Multiuser MIMO Relaying Under Quality of Service ... - IEEE Xplore

IEEE WCNC 2011 - PHY

Multiuser MIMO Relaying Under Quality of Service Constraints Mohamed Fadel, Amr El-Keyi, and Ahmed Sultan Wireless Intelligent Networks Center (WINC), Nile University, Cairo, Egypt. Email: [email protected], {aelkeyi,asultan }@nileuniversity.edu.eg

Abstract—We consider a wireless communication scenario with 𝐾 source-destination pairs communicating through several half-duplex amplify-and-forward relays. We design the relay beamforming matrices by minimizing the total power transmitted from all the relays subject to quality of service constraints on the received signal to interference-plus-noise ratio at each destination node. We propose a novel method for solving the resulting nonconvex optimization problem in which the problem is decomposed into a group of second-order cone programs (SOCPs) parameterized by 𝐾 real parameters. Grid search or nested bisection can be used to search for the optimal values of these parameters. We provide numerical simulations showing the superior performance of the proposed algorithms compared to earlier suboptimal approximations and their ability to approach the globally optimal solution of the non-convex problem. Index Terms—Cooperative communications, MIMO amplify and forward relaying, convex optimization.

I. I NTRODUCTION Wireless relays have received considerable attention in the last decade due to their ability to improve the coverage and capacity of wireless communication systems [1]. In multiuser communication scenarios where multiple sources are targeting one or more destination nodes, cooperative relaying can also be used to provide spatial multiplexing [2]. Spatial multiplexing is essential for achieving the extreme bandwidth efficiency of future wireless systems. It can be attained with the use of relays employing receive and transmit beamformers that redirect each source signal towards its targeted destination node. Relay beamforming requires full knowledge of the channels from the sources to the relays and from the relays to the destination nodes. This channel information can be obtained using orthogonal pilot sequences broadcasted from the source and destination nodes to the relays [3]. We consider a system of multiple source-destination pairs communicating through multiple cooperative MIMO relays. The relays operate in half duplex amplify-and-forward mode in which communication between the source and destination nodes is performed in two phases. In the first one, the sources transmit their signals to the relays. Each relay linearly processes its received signal vector by a beamforming matrix and transmits the processed vector to the destination nodes in the second phase. We design the relay beamforming matrices jointly such that the total power transmitted by the relays is minimized subject to quality of service (QoS) constraints This work was supported by the Egyptian National Telecommunications Regulatory Authority (NTRA).

978-1-61284-254-7/11/$26.00 ©2011 IEEE

on the received signal to interference-plus-noise ratio (SINR) at each destination node. This optimization problem is not convex and, hence, several techniques have appeared in the literature to find approximate solutions for it. For example, semidefinite relaxation was used in [4] to convert the nonconvex problem to a semi-definite program (SDP) that can be solved using interior-point methods [5]. In general, semi-definite relaxation provides a lower bound on the total transmitted power by the relays. This bound is achievable if the SDP solution is rank-one [4]. On the other hand, the algorithm in [6] provides a suboptimal solution by approximating the problem as a second-order cone program (SOCP). It is worth mentioning that only single-antenna relays were considered in [4] and [6]. In this paper, we propose a novel computationally efficient technique that can approach the global optimal solution of the QoS-constrained relay beamforming problem while avoiding earlier suboptimal approximations in [4] and [6]. The proposed technique decomposes the problem into a group of SOCPs indexed by 𝐾 real parameters; each associated with one of the QoS constraints, where 𝐾 is the number of sourcedestination pairs. We present two algorithms for searching for the optimal values of these parameters and the beamforming matrices. The first algorithm uses a (𝐾 −1)-dimensional grid search to find the optimal values of the parameters while the second algorithm is iterative and is based on nested bisection. We also demonstrate the ability of the proposed iterative algorithm to converge to the global optimal solution of the non-convex problem. Numerical simulations are presented showing the superior performance of the proposed algorithms compared to those in [4] and [6]. However, this is achieved at the expense of increased computational complexity. II. S IGNAL M ODEL AND P ROBLEM F ORMULATION We consider a system of 𝐾 source-destination pairs communicating through 𝑅 relays as shown in Fig. 1. The 𝑟th relay is equipped with 𝑀𝑟 antennas that are used for both receiving from the sources and transmitting to the destination nodes. The relays operate in half-duplex mode. The 𝑀𝑟 × 1 received signal vector at the 𝑟th relay at the 𝑛th time instant is given by (1) 𝒙𝑟 (𝑛) = 𝑯 𝑟 𝒔(𝑛) + 𝜼 𝑟 (𝑛) where 𝜼 𝑟 (𝑛) is the 𝑀𝑟 ×1 vector containing the received noise at the 𝑟th relay. It is assumed to be zero-mean Gaussian 2 with covariance 𝜎𝑟(R) 𝑰 𝑀𝑟 and independent of the source

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equation (4) that the received signal at the 𝑘th destination is composed of three components. The first component is the desired signal, i.e., the signal transmitted by the 𝑘th source. The second one is the interference due to the other 𝐾 − 1 sources. The third component is the noise forwarded by the relays in addition to that generated at the destination node. Hence, the received SINR at the 𝑘th destination is given by  𝑅 2 ∑    𝐻 𝒈 𝑟,𝑘 𝑾 𝑟 𝒉𝑟,𝑘  𝑝𝑘    𝑟=1 SINR𝑘 =  2 𝑅 𝑅 ∑   2 1   ∑ (R)2  𝐻  𝐻 (D)2 ˜ 2𝑯 ˜𝐻 𝑾 𝒈 + 𝜎 𝒈 𝑷   𝑾 𝑟,𝑘 𝑟 𝑟 𝑟 𝑟,𝑘 𝑟,𝑘  +𝜎𝑘 𝑘   𝑟=1 𝑟=1 (5)

Fig. 1.

