. Since there is a feasible solution at the first iteration, RTUs achieve their target SINRs, γk , k = 1 . . . K1, and all NRTUs achieve equal SINRs, 1/λ(1) . In the second iteration, q ˜(1) (1) ˜ . In a has been used with (34) to obtain beamformers U conventional SINR balancing problem, SINR of each user increases monotonically at each iteration [6]. However, in the problem considered in this paper, some users (i.e., RTUs) maintain the same SINR (i.e., target SINR) at each iteration. Hence, it is apparent that, in the second iteration, there is a feasible solution available and the NRTUs achieve SINRs not less than the ones that achieved in the first iteration (i.e., 1/λ(2) ≥ 1/λ(1) ). At the end of the second iteration, the equation (20) is updated as (2)

(1)

(1)

(2)

(1)

(2)

q ˜A

A PPENDIX A. Proof of Lemma 1 If ρ(DA ΨA ) ≤ 1, then limn→∞ (DA ΨA )n = 0 [47]. Hence, (I − DA ΨA ) is a nonsingular P∞ matrix and using Neumann series (I − DA ΨA )−1 = n=0 (DA ΨA )n which is a nonnegative matrix [47]. When (I − DA ΨA )−1 is a nonnegative matrix then matrix D in (24) and vector d in

(2)

(54)

(1)

(1)

(2)

(55)

The inequality (55) results into the following inequality for (2) the normalized qA (1)

A joint SINR balancing and target SINR provision based beamforming technique has been proposed for a spectrum sharing network. The proposed technique optimally designs downlink beamformers and the power allocation for a problem with mixed QoS requirements. An iterative algorithm has been proposed to obtain the solution. Simulation results have confirmed the optimality and the convergence of the algorithm.

(1)

≥ D A ΨA q ˜A .

(2) T

VI. C ONCLUSION

(1)

Since there exists a feasible solution at the end of the second (2) (2) (1) (1) (1) (1) (2) iteration, q ˜A , q ˜B , DA bA and DA ΨB q ˜B are nonnegative. Hence,

1 ≥ qA

the convergence time of the algorithm is inversely proportional to t. However, there is a tradeoff that as t increases, the algorithm converges faster but results into a lower steady state SINR value for the NRTUs as shown in Table VIII. However, as compared to the overwhelming benefit in terms of convergence speed, the loss in the SINR value is insignificant.

(1)

q ˜ A = D A ΨA q ˜A + DA bA + DA ΨB q ˜B ,

(1)

(1)

(2)

(56)

D A ΨA q A ,

(1)

where DA ΨA is a nonnegative matrix. Maximum value of (2) T

(1)

(1) (2)

(1)

(1)

(2)

qA DA ΨA qA is equal to ρ(DA ΨA ) when qA is the eigenvector corresponding to the largest eigenvalue of matrix (1) (1) DA ΨA . From the inequality in (56), (1)

(1)

(57)

1 ≥ ρ(DA ΨA ).

From equation (54), the following inequality holds: (2)

q ˜A

(1)

(1)

(2)

(1)

(1)

≥ D A ΨA q ˜ A + DA b A .

(58)

From inequality (58), we can write the following inequalities using Lemma 1, and the equations (57), (21) and (28): (2)

q ˜A

(1)

(1)

(1) (1)

≥ (I − DA ΨA )−1 DA bA ,

(2)

(1)

(1)

(1) (1)

P≥σ TA q ˜A ≥ σ TA (I − DA ΨA )−1 DA bA = c(1). (59) The required conditions are satisfied in (57) and (59). From ˜ (1) ) is a nonnegative matrix. Lemma 1, Υ(U ¥ C. Proof of Lemma 3 After the convergence of BA described in Table I, ρ(D∗A Ψ∗A ) ≤ 1. Hence, after the convergence, the vector d∗ and the matrix D∗ are nonnegative in (23). The following inequality can be obtained from (23): λ∗ q ˜B

≥

D∗ q ˜B , ⇒ λ∗ ≥ ρ(D∗ ).

