Cooperative beamforming for OFDM-based amplify-and-forward relay networks

Physical Communication 4 (2011) 305–312

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Cooperative beamforming for OFDM-based amplify-and-forward relay networks✩ Wenjing Cheng a,∗ , Mounir Ghogho b,c , Qinfei Huang a , Dongtang Ma a , Jibo Wei a a

National University of Defense Technology, Changsha, 410073, China

b

University of Leeds, Leeds, LS2 9JT, United Kingdom

c

International University of Rabat, Rabat, Morocco

article

info

Article history: Received 7 October 2011 Received in revised form 13 October 2011 Accepted 14 October 2011 Available online 29 October 2011 Keywords: OFDM Beamforming Frequency-selective channels Cooperative

abstract In this paper, we investigate the cooperative beamforming (BF) design problem for amplify-and-forward relay networks that are operating over frequency-selective channels using Orthogonal frequency division multiplexing (OFDM) signaling. We focus on the time-domain (TD) BF due to the lower implementation complexity and lower feedback requirement from the destination. Our aim for the BF design is maximizing the minimum signal-to-noise-ratio (SNR) over all subcarriers at the destination. We show that the BF designs lead to non-convex problems generally. Three approaches to approximate these problems into convex problems are proposed. In the first approach, semidefinite relaxation (SDR) techniques with randomization methods are applied. In the second approach, an existing iterative method for cooperative beamforming of multi-group multicasting (MGM) relay networks is extended to solve our problem. The third approach consists of improving the second approach by appropriately choosing the initial phase rotation. Simulation results demonstrate that the third approach always outperforms the other two approaches. Moreover, the performance of the second approach is close to that of the first approach when the filter length is relatively small. When a longer TD filter is employed, the second approach outperforms the first approach. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Recently, cooperative relay networks have been studied widely as they can exploit spatial diversity to improve network performance with the cooperation of multiple single-antenna terminals [1,2]. Cooperative diversity can be achieved using either amplify-and-forward (AF) or decode-and-forward (DF) protocols. In the AF relaying, the relay nodes simply scale their received signals, while in the

✩ This work is supported in part by National Natural Science Foundation of China (No. 60903206) and in part by Doctoral Fund of Ministry of Education of China (No. 20094307110004). ∗ Corresponding author. E-mail addresses: [email protected] (W. Cheng), [email protected] (M. Ghogho), [email protected] (Q. Huang), [email protected] (D. Ma), [email protected] (J. Wei).

1874-4907/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.phycom.2011.10.003

DF relaying, the relay nodes have to decode and re-encode the source messages. Thanks to its low implementation complexity, AF relaying is of practical interest. With partial or full knowledge of channel state information (CSI) in the AF relaying scheme, distributed beamforming can efficiently improve the capacity and reliability of relay networks [3]. Several distributed BF approaches for AF relay networks have been studied in [3–8] and the references therein. However, only flat fading channels were considered in those papers. With the demand of higher transmission rate, frequency-selectivity of the channel cannot be ignored. Unfortunately, techniques optimized for flat fading channels cannot be directly extended to the case of frequency-selective channels. Thus, designing distributed BF over frequency-selective channels is well motivated.

