3570
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011
MIMO Transceiver Designs for Spatial Sensing in Cognitive Radio Networks Keonkook Lee, Student Member, IEEE, Chan-Byoung Chae, Member, IEEE, Robert W. Heath, Jr., Fellow, IEEE, and Joonhyuk Kang, Member, IEEE
Abstract—We propose transceiver algorithms in cognitive radio networks where the cognitive users are equipped with multiple antennas. Prior work has focused on the design of precoding matrices to suppress interference to the primary receivers. This work considers designs of precoding and decoding matrices for spatial sensing to achieve two objectives: i) to prevent interference to the primary receivers and ii) to remove the interference, due to primary transmissions, at the secondary receiver. With single antenna primary terminals and two antenna cognitive terminals, a linear transceiver design has been introduced under a global channel state information (CSI) assumption [1]. In this letter, multiple antenna primary and cognitive terminals and three different CSI scenarios depending upon the amount of CSI are studied: i) local CSI, ii) global CSI, and iii) local CSI with side information. When local CSI is available, we leverage prior work and employ the projected-channel singular value decomposition (P-SVD). In the global CSI scenario, we propose a joint transmitter-receiver design under the assumption of full CSI of all the users at the secondary transceiver. To reduce the feedback overhead, we also propose a new iterative algorithm that exploits only local CSI with side information. In this algorithm, the secondary transmitter and receiver iteratively update precoding and decoding matrices based on the local CSI and side information (precoding/decoding matrices at the previous iteration step) to maximize the rate of the secondary link while maintaining the zero-interference constraint. Convergence is established in the special case of single stream beamforming. Numerical results confirm that the proposed joint design and the iterative algorithm show better achievable rate performance than the P-SVD technique at the expense, respectively, of CSI knowledge and side information. Index Terms—MIMO, cognitive radios, spectrum sharing, projection matrix.
I. I NTRODUCTION
C
OGNITIVE radio networks make efficient use of a spectrum by allowing the coexistence of secondary networks in the spectrum occupied for the use of a legacy primary network [2], [3]. Using multiple antennas at the cognitive transceiver, cognitive networks can operate more efficiently by leveraging the spatial domain [4]–[9]. Sensing how primary users make use of spatial resources and sharing those resources
Manuscript received July 19, 2010; revised January 22, 2011 and June 27, 2011; accepted August 20, 2011. The associate editor coordinating the review of this paper and approving it for publication was N. Sagias. K. Lee and J. Kang are with the Korea Advanced Institute of Science and Technology, Daejeon, Korea (e-mail:
[email protected],
[email protected]). C.-B. Chae is with the School of Integrated Technology, Yonsei University, Korea (e-mail:
[email protected]). R. W. Heath, Jr. is with the Dept. of Electrical and Computer Engineering, The University of Texas at Austin, TX, USA 78712 (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2011.092711.101305
can provide secondary users more degrees of freedom for sharing the spectrum with primary users. In this letter, we refer to this technique as spatial sensing. In [4], [5], the authors proposed an optimal technique for the multiple-input multiple output (MIMO) channel to maximize, by using convex optimization, the rate of secondary link subject to the required primary protection. Moreover, effective low complexity suboptimal algorithms were introduced in [4], [6]. Prior work in [7] and [8] proposed a multicast beamforming algorithm and an optimal interference-constrained beamforming algorithm, respectively, for multiple-input single-output (MISO) cognitive radio networks. The authors in [9] also investigated the multiple antenna cognitive transmitter design by using pre-whitening filters. The authors in [4]–[9], however, ignored interference from the primary transmitters or treated interference as an additive noise at the secondary receiver; thus they primarily focused on the transmitter design. In practice, when the primary transmitters are active, the secondary receiver suffers interference from the primary transmitters [10], [11]. This type of interference motivates research into joint transceiver designs. In this letter, we consider MIMO transceiver algorithms to meet two objectives: i) to prevent the primary receivers from experiencing interference from the cognitive transmitter and ii) to remove interference caused by primary transmissions at the secondary receiver. The authors in [1], [12] studied how the secondary receiver could be designed to overcome interference from the primary transmitters. In particular, the authors in [12] investigated the design of cognitive receiver depending on interference level received at the cognitive receiver and how the existing interference cancellation techniques such as filter-based approach could be used at the cognitive transceiver. In [1], a linear transceiver design with single antenna primary terminals and two antenna cognitive terminals was introduced. The idea, however, is valid only when the secondary transceivers have two antennas under a global CSI assumption. Unlike in [1], in this letter, we consider multiple antenna primary and cognitive terminals under different CSI scenarios. We study, depending upon the amount of channel state information (CSI) at the secondary transmitter and receiver, three different scenarios, i) local CSI, ii) global CSI, and iii) local CSI with side information. We assume that channel gains directly connected to them are available through reciprocity of the channel in the local CSI scenario. Side information, i.e., information about precoding and decoding matrices, and global CSI, i.e., perfect CSI of all channels in the network are assumed to be obtainable from dedicated signal [13], and
c 2011 IEEE 1536-1276/11$25.00 ⃝
LEE et al.: MIMO TRANSCEIVER DESIGNS FOR SPATIAL SENSING IN COGNITIVE RADIO NETWORKS
F1
PU Tx 1 PU Tx KtPU s
T SU Tx
x
II. S YSTEM M ODEL
G1
PU Rx 1
G K PU
FK PU t
r
H
3571
PU Rx KrPU ys
W
r
SU Rx
Fig. 1. Cognitive radio network model with multiple primary transmitters and receivers.