System model

signals and the noise at the other relays where 𝑰 𝑀 denotes the 𝑀 × 𝑀 identity matrix. The 𝑀𝑟 ×𝐾 matrix 𝑯 𝑟 is given by 𝑯 𝑟 = [𝒉𝑟,1 , ⋅ ⋅ ⋅ , 𝒉𝑟,𝐾 ] where 𝒉𝑟,𝑘 is the vector containing the channel coefficients from the 𝑘th source to the 𝑟th relay and the 𝐾 × 1 vector 𝒔(𝑛) is given by 𝑇 𝒔(𝑛) = [𝑠1 (𝑛), ⋅ ⋅ ⋅ , 𝑠𝐾 (𝑛)] where 𝑠𝑘 (𝑛) is the transmitted symbol from the 𝑘th source at the 𝑛th time instant. The signals transmitted from different sources are assumed to be uncorrelated. The relays operate in amplify-and-forward mode, i.e., in the second phase of the 𝑛th time instant, the 𝑟th relay retransmits the received signal in (1) after multiplication by the beamforming matrix 𝑾 𝑟 . Hence, the signal transmitted from the 𝑟th relay can be written as 𝒕𝑟 (𝑛) = 𝑾 𝑟 𝑯 𝑟 𝒔(𝑛) + 𝑾 𝑟 𝜼 𝑟 (𝑛). The power transmitted from the 𝑟th relay is given by   2 1 2  2 𝑃𝑟 = 𝑾 𝑟 𝑯 𝑟 𝑷 2  + 𝜎𝑟(R) ∥𝑾 𝑟 ∥𝐹 𝐹

(2) (3)

where 𝑷 = diag{𝑝1 , . . . , 𝑝𝐾 } is a diagonal matrix whose 𝑖th diagonal element, 𝑝𝑖 , is the power of the 𝑖th source, and ∥𝑿∥𝐹 denotes the Frobenius norm of the matrix 𝑿. Therefore, the signal received at the 𝑘th destination at the second phase of the 𝑛th time instant can be expressed as 𝑅 𝑅 ∑ ∑ ˜ ˜𝑘 (𝑛) 𝒈𝐻 𝑾 𝒉 𝑠 (𝑛) + 𝒈𝐻 𝒚 𝑘 (𝑛) = 𝑟 𝑟,𝑘 𝑘 𝑟,𝑘 𝑟,𝑘 𝑾 𝑟 𝑯 𝑟,𝑘 𝒔 𝑟=1 𝑅 ∑

+

˜ 𝑘 = diag{𝑝1 , . . . , 𝑝𝑘−1 , 𝑝𝑘+1 , . . . , 𝑝𝐾 }. where 𝑷 In this paper, we will assume that the 𝑟th relay can estimate its channel state information, i.e., {𝒉𝑟,𝑘 , 𝒈 𝑟,𝑘 }𝐾 𝑘=1 [4], [6], [7]. This assumption is well justified in time-division duplex systems where channel reciprocity holds. Furthermore, we will assume that a local processing center is connected to the relays through an error-free channel (possibly a wired connection) [4], [7]. The processing center receives the channel estimates from the relays, computes the beamforming coefficients, and feeds them back to the relays. Many applications demand a minimum QoS for operation, e.g., voice communication. Hence, a possible approach for the design of the relay beamforming matrices is through minimizing the total power transmitted from the relays subject to constraints that guarantee a minimum QoS (measured by the SINR) for each destination node [4], [6], i.e., 𝑅 ∑ 𝑃𝑟 min {𝑾 𝑟 }𝑅 𝑟=1

s.t.

min

{𝒘𝑟 }𝑅 𝑟=1

s.t.

(4)

𝑟=1

where 𝜈𝑘 (𝑛) is the noise generated at the 𝑘th destination. It is 2 assumed to be zero-mean Gaussian with variance 𝜎𝑘(D) and independent of the relay noise and the source signals. The 𝑀𝑟 ×1 vector 𝒈 𝑟,𝑘 contains the complex conjugate of the channel coefficients between the 𝑟th relay and the 𝑘th destination. The (𝐾 − 1)×1 ˜𝑘 (𝑛) contains the signals transmitted by the vector 𝒔 sources that are not targeting the 𝑘th destination, i.e., ˜𝑘 (𝑛) = [𝑠1 (𝑛), ⋅ ⋅ ⋅ , 𝑠𝑘−1 (𝑛), 𝑠𝑘+1 (𝑛), ⋅ ⋅ ⋅ , 𝑠𝐾 (𝑛)]𝑇 and the 𝒔 ˜ 𝑟,𝑘 = [𝒉𝑟,1 , ⋅ ⋅ ⋅ , 𝒉𝑟,𝑘−1 , 𝒉𝑟,𝑘+1 , ⋅ ⋅ ⋅ , 𝒉𝑟,𝐾 ] contains matrix 𝑯 the corresponding source-relay channels. We can see from