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Employing OFDM transmission is an effective approach for cooperative systems to compensate for the multi-path effects. BF design for cooperative OFDM networks over frequency-selective channels was considered in [9–11], but it was limited to the single relay case. Later, Liang and Schober [12] proposed frequency-domain (FD) and time domain (TD) BF schemes for the case of multiple relays. In FD BF scheme, the BF weights are applied in the FD while cyclic BF filters (C-BFFs) are used on the TD signal in TD BF schemes. Further, [12] optimized the FD-BF weights and the TD cyclic BF filters (C-BFFs) by maximizing the average mutual information (AMI) per sub-carrier. However, the optimization was subject to total power constraint (TPC) and the BF vectors were designed by ignoring the effects of source-to-relays CSI on the power constraint, which may undermine the TPC under some particular CSI conditions. Further, since each node in a network has its own battery generally, the per-relay power constraint (PPC) BF for AF relaying is more practical than TPC BF. This paper considers the BF design problem for the relay network with one source, one destination and multiple relays under the assumption of frequencyselective channels between nodes. Since the TD BF scheme requires lower implementation complexity and less feedback from the destination, we mainly focus on the TD BF design. Unlike [12] which used the AMI as design criterion, we design the TD C-BFFs weights by maximizing the minimum (max–min) signal-to-noise-ratio (SNR) at the destination over all subcarriers in order to achieve good quality of service (QoS) performance, subject to TPC and PPC. Moreover, the instantaneous CSIs between the source and the relays are taken into consideration; this will guarantee that the TPC or PPC can hold true for all channel realizations. Both TPC and PPC design problems are non-convex generally. To solve the BF design problem efficiently, we propose two approaches to approximate the original non-convex BF problems into convex problems. The first approach is using the semidefinite relaxation (SDR) technique, which approximately solve the BF design problem by a convex semidefinite programming (SDP) problem. The SDR approach provides an upper bound of the optimum value of the BF design problem but incurs high computational complexity, and the used randomization operation cannot guarantee good performance. The second approach consists of applying the existing iterative method for cooperative beamforming of multi-group multicasting (MGM) relay networks [13] to solve our problem. The third approach consists of improving the second approach by choosing the appropriate initial phase rotation. Simulation results show that the performance of the second approach is close to that of the first approach when the filter length is relatively small. When a longer TD filter is employed, the second approach outperforms the first approach. The third approach always performs better than the second approach with an SNR improvement of about 0.6 dB. The rest of this paper is organized as follows. We describe our model for OFDM based cooperative relay network in Section 2. In Section 3, we formulate the BF design problem based on max–min criterion and propose three approaches to solve it efficiently. In Section 4, we present the numerical results to verify our analysis. Section 5 presents the conclusions of this research.

Notations: Superscripts *, T and H denote the complex conjugate, transpose and Hermitian operators respectively. ⊙ denotes the Hadamard product. diag(x) denotes the diagonal matrix whose diagonal is vector x. diag(A) denotes the vector formed with the diagonal of matrix A. IN denotes the (N × N ) identity matrix. Lmax {A} denotes the maximum eigenvalue of matrix A and P {A} denotes the eigenvector corresponding to Lmax {A}. tr (A) represents the trace of matrix A and A ≽ 0 denotes that matrix A is a symmetric positive semidefinite matrix. |z | denotes the modulus of z and ̸ z denotes the phase of z. Finally, ‖x‖ denotes the L2 -norm of vector x. 2. System model Considering a relay network with a source, a destination and Nr relays as illustrated in Fig. 1. We assume that there is no direct link between the source and destination and Nr relays cooperatively establish the communication link between the source and destination. Each node is equipped with a single half-duplex antenna. The channel between nodes are assumed to be frequency-selective and mutually uncorrelated. Let hi denote the channel impulse response (CIR) from the source to the ith relay and wi represents the CIR from the ith relay to the destination. Without loss of generality, we assume that the frequency-selective channels hi , i = 0, . . . , Nr − 1 are of equal length, denoted by Lh . Similarly, wi , i = 0, . . . , Nr − 1 are of equal length, denoted by Lw . The AF relaying scheme consists of two phases. In the first phase, the source sends data to the relays. In this paper, the signaling is based on OFDM. In the second phase, the received signals at the relays are AF beamformed and then transmitted to the destination. As shown in Fig. 1, the data block d = [d(0), d(1), . . . , d(Nc − 1)]T , which satisfy E {|d(n)|2 } = Ps , are taken to be modulated by IDFT at the source. Therefore, the transmitted symbol vector x = [x(0), x(1), . . . , x(Nc −1)]T can be expressed as x = FH d

(1)

where F = √1N

c

e−j2π mn/Nc



Nc −1

m,n=0

is the (Nc × Nc ) DFT

matrix. After removing √ cyclic prefix (CP) and normalizing the signals by ρ = 1/ Ps + 1, the signal received at the kth relay is given by sk = ρ Hk x + ρ nk

(2)

where Hk is an Nc × Nc circulant matrix whose first column is [hk (0), . . . , hk (Lh − 1), 0, . . . , 0]T , nk is an additive white Gaussian noise (AWGN) vector with correlation matrix INc , where INc denotes the Nc × Nc identity matrix. For TD-BF systems, the above signal is cyclic filtered at each relay by a filter gk of length Lg ≤ Nc . The filtered signal can be expressed as tk = ρ Gk Hk x + ρ Gk nk

(3)

where Gk is an Nc × Nc circulant matrix whose first column is [gk (0), . . . , gk (Lg − 1), 0, . . . , 0]T . More details on the C-BFFs implementation can be found in [14].

