A Practical Transmission Scheme for Half-Duplex Decode ... - MWNL

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A Practical Transmission Scheme for Half-Duplex Decode-and-Forward MIMO Relay Channels ∗ International

∗ Edwin

Monroy, † Sunghyun Choi, and ‡ Bijan Jabbari

IT Policy Program, College of Engineering, Seoul National University, Seoul, Korea of Electrical and Computer Engineering and INMC, Seoul National University, Seoul, Korea ‡ Department of Electrical and Computer Engineering, George Mason University, Fairfax, VA, USA Email: [email protected], [email protected], [email protected]

† Department

Abstract—We propose a novel and practical transmission strategy for decode-and-forward MIMO relay channels operating in half-duplex mode that is based on precoding with time, power, and spatial stream allocation. In the proposed scheme, the message from the source is divided into two parts and sent to the destination, one part directly and the other with the help of the relay. We formulate the problem of finding the rate-maximizing precoding matrices and resource allocation and then transform it into a convex form by employing the proposed suboptimal but simplifying precoding structure and performing other manipulations. The numerical results show that the achievable rate of the proposed scheme is close to a well-known theoretical lower bound and even outperforms it in certain scenarios, despite our transmission strategy being based on a much simpler coding scheme. Additionally, the proposed scheme is able to outperform another scheme based on repetition coding with fixed time slot duration. Index Terms—MIMO Relay, half-duplex relay, partial Decode and Forward, precoding, convex optimization.

I. I NTRODUCTION In wireless networks, the core idea of relaying is to introduce new or designate existing network nodes as helpers that can assist data transmission between sources and destinations. This concept has been around in the literature for quite a long time. One of the earliest and more fundamental works on this subject is the one by Cover and El Gamal [1], where they introduce an upper bound on the capacity of the relay channel, known as the cut-set bound (CSB), along with a very well-known lower bound or achievable rate later called Decode and Forward (DF) [2], among others. Later studies have looked into more practical schemes specifically targeting wireless environments. In the case of MIMO relay channel in DF mode, [3], [4] in the full-duplex case, and [5], [6] in the half-duplex case, provide optimal upper and lower bounds mostly through convex formulations. In particular, [3] introduces two transmission strategies for the full-duplex DF relay channel based on superposition coding (SC) or dirtypaper coding (DPC) at the source. A similar strategy based on SC is followed in [5] for the half-duplex case. However, dirty-paper precoding is known to be very complex and coding schemes that implement such an approach are not readily available in practice. Moreover, the relay operates in full-duplex mode and the resulting problem is not convex. Similarly, the transmission strategies proposed in [5] are still difficult to implement in practice since the codebooks used by

source and relay are correlated, and the correlation matrix is optimized for each channel realization. Some other studies on the half-duplex MIMO relay channel, such as [7], [8], ignore the direct link and, therefore, do not take advantage of the possible increase in diversity or rate that the direct link could offer when available. In this work, we propose a novel partial DF transmission scheme for the MIMO relay channel and study the optimal precoder design with spatial stream (SS), time, and power (we call them “resources” in this paper) allocation at the source and relay. Specifically, in our scheme the source splits its messages in two independent parts in each time slot and sends one of them directly to the destination and the other via the relay. Based on this transmission scheme, we formulate an optimization problem to find the source and relay precoders and resource allocation that maximize the achievable rate, and, through the proposed precoding design, convert the original problem into a convex form that can be solved numerically in an efficient way. We employ several techniques such as linear precoding based on Block Diagonalization (BD) [9], repetition coding at the relay [2], and half duplexing with the purpose of assuring the practicality of our scheme. The remainder of the paper is structured as follows. We describe the system model in Section II. The optimization problem to obtain the achievable rate of the proposed scheme is introduced in Section III. Section IV presents the proposed precoding design with resource allocation, whereas Section V shows the numerical results. Our conclusion follows in Section VI. II. S YSTEM M ODEL We consider a three-node topology with one source, relay and destination, each with NS , NR , and ND antennas, respectively. The MIMO channels from source to relay (SR), source to destination (S-D), and relay to destination (RD) are denoted by HSR ∈ CNR ×NS , HSD ∈ CND ×NS , and HRD ∈ CND ×NR , respectively, and are assumed to be perfectly known at all nodes. These matrices can be further expressed √ ¯ ¯ as Hi = αi H i , i = {SR, SD, RD}, where Hi are random and independent matrices whose elements are independent and identically distributed (i.i.d.) zero mean, unit variance, circular symmetric complex Gaussian random variables. The factor αi contains path loss (it might also account for shadowing, but for simplicity we do not consider it in this paper), and can

