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Balanced Linear Precoding in Decode-and-Forward Based MIMO Relay Communications Jong Yeol Ryu, Student Member, IEEE, and Wan Choi, Member, IEEE Abstract—This paper proposes a linear precoding technique for multiple-input multiple-output (MIMO) decode-and-forward (DF) based relay communications. The proposed precoder is constructed by linearly combining two independently designed precoders that maximize data rates at relay and destination, respectively. Contrary to conventional precoding in DF relay communications, the proposed precoding balances direct and relay links and maximize the achievable rate of the overall system. We also propose a distributed precoder design method using limited information. The proposed algorithms are shown to significantly reduce computation complexity of precoder design. The numerical results show that the proposed precoders with full information and limited information achieve significantly higher data rates than conventional precoding neglecting a direct link and comparable data rate to the optimal precoding even when the coefficients for linear combining are restricted to real numbers. Index Terms—Linear precoding, multiple-input multipleoutput (MIMO), relay communications, iterative waterfilling
I. I NTRODUCTION
R
ECENTLY, relay communication has emerged as a promising technique for the next generation wireless networks because of its benefits. In wireless networks, relays are used for reliable communications, coverage extension and improvement of spectral efficiency with low cost. The traditional role of relays was compensation for signal degradation due to propagation loss, strong shadowing and multipath fading [1]–[6]. In modern wireless communication systems, however, relay communications aim at achieving high data rate as well as reliable communications. To increase data rate, MIMO techniques have been exploited in the relay communications. Relays are able to forward multiple streams through multiple transmit antennas to destinations simultaneously and dramatically increase data rate. The ways of utilizing the benefits of MIMO in relay communications have been intensively studied [9]–[17]. Although information theoretic capacity of a MIMO relay channel remains unknown, several capacity bounds were derived [9]–[12]. In addition, the precoding techniques in conventional MIMO communications have been modified and adapted to the relay networks to attain the identified information theoretic capacity bounds [13]–[17].
Manuscript received October 2, 2010; revised February 6, 2011 and March 31, 2011; accepted April 5, 2011. The associate editor coordinating the review of this paper and approving it for publication was D. Huang. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2009-0076305). Parts of this paper were presented as an invited paper at the IEEE Asilomar Conference on Signals, Systems and Computers, Asilomar, CA, Nov. 2010. J. Y. Ryu and W. Choi are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305701, Korea (e-mail:
[email protected],
[email protected]). Digital Object Identifier 10.1109/TWC.2011.050511.101733
In relay communications, relaying protocols and duplexing methods are major factors that determine capacity and outage performance. The typical relaying protocols are amplify-andforward (AF) relaying and decode-and-forward (DF) relaying. In contrast to noise amplification at AF relay, DF relay forwards noiseless data to destination, thus it usually provides better performance in terms of the data rate if a source to relay (S-R) link is reliable. Relays transmit and receive data at the same time and frequency in a full duplex mode whereas relays transmit and receive data using orthogonal time or frequency slots in the half duplex mode. The full duplex relays are considered practically infeasible due to the difficulty of decoupling transmission and reception on the same antenna. Thus, the half duplex relays are regarded as practical ones from an implementation perspective. In this paper, we focus on the repetition based half duplex DF relays because of its superiority and practicality. Precoders for MIMO relay systems are differently designed from those of conventional MIMO systems [18]–[20]. Since the received signals via the source-to-destination (SD) link and the relay-to-destination (R-D) link are combined at destination to strengthen the received signal in repetition based rely communications, the direct link makes the problem of precoding matrix design complicated and difficult since beamforming needs to simultaneously take into account both the direct and relay links. Prior studies mainly considered relay networks without a direct link to simplify the problem. Among those studies, [11] showed that a matched filter type precoding at relay asymptotically achieves the capacity upper bound for single stream transmission. In [13], a QR based precoding was proposed and shown to achieve the maximum multiplexing gain. The key ideas therein are QR decomposition for the backward and forward channels at relay and successive interference cancelation (SIC) at destination. In [14], the authors proved the optimal relaying matrix for AF relaying is given by a singular value decomposition (SVD) for the backward and the forward channels and waterfilling based power allocation. For DF relaying without a direct link, a joint optimization problem to design precoding matrices at source and relay can be split into two independent optimization problems and their solutions are easily obtained by SVD and waterfilling based power allocation. However, a precoding matrix design for a DF relaying system with a direct link becomes complicated. Although the solutions can be obtained by numerical methods since the joint optimization problem is still convex, the numerical solution requires huge computational complexity and moreover it is hard to understand the effects of precoding matrices. Unfortunately, a closed form or intuitive solution to understand the effects of precoding matrices at source and relay has not been available yet.
c 2011 IEEE 1536-1276/11$25.00 ⃝
RYU and CHOI: BALANCED LINEAR PRECODING IN DECODE-AND-FORWARD BASED MIMO RELAY COMMUNICATIONS
In this paper, we solve the problem of a precoder design at source and relay for MIMO DF relaying systems with a direct link. Since the achievable rate of a DF relaying system is bounded by the minimum of the rates at relay and destination, we balance the achievable data rates at source and destination, i.e., the S-R link and the S-D link, and maximize the overall achievable data rate. Consequently, an explicit form of the proposed precoder is obtained by a linear combination of two independently designed precoders that maximize data rates at relay and destination, respectively. For a MISO case where relay and destination have a single antenna but source has multiple antennas, the proposed linearly combined precoder is proved to be optimal and the optimal coefficients are found. When source, relay, and destination have multiple antennas, the proposed linear combination of two independently designed precoding matrices is slightly suboptimal. Our numerical results show that the proposed precoding significantly outperforms the conventional precoding and suffers only a marginal loss compared to the optimal precoding even when the coefficients of a linear combination are restricted to real numbers in MIMO cases. Our complexity analysis shows that the proposed precoders significantly reduces the computational complexity of precoder design compared to conventional algorithms. The rest of this paper is organized as follows. Section II explains our system model and formulates the achievable rate for our MIMO DF relaying system. The proposed algorithms of precoder design according to available information are presented in Section III. The computational complexity of the proposed precoding is analyzed in Section IV. Section V provides numerical results for performance evaluation of the proposed schemes. Finally, conclusions are drawn in Section VI. Notation: The lower and upper case bold face letters represent a vector and a matrix, respectively. (.)𝐻 denotes the complex conjugate transpose and I is an identity matrix. ∥x∥ denotes the Euclidean norm of a complex vector x and ∣𝑥∣ denotes the norm of a complex number 𝑥. For a square matrix, Tr(X) and ∣X∣ denote (the trace )−1and𝐻 the determinant of X, 𝐻 X represents orthogonal respectively. ΠX ≜ X X X projection on to the column space of X and Π⊥ X ≜ I − ΠX denotes orthogonal projection on to the orthogonal complement of the column space of X. X ∼ 𝒞𝒩 (A, B) denotes the elements of X follow complex Gaussian distribution with mean A and covariance B. II. S YSTEM M ODEL We consider a half-duplex DF relaying system consisting of source, relay and destination nodes as shown in Fig. 1. Each node is assumed to have multiple antennas and the number of antennas at source, relay and destination are 𝑀 , 𝐾 and 𝑁 , respectively. In the first transmission phase, the source broadcasts a data vector x multiplied by a linear precoding matrix W𝑆 over 𝑀 transmit antennas. The data vector x = [𝑥1 , . . . , 𝑥𝑀 ]𝑇 is a 𝑀 × 1 complex Gaussian distributed vector with a covariance matrix 𝔼[xx𝐻 ] = I. The source precoding matrix is a 𝑀 × 𝑀 complex valued matrix and should satisfy a power constraint
Fig. 1.
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System model.
