Linear MMSE Transceiver Design in
Amplify-and-Forward MIMO Relay Systems
Fan-Shuo Tseng, Student Member, IEEE, and Wen-Rong Wu, Member, IEEE
Abstract—We consider the precoding problem in an
amplify-and-forward (AF) multiple-input–multiple-output (MIMO) relay system in which multiple antennas are equipped at the source, the relay, and the destination. Most existing methods for this problem only consider the design of the relay precoder, and some even ignore the direct link. In this paper, we consider a joint source/relay precoder design problem, taking both the direct and the relay links into account. Using a minimum-mean-square-error (MMSE) criterion, we first formulate the problem as a constrained optimization problem. However, it is found that the mean square error (MSE) is a highly nonlinear function of the precoders, and a direct optimization is difficult to conduct. We then design the precoders to diagonalize the MSE matrix in the cost function. To do that, we pose certain structural constraints on the precoders and derive an MSE upper bound. It turns out that minimization with respect to this bound becomes simple and straightforward. Using the standard Lagrange technique, we can finally obtain the solution with an iterative water-filling method. Simulation results show that the proposed method, with an additional precoder, outperforms the existing methods, in terms of either the MSE or the bit error rate (BER).
Index Terms—Amplify and forward (AF), cooperative
com-munication, minimum mean square error (MMSE), multiple input–multiple output (MIMO), precoder.
I. INTRODUCTION
D
IVERSITY is a common technique to overcome the multipath channel fading in wireless communications. Popular diversity schemes include time diversity, frequency diversity, and spatial diversity. Among these techniques, the spatial diversity is particularly attractive. This is because this technique can combine with the other two diversity techniques with no expansion of time and bandwidth. The conventional way to obtain the spatial diversity is the use of multiple transmit or multiple receive antennas. When both multiple transmit and receive antennas are used, the system is referred to as a multiple-input–multiple-output (MIMO) system [1]–[13]. MIMO systems have widely been studied in the literature since they can enhance the diversity or spectral efficiency in an efficient way [2]–[13].User cooperation is an alternative way to obtain spatial diversity [14]–[21]. With the aid of additional relay nodes,
Manuscript received April 13, 2009; revised July 14, 2009 and September 23, 2009. First published October 30, 2009; current version published February 19, 2010. This work was supported by the National Science Council under Grant NSC-98-2221-E-009-046-MY3. The review of this paper was coordinated by Dr. A. Ghrayeb.
The authors are with the Department of Electrical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: [email protected]. tw; [email protected]).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2009.2034971
spatial diversity can be achieved in a distributed manner. In typical cooperative communications, the source and relay nodes share the same spectrum. As a result, half-duplexing is of-ten used for signal transmission between the source and the relays. In a typical three-node system, signal transmission is generally divided into two time phases [15]–[28]. In the first phase, the signal at the source is transmitted to the relay and the destination. In the second time phase, the relay forwards its received signal to the destination. Finally, the destination combines the received signals to achieve the spatial diversity. There are several protocols defining retransmission strategies at the relays. Two relay strategies are well known, i.e., amplify-and-forward (AF) and decode-amplify-and-forward (DF). In AF, the relay receives the signal from the source and retransmits it to the destination with signal amplification only. The system with the protocol is also called a nonregenerate cooperative system [23]–[25], [27]. In DF, the relay decodes the received signals, reencodes the information bits, and retransmits the resultant signal to the destination. The system is also called a regenerative cooperative system. It is simple to see that the DF protocol requires higher computational complexity and a larger processing delay at the relay nodes. In this paper, we only consider the AF-based cooperative system.
Recently, the MIMO technique has been introduced to co-operative systems as a means for further performance enhance-ment. With the multiple antennas equipped at the source and each node, a MIMO relay system is constructed [22]–[28]. Capacity bounds for a single-relay MIMO channel was first addressed in [22]. Similar to conventional MIMO systems, the precoding operation can be conducted in a MIMO relay system. The relay precoder in an AF-based MIMO relay system was first designed in [23] and [24] to enhance the overall channel capacity. In most of those approaches, only the relay link (the source-to-relay and relay-to-destination links) is considered. It was shown that the capacity can further be increased if the direct link (the link between the source and the destination) is taken into account [24]. Apart from the capacity, the link quality is another criterion that has been considered [25], [26]. In these works, a relay precoder is designed using a minimum-mean-square-error (MMSE) criterion. Most of the MIMO relay systems considered above use one relay. Precoding in multiple-relay MIMO systems was investigated in [26]. Note that the aforementioned works all address the spatial-multiplexing sce-nario. Recently, the design for the transmission of a single data stream, which is referred to as beamforming, has also been considered. For example, [28] derives the optimal source and relay beamformers using a maximal-SNR criterion. In this paper, the optimal solution is derived for the relay-link-only
system. In addition to beamforming, antenna selection in MIMO relay systems was also studied. With the MMSE cri-terion, an optimum selection scheme was developed in [27]. In this approach, only one antenna is selected at the source and the relay, respectively, for signal transmission.