perfect CSI feedback, respectively. In the local CSI scenario, we use a projected-channel SVD (P-SVD) technique that was introduced for a precoder design in [4], both at the secondary transmitter and receiver. When global CSI is available at the secondary transceiver, we propose a joint transmit-receive design, which provides an upper-bound of the achievable rate of the system. The authors in [14], [15] introduced a distributed iterative interference alignment technique and a distributed pricing algorithm in interference networks, respectively, to overcome the problem of feedback overhead in the global CSI scenario. Such techniques update the precoding and decoding matrices at every time slot by using only local CSI. In a similar manner, we propose a new iterative algorithm for the secondary link in cognitive radio networks when the local CSI and side information are available. In the proposed iterative algorithm, the secondary transmitter and receiver update precoding and decoding matrices based on the local CSI and side information (precoding/decoding matrices at the previous iteration step) to maximize the rate of the secondary link while maintaining the zero-interference constraint. By using the power iteration method, we prove that the iterative algorithm converges to the solution of the joint design with global CSI when the secondary transmitter uses beamforming such as maximal-ratio transmission. The rest of this letter is organized as follows. Section II describes the system model. Section III introduces and analyzes our proposed cognitive transceiver algorithms. Section IV provides the simulation results and we conclude in Section V. Notation: Lower case and upper case boldface denotes vectors and matrices, respectively. [⋅]−1 , [⋅]𝑇 and [⋅]∗ denote 𝐴∣ inverse, transpose and conjugate transpose, respectively. ∣𝐴 𝑎∥ and ∥𝐴 𝐴∥𝐹 denote the Eurepresents the determinant of 𝐴 . ∥𝑎 clidean norm of 𝑎 and the Frobenius norm of 𝐴 , respectively. 𝐴 (𝑀) is a matrix that consists of the first 𝑀 columns of 𝐴 . ⊥ Φ𝐴 and Φ𝐴 are used to denote the column space of matrix 𝐴 and the orthogonal complement of Φ𝐴 , respectively. 𝕌 (𝑀, 𝑁 ) denotes 𝑀 × 𝑁 complex matrices with orthonormal columns, 𝑚, 𝑉 ) which is known as the Stiefel manifold. 𝑧 ∼ 𝒞𝒩 (𝑚 denotes that the elements of 𝑧 are random variables with the distribution of a circularly-symmetric-complex-Gaussian (CSCG) with mean 𝑚 and covariance 𝑉 .
A block diagram of the cognitive system under consideration in this letter is illustrated in Fig. 1. The system consists of 𝐾𝑡PU primary transmitters, 𝐾𝑟PU primary receivers, and a single secondary transmitter with 𝑀𝑡SU antennas and a single secondary receiver with 𝑀𝑟SU antennas. The total number of antennas at the primary transmitters and the primary receivers ∑𝐾𝑡PU PU𝑖 ∑𝐾𝑟PU PU𝑖 are 𝑀𝑡PU = 𝑖=1 𝑀𝑡 and 𝑀𝑟PU = 𝑖=1 𝑀𝑟 , respectively, where 𝑀𝑡PU𝑖 and 𝑀𝑟PU𝑖 denote number of antennas at each primary transceivers. The channel from the 𝑖-th primary transmitter to the secondary receiver is given as 𝐹 𝑖 where } { 𝑖 ∈ 1, 2, ⋅ ⋅ ⋅ , 𝐾𝑡PU and the channel from the secondary transmitter {to the 𝑘-th primary receiver is expressed as 𝐺 𝑘 } where 𝑘 ∈ 1, 2, ⋅ ⋅ ⋅ , 𝐾𝑟PU . 𝐻 denotes the secondary user’s channel. We assume that the channels are Rayleigh fading, i.e., the entries of channel matrices are distributed according to 𝒞𝒩 (0, 1). Multiple antennas at secondary transceiver are used in two ways: to suppress interference to the primary receivers and from the primary transmitters and as well as to maximize the achievable rate of the secondary link. For each channel use, when the number of data streams is equal to 𝑀𝑑 , a data symbol vector, 𝑠 = [ 𝑠1 ⋅ ⋅ ⋅ 𝑠𝑀𝑑 ]𝑇 , is precoded by an 𝑀𝑡SU ×𝑀𝑑 precoding matrix 𝑇 . The precoding matrix is designed not only to remove the interference to the primary receivers but also to maximize the achievable rate of the secondary link. The interference-power constraint at the 𝑘-th primary receiver 𝐺𝑘𝑥 ∥2𝐹 ≤ Γ𝑘 , where Γ𝑘 denotes the can be expressed as ∥𝐺 predefined interference constraint at the 𝑘-th primary receiver and 𝑥 = 𝑇 𝑠 . The received signal at the secondary receiver is 𝐾𝑡PU
𝑦 𝑠 = 𝐻𝑥 +
∑
𝐹 ℓ𝑞 ℓ + 𝑛 𝑠 ,
(1)
ℓ=1
where 𝑞 ℓ is the transmitted signal from the ℓ-th primary transmitter and 𝑛 𝑠 is the additive noise vector(at the secondary ) receiver, which is assumed that 𝑛 𝑠 ∼ 𝒞𝒩 0, 𝜎𝑛2 𝐼 . At the secondary receiver, an 𝑀𝑟SU × 𝑀𝑑 decoding matrix, 𝑊 , is used to remove the interference from the primary transmitters as well as to maximize the rate of the secondary channel. Thus, the received symbol vector at the secondary receiver is given by ∗
∗
𝑟 = 𝑊 𝑦 𝑠 = 𝑊 𝐻𝑇 𝑠 + 𝑊
∗
𝐾𝑡PU
∑
𝐹 ℓ𝑞 ℓ + 𝑊 ∗ 𝑛 𝑠 .