be written as αi = d−n i , where di represents the distance between the two nodes on link i, and n is the path loss exponent (assumed to be equal to 3 in this paper). The noise at each receiver is an i.i.d. complex Gaussian random vector with zero mean and identity variance, and independent of the transmitted signals. The source and relay nodes are subject to maximum transmit power constraints PS and PR , respectively. We assume that the relay operates in half-duplex mode in time, i.e., any packet transmission is completed in two time slots. The channel matrices are assumed to remain constant during both time slots. A superscript (i) identifies a signal or matrix valid during the first (i = 1) or second (i = 2) time slot. During the first time slot, the source transmits a signal x(1) ∈ CNS ×1 that consists of two parts (that we call S direct and forwarding, respectively): x(1) D , transmitted directly over the direct link, and x(1) that will eventually reach the F destination through the relay, as illustrated in Fig. 1a. We employ a linear precoding scheme to separate the two signals (1) (1) (1) (1) (1) (1) so that x(1) ∈ CNS ×mf and S = FF xF + FD xD . Here, FF (1) F(1) ∈ CNS ×md are the source precoding matrices in the D (1) (1) first time slot; x(1) ∈ Cmf ×1 and x(1) ∈ Cmd ×1 ; and F D (1) m(1) f and md represent the number of SSs used to transmit (1) the two signals, and will be defined later. Signals x(1) F and xD are independent, complex Gaussian vectors with zero mean and unitary covariance Im(1) and Im(1) , respectively. The source d

f

(1)H (1)H precoding matrices shall satisfy tr(F(1) +F(1) F FF D FD ) ≤ P S . The received signals at the relay and destination are given by, respectively,

y(1) R y(1) D

(1) (1) (1) (1) = HSR F(1) F xF + HSR FD xD + nR , (1) (1) (1) (1) = HSD F(1) F xF + HSD FD xD + nD ,

(1) (2)

(1) where n(1) R and nD represent the noise vectors at the relay and destination, with variance INR and IND , respectively. The relay processes y(1) R to obtain an estimate of the source’s transmission, denoted as x(2) F , which is then precoded and forwarded to the destination as signal xR = GR x(2) F , where (2) GR ∈ CNR ×mf is the relay precoding matrix. In processing (1) y(1) R , xD is not decoded and is regarded as interference by (1) the relay. Similar treatment is given to x(1) F in yD at the destination. Hence, the destination has yet to receive a copy of the forwarding part of the source’s message. In order to realize the cooperation between source and relay in transmitting such a message, they have to send the same signal to the destination. Thus, the source also sends x(2) F in the second time slot (repetition coding [2]). In addition, it also transmits a new signal x(2) D , corresponding to the direct part of the source’s message. The two signals are transmitted over the SSs on the direct link, as shown in Fig. 1b. Then, the signal transmitted by the source during the second time slot (2) (2) (2) (2) (2) is x(2) = F(2) ∈ CNS ×mf and F xF + FD xD . Here, FF S (2) F(2) ∈ CNS ×md are the source precoding matrices in the D (2) (2) m(2) d ×1 and x second time slot; and x(2) ∈ Cmf ×1 are D ∈ C F independent, complex Gaussian vectors with zero mean and unit covariance Im(2) and Im(2) , respectively. Moreover, m(2) d and d

f

NR Relay

XF(1)

XD(1)

Source

Destination ND

NS

(a) First time slot NR Relay

XF(2)

XF(2) Source

Destination

XD(2) ND

NS

(b) Second time slot

Fig. 1: Operation of the proposed scheme.

m(2) f represent the number of SSs used to transmit the signals, as defined later. The precoding matrices at the source and relay (2)H (2) (2)H H shall meet tr(F(2) F FF +FD FD ) ≤ PS and tr(GR GR ) ≤ PR . Finally, the signal received at the destination in the second time slot is (2) (2) (2) (2) (2) y(2) D = HSD FD xD + HSD FF xF + HRD xR + nD ,

(3)

where n(2) D is the noise vector at the destination with variance IND . III. ACHIEVABLE R ATE The achievable rate of the proposed scheme can be obtained through the partial cooperation strategy defined by Theorem 7 in [1], similar to the full duplex scenario studied in [3] RP S =

max (i) (i)H

t∈(0,1), FF FF

RF + RD ,

≽0

(4)

(i) (i)H

FD FD ≽0, GR GH R ≽0 (i) m(i) f , md , i=1,2

s.t.