(60)

Since ρ(D∗A Ψ∗A ) ≤ 1, the matrix D in (24) can be written as follows using the Neumann series: ∗ D∗ = D∗B Ψ∗D + D∗B ΨC

∞ X

(D∗A Ψ∗A )n D∗A Ψ∗B .

(61)

n=0

T

For a diagonal matrix A, ρ(AB) = ρ(AB ) [47]. From this property, the following inequalities can be obtained from equation (60): λ∗

≥ ρ[D∗B (Ψ∗D + Ψ∗C =

ρ[D∗B (Ψ∗D

+

Ψ∗C

∞ X n=0 ∞ X

(D∗A Ψ∗A )n D∗A Ψ∗B )],

T

T

(62)

D∗D

However, is a nonnegative matrix. From Lemma 1 and the Neumann series, (λ∗ I − D∗D ) is nonsingular and (λ∗ I − P∞ ∗ −1 ∗ n DD ) = n=0 (DD ) is a nonnegative matrix. ¥ D. Proof of Lemma 4 R In equation (53), DR A bA is a nonnegative vector. Hence, the following inequality holds: R R ≥ DR A ΨA q A .

(63)

From inequality (63), the following inequality holds for the normalized qR A λR A

≥

T

R R R qR A DA ΨA qA .

(64)

R R DR A ΨA is a nonnegative matrix and qA is a nonnegative power allocation vector. The largest eigenvalue of irreducible nonnegative matrix and the corresponding eigenvector are T R R R strictly positive [47]. Largest value of qR A DA ΨA qA is equal R R R to ρ(DA ΨA ) if and only if qA is a positive eigenvector R corresponding to the largest eigenvalue of the matrix DR A ΨA . Hence,

λR A

R ≥ ρ(DR A ΨA ).

R R R R DR A ΨA qA + DA bA , R DR A bA .

(66)

R R R −1 But from Lemma 1 if ρ(DR is A ΨA ) ≤ 1, then (I − DA ΨA ) a nonnegative matrix. Hence, the inequality (66) satisfies the following inequality:

qR A

R −1 R R ≥ (I − DR DA bA . A ΨA )

≥

R −1 R R σ TA (I − DR DA bA = c, A ΨA )

T

n=0

R λR A qA

≥ ≥

which satisfies the second sufficient condition (33) for the non˜ (0) ), and it concludes the proof. negativity of Υ(U ¥

(D∗A Ψ∗A )n D∗A Ψ∗C )T ],

= ρ(D∗D ).

R R (I − DR A ΨA )qA

P ≥ σ TA qR A

n=0

= ρ[D∗B (Ψ∗D + Ψ∗B

qR A

From (21) and (28), we could write

T (D∗A Ψ∗A )n D∗A Ψ∗B ) ],

∞ X

inequalities:

(65)

The problem is feasible, only if λR A ≤ 1 which implies that R ρ(DR A ΨA ) ≤ 1. ˜ (0) ) is a nonnegative In Section III-C, by assuming Υ(U matrix, we have proved the convergence of the BA algorithm in Table I. To complete the proof, we have to show that if we use initial power allocation q(0) then we can obtain ˜ (0) ). For all feasible problems there non-negative matrix Υ(U R exists a feasible power allocation qR A . Note that, when qA and 0K−K1 are used as an initial power allocation for the RTUs and the NRTUs respectively, the same matrices DR A and (0) (0) (0) R ΨR and vector b can be obtained for D , Ψ and bA A A A A at step 6 of the BA algorithm in the first iteration. Hence, (0) (0) R ρ(DA ΨA ) = ρ(DR A ΨA ) ≤ 1, which satisfies the first ˜ (0) ). sufficient condition (32) for the nonnegativity of Υ(U For the feasible problem, equation (53) satisfies the following

R EFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” in Wireless Personal Communications, vol. 6, pp. 311–335, Mar. 1998. [2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” in European Transactions on Telecommunications, vol. 10, pp. 585–595, Nov/Dec. 1999. [3] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 461–471, 2004. [4] M. Bengtsson and B. Ottersten, “Optimal and suboptimal transmit beamforming,” in Handbook of Antennas in Wireless Communications, L. C. Godara, Editor, CRC Press, 2001. [5] F. Rashid-Farrokhi, K. Liu, and L. Tassiulas, “Transmit beamforming and power control for cellular wireless systems,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1437–1450, Oct. 1998. [6] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Trans. Veh. Technol., vol. 53, pp. 18–28, Jan. 2004. [7] E. Matskani, N. D. Sidiropoulos, Z. Q. Luo, and L. Tassiulas, “Convex approximation techniques for joint multiuser downlink beamforming and admission control,” IEEE Trans. Wireless Commun., vol. 7, pp. 2682– 2693, Jul. 2008. [8] W. Yang and G. Xu, “Optimal downlink power assignment for smart antenna systems,” in in Proc. IEEE ICASSP, vol. 6, pp. 3337–3340, Seattle, WA, May 1998. [9] M. Bengtsson and B. Ottersten, “Optimal downlink beamforming using semidefinite optimization,” in 37th Annual Allerton Conference on Communications, Control and Computing, pp. 987–996, Sep. 1999. [10] J. Li and P. Stoica, Robust Adaptive Beamforming. United Kingdom: Wiley Series in Telecommunications and Signal Processing, 2006. [11] K. Cumanan, R. Krishna, V. Sharma, and S. Lambotharan, “Robust interference control techniques for multiuser cognitive radios using worst-case performance optimization,” in Proc. Asilomar Conf. Sign., Syst. and Comp., pp. 378–382, Pacific Grove, CA, Oct. 2008. [12] K. Hamdi, W. Zhang, and K. B. Letaief, “Joint beamforming and scheduling in cognitive radio networks,” in Proc. IEEE GLOBECOM, pp. 2977–2981, Washington, DC, Nov. 2007. [13] K. Cumanan, L. Musavian, S. Lambotharan, and A.B.Gershman, “SINR balancing technique for downlink beamforming in cognitive radio networks,” IEEE Signal Process. Lett., vol. 17, pp. 133–136, Feb. 2010. [14] K. Cumanan, R. Krishna, Z. Xiong, and S. Lambotharan, “SINR balancing technique and its comparison to semidefinite programming based QoS provision for cognitive radios,” in IEEE 69th VTC Spring, pp. 1–5, Apr. 2009. [15] L. Zhang, R. Zhang, Y. C. Liang, Y. Xin, and H. V. Poor, “On Gaussian MIMO BC-MAC duality with multiple transmit covariance constraints,” submitted to IEEE Trans. Inf. Theory, Available online at http://arxiv.org/PS cache/arxiv/pdf/0809/0809.4101v1.pdf. [16] A. Fehske, F. Richter, and G. Fettweis, “SINR balancing for the multiuser downlink under general power constraints,” in IEEE GLOBECOM 2008, pp. 1–6, Dec. 2008. [17] L. Zhang, Y.-C. Liang, and Y. Xin, “Joint beamforming and power allocation for multiple access channels in cognitive radio networks,” IEEE J. Sel. Areas Commun., vol. 26, pp. 38–51, Jan. 2008.