(2)
ℓ=1
In practice, especially in fast-fading channels, the interference to primary users might exceed the predefined interference of the system. To best avoid this problem, we consider a secondary transmission technique that produces no interference to the } primary receivers, i.e., Γ𝑘 = 0 where { 𝑘 ∈ 1, 2, ⋅ ⋅ ⋅ , 𝐾𝑟PU . The work in [4], [6] proposed P-SVD, which was shown to be optimal in the sense of maximizing the rate of secondary channel when the interference constraint was equal to zero. P-SVD, however, focused on only for transmitter design by treating the interference from the primary transmissions to the secondary receiver as an additive noise. Thus, a decoder design, which induces a new corresponding optimal transmitter design, i.e., joint design, to avoid the interference
3572
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011
TABLE I R EQUIRED I NFORMATION AT THE S ECONDARY T RANSCEIVER . CSI assumptions Local CSI Global CSI Local CSI with side information
Secondary Tx 𝐺, 𝐻 𝐹 , 𝐺, 𝐻
Secondary Rx 𝐹, 𝐻 𝐹 , 𝐺, 𝐻
Pilot options Common pilot Common pilot and CSI feedback
𝑊 , 𝐺, 𝐻
𝑇 ,𝐹 , 𝐻
Common pilot and dedicated pilot
is also needed. This need is what motivates the study of joint transmit-receive design. In this letter, we assume that the secondary receiver uses a ZF type decoding matrix to remove the interference from the primary transmitter, i.e., 𝑊 ∗𝐹 𝑖 = 0 { } PU where 𝑖 ∈ 1, 2, ⋅ ⋅ ⋅ , 𝐾𝑡 . An MMSE type decoding matrix can also be used to further maximize a performance of the secondary link, but for simplicity we restrict our interest to ZF. We assume that the system uses time division duplexing (TDD). By exploiting the reciprocity of TDD channels, we investigate three different scenarios: i) local CSI, ii) global CSI, and iii) local CSI with side information. When only local CSI is available, we assume that the secondary transceiver has knowledge of CSI, obtainable by using reciprocity of the channels. In the global CSI scenario, all of CSI is known to the secondary transceiver. Although in practice global CSI is infeasible, it provides an upper-bound on the system’s achievable rate. We assume that the secondary transmitter and receiver can estimate the decoding and precoding matrices, respectively. This can be done, for example, by using a dedicated pilot, also called as a precoded pilot, when local CSI and side information are available [13]. The differences among the three CSI scenarios are summarized in Table I. Throughout the letter, we assume that 𝑀𝑡SU and 𝑀𝑟SU are larger than 𝑀𝑟PU and 𝑀𝑡PU , respectively. III. P ROPOSED T RANSCEIVER D ESIGNS A. P-SVD with Local CSI We use P-SVD at both the secondary transmitter and receiver when only local CSI is available at the secondary transceiver. We assume that 𝐺 and 𝐻 are known to the secondary transmitter and 𝐹 and 𝐻 are known to the secondary receiver, where ]𝑇 [ ] [ , 𝐹 = 𝐹 1 ⋅ ⋅ ⋅ 𝐹 𝐾𝑡PU . 𝐺 = 𝐺 𝑇1 ⋅ ⋅ ⋅ 𝐺 𝑇𝐾𝑟PU To satisfy the zero-interference constraint at the secondary 𝐺𝑇 P−SVD ∥2𝐹 = 0, 𝑇 P−SVD should be transmitter, i.e., ∥𝐺 orthogonal to the column space of the 𝐺 ∗ , which is denoted as Φ𝐺 ∗ in this paper. P-SVD first finds the projection matrix that ⊥ projects the transmit signal onto Φ𝐺 ∗ . The projection matrix ⊥ ⊥ onto Φ𝐺∗ , 𝑃 𝐺 ∗ , and the projection matrix onto Φ𝐺 ∗ , 𝑃 𝐺 ∗ , are −1 𝐺𝐺 ∗ ) 𝐺 and 𝐼 − 𝑃 𝐺∗ , respectively [16]. By known as 𝐺 ∗ (𝐺 ⊥ using 𝑃 𝐺 ∗ that enables the secondary signal not to interfere ⊥ with the primary receivers, 𝐻 can be projected onto Φ𝐺 ∗ and the SVD of the projected channel is expressed as ⊥ ∗ 𝐻 𝑡P−SVD = 𝐻 𝑃 𝐺 ∗ = 𝑈 𝑡Σ 𝑡𝑉 𝑡 .
(3)
Since 𝐻 𝑡P−SVD is orthogonal to Φ𝐺∗ and 𝑉 𝑡 consists of singular vectors of 𝐻 𝑡P−SVD , taking 𝑇 P−SVD as 𝑉 𝑡(𝑀𝑑 ) , i.e., the first 𝑀𝑑 columns of 𝑉 𝑡 , satisfy the zero-interference
constraint. Moreover, water-filling algorithm is used for power allocation based on the singular values in Σ 𝑡 [4]. Similar to the transmitter design, the secondary receiver constructs 𝑊 P−SVD to 𝑈 𝑟(𝑀𝑑 ) by using the projected channel ∗ onto Φ𝐹⊥ , 𝐻 𝑟P−SVD = 𝑃 𝐹⊥ 𝐻 = 𝑈 𝑟Σ 𝑟𝑉 ∗𝑟 . It is worth noting ⊥ that we can use a different method to project 𝐻 onto Φ𝐺 ∗ or ⊥ ⊥ ⊥ Φ𝐹 . Instead of the projection matrices 𝑃 𝐺 ∗ and 𝑃 𝐹 , the basis ⊥ ⊥ vectors of Φ𝐺 ∗ and Φ𝐹 , respectively, can be used [17]. Let ⊥ the SVD of 𝑃 𝐺 ∗ be [ ] ]∗ Σ 0 [ (1) ⊥ 𝑃𝐺 , (4) ∗ = 𝐼 − 𝑃 𝐺∗ = 𝑈 𝑉 𝑉 (0) 0 0 where Σ is the 𝑟p × 𝑟p diagonal matrix whose diagonal terms ⊥ are non-zero singular values of 𝑃 𝐺 ∗ and 𝑟p denotes the rank ⊥ (1) ⊥ consists of the basis of Φ𝐺 of 𝑃 𝐺 ∗ . Because 𝑉 ∗ , the ⊥ (1) 𝑃 𝑉 projection matrix 𝐺∗ can be substituted for . Note that 𝑉 (1) consists of orthonormal columns, thus, it has a unitary property, i.e., 𝑉 (1) ∈ 𝕌 (𝑀𝑡 , 𝑟p ). B. Proposed Joint Tx-Rx Design with Global CSI Now, we assume that the secondary transceiver has full CSI of all users. In [1], when (𝑀𝑡PU , 𝑀𝑟PU , 𝑀𝑡SU , 𝑀𝑟SU , 𝑀𝑑 ) = (1, 1, 2, 2, 1) and global CSI assumption, a linear transceiver design was proposed by using a simple linear algebra. Note that, in our work, we do not limit the number of antennas at terminals and use a new linear algebra concept such as projection matrix to resolve the generalized setup. In the proposed joint design with global CSI, first, the secondary transmitter and receiver find the effective channel matrix, 𝐻 eff , ⊥ ⊥ which lies on the intersection of Φ𝐺 ∗ and Φ𝐹 to satisfy the zero-interference constraint both at the secondary transmitter and receiver as follows: ∗
∗ ⊥ 𝐻 eff = 𝑃 𝐹⊥ 𝐻 𝑃 𝐺 ∗ = 𝑈 𝑒Σ 𝑒𝑉 𝑒 .