W. Cheng et al. / Physical Communication 4 (2011) 305–312

307

Fig. 1. OFDM based AF relaying system model.

After removing the CP, the signal received at the destination can be written as r=ρ

Nr −

Wk Gk Hk x + ρ

Nr −

k=1

Wk Gk n k + η

(4)

k=1

where Wk is an Nc × Nc circulant matrix whose first column is [wk (0), . . . , wk (Lw − 1), 0, . . . , 0]T and η is an AWGN vector with correlation matrix INc . After the DFT, the signal at the destination can be expressed as R = ρ

Nr −

Wk,D Gk,D Hk,D d + ρ

k=1

Nr −

ˆ k + ηˆ Wk,D Gk,D n

(5)

k=1

where Wk,D = diag{Wk }, Wk = [Wk (0), . . . , Wk (Nc − 1)]T , ∑Lw −1 −j2π ln/Nc Wk (n) = , Gk,D = diag{Gk }, Gk = l=0 wk (l)e

3. Time-domain beamforming design In this section, we derive the Nr filters gk that maximize the minimum SNR over all subcarriers subject to TPC. For notation convenience, we define g = [gT0 , . . . , gTNr −1 ]. The max–min SNR problem under TPC can be written as max min SNR(n) n

s.t. Ptotal ≤ Pmax

(7)

where the SNR at the nth subcarrier is defined as SNR(n) = Psig (n)

, Psig (n) and Pnoise (n) being respectively the power of the signal and noise, Ptotal is the total power transmitted from the relays, and Pmax is the maximum allowable total power. From Eq. (3), the transmitted power of the kth relay can be expressed as Pnoise (n)

Pk = E {(tk )H tk }

  = ρ 2 Ps Gk Hk,D HHk,D GHk + Gk GHk .

(8)

∑Lg −1 −j2π ln/Nc [Gk (0), . . . , Gk (Nc − 1)], Gk (n) = , l=0 gk (l)e T Hk,D = diag{Hk }, Hk = [Hk (0), . . . , Hk (Nc − 1)] , Hk (n) = ∑Lh −1 −j2π ln/Nc ˆ k = Fnk , ηˆ = Fη are AWGN and n l=0 hk (l)e

Since Gk can be written as Gk = gQk , where Qk is an Lg Nr × Nc matrix with the ith column given by [01×(k−1)Lg ,

vectors with correlation matrix INc . Therefore, the signal received on the nth subcarrier can now be expressed as

Pk = gTk gH

where Tk = ρ + . Thus, the total power transmitted from all relays can be expressed as

 + ρ G(n) W(n) ⊙ nˆ (n) + η( ˆ n) 

T

channel vector, W(n) = W1 (n), . . . , WNr (n) , H(n) =



T ˆ (n) = nˆ 1 (n), . . . , nˆ Nr (n) . and n

H1 (n), . . . , HNr (n) For FD-BF systems, the relays apply the BF vector in the FD, which requires additional DFT and IDFT operations at the relays. Moreover, for FD-BF scheme, Nr Nc complex weights are required to be fed back (Nc weights per relay) from the destination to the relays to perform BF, whereas for the TD-BF scheme, the feedback amounts to only Nr Lg complex weights. As pointed out by Liang and Schober [12], TD-BF is equivalent to FD-BF when Lg = Nc . Due to the advantage of the TD-BF scheme in terms of complexity and feedback requirements, we will focus on the TD-BF design problem in the following.





H Ps Qk Hk,D HH k,D Qk

Qk QH k



(6)

where G(n) = [G1 (n), . . . , GNr (n)] is the beamforming vector, Heq (n) = W(n) ⊙ H(n) is the equivalent

T

(9) 2

R(n) = ρ G(n)Heq (n)d(n)



1, e−j2π (i−1)/Nc , . . . , e−j2π (Lg −1)(i−1)/Nc , 01×(Nr −k)Lg ]T , Eq. (8) can then be rewritten as

Ptotal =

Nr −

Pk = gTgH

(10)

k=1

∑N

−1

r where T = k=0 Tk . It is worth pointing out that T in the above equation was ignored in [12]. From Eq. (6), the signal power at the nth subcarrier can be written as

H

Psig (n) = ρ 2 Ps G(n)Heq (n) Heq (n)