As mentioned, in the precoder design for spatial-multiplexing AF-based MIMO relay systems, the existing works only consider the precoder at the relay. Furthermore, the direct link is frequently ignored [23]–[26]. In this paper, we propose a new design method to solve the problems. Similar to previous works, we assume a linear receiver at the destination. Our approach is to seek precoders such that the MMSE of the linear receiver is minimized. However, we found that the MMSE is a complicated function of precoding matrices, and a direct minimization is almost impossible to conduct. To overcome the difficulty, we propose a structural constraint on the precoders to diagonalize the mean-square-error (MSE) matrix in the cost function. With the precoders, we can then derive a tractable MSE upper bound. Minimization with this upper bound, instead of the original MMSE, then becomes feasible. The proposed precoders can finally be computed via an iterative water-filling technique [9], [31], [32]. Note that the MSE criterion to minimize is the total MSE of the multiplexed signal streams. With the specially designed structure, the proposed precoders can make the individual MSEs of all signal streams equal, indicating that the bit error rate (BER) of the proposed precoded system will be the minimal among all precoded systems with the same minimum total MSE [5]. The rest of this paper is organized as follows. Section II gives the system model and problem formulation, Section III derives the proposed joint design in detail, Section IV reports simulation results for various applications, and finally, Section V concludes this paper.
II. PROPOSEDSYSTEMMODEL
ANDPROBLEMFORMULATION
A. Precoders for the AF MIMO Relay System
We consider a typical three-node half-duplex cooperative AF MIMO relay system where multiple antennas are placed at each node. Under this scenario, signals can be transmitted from the source to the destination and from the source to the relay and then to the destination. To avoid the interference between direct and relay links, we consider the time-division-duplexing scheme [23]–[26] used in a typical two-phase transmission aforementioned (see Fig. 1). Let N , R, and M denote the number of antennas at the source, the relay, and the destination, and assume that all channels are flat fading. For the first phase, the received signals at the destination and the relay can be expressed as
yD,1= HSDFSs + nD,1 (1)
yR= HSRFSs + nR (2)
respectively, where s∈ CL×1 is the transmitted signal vector, with L being the number of the substreams, FS ∈ CN×L is
the precoding matrix at the source, HSR∈ CR×N and HSD∈
CM×Nare the channel matrices corresponding to the
source-to-Fig. 1. Three-node AF-based MIMO relay system with multiple antennas allocated at each node.
relay and source-to-destination channels, respectively, nD,1∈
CM×1is the first-phase received noise vector at the destination,
and nR∈ CR×1is the received noise vector at the relay. Here,
we assume that L≤ min{N, M} provides sufficient degrees of freedom for signal detection.
In the second phase of the transmission, the relay retransmits the received signal with another precoding matrix. Thus, the received signals at the destination can be expressed as
yD,2= HRDFRyR+ nD,2
= HRDFRHSRFSs + (HRDFRnR+ nD,2) (3)
where FR∈ CR×Ris the precoding matrix at the relay, HRD∈
CM×R is the channel matrix corresponding to the
relay-to-destination channel, and nD,2∈ CM×1 is the second-phase
received noise vector at the destination. Here, we assume that each element in nD,1 has a zero-mean circularly symmetric
Gaussian distribution, and all the elements are independent identically distributed (i.i.d.). The same assumption is applied for nD,2 and nR. As a result, the received signal vectors
yD,1 and yD,2 for the two phases can be combined into a
single vector, which is denoted as yD∈ C2M×1. Consequently,
we have yD:= yD,1 yD,2 = HFSs + n (4) where H = HSD HRDFRHSR (5) n = nD,1 HRDFRnR+ nD,2 . (6)
Here, H is the equivalent channel matrix with rank (H) = N , and n is the equivalent noise vector at the destination. It is noteworthy that the noise received at the relay is amplified by the relay precoder and the relay-to-destination channel. Furthermore, the equivalent channel matrix in (5) is a function of the relay precoder FR. This is quite different from the
sce-nario considered in conventional MIMO systems. The precoder design problem is actually a joint-transceiver-design problem. In other words, the optimum precoders depend on not only
the channels but on the receiver as well. Similar to previous works, we will consider the linear MMSE receiver in our design [25], [26].