(5)
The proposed precoding matrix, 𝑇 joint , and decoding matrix, 𝑊 joint , to maximize the rate of the secondary link are obtained as 𝑉 𝑒(𝑀𝑑 ) and 𝑈 𝑒(𝑀𝑑 ) , respectively, i.e., 𝑇 joint = 𝑉 𝑒(𝑀𝑑 ) , 𝑊 joint = 𝑈 𝑒(𝑀𝑑 ) . A summary of the joint design is given in Algorithm 1. Algorithm 1 Joint precoding matrix and decoding matrix design −1
⊥ ∗ 𝐺𝐺 ∗ −1 𝐺 𝑃 ⊥ 𝑃𝐺 𝐹 ∗𝐹 ) 𝐹 ∗ ) , 𝐹 = 𝐼 − 𝐹 (𝐹 ∗ = 𝐼 − 𝐺 (𝐺 ∗ ⊥ 2: 𝐻 eff = 𝑃 𝐹⊥ 𝐻𝑃 𝐺 = 𝑈 𝑒Σ 𝑒𝑉 𝑒 ∗ , 𝑈𝑒 = ∗ 𝑢1 ⋅ ⋅ ⋅ 𝑢 𝑟 ] 𝑉 𝑒 = [𝑣𝑣 1 ⋅ ⋅ ⋅ 𝑣 𝑟 ] [𝑢 𝑣 1 ⋅ ⋅ ⋅ 𝑣 𝑀𝑑 ] 3: 𝑇 joint = 𝑉 𝑒(𝑀𝑑 ) = [𝑣 𝑢1 ⋅ ⋅ ⋅ 𝑢 𝑀𝑑 ] 4: 𝑊 joint = 𝑈 𝑒(𝑀𝑑 ) = [𝑢
1:
LEE et al.: MIMO TRANSCEIVER DESIGNS FOR SPATIAL SENSING IN COGNITIVE RADIO NETWORKS
𝑊 ∗𝐻𝑇 𝑆𝑇 ∗𝐻 ∗𝑊 log2 𝐼 𝑀𝑟SU + 2 ∗ ∗ ∗ 𝜎𝑛𝑊 𝑊 + 𝑊 𝐹 𝑄𝐹 𝑊 ∗ (𝑎) 𝐻 eff + 𝑃 𝐹∗ 𝐻 + 𝐻𝑃 𝐺∗ − 𝑃 𝐹∗ 𝐻𝑃 𝐺∗ ) 𝑇 𝑆𝑇 ∗ (𝐻 𝐻 eff + 𝑃 𝐹∗ 𝐻 + 𝐻𝑃 𝐺∗ − 𝑃 𝐹∗ 𝐻𝑃 𝐺∗ ) 𝑊 𝑊 ∗ (𝐻 = log2 𝐼 𝑀𝑟SU + 𝜎𝑛2 𝑊 ∗ 𝑊 + 𝑊 ∗𝐹 𝑄𝐹 ∗𝑊 ∗ ∗ ∗ (𝑏) 𝑊 𝐻 eff 𝑇 𝑆𝑇 𝐻 eff 𝑊 = log2 𝐼 𝑀𝑟SU + 𝜎𝑛2 𝑊 ∗ 𝑊
Remark 1: The proposed joint design is optimal under the zero-interference constraint both at the secondary transmitter and receiver. Suppose that the projection matrices at the secondary transmitter and receiver are 𝐼 − 𝑃 𝐺 ∗ and 𝐼 − 𝑃 𝐹 , respectively. Then, the effective channel matrix is expressed as 𝐻 eff
∗ (𝐼𝐼 − 𝑃 𝐹 ) 𝐻 (𝐼𝐼 − 𝑃 𝐺 ∗ ) = 𝐻 − 𝑃 𝐹∗ 𝐻 − 𝐻𝑃 𝐺∗ + 𝑃 𝐹∗ 𝐻𝑃 𝐺∗ .
=
(7)
When 𝑆 and 𝑄 denote transmit covariance matrix of 𝑠 and 𝑞 , respectively, where 𝑞 = [𝑞𝑞 𝑇1 ⋅ ⋅ ⋅ 𝑞 𝑇𝐾 PU ]𝑇 , the achievable 𝑡 rate of the secondary link can be rewritten by (6) at the top of this page, where (𝑎) is from (7) and (𝑏) is due to the zero-interference constraint of the design of precoding and decoding matrices. Finally, the optimal transceiver design to maximize the rate of the secondary link with zero-interference can be obtained by optimizing (6), which is the same as the joint design. ⊥ (1) ∈ aforementioned, )𝑃 𝐺 ∗ can be substituted to 𝑉 (As SU ⊥ SU PU in (4). Similarly, 𝑃 𝐹 can also be 𝕌 𝑀𝑡 , 𝑀𝑡 − 𝑀𝑟 ) ( changed to 𝑈 (1) ∈ 𝕌 𝑀𝑟SU , 𝑀𝑟SU − 𝑀𝑡PU , which is ob𝑃)𝐹 . Therefore, tained from the we can ( ) rewrite 𝐻 eff ( SVD of 𝐼 −𝑃 in (5) as an 𝑀𝑟SU − 𝑀𝑡PU × 𝑀𝑡SU − 𝑀𝑟PU matrix. It is known that the unitary property does not change the statistical Thus, (properties. ) 𝐻(eff has the same) statistical properties as an 𝑀𝑟SU − 𝑀𝑡PU × 𝑀𝑡SU − 𝑀𝑟PU CSCG matrix. Moreover, due to the multiplication of the projection matrix to obtain 𝐻 eff , the rank of 𝐻 is reduced to the rank of 𝐻 eff (hereafter(we use 𝑟joint for the rank of) 𝐻 eff ), which is equal to min 𝑀𝑡SU − 𝑀𝑟PU , 𝑀𝑟SU − 𝑀𝑡PU . Remark 2: When 𝑀𝑑 is equal to min(𝑀𝑡SU − 𝑀𝑟PU , 𝑀𝑟SU − 𝑀𝑡PU ), i.e., the length of data vector is equal to 𝑟joint , and equal power allocation is used, the rate of the P-SVD technique is the same as the rate of the proposed joint design. Because the projection matrix is the identity operator on the ⊥ ⊥ projected subspace, i.e., 𝑃 𝐺 ∗ 𝐴 = 𝐴 , where 𝐴 lies on Φ𝐺 ∗ , ∗ the effective channel gain of P-SVD, 𝑊 P−SVD𝐻 𝑇 P−SVD , ∗ ⊥ can be rewritten as 𝑊 P−SVD ∗𝑃 𝐹⊥ 𝐻 𝑃 𝐺 ∗ 𝑇 P−SVD . By comparison with the effective channel gain of joint design, 𝑊 ∗joint𝐻 eff 𝑇 joint , we can see that the new channel matrix of the P-SVD technique is the same as that of the joint design, ∗ ⊥ 𝑃 𝐹⊥ 𝐻 𝑃 𝐺 ∗ . In addition, it is known that 𝑊 1 ∗ 𝐻 𝑇 1 𝑆𝑇 1 ∗𝐻 ∗𝑊 1 log2 𝐼 + 𝜎𝑛2 𝑊 1∗ 𝑊 1 ∗ ∗ ∗ 𝑊 2 𝐻 𝑇 2𝑆𝑇 2 𝐻 𝑊 2 (8) = log2 𝐼 + , 𝜎𝑛2 𝑊 2∗ 𝑊 2 𝑊 𝑖 }2𝑖=1 have unitary properties with rank 𝑇 𝑖 }2𝑖=1 and {𝑊 if {𝑇 𝑟, where 𝑟 denotes the rank of 𝐻 [18]. Therefore, the rates