(G(n))H

(11)

and the noise power at the nth subcarrier can be obtained as



˜ ( n) W ˜ (n) Pnoise (n) = ρ 2 G(n)W

H

(G(n))H + 1

(12)

˜ (n) , diag{W(n)}. Noting that G(n) = where W gK(n), where K(n) is an Lg Nr × Nr matrix with the ith column given by [01×(i−1)Lg , 1, e−j2π (n−1)/Nc , . . . ,

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W. Cheng et al. / Physical Communication 4 (2011) 305–312

e−j2π (Lg −1)(n−1)/Nc , 01×(Nr −i)Lg ]T , Eq. (11) can be rewritten as Psig (n) = ρ Ps gβ(n) (β(n)) g H

2

H

(13)

3.1. SDR method Using the definition X , g˜ H g˜ , problem (19) can be equivalently rewritten as max

and Eq. (12) can be rewritten as



Pnoise (n) = ρ 2 gJ(n)gH + 1

(14)

Ps tr s.t.

where β(n) = K(n)Heq (n) and



˜ ( n) W ˜ (n) J(n) = K(n)W

H

min n

s.t.

(K(n))H .

Ps gβ(n) (β(n))H gH

max

Ps tr

(15) s.t.

gTgH ≤ Pmax .

X ≽ 0.

s.t.

(16)

g˜ g˜ H ≤ Pmax 1

1

˜ n) = T− 2 β(n) and ˜J(n) = T− 2 J(n)T− 2 . It can be where β( easily proved that the constraint in problem (16) should be met with equality at the optimum solution. Therefore, (16) can be rewritten as 

˜ n) β( ˜ n) Ps g˜ β( max

min

g˜ H (17)

g˜ ¯J(n) g˜ H

n

s.t.

H

H

g˜ g˜ = Pmax

where ¯J(n) = ˜J(n) + ((Ps + 1)/Pmax ) ILg Nr . Noting that any scaling of g˜ does not change the value of the objective function in (17), Eq. (17) is equivalent to the following unconstrained optimization problem



H

˜ n) β( ˜ n) Ps g˜ β( max min n

g˜ ¯J(n) g˜ H

g˜ H

.

(18)

However, the obtained vector from (18) has to be properly scaled to satisfy the power constraint. Introducing a new auxiliary variable τ , Eq. (18) can be equivalently transformed to max s.t.

τ

 H ˜ n) β( ˜ n) g˜ H Ps g˜ β( g˜ ¯J(n) g˜ H

≥ τ,



n = 0,

H

. . . , Nc − 1

(21)

− τ ¯J(n), n = 0, . . . ,

Nc − 1, problem (21) is equivalent to

g˜ ˜J(n)˜gH + Ps + 1

1

 H  ˜ n) β( ˜ n) X β(   ≥ τ, tr ¯J(n)X

˜ n) β( ˜ n) Letting Q(n) = Ps β(

 H ˜ n) β( ˜ n) g˜ H Ps g˜ β( n

τ 

gJ(n)gH + Ps + 1

1

min

n = 0, . . . , Nc − 1 (20)

Problem (20) is still non-convex due to the nonconvexity of the rank-one constraint. By dropping the rank constraint, we get the semidefinite relaxed problem, which can be expressed as

Letting g = g˜ T− 2 , problem (15) can then be rewritten as

max

 H  ˜ n) β( ˜ n) X β(   ≥ τ, tr ¯J(n)X

X≽0 rank (X) = 1.

From Eq. (13) to (14) and Eq. (10), problem (7) can be mathematically formulated as follows: max

τ

(19) n = 0, . . . , Nc − 1.

The problem in (19) is generally non-convex. We will propose three approaches to solve our problem approximately.

max s.t.

τ

tr (Q(n)X) ≥ 0, X ≽ 0.

n = 0, . . . , Nc − 1

(22)

Problem (22) is quasi-convex, because for any value of

τ , it turns to the following convex feasibility problem: find s.t.