(i) (i)H

tr(FF FF

tr(GR GH R)

(i) (i)H

+ FD FD ) ≤ PS , i = 1, 2

≤ PR ,

(5) (6)

where variable t represents the proportion of time allocated to the first slot. Intuitively, the rate RP S of the proposed scheme can be interpreted as the sum of RF , the rate of (i) forwarding part of the source’s message (xF , i = 1, 2), and (i) RD , that of the direct part (xD , i = 1, 2). We assume that the (i) destination decodes signals x(2) F and xD sequentially in each time slot. The order in which this is done can yield different resulting rates, so that we obtain the rates achieved when either signal is decoded first and select the decoding order that gives the maximum RP S . When the destination decodes (2) xF first, by denoting the corresponding rates with subindex

(4) becomes RP S,F = max(RF,F + RD,F ) and { (1) (2) (2) RF,F = min tI(x(1) F ; yR ) + (1 − t)I(xF ; yD | xR ), } (1) (2) (2) tI(x(1) (7) F ; yD ) + (1 − t)I(xF xR ; yD ) ,

(,F ) ,

(1) (2) (2) (2) RD,F = tI(x(1) D ; yD ) + (1 − t)I(xD ; yD | xF xR ).

(8)

The expressions for RF,F are as follows: (1)

I(x(1) F ; yR ) =   H (1) (1)H (1) (1)H det(INR + HSR (FD FD +FF FF )HSR )  , (9) log2  (1) (1)H H det(INR + HSR FD FD HSR ) (2)

(2)

(2)

(2) (2) I(x(2) F ; yD |xR ) = H(yD |xF ) − H(yD |xF )= 0,

(10)

(1)

I(x(1) F ; yD ) = ) ( (1) (1)H (1) (1)H det(IND + HSD (FD FD +FF FF )HH SD ) , log2 (1) (1)H det(IND + HSD FD FD HH SD ) (11) (2)

I(x(2) F xR ; yD ) = ( ) (2) (2)H H det(IND + HSD FD FD HH SD + HD FHD ) log2 , (2) (2)H det(IND + HSD FD FD HH SD ) (12) where HD = [HSD

HRD ] and [ (2) (2)H FF FF F= (2)H GR FF

(2)

FF GH R GR GH R

] .

(13)

Note that the mutual information in (10) is zero due to the repetition coding strategy employed in the second time slot. The corresponding expressions for RD,F are as follows: ( ) (1) (1)H det(IND + HSD FD FD HH (1) (1) SD ) I(xD ; yD ) = log2 , (14) (1) (1)H det(IND + HSD FF FF HH SD ) (2) (2) I(x(2) D ; yD | xF xR ) = (2) (2)H

log2 (det(IND + HSD FD FD

HH SD )). (15)

(i)

The expressions for RF,D and RD,D (when xD is decoded first) are obtained similarly and are omitted. The optimization in (4) could then be carried out as it is but it seems highly nontrivial and nonconvex. Therefore, suboptimal approaches are in order to tackle the problem. In the next section, we propose a linear precoding scheme with resource allocation that allows us to transform problem (4)-(6) so that it can be solved more easily. IV. P RECODING D ESIGN WITH R ESOURCE A LLOCATION In this section, we introduce the proposed precoding and resource allocation scheme that are applied in each time slot, which allow us to reformulate (4)-(6) as a convex optimization problem and solve it efficiently.