[18] N. Jindal and A. Goldsmith, “Capacity and optimal power allocation for fading broadcast channels with minimum rates,” IEEE Trans. Inf. Theory, vol. 49, pp. 2895–2909, Nov. 2003. [19] Y. Zhang and C. Leung, “Cross-layer resource allocation for mixed services in multiuser OFDM-based cognitive radio systems,” IEEE Trans. Veh. Technol., vol. 58, pp. 4605–4619, Oct. 2009. [20] P. Tejera, W. Utschick, J. Nossek, and G. Bauch, “Rate balancing in multiuser MIMO OFDM systems,” IEEE Trans. Commun., vol. 57, pp. 1370–1380, May 2009. [21] Z. Shen, J. Andrews, and B. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. Wireless Commun., vol. 4, pp. 2726–2737, Nov. 2005. [22] E. Jorswieck and H. Boche, “Rate balancing for the multi-antenna Gaussian broadcast channel,” in IEEE 7th Int. Symp. Spread Spectrum Tech. Appl, vol. 2, pp. 545–549, Sep. 2002. [23] H. Boche, M. Schubert, and E. Jorswieck, “Trace balancing for multiuser MIMO downlink transmission,” in 36th Asilomar Conference on Signals, Systems and Computers, vol. 2, pp. 1379–1383, Nov. 2002. [24] S. Shi, M. Schubert, and H. Boche, “Capacity balancing for multiuser MIMO systems,” in IEEE ICASSP, vol. 3, pp. III–397–III–400, Apr. 2007. [25] L. Zhang, Y. Xin, and Y. C. Liang, “Weighted sum rate optimization for cognitive radio MIMO broadcast channels,” IEEE Trans. Wireless Commun., vol. 8, pp. 2950–2959, Jun. 2009. [26] M. Schubert and H. Boche, “Iterative multiuser uplink and downlink beamforming under SINR constraints,” IEEE Trans. Signal Process., vol. 53, pp. 2324–2334, Jul. 2005. [27] G. Caire, R. Muller, and R. Knopp, “Hard fairness versus proportional fairness in wireless communications: The single-cell case,” IEEE Trans. Inf. Theory, vol. 53, pp. 1366–1385, Apr. 2007. [28] V. K. N. Lau, “A proportional fair space-time scheduling for wireless communications,” IEEE Trans. Commun., vol. 53, pp. 1353–1360, Aug. 2005. [29] J. Holtzman, “Asymptotic analysis of proportional fair algorithm,” in 12th IEEE Int. Symp. Pers. Indoor and Mobile Radio Commun., vol. 2, pp. F–33–F–37 vol.2, Sep/Oct 2001. [30] J. Cho and D. Hong, “Tradeoff analysis of throughput and fairness on CDMA packet downlinks with location-dependent QoS,” IEEE Trans. Veh. Technol., vol. 54, pp. 259–271, Jan. 2005. [31] R. Lu, X. Lin, H. Zhu, X. Shen, and B. Preiss, “Pi: A practical incentive protocol for delay tolerant networks,” IEEE Trans. Wireless Commun., vol. 9, pp. 1483–1493, Apr. 2010. [32] M. Van Der Schaar and N. S. Shankar, “Cross-layer wireless multimedia transmission: challenges, principles, and new paradigms,” IEEE Trans. Wireless Commun., vol. 12, pp. 50–58, Aug. 2005. [33] H. Hai Jiang, P. Wang, and W. Weihua Zhuang, “A distributed channel access scheme with guaranteed priority and enhanced fairness,” IEEE Trans. Wireless Commun., vol. 6, pp. 2114–2125, Jun. 2007. [34] P. Viswanath, D. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Trans. Inf. Theory, vol. 48, pp. 1277–1294, Jun. 2002. [35] E. Liu and K. Leung, “Throughput of proportional fair scheduling over rayleigh fading channels,” Electronics Letters, vol. 45, pp. 1180–1182, Nov. 2009. [36] CDMA2000 High Rate Packet Data Air Interface Specification. TIA/EIA/3GPP2 Std. IS-856/3GPP2 C.S.0024, Dec. 2001, v3.0. [37] J. Mitola and G. Q. Maguire, “Cognitive radios: Making software radios more personal,” IEEE Pers. Commun., vol. 6, no. 4, pp. 13–18, Aug. 1999. [38] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [39] S. Srinivasa and S. A. Jafar, “The throughput potential of cognitive radio: A theoretical perspective,” IEEE Commun. Mag., vol. 45, pp. 73– 79, May 2007. [40] A. Jovicic and P. Viswanath, “Cognitive radio: An information-theoretic perspective,” IEEE Trans. Inf. Theory, vol. 55, pp. 3945–3958, Sep. 2009. [41] R. Zhang and Y. C. Liang, “Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks,” IEEE J. Sel. Topics Signal Process., vol. 2, pp. 88–102, Feb. 2008. [42] A. Ghasemi and E. S. Sousa, “Fundamental limits of spectrum-sharing in fading environments,” IEEE Trans. Wireless Commun., vol. 6, pp. 649– 658, Feb. 2007. [43] L. Zhang, Y. Xin, Y. C. Liang, and H. V. Poor, “Cognitive multiple access channels: optimal power allocation for weighted sum rate maximization,” IEEE Trans. Commun., vol. 57, pp. 2754–2762, Sep. 2009.