∀ 𝑘 = 1, . . . , 𝐾

(6)

where 𝛾𝑘 is the prespecified threshold for the SINR (minimum required QoS) at the 𝑘th destination. Let us define the 𝑀𝑟2 × 1 vector 𝒘𝑟 as 𝒘𝑟 = vec {𝑾 𝑟 } where vec {⋅} is the vectorization operator that stacks the columns of a matrix on top ( ) of each other. Using the identity vec{𝑨𝑩𝑪} = 𝑪 𝑇 ⊗ 𝑨 vec{𝑩} where ⊗ denotes the Kronecker product operator and substituting with (3) and (5), we can write the above optimization problem as

𝑟=1

𝒈𝐻 𝑟,𝑘 𝑾 𝑟 𝜼 𝑟 (𝑛) + 𝜈𝑘 (𝑛)

𝑟=1

SINR𝑘 ≥ 𝛾𝑘

𝑅 ∑

∥𝑪 𝑟 𝒘𝑟 ∥2 +

𝑟=1

𝑅 ∑

2

𝜎𝑟(R) ∥𝒘𝑟 ∥2

𝑟=1

 𝑅 2 ∑    𝐻 𝒃𝑟,𝑘 𝒘𝑟  𝑝𝑘    𝑟=1 ≥ 𝛾𝑘  𝑅 2 𝑅 ∑  ∑   (R)2 2 (D)2 𝑨𝑟,𝑘 𝒘𝑟  + 𝜎𝑟 ∥𝑮𝑟,𝑘 𝒘𝑟 ∥ +𝜎𝑘    𝑟=1

𝑟=1

∀ 𝑘 = 1, . . . , 𝐾

(7)

𝑇 𝐻 2 where 𝒃𝐻 𝑟,𝑘 = 𝒉𝑟,𝑘 ⊗ 𝒈 𝑟,𝑘 , the 𝐾𝑀𝑟 × 𝑀𝑟 matrix 𝑪 𝑟 , the 2 (𝐾 −1)×𝑀𝑟 matrix 𝑨𝑟,𝑘 , and the 𝑀𝑟 ×𝑀𝑟2 matrix 𝑮𝑟,𝑘 are 1) ( ( 1 )𝑇 ˜ 2 𝑇 ⊗𝒈 𝐻 , ˜ 𝑟,𝑘 𝑷 given by 𝑪 𝑟 = 𝑯 𝑟 𝑷 2 ⊗𝑰 𝑀𝑟 , 𝑨𝑟,𝑘 = 𝑯 𝑘 𝑟,𝑘 and 𝑮𝑟,𝑘 = 𝑰 𝑀𝑟 ⊗ 𝒈 𝐻 𝑟,𝑘 , respectively. 𝑅 ]𝑇 [ ∑ Let us define the 𝑀𝑟2 ×1 vectors 𝒘 = 𝒘𝑇1 , . . . , 𝒘𝑇𝑅 𝑟=1 𝑅 ]𝐻 ∑ √ [ 𝐻 and 𝒃𝑘 = 𝑝𝑘 𝒃1,𝑘 , . . . , 𝒃𝐻 , the (𝐾 − 1)× 𝑀𝑟2 ma𝑅,𝑘

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𝑟=1

𝑅 ∑

𝑅 ∑

trix 𝑨𝑘 = [𝑨1,𝑘 , . . . , 𝑨𝑅,𝑘 ], the 𝑀𝑟2 × 𝑀𝑟2 diagonal 𝑟=1 } 𝑟=1 { (R) 𝑇 1𝑀𝑅 where 1𝑀 is the matrix Λ = diag 𝜎1(R) 1𝑇𝑀1 , . . . , 𝜎𝑅 𝑀 ×1 vector containing ones, also ⎡ ⎤ 𝑪1 0 ⋅ ⋅ ⋅ 0 ⎢ 0 𝑪2 ⋅ ⋅ ⋅ 0 ⎥ ⎢ ⎥ 𝑹𝑐 = ⎢ . .. ⎥ , .. ⎣ .. . . ⎦ ⎡ ⎢ ⎢ 𝑹 𝐺𝑘 = ⎢ ⎣

0

0

⋅⋅⋅

𝜎1(R) 𝑮1,𝑘 0 .. .

𝑪𝑅

0 (R) 𝜎2 𝑮2,𝑘

⋅⋅⋅ ⋅⋅⋅ .. .

0

⋅⋅⋅

0

0 0 .. .

⎤ ⎥ ⎥ ⎥. ⎦

(8)

(R) 𝑮𝑅,𝑘 𝜎𝑅

Therefore, we can write (7) as 2

2

min ∥𝑹𝑐 𝒘∥ + ∥Λ𝒘∥ 𝒘    𝐻 2 𝒃𝑘 𝒘 s.t. 2 ≥ 𝛾𝑘 ∀𝑘 = 1, . . . , 𝐾 (9) 2 2 ∥𝑨𝑘 𝒘∥ + ∥𝑹𝐺𝑘 𝒘∥ + 𝜎𝑘(D) The optimization problem in (9) is nonconvex due to the absolute value operator in the numerator of the constraints. Several techniques have been proposed to provide approximate solutions for this problem. One of these techniques is semi-definite relaxation [4]. It is based on defining the rankone matrix 𝑾 = 𝒘𝒘𝐻 and rewriting (9) as min tr{𝑻˜ 𝑾 } 𝑾