at source such that Tr(W𝑆 W𝑆𝐻 ) ≤ 𝑃𝑆 . The received signal at relay in the first transmission phase is given by y𝑆𝑅 = H𝑆𝑅 W𝑆 x + n𝑆𝑅 ,
(1)
where the MIMO channel between source and relay, H𝑆𝑅 , is modeled by a 𝐾 × 𝑀 circular symmetric complex Gaussian matrix whose entities are independent and identically distributed (i.i.d.) complex Gaussian random variables with 2 . The vector n𝑆𝑅 is a 𝐾 × 1 zero mean and variance 𝜎𝑆𝑅 additive white Gaussian noise (AWGN) vector at relay, whose entries follow i.i.d. complex Gaussian distributions with zero mean and variance 𝑁0 . Then, the achievable rate at relay is given by (2) ℛ𝑅 = log2 I + 𝜌0 H𝑆𝑅 W𝑆 W𝑆𝐻 H𝐻 𝑆𝑅 , where 𝜌0 = 𝑁10 . The received signal at destination in the first transmission phase is also given by y𝑆𝐷 = H𝑆𝐷 W𝑆 x + n𝑆𝐷 ,
(3)
where H𝑆𝐷 ∈ ℂ𝑁 ×𝑀 denotes the MIMO channel be2 I) and tween source and destination, H𝑆𝐷 ∼ 𝒞𝒩 (0, 𝜎𝑆𝐷 𝑁 ×1 n𝑆𝐷 ∈ ℂ denotes the AWGN vector at destination, n𝑆𝐷 ∼ 𝒞𝒩 (0, 𝑁0 I). The relay decodes the data x from (1) and forwards it to destination by multiplying a precoding matrix W𝑅 in the second transmission phase. The precoding matrix at relay, W𝑅 ∈ ℂ𝐾×𝐾 , should satisfy a power constraint given by 𝐻 Tr(W𝑅 W𝑅 ) ≤ 𝑃𝑅 . The received signal at destination in the second transmission phase is given by y𝑅𝐷 = H𝑅𝐷 W𝑅 x + n𝑅𝐷 ,
(4)
where H𝑅𝐷 ∈ ℂ𝑁 ×𝐾 is the MIMO channel from relay 2 to destination ∼ 𝒞𝒩 (0, 𝜎𝑅𝐷 I) and n𝑅𝐷 ∈ ℂ𝑁 ×1 is an AWGN vector at destination in the second transmission phase ∼ 𝒞𝒩 (0, 𝑁0 I). Since reliable communication between source and relay should be guaranteed to earn the benefits of relay 2 2 2 2 communications, we assume 𝜎𝑆𝑅 ≥ 𝜎𝑆𝐷 and 𝜎𝑆𝑅 ≥ 𝜎𝑅𝐷 . At destination, the received signals in the first and the second transmission phases are combined to improve the received signal-to-noise ratio (SNR) because repetition based relaying is employed. From (1) and (4), the achievable rate at destination for the composite link after combining is given by 𝐻 𝐻 +𝜌 H W W H ℛ𝐷 = log2 I+𝜌0 H𝑆𝐷W𝑆W𝑆𝐻H𝐻 0 𝑅𝐷 𝑅 𝑆𝐷 𝑅 𝑅𝐷 . (5)
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Since the achievable rate for the whole system is bounded by the minimum of (2) and (5), the achievable rate for a MIMO DF relaying system is given by { 1 min log2 I+𝜌0 H𝑆𝑅W𝑆W𝑆𝐻H𝐻 𝑆𝑅 , 2 } 𝐻 𝐻 log2 I+𝜌0H𝑆𝐷W𝑆W𝑆𝐻H𝐻 +𝜌 H W W H 0 𝑅𝐷 𝑅 𝑆𝐷 𝑅 𝑅𝐷 , (6)
ℛ𝐷𝐹 =
where the pre-log factor, 12 , results from a transmission duty cycle loss in half duplex relaying systems.
III. D ESIGN OF BALANCED P RECODING M ATRICES In order to maximize the achievable rate, the optimal precoding matrices at source and relay need to be determined by solving the following joint optimization problem: maximize W𝑆 , W𝑅
s.t.
ℛ𝐷𝐹
(7)
Tr(W𝑆 W𝑆𝐻 ) ≤ 𝑃𝑆 , 𝐻 Tr(W𝑅 W𝑅 ) ≤ 𝑃𝑅
where ℛ𝐷𝐹 is given in (6) and correspondingly, the objective function is concave since the minimum of two concave functions is also concave. The concave objective function along with convex inequalities make the problem in (7) a convex optimization problem whose solution can be obtained by various numerical algorithms [24]. However, although the problem is convex, a numerical joint optimization with respect to W𝑆 and W𝑅 requires high computational complexity for a large number of iterations. It is also difficult to insightfully understand the effects of precoding matrices from the numerical solutions. This reasoning motivates us to design linear precoding matrices for achievable rate maximization. Since the overall achievable rate is determined by the bottleneck of the relay and composite links (i.e., (2) and (5)), we design precoding matrices that balance the achievable rates of the relay and composite links in order to maximize the achievable rate of our DF relaying system. The precoding matrices are obtained by a linear combination of the optimal precoding matrices for the relay and the composite links. Specifically, in the first step, we independently design the optimal source precoding matrices W𝑆,1 and W𝑆,2 that maximize (2) and (5), respectively. Then, we linearly combine W𝑆,1 and W𝑆,2 to balance the rates in (2) and (5). The optimal weighting coefficients are obtained by simple one-dimensional search. For the obtained precoder W𝑆∗ , the optimal relay precoder ∗ W𝑅 is easily obtained based on SVD and the waterfilling algorithm.
A. A Special Case: MISO (𝑁 = 𝐾 = 1) We first consider the special MISO case where both relay and destination have a single antenna but the source has 𝑀 transmit antennas. For the MISO case, the achievable rate of
the system in (6) is given by ( ) { 1 2 1+𝜌 min log = ∣h w ∣ , ℛ𝑀𝐼𝑆𝑂 𝑆 𝑆𝑅 𝑆 𝐷𝐹 2 2
ℛ𝑅 ( )} 2 2 , log2 1+𝜌𝑆 ∣h𝑆𝐷w𝑆 ∣ +𝜌𝑅 ∣ℎ𝑅𝐷 ∣
ℛ𝐷
𝑃𝑆 𝑁0
(8)
𝑃𝑅 𝑁0 .
where 𝜌𝑆 = and 𝜌𝑅 = Note that the parameter to be optimized is a precoding vector at source w𝑆 only since relay and destination have a single antenna, respectively. For the MISO case, ℛ𝑅 for the relay link and ℛ𝐷 for the composite link are maximized, respectively, by adopting maximum-ratio transmission (MRT) beamformers for h𝑆𝑅 and h𝑆𝐷 , respectively, such that h𝐻 𝑀×1 𝑆𝑅 , w𝑆,1 =
h𝐻 ∈ ℂ 𝑆𝑅
h𝐻 𝑀×1 𝑆𝐷 w𝑆,2 = . (9)
h𝐻 ∈ ℂ 𝑆𝐷
The following proposition proves that the optimal precoding vector at source is obtained by a linear combination of w𝑆,1 and w𝑆,2 for a MISO case. Proposition 1: Suppose that h𝑆𝑅 and h𝑆𝐷 are linearly independent complex Gaussian channel vectors and w𝑆∗ is the optimal precoding vector at source which maximizes the achievable rate ℛ𝑀𝐼𝑆𝑂 . Then, there exist complex numbers 𝐷𝐹 𝛼 and 𝛽 such that w𝑆∗ = 𝛼w𝑆,1 + 𝛽w𝑆,2 , where w𝑆,1 and w𝑆,2
(10)
2 are given in (9), and w𝑆∗ = 1.