3573
(6)
of P-SVD and joint design are the same when 𝑀𝑑 is equal to 𝑟joint and equal power allocation is used. It is worth noting that the unitary property of input covariance matrix is preserved only when the equal power allocation is used. Otherwise, if the water-filling algorithm to optimize the power allocation is used, the input covariance matrix is not unitary. Note that this is just the information-theoretic perspective. The proposed joint design technique finds the precoding and decoding matrices from the same channel matrix, 𝐻 eff , while the precoding and decoding matrices are from 𝐻 𝑡P−SVD and 𝐻 𝑟P−SVD , respectively in P-SVD. Therefore, the proposed joint design decomposes the MIMO channel into a set of parallel channels, which can be seen as a more preferable approach. Moreover, when 𝑀𝑑 is less than 𝑟joint , the proposed joint design shows better achievable rate performance than the P-SVD technique at the expense of global CSI. The proposed joint design is easily applicable to the broadcast channel (BC) and multiple-access channel (MAC) for cognitive radios [19]. For brevity, these are not described in the letter. C. Iterative Algorithm with the Local CSI and Side Information In the local CSI with side information scenario, we assume that the secondary transceiver can estimate not only the local CSI but also information about precoding and decoding matrices. Such information can be obtained by using a dedicated pilot and exploiting the channel reciprocity in TDD [13]. Based on the given information, the secondary transmitter and receiver update the precoding and decoding matrices at each iteration to maximize the rate of the secondary link, respectively. When the secondary transceiver uses beamforming, i.e., the length of data vector is equal to one, the convergence of the iterative algorithm is also studied. For the initialization step, the secondary transmitter and the secondary receiver construct the precoding matrix, 𝑇 (0) , and the decoding matrix, 𝑊 (0) , respectively, like (3) by only using local CSI and transmit signals in each time slot. After receiving the side information from the secondary receiver, i.e., decoding matrix used at the previous iteration, 𝑊 (𝑖−1) , the secondary transmitter updates the precoding matrix, 𝑇 (𝑖) . The updated precoding matrix is obtained from the SVD of ∗ ⊥ projected effective channel matrix, 𝑊 (𝑖−1) 𝐻 𝑃 𝐺 ∗ . Similarly, the secondary receiver updates the decoding matrix based on the received side information. The iteration is terminated when it reaches a stable condition. The iterative algorithm would produce better precoding and decoding matrices as the number of iterations increases. A summary of the proposed iterative algorithm is given in Algorithm 2. Note that the equal power allocation is assumed until the iteration converges. After the
3574
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011