B. MMSE Receiver and Related MSE Matrix Let RnD,1 = E[nD,1n
H
D,1] = σn,d2 IM, RnD,2 =
E[nD,2nHD,2] = σn,d2 IM, and RR= E[nRnHR] = σ2n,rIR,
where σ2
n,d and σ2n,r are the noise variances at the destination
and the relay, respectively. Furthermore, the elements of the transmitted symbols are i.i.d. with zero mean and covariance matrix Rs= σs2IL, where σs2is the transmitted symbol power.
Using the setting, we can have the covariance matrix of the equivalent noise vector as
Rn= E [nnH] = σ2 n,dIM 0 0 σ2 n,rHRDFRFHRHHRD+ σ2n,dIM . (7) Let G be the equalization matrix in the receiver. Then, the MSE for recovering s, which is denoted as J , is given by
J = EGyD− s2. (8) The minimization of (8) leads to the optimal equalization matrix [4] as Gopt= σ2sFHSHH σ2sHFSFHSHH+ Rn −1 . (9) Substituting (9) into (8) and invoking the matrix inversion lemma [29], we can then have the MMSE, which is denoted by Jmin, as Jmin= tr{E} (10) where E =σ−2s IL+ ES+ ER −1 . (11) In (11) ES= σn,d−2FHSHHSDHSDFS (12) ER= FHSHHSRFHRHHRD σn,r2 HRDFRFRHHHRD+ σn,d2 IM −1 × HRDFRHSRFS. (13)
As we can see from (11), the MMSE is a function of FS and
FR. It is also simple to see that ES accounts for the MMSE
contributed in the direct link, and ERaccounts for that in the
relay link. If we ignore the direct link and only consider the relay precoder, the problem will be degenerated to the case considered in [25].
C. Problem Formulation
As shown in (10), the MMSE is a function of the two precod-ing matrices, i.e., FS and FR. Our task here is to design these
two matrices such that the MSE in (10) can be minimized. The optimization problem can then be formulated as (14), shown at the bottom of the page. The inequalities in (14) indicate that the precoders have to satisfy the transmit power constraints at both the source and the relay.
From (14), we can readily find that (14) is not a convex optimization. Furthermore, the cost function involves a series of matrix multiplications and inversions, it is a complicated and nonlinear function of FS and FR. The cost function may
have many local minimums, and the optimal solution, even with numerical methods [30], is difficult to derive. We will propose a method, which is described below, to solve these problems.
III. JOINTSOURCE/RELAYPRECODERDESIGN
WITHMMSE RECEIVER
As aforementioned, the optimum solution for (14) is difficult to derive. In this section, we then propose a method to seek for a suboptimum solution. One difficulty in (14) is that the number of unknown parameters in FR and FS can be large.
The first idea of our approach is to use a constrained precoder structure such that the number of unknowns can effectively be reduced. The other difficulty in (14) is that the formulas are too complicated to work with. Our second idea is to derive an MMSE upper bound having a simple expression and conduct minimization with this upper bound. Even though the cost function can dramatically be simplified with the pro-posed method, a closed-form solution is still difficult to obtain. We then use an iterative water-filling method to solve the problem. min FS,FR tr{E} = L i=1 E(i,i) s.t. E = ⎛ ⎜ ⎝σs−2IL+ σ−2n,dFHSHHSDHSDFS :=ES + FHSHHSRFHRHHRDσn,r2 HRDFRFHRHHRD+ σ2n,dIM −1 HRDFRHSRFS :=ER ⎞ ⎟ ⎠ −1 trEFRyRyRHFHR = trFR σn,r2 IR+ σs2HSRFSFHSHHSR FHR ≤ PR,T trFSE[ssH]FHS = σs2trFSFHS ≤ PS,T (14)
A. Proposed Approach
When the direct link is ignored and only a relay precoder is considered, the optimal MMSE precoder can analytically be obtained through an MSE matrix diagonalization procedure [25]. Motivated by this fact, we propose to conduct a similar matrix diagonalization in our design. Indeed, if the error matrix
E in (14) can be diagonalized, the trace operation can easily be
conducted, and the whole problem can be greatly simplified. To do that, we first consider the following singular value decomposition (SVD) for the channel matrices in all links:
HSD= UsdΣsdVsdH (15)
HSR= UsrΣsrVHsr (16)
HRD= UrdΣrdVHrd (17)
where Usd∈ CM×M, Usr∈ CR×R, and Urd∈ CM×M are
the left singular matrices of HSD, HSR, and HRD,
respec-tively; Σsd∈ RM×N, Σsr ∈ RR×N, and Σrd∈ RM×R are
diagonal singular-value matrices of HSD, HSR, and HRD,
respectively; and VH
sd∈ CN×N, VsrH ∈ CN×N, and VHrd∈
CR×Rare the right singular matrices of H
SD, HSR, and HRD,
respectively.