Proof: See Appendix A. Using this proposition, design of the optimal precoding vectors can be completed by just determining the complex values of 𝛼 and 𝛽. When the channel condition between source and relay, h𝑆𝑅 , is relatively worse than h𝑆𝐷 and ℎ𝑅𝐷 , the relay link (or ℛ𝑅 ) becomes a bottleneck of the overall system, and a larger value of 𝛼 than 𝛽 improves the capacity bottleneck by balancing the relay and composite links. If the composite link is better than the relay link (i.e., h𝑆𝐷 and ℎ𝑅𝐷 are better than h𝑆𝑅 ), a larger value of 𝛽 than 𝛼 improves the overall achievable rate of the system. Therefore, the optimal coefficients minimize the difference of achievable rates at relay, ℛ𝑅 , and destination, ℛ𝐷 . This simple observation motivates an iterative algorithm of determining the optimal complex values of 𝛼 and 𝛽, from the proof of Proposition 1 in Appendix A. The details of algorithm are described in Algorithm 1. Note that although coefficients are complex valued numbers, we can obtain them by simple one-dimensional search over [0, 1]. Once either 𝛼 or 𝛽 is determined by a real valued number, the magnitude of the other coefficient is obtained by the constraint ∣𝜖1 ∣2 +∣𝜖2 ∣2 = 1 in Algorithm 1 and the phase of the coefficient is determined by (A.11). B. A General Case: MIMO In this subsection, we investigates the design of the precoders that balance the rates at the relay and composite links
RYU and CHOI: BALANCED LINEAR PRECODING IN DECODE-AND-FORWARD BASED MIMO RELAY COMMUNICATIONS
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Algorithm 1 Precoding for a MISO case
precoding matrix at relay is obtained by (0) (0) (0) 𝐻 1) Initially set 𝑖 = 0, 𝜖1 = 1, 𝜖2 = 0 and w𝑆 = . W𝑅,1= arg max log2 I + 𝜌0 H𝑆𝐷W𝑆,1W𝑆,1 H𝐻 𝑆𝐷 𝐻 )≤𝑃 (𝑖) W𝑅 :Tr(W𝑅 W𝑅 𝑅 2) Define 𝒟𝑀𝐼𝑆𝑂 as a rate difference function at the 𝑖-th 𝐻 𝐻 iteration: (13) + 𝜌0 H𝑅𝐷W𝑅W𝑅 H𝑅𝐷 (𝑖) (𝑖) (𝑖) 1 1) ( 𝐻 − − 𝒟𝑀𝐼𝑆𝑂 = ℛ𝑅,𝑀𝐼𝑆𝑂 − ℛ𝐷,𝑀𝐼𝑆𝑂 2 = arg max log2 I+𝜌0 K1 2 H𝑅𝐷W𝑅W𝑅𝐻H𝐻 K 𝑅𝐷 1 h𝐻 𝑆𝑅 ∥h𝐻 𝑆𝑅 ∥
𝐻 W𝑅 :Tr(W𝑅 W𝑅 )≤𝑃𝑅
where
+ log2 ∣K1 ∣
( ) (𝑖) (𝑖) ℛ𝑅,𝑀𝐼𝑆𝑂 = log2 1 + 𝜌𝑆 ∣h𝑆𝑅 w𝑆 ∣2 ( ) (𝑖) (𝑖) ℛ𝐷,𝑀𝐼𝑆𝑂 = log2 1 + 𝜌𝑆 ∣h𝑆𝐷 w𝑆 ∣2 + 𝜌𝑅 ∣ℎ𝑅𝐷 ∣2 . (𝑖)
(𝑖−1)
3) While 𝒟𝑀𝐼𝑆𝑂 < 𝒟𝑀𝐼𝑆𝑂 ∙ ∙ ∙
(𝑖)
(𝑖−1)
𝐻 H𝐻 where K1 = I + 𝜌0 H𝑆𝐷 W𝑆,1 W𝑆,1 𝑆𝐷 . Since the second term log2 ∣K1 ∣ is not a function of W𝑅 for given W𝑆,1 , the optimal precoding matrix at relay is obtained by 1 ( − 12 )𝐻 (𝑖−1) 𝐻 𝐻 2 and 𝜖∗1 = 𝜖1 , W𝑅,1= arg max log2I+𝜌0 K− H W W H 𝑅𝐷 𝑅 𝑅 𝑅𝐷 K1 1
Stop the loop if 𝒟𝑀𝐼𝑆𝑂 ≥ 𝒟𝑀𝐼𝑆𝑂 (𝑖−1) 𝜖∗2 = 𝜖2 (𝑖+1) (𝑖) 𝑖 = 𝑖 + 1, 𝜖1 = 𝜖1 − 𝛿 (𝑖) Update the precoding vector w𝑆 as (𝑖) w𝑆
=
𝐻 (𝑖) h𝑆𝑅
𝜖1
h𝐻 𝑆𝑅
+
𝐻 )≤𝑃 W𝑅 :Tr(W𝑅 W𝑅 𝑅 1
2 = V𝑒𝑓 𝑓,1 Σ𝑒𝑓 𝑓,1 ,
−1
𝑺𝑹
. h𝑆𝐷 Π⊥ h𝑯 𝑺𝑹
𝐻 h𝐻 ∗ h𝑆𝐷 𝑆𝑅 w𝑆∗ = 𝛼∗ + 𝛽
h𝐻
h𝐻 , 𝑆𝑅 𝑆𝐷
{W𝑆,2 , W𝑅,2 }=
where the optimal complex valued 𝛼∗ and 𝛽 ∗ are given, respectively, by
𝐻
Π 𝑯 h
h 𝑆𝐷 ∗ ∗ ∗ h𝑺𝑹 𝑆𝐷 ∗ ∗
𝛼 = 𝜖1 − 𝜖2 ⊥ , 𝛽 = 𝜖 2
Π⊥𝑯 h𝑆𝐷 . Πh𝑯 h𝑆𝐷 h 𝑺𝑹
for general MIMO cases. In general MIMO cases, source, relay, and destination have multiple antennas and hence the optimal precoding matrices at source and relay need to be determined. The achievable rate of a MIMO DF relaying system is given in (6) and the optimal precoding matrix at source that maximizes ℛ𝑅 is obtained by W𝑆,1 = arg max log2 I+𝜌0 H𝑆𝑅 W𝑆W𝑆𝐻 H𝐻 𝑆𝑅 𝐻 )≤𝑃 W𝑆 :Tr(W𝑆 W𝑆 𝑆 1
2 = V𝑆𝑅 Σ𝑆𝑅 ,
(15) 1
𝐻 2 where H𝑒𝑓 𝑓,1 = K1 2 H𝑅𝐷 = U𝑒𝑓 𝑓,1 Λ𝑒𝑓 𝑓,1 V𝑒𝑓 𝑓,1 and Σ𝑒𝑓 𝑓,1 = diag (𝑃𝑒𝑓 𝑓,1,1 , ⋅ ⋅ ⋅ , 𝑃𝑒𝑓 𝑓,1,𝐾∑ ) which is obtained by waterfilling algorithm with constraint 𝑃𝑒𝑓 𝑓,1,𝑖 = 𝑃𝑅 . On the other hand, the optimal precoding matrices at source and relay that maximize ℛ𝐷 are obtained by solving the following joint optimization problem:
Π⊥ h𝑆𝐷 h𝑯
(𝑖) 𝜖2
√ 𝐻 (𝑖) (𝑖) where 𝜖2 = 1 − 𝜖1 𝑒𝑗⋅∠(h𝑆𝐷 ⋅h𝑆𝑅 ) 4) The precoding vector w𝑆∗ is determined by
𝑺𝑹
(14)
arg max
𝐻 )≤𝑃 , Tr(W W𝐻 )≤𝑃 Tr(W𝑆 W𝑆 𝑆 𝑅 𝑅 𝑅
𝐻 𝐻 log2 I+𝜌0 H𝑆𝐷 W𝑆 W𝑆𝐻 H𝐻 +𝜌 H W W H 0 𝑅𝐷 𝑅 𝑆𝐷 𝑅 𝑅𝐷 . (16) This joint maximization problem is similar to the capacity maximization for a two-user multiple access channel (MAC). If we regard H𝑆𝐷 and H𝑅𝐷 as channels for user 1 and user 2 in MAC, the sum capacity of a Gaussian MAC is known to be achieved by the iterative waterfilling (IWF) algorithm [21] and the optimal precoding matrices W𝑆,2 and W𝑅,2 are obtained from the IWF. The objective function at the 𝑖-th iteration is given by ( )𝐻 ( )𝐻 (𝑖−1) (𝑖−1) (𝑖) (𝑖) 𝐻 W𝑆 W log2 I+𝜌0H𝑆𝐷W𝑆 H𝐻 +𝜌 H W 0 𝑅𝐷 𝑅 𝑆𝐷 𝑅 H𝑅𝐷 (( ( )− 12 ( )𝐻 )−1)𝐻 (𝑖) (𝑖) (𝑖) (𝑖) 2 𝐻 =log2 I+𝜌0 K2 H𝑅𝐷W𝑅 W𝑅 H𝑅𝐷 K2 (𝑖) + log2 K2 , (17)
(11) 1 2
𝐻 where H𝑆𝑅 = U𝑆𝑅 Λ𝑆𝑅 V𝑆𝑅 from the singular value decomposition (SVD). ] Σ𝑆𝑅 is given [ The power allocation matrix by Σ𝑆𝑅 = 𝑃𝑆𝑅,1 , 𝑃𝑆𝑅,2 , ⋅ ⋅ ⋅ , 𝑃𝑆𝑅,𝑀 . The power is allocated by the waterfilling algorithm [22] such as ( )+ 1 𝑁0 − (12) 𝑃𝑆𝑅,𝑖 = 𝜈 𝜆𝑖
where 𝜆𝑖 denotes ∑ the 𝑖-th eigenvalue of+ H𝑆𝑅 and 𝜈 is determined by 𝑃𝑆𝑅,𝑖 = 𝑃𝑆 and (𝑥) := max{𝑥, 0}. For the given source precoding matrix W𝑆,1 , the optimal
(𝑖)
(𝑖−1)
where K2 = I + 𝜌0 H𝑆𝐷 W𝑆 (𝑖−1)
( )𝐻 (𝑖−1) W𝑆 H𝐻 𝑆𝐷 . For the
given W𝑆 , the optimal precoding matrix at relay in the (𝑖) 𝑖-th iteration W𝑅 is obtained by ( ) 12 (𝑖) (𝑖) (𝑖) W𝑅 = V𝑒𝑓 𝑓,2 Σ𝑒𝑓 𝑓,2 , ( )12 ( )𝐻 ( )−12 (𝑖) (𝑖) (𝑖) (𝑖) (𝑖) H𝑅𝐷 = U𝑒𝑓𝑓,2 Λ𝑒𝑓𝑓,2 V𝑒𝑓𝑓,2 where H𝑒𝑓𝑓,2 = K2 (𝑖)
(𝑖)
and Σ𝑒𝑓 𝑓,2 = diag(𝑃𝑒𝑓 𝑓,2,1 , ⋅ ⋅ ⋅ , 𝑃𝑒𝑓 𝑓,2,𝐾 ). Once W𝑅 is (𝑖) determined, the objective function in terms of W𝑆 can be
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represented as ( )𝐻 ( )𝐻 (𝑖) (𝑖) (𝑖) (𝑖) 𝐻 log2 I+𝜌0H𝑆𝐷W𝑆 W𝑆 H𝐻 W +𝜌 H W H 0 𝑅𝐷 𝑆𝐷 𝑅𝐷 𝑅 𝑅 ( ) ( ( )−1 ( )𝐻 )−1 𝐻 (𝑖) 2 (𝑖) (𝑖) (𝑖) 2 = log2 I+𝜌0 K3 K3 H𝑆𝐷W𝑆 W𝑆 H𝐻 𝑆𝐷 (𝑖) (18) + log2 K3 , ( )𝐻 (𝑖) (𝑖) (𝑖) (𝑖) where K3 = I+𝜌0 H𝑅𝐷W𝑅 W𝑅 H𝐻 𝑅𝐷 . For given W𝑅 , (𝑖)
the optimal source precoding matrix, W𝑆 is obtained by ( ) 12 (𝑖) (𝑖) (𝑖) W𝑆 = V𝑒𝑓 𝑓,3 Σ𝑒𝑓 𝑓,3 , ( ) 12 ( )𝐻 ( )−12 (𝑖) (𝑖) (𝑖) (𝑖) (𝑖) V𝑒𝑓𝑓,3 H𝑆𝐷 = U𝑒𝑓𝑓,3 Λ𝑒𝑓𝑓,3 where H𝑒𝑓𝑓,3 = K3 (𝑖) Σ𝑒𝑓 𝑓,3
and = diag(𝑃𝑒𝑓 𝑓,3,1 , ⋅ ⋅ ⋅ , 𝑃𝑒𝑓 𝑓,3,𝑀 ). The waterfilling based power allocation algorithm is applied to the effec(𝑖) (𝑖) tive channels Λ𝑒𝑓 𝑓,2 and Λ𝑒𝑓 𝑓,3 with the power constraints (𝑖) (𝑖) (𝑖) (𝑖) Tr(W𝑅 (W𝑅 )𝐻 ) ≤ 𝑃𝑅 and Tr(W𝑆 (W𝑆 )𝐻 ) ≤ 𝑃𝑆 , (𝑖) respectively. After W𝑆 is determined, the IWF algorithm (𝑖) (𝑖+1) returns to (17) with W𝑆 to design W𝑅 and the iterations continue until the solutions converge. The convergence and optimality of the IWF algorithm have already been proved in [21]. Once we obtain {W𝑆,1, W𝑅,1 } and {W𝑆,2, W𝑅,2 } for ℛ𝑅 maximization and ℛ𝐷 maximization, respectively, the proposed precoding matrix at source is designed by a linear combination of two optimal precoding matrices W𝑆,1 and W𝑆,2 such as W𝑆∗ = 𝛼∗ W𝑆,1 + 𝛽 ∗ W𝑆,2 , ∗
(19)
∗
where 𝛼 and 𝛽 are chosen to satisfy the power constraint Tr(W𝑆∗ W𝑆∗,𝐻 ) ≤ 𝑃𝑆 . For the obtained W𝑆∗ , the precoding matrix at relay is computed as ∗ W𝑅 = arg max log2 I + 𝜌0 H𝑆𝐷W𝑆∗W𝑆∗,𝐻H𝐻 𝑆𝐷 𝐻 )≤𝑃 W𝑅 :Tr(W𝑅 W𝑅 𝑅
𝐻 𝐻 +𝜌0 H𝑅𝐷W𝑅W𝑅 H𝑅𝐷 .