X tr (Q(n)X) ≥ 0, X ≽ 0.

n = 0, . . . , Nc − 1

(23)

In fact, the global optimum value τ ∗ can be obtained by employing bisection search method, which solves the feasibility problem (23) at each step. Assuming that τ ∗ lies in the interval [τl , τr ], the bisection search method procedure can be summarized as follows. 1. Let τ , (τl + τr )/2, solve the feasibility problem (23). 2. If (23) is feasible, then τl = τ , otherwise τr = τ . 3. Repeat step 1 with the new interval until (τr − τl ) < ε , where ε is the error tolerance value. Remark 1. Note that the feasibility problem (23) is a standard semidefinite programming (SDP) problem, which can be solved using interior point methods [15]. Remark 2. It is easy to see from (18) that the maximum SNR achieved at the nth subcarrier is

  H   −1 ˜ n) β( ˜ n) γ (n) = Lmax Ps ¯J(n) β( [16]. Thus the initial interval for the bisection search can be selected as [0, minn γ (n)]. The matrix X∗ , which can achieve the optimum τ ∗ is the global optimum vector for problem (20) but may not be the optimum vector for problem (18). Due to the relaxation, the matrix X∗ will not be of rank one in general. For the cases where the matrix X∗ is rank-one, then τ ∗ is also the global optimum vector for problem (18) and its

W. Cheng et al. / Physical Communication 4 (2011) 305–312

principal component is the optimal solution to problem (18). For the cases where the matrix X∗ has rank higher than one, τ ∗ is an upper bound for problem (18) and randomization techniques have to be employed to get a good approximation of the rank-1 solution for problem (18) from X∗ . Randomization technique: The basic idea of randomization is to use X∗ to generate a set of candidate vectors and then select the best solution g˜ ∗ among these candidates. Three randomization methods are considered in this paper.

• randA: Calculate the eigen-decomposition of X∗ = V3VH and choose g˜ A,k = V31/2 ek , where the elements of ek are independent random variables, uniformly distributed on the unit circle in the complex plane; • randB: Choose g˜ B,k = diag (diag (X∗ ))1/2 ek ; • randC: Choose g˜ C ,k = V31/2 ek , where the elements of ek are zero-mean, unit-variance complex circularly symmetric uncorrelated Gaussian random variables.

˜∗

After obtaining the solution g for problem (18), the optimum vector for problem (15) is g∗ =



1

Pmax g˜ ∗ T− 2 /‖˜g∗ ‖.

(24)

3.2. Iterative method In the SDR approach, the number of design variables is increased from Lg Nr to (Lg Nr )2 , which increases the computational complexity. Moreover, the randomization methods cannot guarantee good solutions. Introducing a new variable t ≥ 0 such that τ = t 2 , Eq. (19) can also be equivalently transformed to max

˜ n) β( ˜ n) Ps g˜ β(

H

g˜ H

g˜ ¯J(n)˜

gH

≥ t 2,

(25) n = 0, . . . , Nc − 1.

which is equivalent to max

t



s.t.

˜ n) β( ˜ n) Ps g˜ β(

H

g˜ H

≥ t 2,

(26) n = 0, . . . , Nc − 1.

g˜ ¯J(n) ˜ ¯ Since J(n) is a Hermitian matrix, its singular value decomposition (SVD) can be expressed as ¯J(n) = U(n)D(n)U(n)H , where D(n) is a unitary matrix and V(n) 1 is a diagonal matrix.1 Introducing S(n) = U(n)D(n)− 2 , gH

problem (26) is equivalent to max s.t.

t

˜ n)| ≥ t ‖˜gS(n)‖, Ps |˜gβ(

n = 0, . . . , Nc − 1.

(27)

The above problem can be solved through the bisection search method by solving the following feasibility problem for some given t at each step: find s.t.

˜ g 

We notice that a problem similar to (28) has been studied in [13] for solving the beamforming design problem for MGM relay networks. Thus the idea proposed in [13] could be extended to solve our problem. Further, in the next section, we will propose a searching procedure for choosing the appropriate phase rotation at the initial iteration, which can improve the performance of this iterative method.   ˜ n)| by Re g˜ β( ˜ n) , we get a restricted Replacing |˜gβ( version of the feasibility problem (28), which can be expressed as find

g˜

s.t.







˜ n) ≥ t ‖˜gS(n)‖, Ps Re g˜ β(

˜ n)| ≥ t ‖˜gS(n)‖, Ps |˜gβ(

n = 0, . . . , Nc − 1.

n = 0, . . . , Nc − 1.

(29)

Introducing the real-valued vector c = [c0 , . . . , cNc −1 ]T , we can formulate a new feasibility problem min

1T c

s.t.

t ‖˜gS(n)‖ −

g˜ ,c







˜ n) ≤ cn , Ps Re g˜ β(

(30)

n = 0, . . . , Nc − 1. Problem (30) is always feasible and can be solved by Algorithm I in [13]. We denote the optimal solution of problem (30) by copt and g˜ opt . If 1T copt = 0, we assume that problem (28) is feasible and vice versa.