Observing (14), we will readily find that a complete diag-onalization of E will be difficult. We then first consider the diagonalization of (σ2n,rHRDFRFHRHHRD+ σn,d2 IM)−1using
FR so that the inverse operation can easily be tackled. Such
an approach, although suboptimal, will considerably simplify our derivation. It also allows us to derive an MSE upper bound and then obtain a scalar-valued optimization problem. With the SVD in (17), an immediate choice for FR to diagonalize
(σ2
n,rHRDFRFHRHHRD+ σn,d2 IM) is
FR= VrdΣrUr (18)
where Σr∈ RR×Ris a diagonal matrix, and Ur∈ CR×Ris a
unitary matrix to be determined. With (18), we have σn,r2 HRDFRFRHHHRD+ σr,d2 IM −1 = Urd σ2n,rΣrdΣ2rΣHrd+ σ2r,dIM −1 UHrd. (19) To further diagonalize FH SHHSRFHRHHRD(σn,r2 HRDFR × FH RHHRD+ σn,d2 IM)−1HRDFRHSRFS, we can select Ur= UHsr (20) FS = VsrΣsUs. (21)
Σs∈ RN×Lis a diagonal matrix, and Us∈ CL×Lis a unitary
matrix yet to be specified. From (18) and (20), we have
FR= VrdΣrUHsr. (22)
After some manipulations, we can obtain the MSE in (14) as (23), shown at the bottom of the page, where
V = VHsdVsr (24)
is a constant matrix related to the channels. Note that the inclusion of the unitary matrix Us in (23) will not change
the cost function at all. However, by an appropriate design of
Us, we can make the diagonal components of E equal. It has
been shown that, under a fixed MSE, i.e., tr{E}, the receiver that makes the MSEs of the MIMO components equal has the lowest BER performance [5]. From (23), we now have some observations in order. First, we see that (23) is obtained with the constrained structure of the precoding matrices specified in (21) and (22). The MMSE obtained with the precoders can serve as an upper bound of the true MMSE. Second, the unknown matrices become Σrand Σs, which are diagonal, and the whole
problem is easier to handle. Finally, the matrix ES cannot be
diagonalized. However, starting from (23) and exploiting the diagonal nature of ER, we can further derive an MSE upper
bound and use it to diagonalize ES.
To proceed, let us use the matrix inverse lemma to rewrite (23) as tr(E) = tr ⎛ ⎜ ⎜ ⎝ ⎡ ⎢ ⎢ ⎣σs−2IL+ER :=A +ΣHs σ−2n,dVHΣHsdΣsdV :=B Σs ⎤ ⎥ ⎥ ⎦ −1⎞ ⎟ ⎟ ⎠ = tr(A−1)− tr A−1ΣHsB−1+ ΣsA−1ΣHs −1 ΣsA−1 . (25) It is noted here that to make sure that the inverse of B exists,
B should be positive definite. To achieve that, we assume that N ≤ M. Based on (25), the desired MSE upper bound can be obtained by the aid of the next lemma.
tr{E} = tr σ−2s IL+ UHsΣHsΣHsrΣHrΣHrd σn,r2 ΣrdΣ2rΣHrd+ σn,d2 IM −1 ΣrdΣrΣsrΣsUs + σn,d−2UHs ΣHsVHΣHsdΣsdVΣsUs −1 = tr ⎧ ⎪ ⎨ ⎪ ⎩ ⎛ ⎜ ⎝σ−2s IL+ ΣHs ΣHsrΣHrΣHrd σn,r2 ΣrdΣ2rΣHrd+ σn,d2 IM −1 ΣrdΣrΣsrΣs :=ER + σ−2n,dΣHs VHΣHsdΣsdVΣs :=ES ⎞ ⎟ ⎠ −1⎫⎪ ⎬ ⎪ ⎭ (23)
Lemma: Let D1 and D2 be diagonal matrices, with the diagonal entries of D2 being positive. Then, for any positive definite matrix X, we have
trDH1 (X + D2)−1D1
≥ trDH1 (diag(X) + D2)−1D1 (26) where diag(X) is obtained from X by setting its off-diagonal entries to zero. The equality in (26) holds if X is diagonal.