iteration converges, the water-filling power allocation is used to further maximize the achievable rate. The update order between the secondary transmitter and receiver can be reversed and changing the order does not affect the result. Algorithm 2 Iterative precoding matrix and decoding matrix design 1: Initialization at the secondary transmitter and receiver 2: 𝑖 = 0 ⊥ ∗ 𝑣1 ⋅ ⋅ ⋅ 𝑣 𝑟] 3: 𝐻𝑃 𝐺 ∗ = 𝑈 Σ𝑉 , 𝑉 = [𝑣 (𝑖) 4: 𝑇 = 𝑉 (𝑀𝑑 ) = [𝑣𝑣 1 ⋅ ⋅ ⋅ 𝑣 𝑀𝑑 ] ⊥∗ ∗ 𝑢1 ⋅ ⋅ ⋅ 𝑢 𝑟 ] 5: 𝑃 𝐹 𝐻 = 𝑈 Σ𝑉 , 𝑈 = [𝑢 (𝑖) 𝑢 1 ⋅ ⋅ ⋅ 𝑢 𝑀𝑑 ] 6: 𝑊 = 𝑈 (𝑀𝑑 ) = [𝑢 7: Iterative algorithm at the secondary transmitter and receiver 8: Ω = Ω0 (Ω0 > 𝜖) 9: while (Ω > 𝜖) do 10: 𝑖=𝑖+1 ∗ (𝑖) ⊥ 11: 𝐻 𝑡iter = 𝑊 (𝑖−1) 𝐻𝑃 𝐺 = 𝑈 Σ𝑉 ∗ , 𝑉 = ∗ [𝑣𝑣 1 ⋅ ⋅ ⋅ 𝑣 𝑟 ] 12: 𝑇 (𝑖) = 𝑉 (𝑀𝑑 ) = [𝑣𝑣 1 ⋅ ⋅ ⋅ 𝑣 𝑀𝑑 ] ∗ 𝑢1 ⋅ ⋅ ⋅ 𝑢 𝑟 ] 13: 𝐻 𝑟iter (𝑖) = 𝑃 𝐹⊥ 𝐻 𝑇 (𝑖) = 𝑈 Σ𝑉 ∗ , 𝑈 = [𝑢 𝑢1 ⋅ ⋅ ⋅ 𝑢 𝑀𝑑 ] 14: 𝑊 (𝑖) = 𝑈 (𝑀𝑑 ) = [𝑢 2 ∗ (𝑖−1) ∗ 𝑊 15: Ω = 𝑊 𝐻𝑇 (𝑖−1) − 𝑊 (𝑖) 𝐻𝑇 (𝑖) 𝐹 16: end while 17: 𝑇 iter = 𝑇 (𝑖) , 𝑊 iter = 𝑊 (𝑖) In the iterative algorithm, it is worth noting that not only the ZF receiver, but also the MMSE receiver can be used. When the precoder 𝑇 is given, MMSE receiver can be found as follows [20]: 𝑊 ∗𝐻𝑇 𝑆𝑇 ∗𝐻 ∗𝑊 . (9) max log2 𝐼 𝑀𝑟SU + 2 ∗ ∗ ∗ 𝑊 𝑊 𝑊 𝐹 𝑄 𝐹 𝑊 𝑊 {𝑊 } 𝜎𝑛 + Then, the maximum of (9) is obtained by applying the generalized with the matrix ( eigenvalue decomposition problem ) 𝑆 𝑇 ∗𝐻 ∗ , 𝜎𝑛2 𝐼 𝑀 SU + 𝐹 𝑄𝐹 ∗ . Since 𝑊 MMSE is pencil 𝐻𝑇 𝐻𝑇𝑆 𝑟 found by using the given 𝑇 , the iterative algorithm between 𝑇 and 𝑊 is needed to find the final solution. Theorem 1: When 𝑀𝑑 is equal to 1, the converged transmit beamforming vector of the iterative algorithm is the same as the beamforming vector obtained in the proposed joint design. Proof: Let the transmit beamforming vector and the receive combining vector be 𝑡 and 𝑤 , respectively. In the beamforming scenario, the updated receive combining vector of the 𝑖-th iteration at the secondary receiver, 𝑤 (𝑖) , which chooses ∗ the principle left singular vector of 𝑃 𝐹⊥ 𝐻 𝑡 (𝑖) , is equivalent ⊥ (𝑖) 𝑃 𝐹 𝐻𝑡 . to the maximal-ratio combiner, 𝑤 (𝑖) = 𝑃 𝐹⊥𝐻𝑡 (𝑖) / 𝑃 (𝑖) = In a similar manner, 𝑡 (𝑖) can be obtained by 𝑡 ⊥ ∗ (𝑖−1) 𝑃 ⊥ ∗ (𝑖−1) / 𝑃 𝐺∗ 𝐻 𝑤 𝑃 𝐺∗ 𝐻 𝑤 . Then, beamforming vector at the 𝑖-th iteration, 𝑡 (𝑖) , is written by replacing 𝑤 (𝑖−1) as follows: )(𝑖) ( ⊥ ∗ ⊥ ⊥ ∗ 𝑃 ⊥ 𝐻 𝑡 (𝑖−1) 𝑡 (0) 𝑃 ∗𝐻 𝑃 𝐹 𝐻 𝐺 𝐻 𝑃 𝐺∗ 𝐹 = ( , 𝑡 (𝑖) = )(𝑖) ⊥ ∗ ⊥ (𝑖−1) ⊥ 𝑃 𝐺∗ 𝐻 𝑃 𝐹 𝐻 𝑡 (0) 𝑃 𝑃 ∗ 𝐻 ∗𝑃 ⊥𝐻 𝑡 𝐹 𝐺
Fig. 2. Achievable rate comparison of the proposed joint design, iterative solution with various iterations, non-iterative solution (from Theorem 1), and the extension of P-SVD when beamforming transmission is assumed, i.e., 𝑠𝑠∗ ) = 1. Scenario 1: (𝑀𝑟SU , 𝑀𝑡SU , 𝑀𝑟PU𝑖 , 𝐾𝑟PU , 𝑀𝑡PU𝑖 , 𝐾𝑡PU ) = rank(𝑠𝑠 (4, 4, 1, 1, 1, 1).
where 𝑡 (0) is an initial vector. It is known that 𝑏 (𝑖+1) = 𝐴𝑏 (𝑖) / 𝐴𝑏 (𝑖) produces the principal eigenvector of 𝐴 as 𝑖 → ∞ unless the initial vector is orthogonal to the principal eigenvector in the power iteration method [21]. Therefore, the iterative algorithm converges to the principal eigenvector ⊥ ∗ ⊥ of 𝑃 𝐺 ∗ 𝐻 𝑃 𝐹 𝐻 , which, hereinafter, is called a non-iterative solution. The convergence speed depends on the ratio of two ⊥ ∗ ⊥ largest eigenvalues of 𝑃 𝐺 ∗𝐻 𝑃 𝐹 𝐻 . Because the projection matrix is the identity operator on the projected subspace and the square of the projection matrix is again the projection matrix, i.e., ⊥2 ⊥ = 𝑃𝐴 , we can rewrite the non-iterative solu𝑃𝐴 ⊥ ⊥ ⊥ ∗ ⊥ ⊥ tion as 𝑃 𝐺 ∗ 𝐻 ∗𝑃 𝐹⊥𝐻 = 𝑃 𝐺 ∗ 𝐻 𝑃 𝐹 𝑃 𝐹 𝐻 𝑃 𝐺 ∗ . Moreover, since 𝑃 𝐺∗ and 𝑃 𝐹 are Hermitian matrices, the projection matrices, 𝐼 − 𝑃 𝐺∗ and 𝐼 − 𝑃 𝐹 , are also Hermitian matrices. Therefore, the transmit beamforming vector of the iterative algorithm converges to the principal eigen⊥ ⊥ ⊥ ∗ ∗ ⊥ ⊥∗ ⊥ ∗ ⊥ ⊥ vector ( ∗ of 𝑃 𝐺)∗∗𝐻( 𝑃 𝐹∗𝑃 𝐹 𝐻 𝑃)𝐺 ∗ = 𝑃 𝐺∗ 𝐻 𝑃 𝐹 𝑃 𝐹 𝐻 𝑃 𝐺 ∗ = ⊥ ⊥ 𝑃 𝐹⊥ 𝐻 𝑃 𝐺 𝑃 𝐹⊥ 𝐻 𝑃 𝐺 ∗ ∗ , which is equal to the principal ∗
⊥ right singular vector of 𝐻 eff = 𝑃 𝐹⊥ 𝐻 𝑃 𝐺 ∗.