2

s.t. tr{𝑻˜ 𝑘 𝑾 } ≥ 𝛾𝑘 𝜎𝑘(D)

rank {𝑾 } = 1, 𝑾 ≽ 0

∀ 𝑘 = 1, . . . , 𝐾 (

(10) )

𝐻 𝐻 where the matrix 𝑻˜ 𝑘 = 𝒃𝑘 𝒃𝐻 𝑘 − 𝛾 𝑘 𝑨 𝑘 𝑨 𝑘 + 𝑹 𝐺𝑘 𝑹 𝐺𝑘 𝐻 and 𝑻˜ = 𝑹𝐻 𝑐 𝑹𝑐 + Λ Λ. The optimization problem in (10) is still non-convex due to the rank constraint. This constraint is dropped resulting in an SDP problem that can be solved using interior-point { methods with a }worst-case 𝑅 )6.5 3 (∑ [8]. The 𝑀𝑟2 computational complexity of 𝒪 𝐾 2 𝑟=1

optimal beamforming matrix 𝑾 ★ obtained via semidefinite relaxation will not be rank one in general, and hence, the semidefinite relaxation technique provides a lower bound to the original problem in (9). If 𝑾 ★ happens to be rank-one, then its principal component will be the optimal solution to the original problem in (9). Otherwise, one has to resort to randomization techniques developed in [9] to obtain a suboptimal rank-one solution from 𝑾 ★ . In [6], a suboptimal method with reduced computational complexity was proposed for solving (9). In this technique, the absolute value operator in the numerator of each constraint is replaced by the real-part (or imaginary part) operator. The resulting problem is given by 2

2

min ∥𝑹𝑐 𝒘∥ + ∥Λ𝒘∥ 𝒘

⎡ ⎤  𝑨𝑘 𝒘 { } √    𝐻  ⎣ ⎦ s.t. ℜ 𝒃𝑘 𝒘 ≥ 𝛾𝑘  𝑹𝐺𝑘 𝒘   𝑘 = 1, . . . , 𝐾(11)   𝜎𝑘(D)

where ℜ{⋅} and ℑ{⋅} denote the real and imaginary parts of a complex number, respectively. The above problem can be written as an SOCP that can also be solved using interiorpoint with} a worst-case computational complexity of { methods 𝑅 )3 3 (∑ 𝑀𝑟2 𝒪 𝐾2 [8]. For any complex number 𝑧, ∣𝑧∣ ≥ 𝑟=1

ℜ{𝑧}, and hence, the feasible set of (11) is smaller than or equal to that of the original problem in (9). Therefore, the optimal solution of (11) is in general inferior to that of (9), i.e., it yields a higher value of the cost function. Nevertheless, in the case of 𝐾 = 1, there is no loss in optimality in the above approximation. This is due to the fact we can always phase-rotate the vector 𝒘 such that { that } 𝐻 ℜ 𝒃1 𝒘 = ∣𝒃𝐻 1 𝒘∣ without affecting the cost function or the R.H.S. of the constraint. However, if 𝐾 > 1, the complex 𝐾 numbers {𝒃𝐻 𝑘 𝒘}𝑘=1 do not necessarily have the same phase, and hence, (11) provides a suboptimal solution of the original problem in (9). III. P ROPOSED M ETHOD In this section, we provide a novel approach to solve the relay beamforming problem in (9) using the polyhedral approximation of the complex-valued numerator of the constraints [10]. In this approach, we consider the phase angles 𝐾 of the complex numbers {𝒃𝐻 𝑘 𝒘}𝑘=1 as optimization variables and provide two methods for searching for the optimal phases of these variables. The first method uses a grid search over 𝐾 −1 phase angles while the second one uses nested bisection to search for these angles. Let 𝜃𝑧 = arg{𝑧} denote the {phase of} the complex number 𝑧. Thus, we can write ∣𝑧∣ = ℜ 𝑧𝑒−𝑗𝜃𝑧 , and, as a result, the absolute value of the complex number in the numerator of the 𝑘th constraint in (9) can be written as   { }  𝐻  𝒘 (12) 𝒃𝑘 𝒘 = ℜ 𝑒−𝑗𝜃𝑘 𝒃𝐻 𝑘 𝐾 where 𝜃𝑘 = arg{𝒃𝐻 𝑘 𝒘}. Note that if the values of {𝜃𝑘 }𝑘=1 are given, the R.H.S. of (12) becomes linear in 𝒘, and hence, the optimization problem resulting from replacing the numerators of the 𝐾 constraints in (9) by the R.H.S. of (12) can be written as an SOCP. However, the values of the optimal phase angles, {𝜃𝑘∗ }𝐾 𝑘=1 , are unknown prior to solving (10) as they depend on the optimal beamforming vector 𝒘∗ . The following proposition depicts the relationship between the optimal phase angles and the optimal beamforming vector.