˜ n)1 = β( ˜ n), n = 0, . . . , Nc − Algorithm I. Initialization: β( 1 for i = 1: I ˜ n) = β( ˜ n)i and the optimum Solve problem (30) with β( solution is denoted by g˜ opt . i ˜ n)i+1 = β( ˜ i Perform the rotation: β(  n) exp  (−jα(n) ) n = 0, . . . , Nc − 1, where α(n)i = ̸

t2



s.t.

309

˜ n)i . end g˜ opt β(

Thus the whole bisection search method procedure for solving problem (27) can be summarized as follows. Assuming that t ∗ lies in the interval [tl , tr ], 1. let t , (tl + tr )/2, solve the problem (30) by Algorithm I; 2. if 1T copt = 0, then tl = t, otherwise tr = t; 3. repeat step 1 with the new interval until (tr − tl ) < ε , where ε is the error tolerance value. Then tl is the optimum value for the objective function of problem (27). Let t = tl , solve problem (30) by ˜ n)I +1 is obtained, replace β( ˜ n) of Algorithm I. After β( ˜ n)I +1 and solve this new problem by problem (29) with β( Algorithm I. Then the obtained g˜ opt is the optimum solution for problem (27). Remark 3. Note that problems (29) and (30) are standard second-order cone programming (SOCP) problems with Nc second-order cone constraints, which can be solved using interior point methods [15]. Remark 4. The initial √ interval for the bisection search can be selected as [0, minn γ (n)].

(28)

1 Since ¯J(n) is not always positive definite, the Cholesky factorization of ¯J(n) may not exist.

3.3. Initial phase rotation searching algorithm The iterative algorithm described above is based on the ˜ n)| with its real part, initially, and approximation of |˜gβ(

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W. Cheng et al. / Physical Communication 4 (2011) 305–312

then improve this approximation successively by applying an appropriate rotation to the individual channel vectors ˜ n) is not in each iteration. However, the real part of g˜ β( always a good initial approximation of its modulus. In this subsection, we propose a searching procedure to obtain good initial approximation. We introduce the following lemma first. Lemma (Linear Approximation of Modulus [17]). The modulus of a complex number Z ∈ C can be linearly approximated with the polyhedral norm given by



pL (Z ) = max Re{Z } cos



lπ



L

l∈L

 + Im{Z } sin

lπ L

pL (Z ) ≤ |Z | ≤ pL (Z ) sec

lπ

(31)

 (32)

L

and L is a positive integer such that L ≥ 2.



It can be seen from Lemma that the modulus of a complex number can be approximated by maxl=0,1,...,2L−1 −j lπL

˜ n)| with Re(˜gβ( ˜ n) }. Thus we can replace |˜gβ( ), l = 0, . . . , 2L − 1 in problem (27) and choose

Re{Ze −j lπL

e the best solution lopt among these 2L candidate solutions. However, when L is large, the complexity may be problematic in practical systems. The proposed searching algorithm for the best lopt is described as follows.

Algorithm II. Initialization: Set feasible section = [0, 1, . . . , 2L − 1], search section = [0, 1, . . . , 2L − 1]. Set num = 0; Iteration k + 1: (a) If length (feasible section) = 1, then terminate; the only element of feasible section is the optimal lopt . (b) If num > nummax , then terminate; randomly choose lopt in feasible section. (c) If length (feasible section) ̸= 1 and num <= nummax , then

• Set t , (tl + tr )/2;   ˜ n)) with Re g˜ β( ˜ n)e−j lπL , l = 0, . . . , • Replace Re(˜gβ( 2L − 1 in problem (30) and solve these 2L problems by Algorithm I; • If neither of the optimum objective function values of these 2L problems is zero, set tr = t and search section = feasible section; • If some of the optimum objective function values of these 2L problems are zero, the corresponding values of l combines new feasible section. Set tl = t and search section = feasible section; • increase num by 1 and repeat the iteration process.