Proof: See the Appendix. By the lemma, it follows that tr A−1ΣHs B−1+ ΣsA−1ΣHs −1 ΣsA−1 ≥ trA−1ΣHs diag(B−1) + ΣsA−1ΣHs −1 ΣsA−1 . (27) Using (25) and (27), we can have the following key result:
tr(E)≤ tr(A−1) −trA−1ΣHsdiag(B−1) + ΣsA−1ΣHs −1 ΣsA−1 = L i=1 1 σs−2+ σ 2
s,iσ2r,iσ2sr,iσ2rd,i
σ2 n,rσ2r,iσ 2 rd,i+σ 2 n,d + σ2
s,i(B−1(i, i))−1
. (28) Compared with the original MSE function (14), the upper bound in (28) admits a much simpler form and is analytically tractable. Hence, we propose to design the precoder by min-imizing the upper bound in (28). For convenience, let ps,i =
σ2
s,i and pr,i= σr,i2 in (28). The optimization can finally be
formulated as min ps,i,pr,i, i=1,···,L L i=1 1 σ−2s + ps,ipr,iσ 2 sr,iσ 2 rd,i σ2
n,rpr,iσrd,i2 +σn,d2 + ps,i(B
−1(i, i))−1 s.t. trΣr σ2n,rIR+ σs2ΣsrΣsΣHs ΣHsr ΣHr = L i=1 pr,i σn,r2 + σs2ps,iσ2sr,i ≤ PR,T σs2tr ΣsΣHs = σs2 L i=1 ps,i≤ PS,T,
ps,i ≥ 0, pr,i≥ 0, ∀i. (29)
It is simple to see that the problem in (29) is not a convex optimization problem either, and the optimum solution is still difficult to find. However, note that if one of pr,i and ps,i is
given, (29) will become a convex optimization problem. This suggests a method, which is referred to as the iterative water-filling method [9], [31], [32], to find a suboptimum solution. For a given ps,i, the optimum pr,ican be expressed as (30) (see
Appendix B), shown at the bottom of the page, where [y]+= max[0, y], and μris the water level chosen to satisfy the power
constraint at the relay, i.e., &Li=1pr,i(σn,r2 + σs2ps,iσ2sr,i) =
PR,T. With pr,i= σ2r,i in (30), the relay precoder can be
obtained by (22). For a given pr,i, the optimum ps,i can be
expressed as (31), shown at the bottom of the page, where μs
is the water level chosen to meet the power constraint at the source, i.e.,&Li=1ps,i = PS,T, and
βi=
σn,d2 + pr,iσ2n,rσrd,i2
×B−1(i, i)−1σn,d2 + pr,iσn,r2 σrd,i2
+ pr,iσ2sr,iσ2rd,i
. (32) Thus, we can use (30) and (31) to iteratively solve pr,iand ps,i.
To determine the Us, we first substitute (30) and (31) into (22)
and (21), respectively, and express the error matrix in (11) as
E = σ−2s IL+ UHsEU˜ s −1 (33) where ˜ E = ΣHsΣHsrΣHr ΣHrdσ2n,rΣrdΣ2rΣHrd+ σ2n,dIM −1 × ΣrdΣrΣsrΣs+ σ−2n,dΣ H sVHΣHsdΣsdVΣs. (34)
Our task now is to design Ussuch that (33) has equal diagonal
MSE values. To do that, we consider the following eigendecom-position:
˜
E = VE˜DE˜VEH˜ (35) where V˜E∈ CL×L is a matrix with the eigenvectors of ˜E as
its columns, and DE˜ ∈ RL×L is a diagonal matrix with the eigenvalues of ˜E as its diagonal components. Therefore, if
we let Us= VE˜FL (36) pr,i= ⎡ ⎣μrσn,d √p
s,iσsr,iσrd,i
σ2 sps,iσ2sr,i+ σ2n,r −1/2− σ2 n,d σ−2s + ps,i B−1(i, i)−1 σ2 rd,i σ2 n,r
σs−2+ ps,i(B−1(i, i))−1
+ ps,iσ2sr,i ⎤ ⎦ + (30) ps,i = ⎡ ⎣ μs √ βi− σ−2s σ2 n,d+ pr,iσn,r2 σrd,i2 (B−1(i, i))−1 σ2 n,d+ pr,iσn,r2 σrd,i2
+ pr,iσ2sr,iσ2rd,i
⎤ ⎦ +
TABLE I
COMPLEXITY OF THEPROPOSEDJOINTPRECODERS
where FLis the L-point discrete-Fourier-transform matrix, (33)
can be reexpressed as
E = FHL σs2IL+ DE˜ −1
FL (37)
which reveals that E is a circulant matrix with equal diago-nal elements. It is simple to check the unitary property that
UsUHs = UHs Us= IL.
The proposed scheme mainly involves the operations of the SVD in (15)–(17) and (35) and the inversion of the matrix B in (28). The computational complexity of the proposed scheme, which is measured in terms of floating-point operations (FLOPs), is summarized in Table I.