Proposition 1: The optimum solution (𝒘∗ , {𝜃𝑘∗ }𝐾 𝑘=1 ) of min ∥𝑹𝑐 𝒘∥2 + ∥Λ𝒘∥2

𝒘,{𝜃𝑘 }𝐾 𝑘=1

s.t. ℜ

{

𝑒−𝑗𝜃𝑘 𝒃𝐻 𝑘 𝒘

}

≥

√

⎡ ⎤  𝑨𝑘 𝒘    ⎣ ⎦ ∀𝑘 = 1,. . . ,𝐾(13) 𝛾𝑘  𝑘𝒘   𝑹𝐺(D)   𝜎𝑘

} { ∗ ∀𝑘 = 1,. . . ,𝐾. is achieved at 𝜃𝑘∗ = arg 𝒃𝐻 𝑘 𝒘

Proof: We will prove Proposition 1 by contradiction. Let us assume that the optimal solution of (13) is achieved at

1625

{

}}𝐾 { ∗ 𝒘 . Let us consider the following 𝜃𝑘 = 𝜃˜𝑘 ∕= arg 𝒃𝐻 𝑘 𝑘=1 optimization problem with respect to the variable 𝒘 only min ∥𝑹𝑐 𝒘∥2 + ∥Λ𝒘∥2 𝒘

{

}

s.t. ℜ 𝑒−𝑗𝜃𝑘 𝒃𝐻 𝑘 𝒘 ≥

√

⎡ ⎤  𝑨𝑘 𝒘    ⎣ ⎦ ∀𝑘 = 1, . . . , 𝐾 (14) 𝛾𝑘  𝑘𝒘   𝑹𝐺(D)   𝜎𝑘

∗ whose optimal solution at {𝜃𝑘 = 𝜃˜𝑘 }𝐾 𝑘=1 is given by 𝒘 according to our assumption. {Now, Let us consider the { 𝐻 ∗ }}𝐾 ∗ . optimization problem in (14) at 𝜃𝑘 = 𝜃𝑘 = arg 𝒃𝑘 𝒘 𝑘=1 ∗ We can see that setting 𝜃𝑘 = 𝜃𝑘 in the L.H.S. of the 𝑘th constraint in (14), increases the L.H.S. of this constraint without affecting its R.H.S. or any of the other 𝐾 − 1 constraints. Thus, the feasible set for (14) at {𝜃𝑘 = 𝜃˜𝑘 }𝐾 𝑘=1 is a proper (strict) subset of the feasible set for (14) at 𝐾 {𝜃𝑘 = 𝜃𝑘∗ }𝑘=1 . Since the optimal solution of (14) is always on the boundary of the feasible set1 , the cost function of (14) 𝐾 can be further minimized by selecting {𝜃𝑘 = 𝜃𝑘∗ }𝑘=1 instead of {𝜃𝑘 = 𝜃˜𝑘 }𝐾 𝑘=1 . This contradicts our assumption{ that the } ∗ optimal solution of (13) is achieved at 𝜃˜𝑘 ∕= arg 𝒃𝐻 . 𝑘 𝒘 𝐾 Therefore, the optimal choice for {𝜃 } is given by 𝑘 𝑘=1 } { ∗ for all 𝑘 = 1,. . . ,𝐾. 𝜃𝑘∗ = arg 𝒃𝐻 𝑘 𝒘

In the next two subsections, we will introduce two algorithms for searching for the optimal values of the parameters {𝜃𝑘 }𝐾 𝑘=1 . The first algorithm uses a multidimensional search over a uniform (𝐾 − 1)-dimensional grid whereas the second uses nested bisection. A. Grid Search Algorithm We can use the grid search technique to find the optimal values of the parameters {𝜃𝑘 }𝐾 𝑘=1 , i.e., we choose the possible where 𝑙 = 0, 1, . . . , 2𝐿 − 1. For values for 𝜃𝑘 as 𝜃𝑘 = 𝜋𝑙 𝐿 each of the (2𝐿)𝐾 possible combinations of the parameters {𝜃𝑘 }𝐾 𝑘=1 , we solve the SOCP problem in (14). From Proposition 1, the optimum beamforming vector that minimizes the cost function of (9) is the one that solves (14) for the choice of {𝜃𝑘 }𝐾 𝑘=1 that yields the minimum value of the objective function among all (2𝐿)𝐾 SOCPs. In this case, each } { parameter ∗ . As 𝜃𝑘 corresponds to the best approximation of arg 𝒃𝐻 𝑘 𝒘 we increase the value of 𝐿, the accuracy of the approximation is increased, but on the other hand the complexity increases. Note that we can reduce the complexity without any loss of optimality by setting 𝜃1 = 0 and searching only over the remaining 𝐾−1 parameters, i.e., by{ phase } rotating the optimal ∗ . 𝒘 beamforming vector of (9) by arg 𝒃𝐻 1 B. Nested Bisection Algorithm The grid search technique involves solving (2𝐿)𝐾−1 SOCPs. This might be prohibitive especially for large values of 𝐿. In this subsection, we will present a more 1 At the optimal solution of (14), at least one of the constraints has to be active, i.e., satisfied with equality. This can be easily proved by noticing that if all the constraints are inactive, we can always scale down the optimal beamforming vector so that the cost function is further minimized while satisfying the constraints.

computationally efficient method for searching for the optimal values of {𝜃𝑘 }𝐾 𝑘=1 using nested bisection. We will start by examining the quasi-concavity of the L.H.S. of the 𝑘th constraint in the parameter 𝜃𝑘 . This is established through the following proposition. Proposition 2: For a complex 𝑧 with arg{𝑧} ∈ [0, 𝜋], the function } { number 𝑓 (𝜙) = ℜ 𝑧𝑒−𝑗𝜙 is quasi-concave in [0, 𝜋] and has even symmetry around 𝜙 = arg{𝑧}. Furthermore, argmax 𝑓 (𝜙) is either 𝜙 = 𝜋 or 𝜙 = 2𝜋. 𝜙∈[𝜋,2𝜋]