IV. N UMERICAL R ESULTS In this section, we compare the achievable rates of the proposed iterative algorithm and the joint design with the extension of P-SVD technique. Moreover, the convergence of the iterative algorithm is investigated. As mentioned in ) ( channel with Section III.B, an 𝑀(𝑟SU × 𝑀𝑡SU cognitive ) ( ) joint design results in an 𝑀𝑟SU − 𝑀𝑡PU × 𝑀𝑡SU − 𝑀𝑟PU pointto-point MIMO channel. To provide a reliable reference for performance comparison purposes, ( SU ) ( ) the achievable rate of 𝑀𝑟 − 𝑀𝑡PU × 𝑀𝑡SU − 𝑀𝑟PU MIMO channel is compared to the theoretical bound of the proposed algorithm. For simulations, we consider two scenarios, 1) a 4 × 4 secondary link and a single primary transmitter with a single antenna and a single primary receiver with a single antenna, and 2) a 8 × 8
LEE et al.: MIMO TRANSCEIVER DESIGNS FOR SPATIAL SENSING IN COGNITIVE RADIO NETWORKS
3575
TABLE II C OMPLEXITY C OMPARISONS OF THE P ROPOSED T ECHNIQUES . Techniques P-SVD
1
Joint Tx-Rx design
Iterative algorithm
Initialization Every iteration
1 1 1
Secondary Tx 1 matrix projection 1 matrix multiplication singular value decomposition 2 matrix projections 2 matrix multiplications singular value decomposition 1 matrix projection 1 matrix multiplication singular value decomposition 2 matrix multiplications singular value decomposition
Fig. 3. Achievable rate comparison of the proposed joint design, iterative solution with various iterations, and the extension of P-SVD. Scenario 1: (𝑀𝑟SU , 𝑀𝑡SU , 𝑀𝑟PU𝑖 , 𝐾𝑟PU , 𝑀𝑡PU𝑖 , 𝐾𝑡PU ) = (4, 4, 1, 1, 1, 1) with 𝑀𝑑 = 2 and scenario 2: (𝑀𝑟SU , 𝑀𝑡SU , 𝑀𝑟PU𝑖 , 𝐾𝑟PU , 𝑀𝑡PU𝑖 , 𝐾𝑡PU ) = (8, 8, 2, 2, 2, 2) with 𝑀𝑑 = 3.
secondary link and two primary transmitters each with two antennas and two primary receivers each with two antennas. Water-filling power allocation is used to obtain the simulation results, including theoretical bound, when 𝑀𝑑 is equal to or larger than two. Note that the power allocation is not needed when 𝑀𝑑 = 1. For a termination of the iterative algorithm, we set 𝜖 as 10−3 . Fig. 2 shows the average achievable rate of the proposed algorithms and P-SVD with scenario 1 when 𝑀𝑑 is equal to 1. The proposed joint design and the iterative algorithm show better performance than P-SVD at the expense of knowledge of global CSI and side information, respectively. We use 1, 2 and 10 iterations to show the convergence of the iterative algorithm and it is observed that as the number of iterations increases the rate of the iterative algorithm approaches to the rate of the joint design. Note that the iterative algorithm converges to the final solution rapidly with only a few iterations. In accordance with Theorem 1, the rate of the non-iterative solution, which is obtained from the principal eigenvector of ⊥ ∗ ⊥ 𝑃𝐺 ∗ 𝐻 𝑃 𝐹 𝐻 , is the same as the rate of the joint design. Moreover, it is observed that the rate of the joint design is equal to the rate of the theoretical bound, in this case a
1 1 1 1
Secondary Rx 1 matrix projection 1 matrix multiplication singular value decomposition 2 matrix projections 2 matrix multiplications singular value decomposition 1 matrix projection 1 matrix multiplication singular value decomposition 2 matrix multiplications singular value decomposition
3 × 3 MIMO channel with a single data stream1 . Note that the performance with the non-iterative MMSE receiver is left for future work 2 . In Fig. 3, we compare the average achievable rate of the proposed algorithms with the rate of P-SVD when 𝑀𝑑 are equal to 2 or 3. In scenario 1 and scenario 2, the secondary transmitter sends two data streams, 𝑀𝑑 = 2, and three data streams, 𝑀𝑑 = 3, to the secondary receiver, respectively. Similar with the beamforming case, the proposed joint design and the iterative algorithm outperforms the P-SVD technique. Moreover, the achievable rate of the iterative algorithm converges to the joint design with a few iterations, too. The achievable rates of the theoretical bounds of 3 × 3 and 4 × 4 MIMO channels for scenario 1 and scenario 2, respectively, are the same as the achievable rates of the joint design. Note that we explained that the performance of the P-SVD technique and joint design are the same when the maximum number of data streams and equal power allocation are used, i.e., 𝑀𝑑 = 3 in scenario 1 and 𝑀𝑑 = 4 in scenario 2, in Remark 2. The performance gap between the two techniques is also quite marginal even if water-filling power allocation is used, i.e., joint design marginally outperforms the P-SVD technique in the low SNR regime. Fig. 4 illustrates the average achievable rate of the proposed iterative algorithm as a function of the number of iterations when SNR is equal to 10 dB and scenario 2 is used for simulation. As the number of iteration increases, the rate of the iterative algorithm converges rapidly. Note that the case of zero iteration indicates the P-SVD technique; thus, the iterative algorithm starts from the P-SVD technique. Various numbers of data streams are used to show the effect of the number of data streams on the convergence speed and it is observed that the convergence of the proposed iterative algorithm is not 1 The authors of [1] also considered the zero-interference constraint, thus, the idea proposed in [1] is comparable with our proposed algorithms. In particular, the idea in [1] shows the same performance as our joint design technique with global CSI when (𝑀𝑟SU , 𝑀𝑡SU , 𝑀𝑟PU𝑖 , 𝐾𝑟PU , 𝑀𝑡PU𝑖 , 𝐾𝑡PU ) = (2, 2, 1, 1, 1, 1). The idea proposed in [1], however, is limited to a two antenna cognitive transceiver, global CSI scenario, and single data stream transmission. In this letter, to show the simulation results under various environments, we do not include the comparison results with [1]. Instead, we provide the achievable rate of MIMO channel as a reference curve. Note that the proposed joint design shows the same performance as the technique in [1] regardless of channel accuracy. 2 Since we studied the joint transceiver design, the precoding matrix with MMSE receiver is different with that with ZF receiver. Therefore, there is a constant gap between MMSE design and ZF design. This will be studied in future work also.