B. Special Case: Cooperative Beamforming
In this section, we consider the cooperative beamforming in a two-hop cooperative system. This is a special case of our
precoding problem in which L = 1 and the direct link is not considered (i.e., HSD= 0).
For a given source beamforming vector fS, the optimal relay
precoder can be derived by [25]
FR= VrdΣrUHsr (38)
where Σr= diag{σr,1, . . . , σr,R}, with σr,1≥ · · · ≥ σr,R.
Let fS =√αsvS∈ CN×1, where vS is a unit vector, and
fS satisfies the transmit power constraint, i.e., σ2sαsvS2≤
PS,T. Substituting the beamformer and (38) into (10) with
HSD= 0, we have (39), shown at the bottom of the page,
where wsr = VsrHvS= [wsr,1, . . . , wsr,N]T andwsr2= 1,
Σsr, Σrd, Σr are diagonal matrices, with their diagonal
ele-ments arranged in decreasing order. The beamforming problem can then be formulated as (40), shown at the bottom of the page. Theorem 1: The optimal beamforming vector denoted by fS∗ and the optimal relay precoder denoted
Jmin= tr σ−2s + fSHVsrΣHsrΣHrΣHrd σn,d2 IM+ σ2n,rΣrdΣrΣHrΣHrd −1 ΣrdΣrΣsrVHsrfS −1 = tr ⎧ ⎪ ⎨ ⎪ ⎩ ⎛ ⎜ ⎝σ−2s + αsv HSVsr :=wH sr ΣHsrΣHr ΣHrdσn,d2 IM + σn,r2 ΣrdΣrΣHrΣHrd −1 ΣrdΣrΣsrV HsrvS :=wsr ⎞ ⎟ ⎠ −1⎫⎪ ⎬ ⎪ ⎭ = 1 σs−2+ αs &min{N,M,R} i=1 |wsr,i|2 σ2
r,iσsr,i2 σrd,i2
σ2
n,d+σ2n,rσ2r,iσ2rd,i
(39)
min
αs,wsr,i,σr,i,∀i
1 σs−2+ αs &min{N,M,R} i=1 |wsr,i|2 σ2 r,iσ 2 sr,iσ 2 rd,i σ2 n,d+σ2n,rσ2r,iσrd,i2 s.t. σs2fS2= σs2αsvS2≤ PS,T, N i=1 |wsr,i|2= 1, trΣr σ2n,rIR+ σs2ΣsrVHsrfSfSHVsrΣHsr ΣHr ≤ PR,T (40)
by F∗R for (40) are '(PS,T/σ2s)Vsr(:, 1) and
(
(PR,T/(σ2n,r+ PS,Tσsr,12 ))Vrd(:, 1)[Usr(:, 1)]H, where
Vrd(:, i) and Usr(:, i) denote the ith columns of Vrdand Usr,
respectively.
Proof: We first derive the optimal wsr for given αs
and σr,i, i = 1, . . . , R. From (40), it is simple to see that
the optimal wsr can be derived by the following equivalent
problem: max wsr min{N,M,R} i=1 |wsr,i|2 σ2
r,iσsr,i2 σrd,i2
σ2 n,d+ σn,r2 σr,i2 σ2rd,i s.t. N i=1 |wsr,i| = 1. (41)
From (41), it is obvious that the optimum wsris [1, 0, . . . , 0]T.