} { Proof: The derivative of 𝑓{(𝜙) is } given by 𝑓 ′ (𝜙) = ℑ 𝑧𝑒−𝑗𝜙 . −𝑗𝜙 > 0 for 𝜙 ∈ [0, arg{𝑧}[ and Since } ∈ [0, 𝜋], ℑ 𝑧𝑒 { arg{𝑧} ℑ 𝑧𝑒−𝑗𝜙 < 0 for 𝜙 ∈] arg{𝑧}, 𝜋]. Hence, 𝑓 (𝜙) is increasing for 𝜙 ∈ [0, arg{𝑧}[ and decreasing for 𝜙 ∈] arg{𝑧}, 𝜋] which establishes the quasi-concavity of 𝑓 (𝜙) in [0, 𝜋] [5]. The even symmetry of 𝑓 (𝜙) around the vertical line 𝜙 = arg{𝑧} can be seen by substituting 𝑧 = ∣𝑧∣𝑒𝑗 arg{𝑧} in 𝑓 (𝜙) to get 𝑓 (𝜙) = ∣𝑧∣ cos(arg{𝑧} − 𝜙). In order to prove the second part of Proposition 2, we note that since arg{𝑧} ∈ [0, 𝜋], the value of 𝜙 ∈ [𝜋, 2𝜋] that maximizes 𝑓 (𝜙) is the one that minimizes ∣ arg{𝑧} − 𝜙∣, i.e., either 𝜙 = 2𝜋 if arg{𝑧} ∈ [0, 𝜋/2] or 𝜙 = 𝜋 if arg{𝑧} ∈ [𝜋/2, 𝜋]. From Proposition 2, we can use the bisection method to search for the optimal 𝜙 that maximizes 𝑓 (𝜙). This can be achieved by dividing the interval [0, 2𝜋] into two subintervals, ℐ1 = [0, 𝜋] and ℐ2 = [𝜋, 2𝜋], and performing a bisection search in each interval. The steps of the bisection search for 1 over the interval ℐ1 are given by: 𝜙ℐmax Initialize: 𝜙(l) = 0 and 𝜙(u) = 𝜋, While 𝜙(u) − 𝜙(l) > 𝜀 do: (u) (l) if 𝑓 (𝜙(u) ) > 𝑓 (𝜙(l) ), then: 𝜙(l) = 𝜙 +𝜙 2 (u) (l) else 𝜙(u) = 𝜙 +𝜙 . 2 end (u) (l) 1 = 𝜙 +𝜙 . 𝜙ℐmax 2 2 over the The same steps can be used for searching for 𝜙ℐmax interval ℐ2 by replacing the initialization step by Initialize: 𝜙(l) = 𝜋 and 𝜙(u) = 2𝜋 The maximum of 𝑓 (𝜙) over [0, 2𝜋], 𝜙∗ , is obtained as the maximum of the two bisection searches over ℐ1 and ℐ2 . Without any loss of generality, let us assume that the optimal value 𝜙∗ = argmax 𝑓 (𝜙) ∈ ℐ1 . Applied on the interval 𝜙∈[0,2𝜋]

ℐ2 , the above algorithm will always converge to one of the 2 ) for all 𝜙 ∈ ℐ2 . boundary points of ℐ2 where 𝑓 (𝜙) ≤ 𝑓 (𝜙ℐmax For the interval ℐ1 , since 𝑓 (𝜙) is quasi-concave in ℐ1 and is symmetric around the vertical line at its maximum point, i.e., 1 1 − 𝜙(l) ∣ ≶ ∣𝜙ℐmax − 𝜙(u) ∣, the if 𝑓 (𝜙(l) ) ≷ 𝑓 (𝜙(u) ) then ∣𝜙ℐmax ℐ1 above algorithm will always converge to 𝜙max . Since 𝜙∗ ∈ ℐ1 , 1 2 1 and 𝜙ℐmax will yield 𝜙∗ = 𝜙ℐmax . comparing 𝜙ℐmax From the proof of Proposition 1, we can see that changing 𝜃𝑘 such that the L.H.S. of the 𝑘th constraint in (14) increases leads to increasing the size of the feasible set of (14), and hence, the cost function is further minimized. From the results on the quasi-concavity of the L.H.S. of the 𝑘th

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1. Do for 𝜃1 ∈ [0, 𝜋] and 𝜃2 ∈ [0, 𝜋]: ∗ ∗ ˆ ˆ in [0, 𝜋]. Initialize  𝜃1 , 𝜃2 , 𝜃1 , [𝜃2 randomly ]𝑇   ∗ ∗𝑇  ˆ ˆ   While [𝜃1 , 𝜃2 ] − 𝜃1 , 𝜃2  > 𝜀(O) do: Set 𝜃ˆ1 = 𝜃∗ , and 𝜃ˆ2 = 𝜃∗ . 1

2

Initialize 𝜃2(l) = 0 and 𝜃2(u) = 𝜋. While 𝜃2(u) − 𝜃2(l) > 𝜀(I) do: Solve (14) at 𝜃1 = 𝜃1∗ and 𝜃2 = 𝜃2(l) Solve (14) at 𝜃1 = 𝜃1∗ and 𝜃2 = 𝜃2(u) 𝜃 (u) +𝜃 (l) If 𝑃2(l) < 𝑃2(u) , then 𝜃2(u) = 2 2 2 . 𝜃 (u) +𝜃 (l) else 𝜃2(l) = 2 2 2 .