(20)
The coefficients 𝛼∗ and 𝛽 ∗ can be matrices, vectors, or complex valued numbers but we restrict the coefficients as nonnegative real valued numbers for simplicity in our proposed algorithm. Since the coefficients are confined to real valued numbers, there is a rate loss due to its suboptimality. However, the computation complexity is significantly reduced by onedimensional search and it is shown in Section V that the achievable rate loss caused by the restricted search dimension is marginal. The details of the proposed algorithm is depicted in Algorithm 2. Similarly to the precoding vector design in the MISO case, the coefficients are determined to minimize the difference of achievable rates at relay, ℛ𝑅 , and destination, ℛ𝐷 , by balancing those rates. It should be noted that since the range of coefficient 𝜖 is restricted as 0 ≤ 𝜖 ≤ 1 in our algorithm, the coefficient can be found with a small number of iterations for a proper step size (i.e. 𝛿 = 0.1). Although this restriction makes the proposed algorithm suboptimal, it is shown in numerical results that the proposed algorithm achieves near optimal performance in terms of achievable rate.
Algorithm 2 Precoding for the MIMO case (0)
1) Initially set 𝑖 = 0, 𝜖(0) = 0 and W𝑆 = W𝑆,2 . (𝑖) 2) Define 𝒟𝑀𝐼𝑀𝑂 as a rate difference function at 𝑖-th iteration: (𝑖) (𝑖) (𝑖) 𝒟𝑀𝐼𝑀𝑂 = ℛ𝑅,𝑀𝐼𝑀𝑂 − ℛ𝐷,𝑀𝐼𝑀𝑂 where
(𝑖) (𝑖) ( (𝑖) )𝐻 𝐻 ℛ𝑅,𝑀𝐼𝑀𝑂 = log2 I + 𝜌0 H𝑆𝑅 W𝑆 W𝑆 H𝑆𝑅 , (𝑖) (𝑖) ( (𝑖) )𝐻 𝐻 H𝑆𝐷 ℛ𝐷,𝑀𝐼𝑀𝑂 = log2 I + 𝜌0 H𝑆𝐷 W𝑆 W𝑆 ( ) (𝑖) (𝑖) 𝐻 𝐻 + 𝜌0 H𝑅𝐷 W𝑅 W𝑅 H𝑅𝐷 . (𝑖)
(𝑖−1)
3) While 𝒟𝑀𝐼𝑀𝑂 < 𝒟𝑀𝐼𝑀𝑂 ∙ ∙ ∙ ∙
∙
(𝑖)
(𝑖−1)
Stop the loop if 𝒟𝑀𝐼𝑀𝑂 ≥ 𝒟𝑀𝐼𝑀𝑂 and 𝜖∗ = 𝜖(𝑖−1) Stop the loop if 𝜖(𝑖) = 1 and 𝜖∗ = 1 𝑖 = 𝑖 + 1, 𝜖(𝑖+1) = 𝜖(𝑖) + 𝛿 (𝑖) Update the precoding matrix at source W𝑆 as ) ( 𝜖(𝑖) W𝑆,1 + 1 − 𝜖(𝑖) W𝑆,2 (𝑖) W𝑆 = 𝑃𝑆 ⋅ (𝑖) (𝑖) Tr(W𝑆 (W𝑆 )𝐻 ) (𝑖)
(𝑖)
For given W𝑆 , the precoding matrix at relay W𝑅 is obtained by (𝑖) (𝑖)( (𝑖))𝐻 W𝑅 = arg max log2 I+𝜌0H𝑆𝐷W𝑆 W𝑆 H𝐻 𝑆𝐷 𝐻 )≤𝑃 W𝑅 :Tr(W𝑅 W𝑅 𝑅
(𝑖) ( (𝑖) )𝐻 𝐻 + 𝜌0 H𝑅𝐷 W𝑅 W𝑅 H𝑅𝐷 ( ) 12 (𝑖) (𝑖) , = V𝑒𝑓 𝑓 Σ𝑒𝑓 𝑓
where
)−12 ( (𝑖) (𝑖) ( (𝑖) )𝐻 𝐻 H𝑒𝑓 𝑓 = I+H𝑆𝐷 W𝑆 W𝑆 H𝑆𝐷 H𝑅𝐷 1( ( ) ) (𝑖) (𝑖) (𝑖) 𝐻 = U𝑒𝑓 𝑓 Λ𝑒𝑓 𝑓 2 V𝑒𝑓 𝑓 .
4) The precoding matrix at source W𝑆∗ is determined by W𝑆∗ = 𝛼∗ W𝑆,1 + 𝛽 ∗ W𝑆,2 , where 𝛼∗ and 𝛽 ∗ are given, respectively, by 𝛼∗ = 𝛽∗ =
𝑃𝑆 ⋅ 𝜖∗
𝐻
,
𝐻
,
Tr(W𝑆∗ (W𝑆∗ ) ) 𝑃𝑆 ⋅ (1 − 𝜖∗ ) Tr(W𝑆∗ (W𝑆∗ ) )
∗ is obtained by and the precoding matrix at relay W𝑅 𝐻 ∗ = arg max log2 I+𝜌0 H𝑆𝐷W𝑆∗ (W𝑆∗ ) H𝐻 W𝑅 𝑆𝐷 𝐻 )≤𝑃 W𝑅 :Tr(W𝑅 W𝑅 𝑅
𝐻 𝐻 + 𝜌0 H𝑅𝐷 W𝑅 W𝑅 H𝑅𝐷 .