3576
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011
(Korea Communications Commission), Korea, under the R&D program supervised by the KCA (Korea Communications Agency) (KCA-2011-(09913-05003)). R EFERENCES
Fig. 4. Achievable rate vs. number of iterations with various number of data streams at SNR = 10 dB. Scenario 2: PU PU (𝑀𝑟SU , 𝑀𝑡SU , 𝑀𝑟 𝑖 , 𝐾𝑟PU , 𝑀𝑡 𝑖 , 𝐾𝑡PU ) = (8, 8, 2, 2, 2, 2).
affected by the number of data streams, i.e., 𝑀𝑑 . In Table II, we compare computational complexities of the proposed techniques, P-SVD with local CSI, joint transmitreceive design with global CSI, and iterative algorithm with local CSI and side information. Although the speed of the convergence of the iterative algorithm varies depending on the instantaneous channel information, as shown in Fig. 4, the iterative algorithm shows fast convergence to the final solution (within 3 iterations under Rayleigh fading channels). For this case, the complexity of the iterative algorithm is 1 matrix projection, 7 matrix multiplications, 4 SVDs, thus the difference between the complexities of P-SVD, joint Tx-Rx design, and the iterative algorithm is quite marginal. Note that the iterative algorithm does not require global CSI. V. C ONCLUSION We proposed three transceiver algorithms for spatial sensing in cognitive radio networks depending upon the level of channel state information (CSI) at the secondary transceiver, i) P-SVD with local CSI, ii) joint Tx-Rx algorithm with global CSI, and iii) iterative algorithm with local CSI and side information. When beamforming is considered, we showed that the iterative algorithm with local CSI converges to the joint Tx-Rx algorithm with global CSI. We also investigated the convergence of the iterative algorithm. In future work, we will consider limited feedback and a quantization error for the side information to study the performance degradation. ACKNOWLEDGEMENT The authors would like to thank Prof. Osvaldo Simeone for his valuable discussion. The work of C.-B. Chae was in part supported by the Ministry of Knowledge Economy under the “IT Consilience Creative Program” (NIPA-2010-C1515-10010001) and the Yonsei University Research Fund of 2011. The work of K. Lee and J. Kang was in part supported by the KCC
[1] L. Bixio, G. Oliveri, M. Ottonello, M. Raffetto, and C. S. Regazzoni, “Cognitive radios with multiple antennas exploiting spatial opportunities,” IEEE Trans. Signal Process., vol. 58, no. 8, pp. 4453–4459, Aug. 2010. [2] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [3] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, “Breaking spectrum gridlock with cognitive radios: an information theoretic perspective,” Proc. IEEE, vol. 97, no. 5, pp. 894–914, May 2008. [4] 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, no. 1, pp. 88–102, Feb. 2008. [5] R. Zhang, Y.-C. Liang, and S. Cui, “Dynamic resource allocation in cognitive radio networks: a convex optimization perspective,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 102–114, May 2010. [6] G. Scutari, D. P. Palomar, and S. Barbarossa, “Cognitive MIMO radio,” IEEE Signal Process. Mag., vol. 25, no. 6, pp. 46–59, Nov. 2008. [7] 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, no. 6, pp. 2323–2335, June 2009. [8] J. Zhou and J. Thompson, “Linear precoding for the downlink of multiple input single output coexisting wireless systems,” IET Commun., vol. 2, no. 6, pp. 742–752, July 2008. [9] M. Kang, B. Jung, D. Sung, and W. Choi, “A pre-whitening scheme in a MIMO-based spectrum-sharing environment,” IEEE Commun. Lett., vol. 12, no. 11, pp. 831–833, Nov. 2008. [10] S. Sridharan and S. Vishwanath, “On the capacity of a class of MIMO cognitive radios,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp. 103–117, Feb. 2008. [11] C.-B. Chae, S. Kim, and R. W. Heath, Jr., “Network coordinated beamforming for cell-boundary users: linear and non-linear approaches,” IEEE J. Sel. Topics Signal Process., vol. 3, no. 6, pp. 1094–1105, Dec. 2009. [12] X. Hong, Z. Chen, C.-X. Wang, S. Vorobyov, and J. Thompson, “Cognitive radio networks,” IEEE Veh. Technol. Mag., vol. 4, no. 4, pp. 76–84, Dec. 2009. [13] C.-B. Chae, D. Mazzarese, T. Inoue, and R. W. Heath, Jr., “Coordinated beamforming for the multiuser MIMO broadcast channel with limited feedforward,” IEEE Trans. Signal Process., vol. 56, no. 12, pp. 6044– 6056, Dec. 2008. [14] K. Gomadam, V. R. Cadambe, and S. A. Jafar, “Approaching the capacity of wireless networks through distributed interference alignment,” in Proc. IEEE Glob. Telecom. Conf., Dec. 2008, pp. 1–6. [15] D. A. Schmidt, C. Shi, R. A. Berry, M. L. Honig, and W. Utschick, “Distributed resource allocation schemes,” IEEE Signal Process. Mag., vol. 26, no. 5, pp. 53–63, Sep. 2009. [16] G. Strang, Linear Algebra and Its Applications, 4th edition. Brooks Cole, 2005. [17] R. Chen, R. W. Heath, Jr., and J. G. Andrews, “Transmit selection diversity for unitary precoded multiuser spatial multiplexing systems,” IEEE Trans. Signal Process., vol. 55, no. 3, pp. 1159–1171, Mar. 2007. [18] D. J. Love and R. W. Heath, Jr., “Limited feedback unitary precoding for spatial multiplexing systems,” IEEE Trans. Inf. Theory, vol. 51, no. 8, pp. 2967–2976, Aug. 2005. [19] R. Zhang and Y.-C. Liang, “Investigation on multiuser diversity in spectrum sharing based cognitive radio networks,” IEEE Commun. Lett., vol. 14, no. 2, pp. 133–135, Feb. 2010. [20] C.-B. Chae, I. Hwang, R. W. Heath, Jr., and V. Tarokh, “Interference aware-coordinated beamform system in a two-cell environment,” Technical report. Available: http://dash.harvard.edu/handle/1/3293263. [21] C.-B. Chae, D. Mazzarese, N. Jindal, and R. W. Heath, Jr., “Coordinated beamforming with limited feedback in the MIMO broadcast channel,” IEEE J. Sel. Areas Commun., vol. 26, no. 8, pp. 1505–1515, Oct. 2008.