yielding 𝑃2(l) . yielding 𝑃2(u) .

end Set 𝜃2∗ =

𝜃2(u) +𝜃2(l) . 2 (l) Initialize 𝜃1 = 0 and 𝜃1(u) = 𝜋. While 𝜃1(u) − 𝜃1(l) > 𝜀(I) do: Solve (14) at 𝜃2 = 𝜃2∗ and 𝜃1 = 𝜃1(l) Solve (14) at 𝜃2 = 𝜃2∗ and 𝜃1 = 𝜃1(u) 𝜃 (u) +𝜃 (l) If 𝑃1(l) < 𝑃1(u) , then 𝜃1(u) = 1 2 1 . 𝜃 (u) +𝜃 (l) else 𝜃1(l) = 1 2 1 .

end Set 𝜃1∗ =

yielding 𝑃1(l) . yielding 𝑃1(u) .

𝜃1(u) +𝜃1(l) . 2

end Solve (14) at 𝜃1 = 𝜃1∗ and 𝜃2 = 𝜃2∗ yielding 𝑃Sub1 2. Repeat for 𝜃1 ∈ [𝜋, 2𝜋] and 𝜃2 ∈ [0, 𝜋] and get 𝑃Sub2 . 3. Repeat for 𝜃1 ∈ [0, 𝜋] and 𝜃2 ∈ [𝜋, 2𝜋] and get 𝑃Sub3 . 4. Repeat for 𝜃1 ∈ [𝜋, 2𝜋] and 𝜃2 ∈ [𝜋, 2𝜋] and get 𝑃Sub4 .

12 Proposed technique (grid search) Proposed technique (bisection) SOCP approximation technique of [6] Semi−definite relaxation technique of [4]

10 Minimum transmitted relay power (dB)

constraint in the { parameter 𝜃𝑘} established in Proposition 2 −𝑗𝜃𝑘 where 𝑓 (𝜃𝑘 ) = ℜ 𝒃𝐻 , and the relationship between 𝑘 𝒘𝑒 the L.H.S. of the constraints of (14) and its cost function, we can use the nested bisection method to find the optimal values of {𝜃𝑘 }𝐾 𝑘=1 that yield the minimum value of the cost function of (14). In this case, we divide the (𝐾 − 1)-dimensional interval [0, 2𝜋]𝐾−1 into 2𝐾−1 subintervals (where we have set 𝜃1 = 0 without loss of optimality). In each subinterval, a nested bisection search is performed over the 𝐾−1 parameters 𝐾 {𝜃𝑘 }𝐾 𝑘=2 and the optimal choice of the parameters {𝜃𝑘 }𝑘=2 is the one that yields the lowest value of the cost function of (14) among the results of all the 2𝐾−1 bisection searches. For example, for the case of 𝐾 = 2, we perform two bisection searches over the intervals [0, 𝜋] and [𝜋, 2𝜋]. In each iteration, we solve the SOCP in (14) for two values of 𝜃2 ; 𝜃2(l) and 𝜃2(u) . Since a lower value of the cost function of (14) corresponds to a value of 𝜃2 that further increases the L.H.S. of the constraint of (14), we replace the value of 𝜃2 that yields the higher value 𝜃 (l) +𝜃 (u) of the cost function by 2 2 2 . The convergence of the nested bisection algorithm for the case of 𝐾 = 2 can be shown using the relationship between the L.H.S. of the constraint and the cost function of (14). However, for 𝐾 > 2 further investigation is required to prove the global optimality of the algorithm. The nested bisection algorithm for searching for two parameters 𝜃1∗ and 𝜃2∗ is given by:

8

6

4

2

0

5

6

7

8

9

10

11

12

γ (dB)

Fig. 2. QoS.

Minimum power transmitted from the relays versus the required

5. Find minimum relay power =

min 𝑃Sub𝑖 and get the

𝑖=1,...,4

corresponding optimal beamforming vector. IV. N UMERICAL S IMULATIONS We consider a system of 𝐾 = 3 sources each transmitting with 10 dB power to 𝑅 = 2 relays each equipped with 𝑀𝑟 = 5 antennas. The QoS threshold 𝛾 is the same for all the 𝐾 destinations. The variance of the relay and destination noise is selected as 0 dB and all channel coefficients are generated as independent standard circular Normal random variables. We compare the performance of the proposed algorithms with that of the semidefinite relaxation algorithm [4] and the SOCP algorithm in (11) [6]. We select the parameter 𝜃1 = 0 and employ both the grid search technique with 𝐿 = 4 and the nested bisection method with 𝜀(O) = 𝜀(I) = 1∘ to find the two parameters 𝜃2 and 𝜃3 . Simulation results are averaged over 100 Monte Carlo runs. In each run, the channel coefficients and the relay and destination noise are independent. Fig. 2 shows the minimum transmitted power from the relays obtained by solving different problem formulations versus different values of 𝛾. In our simulation, the beamforming matrix 𝑾 obtained by solving the SDP formulation of [4] has rank one, and hence, the SDP formulation yields the optimal solution of (9). The SOCP formulation of [6] provides a solution with lower complexity than that of [4] at the expense of increased relay transmission power. We can see from Fig. 2 that the proposed grid search and bisection techniques yield a relay power very close to that of the SDP formulation. Fig. 3 shows the minimum power transmitted from the relays for the above system versus different values of the grid size 2𝐿 at 𝛾 = 12 dB. We can see from this figure that by increasing the value of 𝐿 the accuracy of the proposed grid search technique increases and its performance approaches that of the SDP technique. However, this is achieved at the expense of increased computational complexity. In the next simulation, we investigate the effect of the number of source-destination pairs on the performance of different relay beamforming algorithms. We consider a system