C. Precoding Matrix Design with Limited Information In the previous subsections, we have proposed precoding matrix design based on full CSI at source. Therefore, source requires full CSI including CSI of the R-D link which is not
RYU and CHOI: BALANCED LINEAR PRECODING IN DECODE-AND-FORWARD BASED MIMO RELAY COMMUNICATIONS
directly connected with source. This centralized precoding design with full CSI entails additional feedback to provide the source with CSI of the R-D link. To reduce the burden of feedback, this subsection develops a distributed algorithm of precoding matrix design that is practically implementable at source and relay with limited CSI. Contrary to previous subsections, this subsection assumes that source has perfect knowledge of the channels from it to other nodes – S-R link and S-D link – but has the knowledge of the average channel 2 , and one additional real power gain for the R-D link, 𝜎𝑅𝐷 value which will be identified in the following part. ∗ In the first step, the precoding matrices W𝑆,1 and W𝑅 are independently designed at source and relay without any cooperation. Since the source has full CSI of the S-R link, the optimal precoding matrix W𝑆,1 that maximizes ℛ𝑅 is obtained by (11). The precoding matrix at relay is designed based on CSI of the R-D link only such that ∗ 𝐻 𝐻 W𝑅 = arg max log2 I+𝜌0 H𝑅𝐷W𝑅W𝑅 H𝑅𝐷 𝐻 )≤𝑃 W𝑅 :Tr(W𝑅 W𝑅 𝑅 1 2
= V𝑅𝐷 Σ𝑅𝐷 ,
(21) 1
𝐻 2 = U𝑅𝐷 Λ𝑅𝐷 V𝑅𝐷 and Σ𝑅𝐷 = where H𝑅𝐷 diag(𝑃𝑅𝐷,1 , ⋅ ⋅ ⋅ , 𝑃𝑅𝐷,𝑁 ). Then, the relay only feeds the average channel power gain 2 of the R-D link 𝜎𝑅𝐷 instead of full channel information of ∗ H𝑅𝐷 and W𝑅 back to the source. With the knowledge of 2 𝜎𝑅𝐷 , source designs a precoding matrix W𝑆,2 to maximize the upper bound on the expectation of ℛ𝐷 given by [ ] [ 𝔼 ℛ𝐷 = 𝔼H𝑅𝐷 log2 I + 𝜌0 H𝑆𝐷 W𝑆 W𝑆𝐻 H𝐻 𝑆𝐷 ] 𝐻 𝐻 + 𝜌0 H𝑅𝐷W𝑅W𝑅 H𝑅𝐷 (22) ≤ log2 𝜌0 H𝑆𝐷 W𝑆 W𝑆𝐻 H𝐻 𝑆𝐷 ] [ 𝐻 𝐻 (23) + 𝜌0 𝔼 𝑁0 I+H𝑅𝐷W𝑅W𝑅 H𝑅𝐷 ,
where (23) comes from Jensen’s inequality. 𝐻 𝐻 Proposition 2: Let R = 𝑁0 I + H𝑅𝐷 W𝑅 W𝑅 H𝑅𝐷 , then the expectation of R is approximated as a function of a powerlevel constant at relay, 𝜈𝑅 , such as [ ] 𝐻 𝐻 𝔼 𝑁0 I + H𝑅𝐷 W𝑅 W𝑅 H𝑅𝐷 ( ) ( √ ) ( 2 2 1 𝜈𝑅 ) 𝜋 4 𝑁 𝜎𝑅𝐷 −1 1− + ≈ −1 −1 +sin I, 𝜋𝜈𝑅 4 𝜈𝑅 𝜈𝑅 2 2 (24)
where 𝜆 is an unordered eigenvalue distribution of a Wishart matrix and 𝜈𝑅 is a power-level ∑ constant for the waterfilling algorithm and determined by 𝑃𝑅𝐷,𝑖 = 𝑃𝑅 . Proof: See Appendix B. From Proposition 2, we note that the required information to design the precoding matrix maximizing the upper bound 2 2 . Since 𝜎𝑅𝐷 in (23) are a power level constant 𝜈𝑅 and 𝜎𝑅𝐷 can be obtained at source by a long term feedback or channel reciprocity, relay is required to feed only a single real-valued constant 𝜈𝑅 back to source every coherence time. Then the
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Algorithm 3 Precoding with limited information (0)
1) Initially set 𝑖 = 0, 𝜖(0) = 0 and W𝑆 = W𝑆,2 . (𝑖) 2) Define 𝒟𝑙𝑖𝑚 as a rate difference function at 𝑖-th iteration: (𝑖) (𝑖) (𝑖) 𝒟𝑙𝑖𝑚 = ℛ𝑅,𝑙𝑖𝑚 − ℛ𝐷,𝑙𝑖𝑚 where
( )𝐻 (𝑖) (𝑖) (𝑖) ℛ𝑅,𝑙𝑖𝑚 = log2 I + 𝜌0 H𝑆𝑅 W𝑆 W𝑆 H𝐻 𝑆𝑅 ( )𝐻 (𝑖) (𝑖) (𝑖) ℛ𝐷,𝑙𝑖𝑚 = log2 𝛾I + 𝜌0 H𝑆𝐷 W𝑆 W𝑆 H𝐻 𝑆𝐷 . (𝑖)
(𝑖−1)
3) While 𝒟𝑙𝑖𝑚 < 𝒟𝑙𝑖𝑚 ∙ ∙ ∙ ∙
(𝑖)
(𝑖−1)
Stop the loop if 𝒟𝑙𝑖𝑚 ≥ 𝒟𝑙𝑖𝑚 and 𝜖∗ = 𝜖(𝑖−1) Stop the loop if 𝜖(𝑖) = 1 and 𝜖∗ = 1 𝑖 = 𝑖 + 1, 𝜖(𝑖+1) = 𝜖(𝑖) + 𝛿 (𝑖) Update the precoding matrix at source W𝑆 as ) ( 𝜖(𝑖) W𝑆,1 + 1 − 𝜖(𝑖) W𝑆,2 (𝑖) W𝑆 = 𝑃𝑆 ⋅ (𝑖) (𝑖) Tr(W𝑆 (W𝑆 )𝐻 )
4) The precoding matrix at source W𝑆∗ is determined by W𝑆∗ = 𝛼∗ W𝑆,1 + 𝛽 ∗ W𝑆,2 , where 𝛼∗ and 𝛽 ∗ are given, respectively, by 𝛼∗ =
𝑃𝑆 ⋅ 𝜖∗
𝐻
Tr(W𝑆∗ (W𝑆∗ ) )
, 𝛽∗ =
𝑃𝑆 ⋅ (1 − 𝜖∗ )
𝐻
Tr(W𝑆∗ (W𝑆∗ ) )
.
precoding matrix at source which maximizes (23) is obtained by W𝑆,2 = arg max log2 𝛾 ⋅ I+𝜌0 H𝑆𝐷W𝑆W𝑆𝐻H𝐻 𝑆𝐷 𝐻 )≤𝑃 W𝑆 :Tr(W𝑆 W𝑆 𝑆 1
2 = V𝑆𝐷 Σ𝑆,2 ,
(25)
where 𝛾 is a solution of (24), Σ𝑆,2 = ] [ diag 𝑃𝑆,2,1 , ⋅ ⋅ ⋅ , 𝑃𝑆,2,𝑁 , and the power allocation is determined by ( )+ 1 𝛾 𝑃𝑆,2,𝑖 = − (26) 𝜈𝑆 𝜆𝑆,𝑖 where 𝜆𝑆,𝑖 denotes the 𝑖-th eigenvalue of H𝑆𝐷 and ∑ 𝜈𝑆 is a power level constant at source and determined by 𝑃𝑆,2,𝑖 = 𝑃𝑆 . Once W𝑆,1 and W𝑆,2 are determined as described above, the precoding matrix at source is designed by W𝑆∗ = 𝛼∗ W𝑆,1 + 𝛽 ∗ W𝑆,2 .
(27)
Similarly to previous subsections, the non-negative real valued weighting coefficients are obtained by Algorithm 3. In this way, the precoders at the source and relay are designed in a distributed manner and the amount of feedback information from relay to source is much reduced. IV. C OMPLEXITY A NALYSIS In this section, we analyze computational complexity of the proposed algorithms and compare it with that of conventional
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achievable rates within 1% error. Fig. 2 shows that the number of required iterations for the proposed algorithm (Algorithm 2) is much less than that for the conventional algorithm. The computational complexity of the proposed algorithm with limited information (Algorithm 3) is further reduced. Since waterfilling power allocation is performed twice at source and once at relay, the total computational complexity corresponds to 3⋅𝒪(𝑛3 ) with some finite numbers of additions.
50 45 40 Proposed Algorithm (Algorithm 2) Joint Optimization
iterations
35 30 25 20
B. Complexity of Algorithm 1 (MISO cases)
15 10 5 0
2
3
4
5 6 7 number of antennas
8
9
10
Fig. 2. The number of outer iteration of the proposed scheme (Algorithm 2) 2 , 𝜎 2 , 𝜎 2 } = {−10, −15, −20}dB, the average SNR=10dB with {𝜎𝑆𝑅 𝑅𝐷 𝑆𝐷 and 𝛿 = 0.1.
For MISO cases, the complexity reduction by the proposed algorithm (Algorithm 1) is more significant. Since the optimization problem is formed as a second-order cone program (SOCP), the computational complexity of the conventional algorithm grows cubically with respect to the dimensionality of an input space, i.e., 𝒪(𝑛3 ). On the other hand, the proposed algorithm (Algorithm 1) does not depend on the dimensionality of an input space i.e., 𝒪(1) since the proposed algorithm requires only simple one-dimensional search for given channel realization.
algorithm that numerically solve the joint optimization problem given in (7).
V. S IMULATION R ESULTS
A. Complexity of Algorithm 2 (MIMO cases) and Algorithm 3 (Limited information)
This section evaluates the achieved rate by the proposed linear precoding method in our system model described in Section II. We compare the achievable rate of the proposed scheme with those of the optimal and conventional precoding schemes.