A. Precoding and Power Allocation in the First Time Slot It can be seen from (1) and (2) that, in order to nullify interference at the relay and destination, we can employ BD (1) to let F(1) F and FD lie in the nullspaces of HSD and HSR , respectively. This can be achieved by following the method in [9]. Let NS ≥ ND , so that the singular value decomposition (SVD) of HSD reads as [ ]H . (16) HSD = USD [ΣSD 0] VSD V(0) SD Defined in such a way, V(0) SD ∈ Null (HSD ), so that if we (0) B , we are able to meet the zero-interference make F(1) = V F SD F requirement. Matrix BF is used for eigenmode decomposition and power allocation of the equivalent channel as explained later. An identical procedure is performed to obtain F(1) D = V(0) B . After this, the equivalent channels on the S-R and SSR D D links are, respectively, the NR ×(NS −ND ) matrix HSR V(0) SD and the ND × (NS − NR ) matrix HSD V(0) SR . Water filling can now be separately performed over each link to optimize the sum rate from source to relay and destination. Then, we can 1 1 ˜ SR P 2 and BD = V ˜ SD P 2 , where V ˜ SR and V ˜ SD are make BF = V F D the matrices of left singular vectors of each equivalent channel 1 1 while PF2 and PD2 are the diagonal power allocation matrices with tr (PF + PD ) ≤ PS . From this, the precoding matrices can now be written as 1

1

(0) ˜ (1) (0) ˜ 2 2 F(1) F = VSD VSR PF , FD = VSR VSD PD ,

(17)

the dimensions of which are NS ×(NS −ND ) and NS ×(NS − NR ), respectively. In Section II, these matrices were defined (1) as having dimensions NS × m(1) f and NS × md , respectively. (1) Therefore, we now have mf ≤ min (NR , NS − ND ) and m(1) d ≤ min (ND , NS − NR ). It can be seen from (16) that BD requires NS > max (NR , ND ) in order to guarantee that the (0) matrices V(0) SD and VSR exist. Thus, if we make NS ≥ NR +ND (1) then mf ≤ NR and m(1) d ≤ ND and we are able to use up to all the available spatial streams on the S-R and S-D links. B. Precoding and Power Allocation in the Second Time Slot Let mSD ≤ min (NS , ND ) and mRD ≤ min (NR , ND ) be the number of SSs on the S-D link and R-D link, respectively. (2) (2) ≤ mRD . Then, it is clear that m(2) f + md ≤ mSD and mf For simplicity, let us assume that NR ≥ ND , which is a reasonable assumption in the downlink case, as in real systems it is expected that network terminals have more antennas than user devices. Consequently, by the conditions imposed by BD and since the channel matrices are full rank, we have (2) m(2) f +md ≤ ND . Next, the precoding matrices for the second time slot can be written as (2) ˜ ˜ ˜ F(2) F = FF WF , FD = FD WD , GR = GR WR ,

(18)

˜ F and F ˜ D are NS × where F and NS × matrices, respectively, that separate the two signals sent by the source and assign the largest m(2) SSs to xD and the remaining d (2) m(2) to x . This is so because the largest SSs on the direct F f link have better quality and do not need relaying. At the m(2) f

m(2) d

˜ R is a NR × m(2) matrix that chooses the largest relay, G f (2) mf out of ND available on the R-D link. The remaining mRD − m(2) f SSs are not used. These matrices can be easily found from the SVD of their respective channel matrices: H HSD = USD ΣSD VH SD , and HRD = URD ΣRD VRD . Then, assuming that the singular values are given in descending order, we (2) (2) (2) (2) f +md ˜ R = VRD |mf , ˜ F = VSD |m(2) ˜ D = VSD |md , F can define F ,G 1 1 m +1 d

where A|ba represents columns a to b of matrix A. Matrices WF , WD , and WR on the right-hand side of (18), (2) (2) (2) whose dimensions are, respectively, m(2) f × mf , md × md , (2) (2) and mf × mf , determine the power allocation and will be found in the next subsection. C. Convex Formulation for Obtaining Achievable Rate of Proposed Scheme We are now ready to evaluate the achievable rate of the proposed scheme. By plugging (17) and (9)-(15) in (7) and (8), it can be easily verified that the achievable rate when (2) xF is decoded first is always greater than the rate in the other order. We can further simplify the problem by noting that mSD , the number of spatial streams over the S-D link, is likely to be limited to only a few in practice, since (2) mSD ≤ ND . Thus, we could find the optimal m(2) f and md in problem (4)-(6) by performing a full seach over all the possible (2) combinations such that m(2) f + md = mSD . This amounts to mSD +1 possible combinations (excluding the trivial case when (2) m(2) f = md = 0), so that, for instance, when the destination has two antennas and if the channel matrices are full rank, we only have to search over three different combinations. Note that when m(2) f = 0, the relay would be silent and the source (2) only transmits x(2) D . Instead, we let the relay transmit xF using up to all the SSs on the R-D link. and m(2) Thus, for given m(2) d , problem (4)-(6) can be f rewritten as RP S = = s.t.