˜ n)| is Once the best lopt is obtained by Algorithm II, |˜gβ( 

˜ n)e−j replaced with Re g˜ β(

lopt π L



max min SNR(n) n

s.t. Pk ≤ Pk,max , k = 0, 1, . . . , Nr − 1

(33)

where Pk is the total power transmitted from the kth relay, and Pk,max is the maximum allowable power of the kth relay. From Eq. (13) to (14) and (9), problem (33) can be mathematically formulated as follows:



where L = {1, 2, . . . , 2L}, Re{Z } and Im{Z } denote the real and imaginary parts of Z and the polyhedral norm pL (Z ) is bounded by



PPC case: For the PPC case, the max–min SNR problem can be written as

max

Ps gβ(n) (β(n))H gH

min

gJ(n)gH + Ps + 1

n

s.t.

gTk gH ≤ Pk,max ,

(34) k = 0, 1, . . . , Nr − 1.

Problem (34) is also non-convex in general. However, it can be efficiently solved by the three approaches proposed before. Using the definition X , gH g, problem (34) can be equivalently rewritten as max

τ Ps tr β(n) (β(n))H X



s.t.



≥ τ , n = 0, . . . , Nc − 1 tr (J(n)X) + Ps + 1 (35) tr (Tk X) ≤ Pk,max , k = 0, 1, . . . , Nr − 1 X≽0 rank (X) = 1.

Problem (35) is similar to (20) except that problem (35) has extra Nr semidefinite matrix constraints. Thus the optimization problem for the TD-BF scheme subject to PPC can be solved similarly by the SDR method proposed in Section 3.1. Letting

˘J(n) =

[

Ps + 1

01×Nr Lg

0Nr Lg ×1

J(n)

] (36)

problem (34) can also be equivalently rewritten as max

t

s.t.



˘ n)| ≥ t ‖˘gM(n)‖, Ps |˘gβ(

‖˘gV˘ k ‖ ≤

Pk,max ,



n = 0, . . . , Nc − 1

k = 0, . . . , Nr − 1

(37)

g˘ e1 = 1,

˘ k = [0Nr Lg ×1 VTk ]T , e1 is the first where g˘ = [1 g], V ˘ n) = [0 β(n)T ]T , Tk = Vk VH and column of ILg Nr , β( k ˘J(n) = M(n)M(n)H .2 Problem (37) is similar to (27) except that problem (37) has extra one linear constraint and Nr second-order conic constraints. Thus the optimization problem for the TD-BF scheme subject to PPC can also be solved similarly by the iterative method proposed in Sections 3.2 and 3.3.

in problem (27) and then

the optimum solution can be obtained.

2 V and M(n) can be obtained from the SVD of T and ˘J(n), respectively. k k

W. Cheng et al. / Physical Communication 4 (2011) 305–312

311

4. Simulation results In this section, the performance of TD BF by three approaches are evaluated for a cooperative network with Nr = 2 relays. The total number of data (subcarriers) of each block is Nc = 64 and the data symbols are drawn from BPSK constellations. Lh and Lw are set to 10 in the simulations. We also assume that the channel taps hi (l), wi (l) are uncorrelated zero-mean complex Gaussian random variables with exponential power delay profile pl = λ exp(−0.2l), where λ is a scalar that ensures that the total energy of the channel taps is normalized to unity. The channels corresponding to different relays are assumed uncorrelated and known perfectly at the destination. The latter performs BF design and feedback a BF vector to each relay. In this paper, we assume that this feedback is error-free. In the first approach, 100 candidates are generated for each randomization technique. The value of I in Algorithms I and II is chosen to be 5. In Algorithm II, num is set to be 5 and L is set to be 2. Fig. 2 depicts the achieved minimum SNR over all subcarriers under TPC versus the total power of a data block per relay for TD BF with Lg = 4. The label ‘bound’ represents the non rank-one solution from the SDR method, which cannot be achieved. The label ‘randA’, ‘randB’ and ‘randC’ represent the rank-one solution from the three randomization methods mentioned in Section 3.1, respectively. The label ‘iterative’ represents the solution from the iterative method proposed in 3.2 and ‘iterative with search’ represents the solution from the method proposed in 3.3. It is shown that the ‘iterative with search’ method outperforms other algorithms, with about 0.6 dB SNR improvement, compared to the ‘iterative’ method. The ‘randC’ method performs closely to the ‘iterative’ method and performs better than other two randomization techniques. Fig. 3 depicts the achieved minimum SNR under TPC versus the total power of a data block per relay for TD BF with Lg = 6. As can be observed, the ‘iterative with search’ method also outperforms other algorithms, with about 0.6 dB SNR improvement, compared to the ‘iterative’ method. The ‘iterative’ method performs better than SDR techniques with randomization methods and ‘randC’ has the best performance among the three randomization techniques. Fig. 4 shows the performance of the BF schemes under TPC versus filter length Lg with Pmax /Nr = 21 dB. From Fig. 4, we can find that the minimum SNR over all subcarriers is improved with the increase of the filter length. However, the performance improvement is limited when Lg ≥ 10. It implies that employing relatively short filter is sufficient to achieve the best performance of TD-BF.