In our proposed algorithm for MIMO cases (Algorithm 2), SVD and waterfilling power allocation are performed at every iteration step. If we assume that all channels are modeled as 𝑛 × 𝑛 square matrices for simplicity, the computational complexity for SVD is known to be 𝒪(𝑛3 ) and the waterfilling algorithm has complexity of 𝒪(log 𝑛) since the water-level is determined by binary search. Therefore, the computational complexity at each iteration step becomes 𝒪(𝑛3 ) + 𝒪(log 𝑛). Since one dimensional search does not depend on the dimensionality of an input space, the total computational complexity of the proposed algorithms is 𝐿1 ⋅ 𝒪(𝑛3 ), where 𝐿1 is the number of iterations of the proposed algorithm and it contains iterations for one-dimensional search and convergence of iterative waterfilling. The conventional algorithm to solve a joint convex optimization problem with two variables is known to have computational complexity of 𝒪(2𝑛6 ) with respect to 𝑛 [24]. For a standard interior point method, the computational complexity to solve optimization problem given in (7) is 𝐿2 ⋅𝒪(2𝑛6 ) where 𝐿2 is the number of iterations which includes inner Newton iterations to solve the joint optimization problem. Therefore, the computational complexity of precoder design reduces from 𝒪(𝑛6 ) to 𝒪(𝑛3 ) by using the proposed algorithm. If 𝑛 is not large, the number of required iterations is rather a dominant factor than the asymptotical complexity growth in terms of 𝑛. The number of required iterations for the conventional algorithm, 𝐿2 , is also much larger than that of the proposed algorithm, 𝐿1 , since the proposed algorithm requires only simple one-dimensional search. Fig. 2 shows the numbers of required iterations for the proposed algorithm (Algorithm 2) and the conventional algorithm adopted in [23]. For a fair comparison, both algorithms are required to converge the
A. A Special Case: MISO (𝑁 = 𝐾 = 1) We first evaluate the achievable rate for a MISO case. The achievable rate of the conventional precoding vector, which does not take into account the direct link h𝑆𝐷 and thus is a MRT beamformer for h𝑆𝑅 given in (9), is compared with the proposed scheme as a reference scheme. Fig. 3 presents the achievable rate of the propose algorithm (Algorithm 1) and conventional precoding scheme according to average SNR values when the average channel gains are 2 2 2 given by {𝜎𝑆𝑅 , 𝜎𝑅𝐷 , 𝜎𝑆𝐷 } = {−10, −15, −20}dB, respectively. The numbers of antennas at source, relay and destination are 2, 1 and 1, respectively. This figure shows that the proposed scheme (Algorithm 1) outperforms the conventional precoding scheme in the whole SNR regime. Similar results are shown in Fig. 4 where the achievable rate versus average SNR values is plotted when the antenna configuration is given by {𝑀, 𝐾, 𝑁 } = {4, 1, 1}. In this configuration, the proposed scheme based on Algorithm 1 far outperforms the conventional scheme. Since the S-R link is strengthened by multiple antennas at source, the precoding vector aims at enhancing ℛ𝐷 to balance ℛ𝑅 and ℛ𝐷 . Consequently, the performance gap in terms of achievable rate between the proposed and conventional schemes becomes larger than that in Fig. 3. The impact of the link gain for the proposed scheme is shown in Fig. 5. The capacity improvement by the proposed technique largely depends on the link gains, especially the channel gain of the S-D link since the proposed precoder is designed by taking account into the direct link. Fig. 5 shows
RYU and CHOI: BALANCED LINEAR PRECODING IN DECODE-AND-FORWARD BASED MIMO RELAY COMMUNICATIONS
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1
1.4 Proposed (Optimal) Conventional
0.9
1.2
Proposed (Optimal) Coventional
Achievable rate [bps/Hz]
Achievable rate [bps/Hz]
0.8 1
0.8
0.6
0.4
0.7 0.6 0.5 0.4 0.3
0.2
0
0.2
0
5
10 average SNR [dB]
15
0.1 −40
20
Fig. 3. Achievable rate versus average SNR for the proposed linear precoding algorithm (Algorithm 1) with {𝑀, 𝑁, 𝐾}={2, 1, 1} and 2 , 𝜎 2 , 𝜎 2 } = {−10, −15, −20}dB. {𝜎𝑆𝑅 𝑅𝐷 𝑆𝐷
−35
−30 −25 −20 average channel gain of S−D link [dB]
−15
−10
Fig. 5. Achievable rate versus average channel gain of S-D link 2 ) for the proposed linear precoding algorithm (Algorithm 1) with (𝜎𝑆𝐷 2 , 𝜎 2 } = {−10, −15}dB and average {𝑀, 𝑁, 𝐾}={4, 1, 1}, {𝜎𝑆𝑅 𝑅𝐷 SNR=10dB.
1.5 Proposed (Optimal) Conventional
3 Optimal Proposed (Full Information) Proposed (limited Information) Conventional
1 Achievable rate [bps/Hz]
Achievable rate [bps/Hz]
2.5
0.5
2
1.5
1
0.5 0
0
5
10 average SNR [dB]
15
20
Fig. 4. Achievable rate versus average SNR for the proposed linear precoding algorithm (Algorithm 1) with {𝑀, 𝑁, 𝐾}={4, 1, 1} and 2 2 2 } = {−10, −15, −20}dB. {𝜎𝑆𝑅 , 𝜎𝑅𝐷 , 𝜎𝑆𝐷
that the proposed scheme achieves significantly higher rate than the conventional scheme as the S-D link gain becomes larger. When the S-D link gain is negligible, the proposed scheme achieves the same rate with the conventional scheme. B. A General Case: MIMO In this subsection, MIMO scenarios where source, relay, and destination have multiple antennas are considered. The proposed schemes with perfect CSI (Algorithm 2) and with limited information (Algorithm 3) are compared with the optimal scheme and the conventional scheme neglecting the S-D link in precoding matrix design at source and relay. The achievable rate by the optimal precoding matrices is numerically obtained from (7). The precoding matrices for the conventional scheme are given in (11) and (21). In Fig. 6, the achievable rates of the proposed and reference schemes are plotted according to the average SNR values
0
0
5
10 average SNR [dB]
15
20
Fig. 6. Achievable rate versus average SNR for the proposed linear precoding algorithms (Algorithm 2, Algorithm 3) with {𝑀, 𝑁, 𝐾}={2, 2, 2} and 2 , 𝜎 2 , 𝜎 2 } = {−5, −15, −20}dB. {𝜎𝑆𝑅 𝑅𝐷 𝑆𝐷
when the average channel gains and the antenna configurations 2 2 2 are given by {𝜎𝑆𝑅 , 𝜎𝑅𝐷 , 𝜎𝑆𝐷 } = {−5, −15, −20}dB and {𝑀, 𝐾, 𝑁 } = {2, 2, 2}, respectively. In the whole SNR regime, the proposed algorithms always outperforms the conventional precoding scheme due to the gain from balancing ℛ𝑅 and ℛ𝐷 . The performance gap between the proposed scheme with full information (Algorithm 2) and the optimal scheme is marginal. It is also shown that the proposed scheme with limited information (Algorithm 3) is slightly worse than the scheme with full information. Fig. 7 shows achievable rate of the proposed schemes for an asymmetric antenna configuration that {𝑀, 𝑁, 𝐾} = {4, 2, 2}. The achievable rates of the optimal and the conventional schemes are also plotted for comparison. Since two streams are transmitted through four antennas at source, the composite channel, i.e., ℛ𝐷 , is strengthened owing to multiple
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3
2.5
Achievable rate [bps/Hz]
A PPENDIX A P ROOF OF P ROPOSITION 1
Optimal Proposed (Full Information) Proposed (Limited Information) Conventional
ℛ𝑀𝐼𝑆𝑂 in (8) is maximized according to the following 𝐷𝐹 three possible cases.
2
1) ℛ𝑅 (w𝑆 ) < ℛ𝐷 (w𝑆 ), ∀w𝑆 𝑠.𝑡. ∥w𝑆 ∥2 ≤ 1: In this case, the maximized value of ℛ𝑅 is always less then ℛ𝐷 regardless of precoding vectors w𝑆 . So the optimal precoding vector is
1.5
1
h𝐻 𝑆𝑅 w𝑆∗ ≜ w𝑆,1 = arg max ℛ𝑅 (w𝑆 ) =
h𝐻 ∥w𝑆 ∥2 ≤1 𝑆𝑅
0.5
0
0
5
10 average SNR [dB]
15
20
Fig. 7. Achievable rate versus average SNR for the proposed linear precoding algorithms (Algorithm 2, Algorithm 3) with {𝑀, 𝑁, 𝐾}={4, 2, 2} and 2 , 𝜎 2 , 𝜎 2 } = {−10, −15, −20}dB. {𝜎𝑆𝑅 𝑅𝐷 𝑆𝐷
transmit antennas at source. As a result, the proposed scheme with full information far outperforms the conventional scheme which neglects the S-D link in the precoding matrix design. Although the performance gap between the proposed scheme with full information and the optimal scheme is larger than that in previous results, it is still marginal. It is also shown that the gain of the proposed scheme using limited information over the conventional scheme grows with average SNR because the upper bound in (A.10) based on which the precoding matrix is designed is accurate at the high SNR regime. VI. C ONCLUSION We proposed a novel linear precoding scheme to balance the S-R link and the S-D link and hence maximize the achieved rate in MIMO DF relaying systems. The proposed precoders are obtained by linearly combining two independently designed precoders that maximize data rates at relay and destination, respectively. For the MISO case, we proved that the optimal precoding vector is a linear combination of MRT beamformers for the S-R link and the S-D link and derived the optimal precoding vector by determining the optimal coefficients for the linear combining. The precoding design approach of a linear combination was extended to MIMO cases. The numerical results showed that the proposed precoding significantly outperforms the conventional precoding and showed only a marginal loss compared to the optimal precoding even when the coefficients of a linear combination are restricted to real numbers in MIMO cases. We also proposed a distributed algorithm to determine the precoders in limited feedback environments and showed that the proposed method using limited feedback achieves comparable rate to the optimal precoding scheme with full channel information. To justify the merits of the proposed precoding design methods, it was also shown that the proposed design significantly reduces computational complexity.