max

RF + RD

max

min (RA , RB )

t∈(0,1), QF ≽0, QD ≽0, QR ≽0 t∈(0,1), QF ≽0, QD ≽0, QR ≽0

(19)

tr(QF ) + tr(QD ) ≤ PS ,

(20)

tr (QR ) ≤ PR ,

(21)

H H where QF = WF WH F , QD = WD WD , QR = WR WR , and { ˜ 2 PF ))+ RA =t log2 (det(INR + Σ SR } 2 ˜ log2 (det(IND + ΣSD PD )) + H

˜ SD )), ˜ SD QD H (1 − t) log2 (det(IND + H 2 ˆ PD ))+ RB =t log2 (det(IND + Σ SD

[ ˜D = H ˆ SD H

] ˜ RD , and H [ WF WFH Q= WR WFH

]

H WF WR WR WH R

.

(24)

Although the problem has been greatly simplified, it is still not jointly convex in all its parameters. However, similar to [5], we introduce the following new variables: τ1 = t, τ2 = 1 − t, CD = τ2 QD , C = τ2 Q. We also define DF = [Im(2) 0m(2) ×m(2) ] and DR = [0m(2) ×m(2) Im(2) ], which, f f f f f f along with (24), allow us to express QF =

1 DF CDH F, τ2

QR =

1 DR CDH R. τ2

(25)

Since the two first terms in the right hand-side of (22) and the first term in the right-hand side of (23) are fixed and not to be optimized, we can also de˜ 2 PF )) + log2 (det(IN + fine a = log2 (det(INR + Σ D SR 2 ˜ ˜ 2 PD )). Finally, the ΣSD PD )), and b = log2 (det(IND + Σ SD resulting problem now reads as RP S =

max

τ1 >0,τ2 >0 C≽0,CD ≽0

min (RA1 , RB1 )

˜ SD RA1 = aτ1 + τ2 log2 (det(IND + H

(26)

CD ˜ H H )), τ2 SD

(27)

RB1 = bτ1 + ˜ SD τ2 log2 (det(IND + H

CD ˜ H ˜D CH ˜ H )) H +H τ2 SD τ2 D

(28)

s.t. tr(DF CDH F ) + tr(CD ) ≤ τ2 PS tr(DR CDH R)

(29)

≤ τ2 PR , τ1 + τ2 ≤ 1.

(30)

Note that function g(CD , τ2 ) = ˜ SD CD H ˜H τ2 log2 (det(IND + H )), C ≽ 0, τ > 0 D 2 SD τ2 is the perspective function (see [10]) of f (CD ) = ˜ SD CD H ˜H log2 (det(IND + H SD )), CD ≽ 0, which is concave in CD . Therefore, g is also concave, in both CD and τ2 . The same result holds for the other log det term in (28). Since all constraints are affine, we conclude that problem (26)-(30) is jointly convex in all its parameters, namely, τ1 , τ2 , CD , C. After solving the problem, we need to recover the optimal (2) precoding matrices F(2) F , FD , and GR from the resulting C and CD . Since QF and QR are given by (25), then, using their respective eigenvalue decompositions, we have H 1 H H that τ12 DF CDH F = UF ΛF UF , τ2 DR CDR = UR ΛR UR , while H 1 QD = τ2 CD = UD ΛD UD . From this, we can easily obtain 1

1

1

WF = UF ΛF2 , WR = UR ΛR2 , WD = UD ΛD2 , and finally, find the source and relay precoding matrices from (18). (22)

˜ ˜H ˜ SD QD H ˜H (1 − t) log2 (det(IND + H SD + HD QHD )), (23) ˜ R, H ˜ SD = HSD F ˜D, H ˜ RD = HRD G ˆ SD = HSD F ˜F, where H

V. N UMERICAL R ESULTS In this section we provide some numerical results to evaluate the performance of the proposed scheme in terms of its achievable rate. We consider a simple yet commonly used scenario where the S-D distance, dSD , is normalized to one and the relay is located along the line connecting the two nodes. Thus, dRD = 1−dSR , dSR = d, and we let d = [0.1, 0.9]. We compare

13

based on repetition coding [11] (Ryu&Choi), as well as its rate with twice as much transmit power (2xPS ), as the source is silent in the second time slot. The figure shows that the rate of DF is greater than that of the proposed scheme, although they become very close when the relay is far from the source. This is expected since it is known that DF is capacity achieving when the S-D link is much worse than the other two links. In contrast, the rate of the proposed scheme suffers because (i) it relies on the direct link to deliver xD to the destination. Nonetheless, our scheme is still able to outperform that of [11].