Fig. 2. SNR versus average relay power, Lg = 4.

Fig. 3. SNR versus average power, Lg = 6.

5. Conclusion In this paper, we addressed the problem of BF design for OFDM based cooperative networks over frequencyselective channels. Based on max–min SNR criterion, the TD BF vectors were obtained through three methods. The first method was based on SDR technique with

Fig. 4. SNR versus filter length, Pmax /Nr = 21 dB.

randomization methods. The second approach was an iterative method which was further improved in the third

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Wenjing Cheng received the B.Sc. degree in Communication Engineering from the National University of Defense Technology (NUDT), Changsha, P.R. China in 2005. She is currently pursuing the Ph.D. degree with the School of Electronic Science and Engineering, NUDT. Her research interests are in the area of signal processing for wireless communications with emphasis on distributed space time coding and beamforming for cooperative systems.

Mounir Ghogho received the M.S. degree in 1993 and the Ph.D. degree in 1997, both in Signal and Image Processing from the National Polytechnic Institute (INP), Toulouse, France. He was an EPSRC Research Fellow with the University of Strathclyde, Glasgow, from September 1997 to November 2001. Since December 2001, he has been a faculty member with the school of Electronic and Electrical Engineering at the University of Leeds, where he is currently a Professor. He is also currently a Professor at the International University of Rabat in Morocco. He served as an Associate Editor of the IEEE Signal Processing Letters from December 2001 to December 2004. He is currently an associate editor of the IEEE Transactions on Signal Processing and a member of both the IEEE Signal Processing Society SPCOM and the IEEE Signal Processing Society SPTM Technical Committees. He was the technical co-chair of the MIMO symposium of IWCMC 2007 and IWCMC 2008, a technical area co-chair of Eusipco 2008, Eusipco 2009 and ISCCSP’05, and a guest co-editor of the EURASIP Journal on Wireless Communications and Networking special issue on ‘‘Synchronization for wireless communications’’. He is the general co-chair of the eleventh IEEE workshop on Signal Processing for Advanced Wireless Communications (SPAWC’2010). His research interests are in signal processing for communications and networking. Prof. Ghogho was awarded a five-year Royal Academy of Engineering Research Fellowship in September 2000.

Qinfei Huang received the B.Sc. (first class) degree in Communication Engineering from the National University of Defense Technology (NUDT), Changsha, P.R. China in 2004. He is currently pursuing the Ph.D. degree with the School of Electronic Science and Engineering, NUDT. His research interests are in the area of signal processing for wireless communications with emphasis on receiver design for cooperative systems.

Dongtang Ma received the B.S. degrees in applied physics in 1990, and the M.S. and Ph.D. in information and communication engineering in 1997 and 2004 respectively, all from NUDT. From 2004 to 2009, he was an Associate Professor of the school of electronic science and engineering in NUDT and member of the Broadband Communication and Network Research Group, where he worked on the system analysis, resource management and system evaluation for constellation satellite communication system. Since January 2010 he has been a professor. His research interests include several aspects of cooperative communication and network: cooperative MIMO, cooperative MAC protocol, multiuser MIMO, cooperative transmit beam forming, and dynamic resource allocation as well as high altitude platform station communication and satellite communication techniques.

Jibo Wei received his B.Sc. degree and M.Sc. degree from the National University of Defense Technology (NUDT), Changsha, P.R. China, in 1989 and 1992, respectively, and the Ph.D. degree from Southeast University, Nanjing, P.R. China, in 1998, all in electronic engineering. He is currently the director and a professor of the Department of Communication Engineering of NUDT. His research interests include wireless network protocol and signal processing in communications, more specially, the areas of MIMO, multicarrier transmission, cooperative communication, and cognitive network. Prof. Wei is a member of the IEEE Communication Society and also a member of the IEEE VTS. He also works as one of the editors of Journal on Communications and the senior member of China Institute of Communications and Electronics respectively.