(A.1)
and corresponding achieved rate is given by ℛ𝑀𝐼𝑆𝑂 = 𝐷𝐹 ℛ𝑅 (w𝑆,1 ). 2) ℛ𝑅 (w𝑆 ) > ℛ𝐷 (w𝑆 ), ∀w𝑆 𝑠.𝑡. ∥w𝑆 ∥2 ≤ 1: When ℛ𝐷 is always less than ℛ𝑅 regardless of precoding vectors w𝑆 , the optimal precoding vector becomes h𝐻 𝑆𝐷 w𝑆∗ ≜ w𝑆,2 = arg max ℛ𝐷 (w𝑆 ) =
h𝐻 ∥w𝑆 ∥2 ≤1 𝑆𝐷
(A.2)
= ℛ𝐷 (w𝑆,2 ). and the achieved rate is ℛ𝑀𝐼𝑆𝑂 𝐷𝐹 3) ℛ𝑅 (w𝑆) = ℛ𝐷 (w𝑆) for some w𝑆 𝑠.𝑡. ∥w𝑆 ∥2 ≤ 1 : The last possible case is that there exists a precoding vector w𝑆 such that ℛ𝑅 (w𝑆 ) = ℛ𝐷 (w𝑆 ). Since ℛ𝑅 and ℛ𝐷 are concave and monotonic increasing functions with respect to w𝑆 , min{ℛ𝑅 , ℛ𝐷 } is maximized when ℛ𝑅 = ℛ𝐷 = ℛ𝑜𝑝𝑡 and ∥w𝑆 ∥2 = 1. Therefore, the optimal precoding vector is obtained by w𝑆∗ = arg max {ℛ𝑅 (w𝑆 ) = ℛ𝐷 (w𝑆 )} , w𝑆 ∈ ℂ𝑀×1 . ∥w𝑆 ∥2 =1
(A.3)
The 𝑀 dimensional optimal precoding vector can be represented by a linear combination of 𝑀 orthonormal basis vectors such as h𝐻 𝑆𝑅 w𝑆∗ = 𝛼′
h𝐻 + 𝛼1 𝝂1 + ⋅ ⋅ ⋅ + 𝛼𝑀−1 𝝂𝑀−1 , (A.4) 𝑆𝑅 where 𝝂𝑚 , 𝑚 = 1, . . . , 𝑀 − 1, are orthonormal column vectors satisfying h𝑆𝑅 ⋅ 𝝂𝑚 = 0 and ∣𝛼′ ∣2 + ∑𝑀−1 2 ′ 𝑚=1 ∣𝛼𝑚 ∣ = 1, and the coefficients 𝛼 and {𝛼𝑖 } are all complex numbers. For the optimal precoding vector in (A.4), the achievable rate at relay ℛ𝑅 is given by ( ) 2 ℛ𝑅 = log2 1 + 𝜌𝑆 ∣h𝑆𝑅 w𝑆∗ ∣ ( ) = log2 1 + 𝜌𝑆 ⋅ ∣𝛼′ ∣2 ∥h𝑆𝑅 ∥2
(A.5)
since h𝑆𝑅 ⋅ 𝝂𝑚 = 0, ∀𝑚 ∈ {1, ⋅ ⋅ ⋅ , 𝑀 − 1}. On the other hand, ℛ𝐷 is given by ( ) 2 2 ℛ𝐷 = log2 1+𝜌𝑆 ∣h𝑆𝐷 w𝑆∗ ∣ +𝜌𝑅 ∣ℎ𝑅𝐷 ∣
(A.6)
RYU and CHOI: BALANCED LINEAR PRECODING IN DECODE-AND-FORWARD BASED MIMO RELAY COMMUNICATIONS
where h𝑆𝐷 w𝑆∗ is given by h𝑆𝐷 w𝑆∗ 𝛼′ h𝐻 𝑆𝑅
+h𝑆𝐷 (𝛼1 𝝂1 +⋅ ⋅ ⋅ + 𝛼𝑀−1 𝝂𝑀−1) = h𝑆𝐷
h𝐻 𝑆𝑅
(A.7) ) ( 𝛽′ Π h 𝛼′ h𝐻 𝑆𝑅
+ h𝑆𝐷 {𝝂𝑖 } 𝑆𝐷
(A.8) = h𝑆𝐷 𝐻
h
Π{𝝂 } h𝑆𝐷 𝑖 𝑆𝑅 ) ( Π⊥ h𝑆𝐷 𝛼′ h𝐻 h𝑯 ′ 𝑆𝑅 𝑺𝑹
+ h𝑆𝐷 𝛽
(A.9) = h𝑆𝐷
h𝐻
Π⊥𝑯 h𝑆𝐷 𝑆𝑅 h𝑺𝑹 ) ( ⊥ 𝐻 Πh𝑯 h𝑆𝐷 h𝑆𝑅 ′ ′ 𝑺𝑹
≤∣𝛼 ∣⋅ h𝑆𝐷
h𝐻 +∣𝛽 ∣⋅ h𝑆𝐷 Π⊥𝑯 h𝑆𝐷 , 𝑆𝑅 h𝑺𝑹 (A.10) √∑ 𝑀−1 where ∣𝛽 ′ ∣ = ∣𝛼𝑚 ∣2 and note again that ΠX ≜ ( 𝐻 )−1 𝐻 𝑚=1 X denotes orthogonal projection on to X X X the column space of X and Π⊥ X ≜ I − ΠX denotes orthogonal projection on to the orthogonal complement h𝑆𝐷 ) = of the column space of X. Since (h𝑆𝐷 ⋅ Π⊥ h𝑯 𝑺𝑹 ⊥ 2 ∥Πh𝑯 h𝑆𝐷 ∥ , the equality in (A.10) holds when 𝑺𝑹 ( ) ′ ∠ 𝛼′ h𝑆𝐷 ⋅ h𝐻 (A.11) 𝑆𝑅 = ∠ (𝛽 ) . So the optimal precoding vector w𝑆∗ is rewritten by w𝑆∗ =
h𝐻 𝑆𝑅 𝛼′
h𝐻 𝑆𝑅
Π⊥ h𝑆𝐷 h𝑯 𝑺𝑹
+ 𝛽′ ⊥
Π 𝑯 h𝑆𝐷 h
(A.12)
𝑺𝑹
and 𝛼′ and 𝛽 ′ are determined to satisfy (A.11). Since h𝐻 𝑆𝐷 can be represented as h𝐻 h𝑆𝐷 + Π⊥ 𝑆𝐷 = Πh𝑯 h𝑯 h𝑆𝐷 𝑺𝑹
(A.13)
𝑺𝑹
h𝐻 ⊥ 𝑆𝑅 = Πh𝑯 h𝑆𝐷 ⋅ h𝑆𝐷 , 𝑺𝑹
h𝐻 +Πh𝑯 𝑺𝑹 𝑆𝑅
(A.14)
the optimal precoding vector in (A.12) is again rewritten by ( )
h𝐻 𝛼′ h𝐻 𝛽′ ∗ 𝐻 𝑆𝑅 𝑆𝑅
w𝑆 = h𝑆𝐷 ⋅ 𝑺𝑹
h𝐻 + Π⊥𝑯 h𝑆𝐷 h𝑆𝐷− Πh𝑯
h𝐻 𝑆𝑅 𝑆𝑅 h𝑺𝑹 (A.15)
) (
Πh𝑯 h𝑆𝐷 h𝐻 𝐻 ′ h𝑆𝐷 𝑺𝑹
𝑆𝑅
= 𝛼′ −𝛽 ′
Π⊥𝑯 h𝑆𝐷 h𝐻 +𝛽 Π⊥𝑯 h𝑆𝐷 𝑆𝑅 h𝑺𝑹 h𝑺𝑹 (A.16) h𝐻 h𝐻 𝑆𝑅
𝑆𝐷 =𝛼 + 𝛽
h𝐻
h𝐻 𝑆𝑅 𝑆𝐷 =𝛼w𝑆,1 + 𝛽w𝑆,2 ,
(A.17) (A.18)
A PPENDIX B P ROOF OF P ROPOSITION 2 ¯ 𝑅𝐷 such that H𝑅𝐷 = 𝜎𝑅𝐷 H ¯ 𝑅𝐷 ∼ ¯ 𝑅𝐷 and H Define H 𝒞𝒩 (0, I). Then the expectation of R with respect to H𝑅𝐷 is given by [ ] 𝐻 𝐻 H𝑅𝐷 𝔼H𝑅𝐷 𝑁0 I+ H𝑅𝐷 W𝑅 W𝑅 [ ] 2 ¯ 𝑅𝐷 W𝑅 W𝐻 H ¯𝐻 =𝔼H𝑅𝐷 𝑁0 I+ 𝜎𝑅𝐷 H 𝑅 𝑅𝐷 [ ] 1 1 2 ¯ ¯ 2 Σ𝑅𝐷 Λ ¯ 2 ,𝐻 U ¯𝐻 U𝑅𝐷 Λ =𝔼H𝑅𝐷 𝑁0 I+ 𝜎𝑅𝐷 𝑅𝐷 𝑅𝐷 𝑅𝐷 [ 𝑁 ] ∑( ) 2 𝐻 =𝔼U 𝑁0 +𝜎𝑅𝐷 ⋅𝜆𝑛 𝑃 (𝜆𝑛 ) u𝑛 u𝑛 , ¯ 𝑅𝐷 ¯ 𝑅𝐷 , Λ
h𝑺𝑹
h𝑺𝑹
Consequently, the optimal precoding vector for all three possible cases ((1), (2), and (3)) can be represented as a linear combination of w𝑆,1 and w𝑆,2 . ■
(A.19) (A.20) (A.21)
𝑛=1
¯ 𝑅𝐷 and u𝑛 is the 𝑛-th where 𝜆𝑛 is the 𝑛-th eigenvalue of H ¯ 𝑅𝐷 . Since U ¯ 𝑅𝐷 and Λ ¯ 𝑅𝐷 are statistically column vector of U [ ] 𝐻 = I, independent and each u𝑛 is i.i.d with 𝔼U ¯ 𝑅𝐷 u𝑛 u𝑛 (A.21) can be rewritten by [ ] 𝐻 𝐻 𝔼H𝑅𝐷 𝑁0 I + H𝑅𝐷 W𝑅 W𝑅 H𝑅𝐷 ] [ 𝑁 ∑( ) 2 =𝔼Λ 𝑁0 + 𝜎𝑅𝐷 ⋅ 𝜆𝑛 𝑃 (𝜆𝑛 ) ⋅ I ¯ 𝑅𝐷
(A.22)
[ ] 2 =𝑁 ⋅ 𝔼Λ ¯ 𝑅𝐷 𝑁0 + 𝜎𝑅𝐷 ⋅ 𝜆𝑃 (𝜆) ⋅ I,
(A.23)
𝑛=1
where 𝜆 is an[ unordered eigenvalue ] of a Wishart matrix. In 2 (A.23), 𝔼Λ ¯ 𝑅𝐷 𝑁0 + 𝜎𝑅𝐷 ⋅ 𝜆𝑃 (𝜆) can be rewritten by ] [ 2 𝔼Λ ¯ 𝑅𝐷 𝑁0 + 𝜎𝑅𝐷 ⋅ 𝜆𝑃 (𝜆) ∫ ∞ ( ) 2 = 𝑁0 + 𝜎𝑅𝐷 ⋅ 𝜆𝑃 (𝜆) ⋅ 𝑓𝜆 (𝜆) 𝑑𝜆 (A.24) ( )) ∫0 ∞ ( 1 𝑁0 2 = ⋅𝜆 − 2 𝑁0 + 𝜎𝑅𝐷 ⋅ 𝑓𝜆 (𝜆) 𝑑𝜆 𝜈 𝜎 𝑅 𝜈𝑅 𝑅𝐷 ⋅ 𝜆 (A.25) ∫ ∞ 𝜆 2 = 𝜎𝑅𝐷 ⋅ ⋅ 𝑓𝜆 (𝜆) 𝑑𝜆, (A.26) 𝜈 𝑅 𝜈𝑅 )+ ( is obtained by the waterfilling where 𝑃 (𝜆) = 𝜈1𝑅 − 𝜎2𝑁0⋅𝜆 𝑅𝐷 algorithm for 𝜈𝑅 > 0 and 𝑓𝜆 (𝜆) is a probability density function (pdf) of an unordered eigenvalue 𝜆. Telatar derived a pdf of 𝜆 by integrating out all other eigenvalues in the unordered joint eigenvalue distribution in [8]. However, the derived distribution is too complicated to handle in closed form. Thus, he derived √ an empirical limiting distribution of the eigenvalue of H/ 𝑛 and it is given by
where
𝐻
Π 𝑯 h
h ′ ′ h𝑺𝑹 𝑆𝐷 ′ 𝛼=𝛼 −𝛽 ⊥ , 𝛽 = 𝛽 ⊥ 𝑆𝐷 .
Π 𝑯 h𝑆𝐷 Π 𝑯 h𝑆𝐷
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1 𝑓𝜆 (𝜆) ≈ 𝜋
√
1 1 − , 𝜆 4
𝜆 ∈ (0, 4)
(A.27)
as 𝑁 goes infinity [8]. Fortunately, the simulation results [8] showed that this approximation holds well even for medium number of antennas (i.e., 𝑁 = 4). By applying an empirical