[44] K. Phan, S. Vorobyov, N. Sidiropoulos, and C. Tellambura, “Spectrum sharing in wireless networks via QoS-aware secondary multicast beamforming,” IEEE Trans. Signal Process., vol. 57, pp. 2323–2335, Jun. 2009. [45] G. Zheng, K.-K. Wong, and B. Ottersten, “Robust cognitive beamforming with bounded channel uncertainties,” IEEE Trans. Signal Process., vol. 57, pp. 4871–4881, Dec. 2009. [46] S. Boyd, L. Xiao, and A. Mutapici, Subgradient methods. Available online at http://mit.edu/6.976/www/notes/subgrad-method.pdf, Oct. 2003. [47] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. SIAM, 2000. [48] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Trans. Commun., vol. 54, pp. 1310–1322, Jul. 2006. [49] V. Y. Pan and Z. Q. Chen, “The complexity of the matrix eigenproblem,” in 31st Ann. ACM Symp. on Theory of Computing, ACM Press, New York, 1999.

Yogachandran Rahulamathavan (S’11) graduated with a First Class Honours degree in Electronic and Telecommunication Engineering from University of Moratuwa, Sri Lanka in 2008. He is currently pursuing the Ph.D. degree in Signal Processing for Wireless Communication from Loughborough University, UK. He received a scholarship from Loughborough University to pursue his Ph.D. degree. From April 2008 to September 2008, he was an Engineer at Sri Lanka Telecom, Sri Lanka. His current research interests include resource optimization techniques, MIMO and beamforming for wireless networks.

Kanapathippillai Cumanan (M’10) received his B.Sc. degree with First Class Honours in Electrical and Electronic Engineering from the University of Peradeniya, Sri Lanka in 2006 and his Ph.D. degree in Signal Processing from Loughborough University, Loughborough, UK in 2009. He received an overseas research students (ORS) award from Cardiff University, UK where he was a research student between September 2006 and July 2007. From January 2006 to August 2006, he was a teaching assistant at the Department of Electrical and Electronic Engineering at the University of Peradeniya, Sri Lanka. He is currently working as a research associate in the Advanced Signal Processing Group at Loughborough University, UK. Dr. Cumanan’s research interests include cognitive radio networks, relay networks, convex optimization techniques and resource allocation techniques.

Sangarapillai Lambotharan (SM’06) received his Ph.D. degree in signal processing from Imperial College, U.K., in 1997 and he remained there until 1999 as a Postdoctoral Research Associate. In 1996, he was a Visiting Scientist with the Engineering and Theory Center of Cornell University, USA. From 1999 to 2002, he was with the Motorola Applied Research Group, U.K. as a Research Engineer and investigated various projects, including physical-linklayer modelling and performance characterization of GPRS, EGPRS, and UTRAN. From 2002 to 2007, he was with the King’s College London, UK., and the Cardiff University, U.K., as a Lecturer and Senior Lecturer, respectively. In Sept. 2007, he joined the Advanced Signal Processing Group at Loughborough University, U.K as a Reader and he was promoted to Professor of Digital Communications in Sept. 2011. His current research interests include MIMO, wireless relay networks, cognitive radio networks and smart grids and he has published more than 100 conference and journal articles in these areas. He serves as an Associate Editor for the EURASIP Journal on Wireless Communications and Networking.