This can easily be checked by σ2r,iσsr,i2 σ2rd,i
σ2 n,d+ σn,r2 σr,i2 σrd,i2 ≥ σ 2 r,jσsr,j2 σ2rd,j σ2 n,d+ σ2n,rσ2r,jσrd,j2 , i≥ j. (42) The solution implies that the optimal vS, which is denoted by
v∗S, is
v∗S= Vsr(:, 1). (43)
As a result, the design problem can therefore be expressed as max αs,σr,i,∀i αs σr,12 σsr,12 σ2rd,1 σ2 n,d+ σn,r2 σr,12 σrd,12 s.t. 0≤ αs≤ PS,T σ2 s trΣr σn,r2 IR+αsσ2sΣsrVHsrvSvHSVsrΣHsr ΣHr = ) σn,r2 R i=1 σ2r,i * + αsσs2σ2r,1σ2sr,1≤ PR,T. (44) Taking a close look at (44), we first find that the cost function is only related to αsand σ2r,1. Then, we can have the following
observation: αs σr,12 σsr,12 σ2rd,1 σ2 n,d+ σn,r2 σr,12 σrd,12 is monotonous in σr,12 and αs. (45) From (45) and (44), we can rewrite the power constraint for the relay as
σ2r,1σn,r2 + α2sσ2sσsr,12 ≤ PR,T (46)
and consequently have the following relationship: σ2r,1≤ PR,T
σ2
n,r+ α2sσs2σ2sr,1
. (47)
It is noteworthy that (46) also implies that the optimal Σr,
which is denoted by Σ∗r, is Σ∗r= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ σr,1 0 · · · 0 0 σr,2 . .. ... .. . . .. ... 0 0 · · · 0 σr,R ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ σr,1 0 · · · 0 0 0 . .. ... .. . . .. ... 0 0 · · · 0 0 ⎤ ⎥ ⎥ ⎥ ⎦. (48)
Substituting (47) into the cost function in (44), we have αs σ2 r,1σsr,12 σ2rd,1 σ2 n,d+ σn,r2 σr,12 σrd,12 ≤ αsσ 2 sr,1σrd,12 σ2 n,d (σ2 n,r+αsσ2sσ2sr,1) PR,T + σ 2 n,rσ2rd,1 (49) where the upper bound of the cost function can be achieved if
σr,12 = PR,T σ2 n,r+ αsσs2σsr,12 . (50)
Therefore, via (50), the problem can finally be expressed as the minimization of a function of αsgiven by
max αs σ2 sr,1σ2rd,1αs σ2 n,dσs2σsr,12 PR,T αs+ σ 2 n,rσrd,12 + σ2 n,dσ2n,r PR,T s.t. 0≤ αs≤ PS,T σ2 s . (51)
Since the cost function in (51) is monotonically increasing in αs, it is clear that the optimal αs, which is denoted by α∗s, is
α∗s=PS,T σ2
s
. (52)
Combining (52) and (43), we finally obtain the optimal beam-forming vector as fs∗= + PS,T σ2 s Vsr(:, 1). (53)
Substituting (52) into (50) and combining (48) and (38), we have the optimal relay precoder as
+ PR,T σ2 n,r+ PS,Tσ2sr,1 Vrd(:, 1) [Usr(:, 1)]H. (54)
It is noteworthy that the result of Theorem 1 is the same as that in [28], in which the criterion for the beamformer design is the maximization of the received SNR. Here, we use the MMSE criterion and obtain the same solution.
IV. APPLICATIONS
The proposed precoding scheme can be used in many sce-narios. In this section, we conduct simulations to evaluate the
Fig. 2. MSE performance comparison for the unprecoded and proposed precoded schemes in an AF-based SISO OFDM cooperative system.
performance of the proposed scheme in three different applica-tions, namely, a single-input–single-output (SISO) orthogonal-frequency-division-multiplexing (OFDM) system, a two-hop MIMO relay system (where only the relay link is considered), and a general MIMO relay system. Assume that all channel state information (CSI) of all the links are available at all nodes and that perfect synchronization can be achieved. For the first case, the channel is assumed to be frequency-selective fading, and for the rest of the cases, the channel is assumed to be flat fading. Furthermore, the modulation scheme is quaternary phase-shift keying.
A. SISO OFDM Relay System
Assume that the cyclic prefix length is longer than the overall channel delay spread such that intersymbol interference will not occur. Furthermore, the channel is assumed to be quasi-static, which means that its response remains constant during each OFDM symbol. Note that each node only has one antenna. As a result, the equivalent frequency-domain channel matrices of all links are diagonal. The proposed pre-coders in (21) and (22) therefore become FS = FHLΣsFL
and FR= Σr, where the relay precoder becomes a subcarrier
power-allocation problem. Let hsr(l), hrd(l), and hsd(l) be
the channel impulse responses for the source-to-relay, relay-to-destination, and source-to-destination channels, respectively. The channel taps hsr(l), hrd(l), and hsd(l), 0≤ l ≤ 5, are
gen-erated from i.i.d. complex Gaussian random variables with zero mean and a variance of 1/6, such that E{&5l=0|hsr(l)|2} =
E{&5
l=0|hsd(l)|2}E{
&5
l=0|hrd(l)|2} = 1. Furthermore, let
N = 64, the total available powers at the source and the relay be equal, and SNRsr, SNRrd, and SNRsd be defined as the
received SNR at the source-to-relay, relay-to-destination, and source-to-destination links. Here, we let SNRsr= SNRrd=
SNRsd= SNR. Figs. 2 and 3 show the MSE and BER
com-parisons for the unprecoded and proposed precoded systems,
Fig. 3. BER performance comparison for the unprecoded and proposed precoded schemes in an AF-based SISO OFDM cooperative system.