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limiting distribution in (A.26), we have [ ] 2 𝔼Λ ¯ 𝑅𝐷 𝑁0 + 𝜎𝑅𝐷 ⋅ 𝜆𝑃 (𝜆) √ ∫ 4 1 𝜆 1 1 2 − 𝑑𝜆 (A.28) 𝜎𝑅𝐷 ⋅ ⋅ ≈ 𝜈𝑅 𝜋 𝜆 4 𝜈𝑅 √ ( ( ) ) ( 𝜎2 1 2 𝜈𝑅 ) 𝜋 4 = 𝑅𝐷 −1 −1 +sin−1 1− + . 𝜋𝜈𝑅 4 𝜈𝑅 𝜈𝑅 2 2 (A.29) R EFERENCES [1] T. M. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572-584, Sep. 1979. [2] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless network,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415-2425, Oct. 2003. [3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004. [4] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity, part I: system description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927-1938, Nov. 2003. [5] ——, “User cooperation diversity, part II: implementation aspects and performance anaysis,” IEEE Trans. Commun., vol. 51, no. 11, pp. 19391948, Nov. 2003. [6] A. Nosratinia, T. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun. Mag., vol. 42, no. 10, pp. 74-80, Oct. 2004. [7] T. W. Ban, W. Choi, B. C. Jung, and D. K. Sung, “A cooperative phase steering scheme in multi relay node environments,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 77-78, Jan. 2009. [8] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecom. ETT, vol. 10, no. 6, pp. 585-596, 1999. [9] B. Wang, J. Zhang, and A. Høst-Madsen, “On the capacity of MIMO relay channels,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 29-43, Jan. 2005. [10] C. K. Lo, S. Vishwanath, and R. W. Heath, Jr., “Sum-rate bounds for MIMO relay channels using precoding,” in Proc. Glob. Telecom. Conf., Nov. 2005. [11] H. B. Solcskei, R. Nabar, O. Oyman, and A. Paulraj, “Capacity scaling laws in MIMO relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1433-1444, June 2006. [12] S. Jin, M. R. McKay, C. Zhong, and K.-K. Wong, “Ergodic capacity analysis of amplify-and-forward MIMO dual-hop systems,” IEEE Trans. Inf. Theory, vol. 56, no. 5, pp. 2204-2224, May 2010. [13] H. Shi, T. Abe, T. Asai, and H. Yoshino, “A relaying scheme using QR decomposition with phase control for MIMO wireless networks,” in Proc. IEEE Int. Conf. Commun., May 2005. [14] X. Tang and Y. Hua, “Optimal design of non-regenerative MIMO wireless relays,” IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 13981407, Apr. 2007. [15] C. Chae, T. Tang, R. W. Heath, Jr., and S. Cho, “MIMO relaying with linear processing for multiuser transmission in fixed relay networks,” IEEE Trans. Signal Process., vol. 56, no. 2, pp. 727-738, Feb. 2008. [16] N. Varanese, O. Simeone, Y. Bar-Ness, and U. Spagnolini, “Achievable rates of multi-hop and cooperative MIMO amplify-and-forward relay systems with full CSI,” in Proc. IEEE Signal Process. Adv. in Wireless Commun., July 2006.
[17] B. Khoshnevis, W. Yu, and R. Adve, “Grassmannian beamforming for MIMO amplify-and-forward relaying,” IEEE J. Sel. Areas. Commun., vol. 26, no. 8, pp. 1397-1407, Oct. 2008. [18] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 528-541, Mar. 2006. [19] Q. 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, Feb. 2004. [20] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 506-522, Feb. 2005. [21] W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi, “Iterative waterfilling for Gaussian vector multiple access channels,” IEEE Trans. Inf. Theory, vol. 50, pp. 145-152, Jan. 2004. [22] T. Cover and J. A. Thomas, Elements of Information Theory. Wiley, 1991. [23] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, Mar. 2008. Available: http://stanford.edu/boyd/cvx. [24] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. Jong Yeol Ryu received the B.S. degree in electrical engineering from Chungnam National University (CNU), Daejeon, Korea in 2008. In February 2010, he received the M.S. degree in electrical engineering from Korea Advance Institute of Science and Technology (KAIST), Daejeon, Korea. He is currently working towards the ph.D. degree at Korea Advance Institute of Science and Technology (KAIST), Daejeon, Korea. His current research interests include interference management and signal processing in MIMO communications. Wan Choi is an associate professor of the Department of Electrical Engineering, Korea Advance Institute of Science and Technology (KAIST), Daejeon, Korea. He was in the School of Engineering, Information and Communications University (ICU), Daejeon, Korea, from Feb. 2007 to Feb. 2009 as an assistant professor. He received his B. Sc. and M. Sc. degrees from the School of Electrical Engineering and Computer Science (EECS), Seoul National University (SNU), Seoul, Korea, in 1996 and 1998, respectively, and the Ph.D. degree in the Department of Electrical and Computer Engineering at the University of Texas at Austin in 2006. He was a Senior Member of the Technical Staff of the R&D Division of KT Freetel Co. Ltd., Korea from 1998 to 2003 where he researched 3G CDMA systems. He also researched at Freescale Semiconductor and Intel Corporation during 2005 and 2006 summers, respectively, where he collaborated with them on practical wireless communication issues. He is the recipient of IEEE Vehicular Technology Society Jack Neubauer Memorial Award which recognized the best paper published in IEEE Transactions on Vehicular Technology for 2001. He also received IEEE Vehicular Technology Society Dan Noble Fellowship Award in 2006 and IEEE Communication Society Asia Pacific Young Researcher Award in 2007. While at the University of Texas at Austin, he was the recipient of William S. Livingston Graduate Fellowship and Information and Telecommunication Fellowship from Ministry of Information and Communication (MIC), Korea. He serves as an associate editor for IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS and was a co-chair of PHY track in IEEE WCNC 2011. He was a publication chair of IEEE WiOpt 2009 and a co-chair of MIMO symposium in IEEE IWCMC 2008.