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3

11.5 Proposed technique (grid search) SOCP approximation technique of [6] Semi−definite relaxation technique of [4]

SOCP approximation technique of [6] Semi−definite relaxation technique of [4] (Lower bound) Randomization technique of [9] Proposed method (bisection)

2

11.3

Minimum transmitted relay power (dB)

Minimum transmitted relay power (dB)

11.4

11.2 11.1 11 10.9 10.8

1

0

−1

−2

−3

10.7 −4

10.6 10.5

Fig. 3.

−5 1

2

3

4

5 6 Grid size (2L)

7

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9

10

Minimum power transmitted from the relays versus grid size (2𝐿).

with 𝑅 = 2 relays each equipped with 𝑀𝑟 = 5 antennas. The transmitted powers of the sources are given by 10 dB and the required QoS is 𝛾 = 0 dB. The noise variances at all the relays and destinations are 0 dB. Simulation results are averaged over 25 Monte Carlo runs. Fig. 4 shows the average power transmitted from the relays versus the number of source-destination pairs. We compare the performance of the proposed nested bisection algorithm with that of the SOCP formulation of [6] and the SDP formulation of [4]. The parameters of the nested bisection algorithm are selected as 𝜀(O) = 10−4 and 𝜀(I) = 1∘ . In some of the simulation runs, the optimal solution of the semidefinite relaxation technique has a rank higher than one. In these cases, we employ the randomization algorithms proposed in [9] to obtain a suboptimal rank-one solution. For each randomization technique, we generate 100 beamforming vector and select the one which yields the lowest value of the cost function while satisfying the constraints. It is worth mentioning that for some channel realizations the randomization technique fails to obtain a beamforming vector that satisfies all the constraints. In this case, the channel realization is dropped and we generate a new one. We can see from Fig. 4 that the performance of the proposed algorithm is close to the lower bound provided by the SDP algorithm. We can also see that the SDP algorithm does not always return a rank one solution especially as the number of users increases and that the performance of the randomization techniques in [9] is far from the lower bound provided by the SDP algorithm. Also, we can see from Fig. 4 that the performance of the SOCP algorithm in [6] is much worse than the performance of the proposed algorithm. These performance improvements justify the computational complexity associated with the proposed algorithms when the semi-definite relaxation technique does not yield a rank-one solution. V. CONCLUSION In this paper, we have considered the problem of designing the beamforming matrices for multiple cooperating MIMO relays that amplify the signals received from multiple sources

3

3.5

4 4.5 5 Number of source−destination pairs (K)

5.5

6

Fig. 4. Minimum relay power versus the number of source-destination pairs.

and forward them to the destination nodes. The beamforming matrices are deigned such the total power transmitted from the relays is minimized subject to constraints on the received SINR at each destination node. We have presented a novel approach for solving this non-convex problem by posing it a group of SOCPs parameterized by 𝐾 real parameters where 𝐾 is the number of source-destination pairs. We have employed both the grid search and nested bisection techniques for searching for these parameters. Simulation results have been presented showing the ability of the proposed technique to provide a solution superior to that of earlier suboptimal algorithms with moderate computational complexity. R EFERENCES [1] A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. B¨olcskei, “An overview of MIMO communications – A key to Gigabit wireless,” Proc. IEEE, vol. 92, pp. 198–218, Feb. 2004. [2] T. Abe, H. S. Hi, T. Asai, and H. Yoshino, “A relaying scheme for MIMO wireless networks with multiple source and destination pairs,” in Proc. Asia-Pacific Conf. on Communications, Perth, Australia, Oct. 2005, pp. 77–81. [3] A. Wittneben and B. Rankov, “Distributed antenna systems and linear relaying for Gigabit MIMO wireless,” in Proc. IEEE Veh. Technol. Conf., Los Angeles, CA, Sept. 2004, vol. 5, pp. 3624–3630. [4] S. Fazeli-Dehkordy, S. Gazor, and S. Shahbazpanahi, “Distributed peerto-peer multiplexing using ad hoc relay networks,” in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, Las Vegas, NV, Apr. 2008. [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004. [6] H. Chen, A. B. Gershman, and S. Shahbazpanahi, “Distributed peerto-peer beamforming for multiuser relay networks,” in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, Taipei, Taiwan, Apr. 2009. [7] A. El-Keyi and B. Champagne, “Cooperative MIMO-beamforming for multiuser relay networks,” in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, Las Vegas, NV, March 2008. [8] M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone programming,” Linear Algebra and Its Applications, vol. 284, pp. 193–228, 1998. [9] N. D. Sidiropoulos, T. N. Davidson, and Z.-Q. Luo, “Transmit beamforming for physical-layer multicasting,” IEEE Trans. Signal Processing, vol. 54, pp. 2239–2252, June 2006. [10] R. Zhang, C. C. Chai, , and Y. C. Liang, “Joint beamforming and power control for multiantenna relay broadcast channel with QoS constraints,” IEEE Trans. Signal Processing, vol. 57, pp. 726–737, Feb. 2009.

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