Achievable Rate (b/s/Hz)

12.5 12 11.5 11

Upper bound DF Proposed Direct link

10.5 10 9.5 9 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d

Fig. 2: Achievable rate vs. source-to-relay distance (PS = PR = 10). 8

Achievable Rate (b/s/Hz)

7.5 7 6.5 6 5.5

Upper bound DF Proposed Ryu&Choi 2xPS

5 4.5

Ryu&Choi Direct Link

4 3.5 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d

Fig. 3: Achievable rate vs.( source-to-relay distance with time ) slot duration equal to 0.5 PS = PR = 10, αSD = 0.1d−n SD . our scheme with the theoretical DF lower bound (as in [5]) and direct transmission from source to destination, i.e., without using the relay at all. We also show the CSB upper bound (also from [5]) as reference. In all cases NS = 6, NR = 2, and ND = 2; and the results are averaged over 50 independent channel realizations. Fig. 2 shows the achievable rate of the proposed scheme compared with the other considered strategies. We observe a well-known behavior of DF, i.e., that its rate is very close to the upper bound when the relay is near the source, whereas it becomes worse as the relay moves close to the destination. It is in this region, where the relay is far from the source, that the proposed scheme outperforms DF. This is so even though the coding strategy of our scheme is much simpler. Recall that for DF, the correlation between the codebooks at source and relay in the second time slot is optimized for each channel realization; the proposed scheme, in contrast, always uses the much simpler repetition coding scheme. Although optimizing the time slot duration maximizes the achievable rate, it may be desirable to keep it fixed to reduce complexity in some systems. This is shown in Fig. 3 for the same scenario as before, except that the S-D link is made worse by introducing a 10 dB loss, and the time slot duration is equal to 0.5. We also plot the rate of a simple DF relay protocol

VI. C ONCLUSION We have proposed a novel transmission strategy for the MIMO half-duplex relay channel based on simple/practical techniques such as linear precoding, repetition coding, and half duplexing that yields good performance in terms of achievable rate compared with other practical and theoretical schemes. We have formulated the corresponding optimization problem to maximize the achievable rate and transformed it into a convex form by making use of several simplifying assumptions. The numerical results reflect that the proposed scheme works better when the S-D link is in good condition relative to the S-R link, unlike the theoretic DF lower bound, which performs well just in the opposite case. This is due to the fact that the proposed scheme relies on the direct link to transfer part of the message to the destination, and therefore, whenever this link is in good condition with respect to the S-R link, more data can be transmitted and a higher rate can be achieved. ACKNOWLEDGMENT This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 0423-20120017). R EFERENCES [1] T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572–584, Sept. 1979. [2] J. N. Laneman, D. 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. [3] C. K. Lo and R. W. Heath, “Rate bounds for MIMO relay channels,” J. Commun. and Net., vol. 10, no. 2, pp. 194–203, June 2008. [4] B. Wang, J. Zhang, and A. H. Madsen, “On the capacity of MIMO relay channels,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 29–43, Jan. 2005. [5] L. Gerdes and W. Utschick, “Optimized capacity bounds for the halfduplex Gaussian MIMO relay channel,” in Proc. Intl. ITG Workshop Smart Antennas, Aachen, Germany, Feb. 2011. [6] C. T. K. Ng, and G. J. Foschini, “Transmit signal and bandwidth optimization in multiple-antenna relay channels,” IEEE Trans. Commun., vol. 59, no. 11, pp. 2987–2992, Nov. 2011. [7] C. B. 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. [8] W. Xu, X. Dong, and W.-S. Lu, “MIMO relaying broadcast channels with linear precoding and quantized channel state information feedback,” IEEE Trans. Signal Process., vol. 58, no. 10, pp. 5233–5245, Oct. 2010. [9] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 461–471, Feb. 2004. [10] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [11] J. Y. Ryu and W. Choi, “Balanced linear precoding in decode-andforward based MIMO relay communications,” IEEE Trans. Wireless Commun., vol. 10, no. 7, pp. 2390–2400, July 2011.