Fig. 4. MSE performance comparison for existing the unprecoded/precoded and proposed precoded schemes in an AF-based two-hop MIMO relay system.
respectively. As shown in the figures, the proposed precoded system significantly outperforms the unprecoded system be-cause the proposed system considers all the link resources and properly allocates the power.
B. Two-Hop MIMO Relay System
In this scenario, the channel condition in the direct link is poor such that the destination only receives the signal from the relay link. Here, we first consider the case that N = R = M = L = 4. Let the elements of each channel matrix be i.i.d. complex Gaussian random variables with zero mean and unity variance. Let SNRsr and SNRrd denote,
respec-tively, the SNR per receive antenna of the source-to-relay and relay-to-destination links. Here, we set SNRsr= 20 dB
and vary SNRrd. Figs. 4 and 5 show the MSE and BER
comparisons, respectively, for 1) an unprecoded system with a zero-forcing (ZF) receiver; 2) an unprecoded system with
Fig. 5. BER performance comparison for the existing unprecoded/precoded and proposed precoded schemes in an AF-based two-hop MIMO relay system.
Fig. 6. BER performance comparison for the antenna selection [27] and proposed beamforming schemes in an AF-based two-hop MIMO relay system (L = 1 and N = R = M = 4).
an MMSE receiver; 3) a precoded system with the optimal relay precoder [25]; and 4) the proposed precoded system. From those figures, we can see that the proposed precoded system outperforms not only the unprecoded system but the precoded system in [25] as well. This is because the proposed method incorporates an additional source precoder such that the performance can be enhanced even if the direct link is not considered.
We also report the simulation result for cooperative beam-forming, i.e., L = 1. As discussed in Theorem 1, our design for this case is optimal. We let N = R = M = 4 and SNRsr =
5 dB. Fig. 6 shows the BER comparison for the antenna-selection method in [27] and the proposed method. From the figure, we can see that the proposed method is superior to the antenna selection. This is expected since the proposed beamforming scheme is optimal.
Fig. 7. MSE performance comparison for the existing unprecoded/precoded and proposed precoded schemes in an AF-based MIMO relay system.
Fig. 8. BER performance comparison for the existing unprecoded/precoded and proposed precoded schemes in an AF-based MIMO relay system.
C. General MIMO Relay Channel
In this scenario, we consider a symmetric MIMO relay system, i.e., N = M = R = L = 4. As in the previous case, each element of the channel matrices is assumed to be i.i.d. complex Gaussian random variables with zero mean and the same variance. We let SNRsrand SNRrdbe the same as those
defined in Section IV-B and SNRsdbe the SNR per receive
an-tenna for the source-to-destination link. Here, we set SNRsr=
15 dB and SNRrd= 10 dB and vary SNRsd. Figs. 7 and 8
show the MSE and BER comparisons, respectively, for the proposed system and other systems described in Section IV-B. Note that the optimal relay precoder in [25] only considers the two-hop relay system. For fair comparison, we include the direct link at the destination when implementing the MMSE receiver. As expected, the proposed method outperforms all the other systems.
V. CONCLUSION
In this paper, we have proposed a joint design method for precoders in a half-duplex AF-based MIMO relay system. In the system, the MMSE receiver is used at the destination, and the precoders at the source and at the relay are determined to minimize the MMSE. Since the MMSE is a complicated function of the precoding matrices, a direct minimization is not feasible. To solve the problem, we have first used a con-strained precoder structure and derived an MMSE upper bound. Since the upper bound has a simple expression, minimiza-tion with the upper bound becomes feasible. Resorting to the Karush–Kuhn–Tucker (KKT) optimality conditions, we have used an iterative water-filling algorithm to obtain a subopti-mal solution. The proposed scheme can be applied in various kinds of communication systems as long as the channel effect can be described with an equivalent channel matrix. We have then considered the applications of the proposed scheme in the SISO OFDM, two-hop MIMO relay channel, and general MIMO relay systems. Simulations show that the proposed scheme outperforms the existing unprecoded/precoded systems in terms of either the MSE or the BER. In practical sys-tems, perfect CSI may not be available. How to design the robust precoders for imperfect CSI can be a topic for further research.
APPENDIXA PROOF OFLEMMA(26) Let us first rewrite
trDH1 (X + D2)−1D1 = L i=1 |D1(i, i)|2(X + D2)−1(i, i) (55) where D1∈ RN×L and D2∈ RN×N, N≥ L, are diagonal matrices with positive elements; X∈ CN×N is a Hermitian matrix. Therefore, (X + D2) is a positive definite matrix.
We claim that
(X + D2)−1(i, i)≥
1
(X + D2)(i, i) (56) which will be proved in the next paragraph. From (55) and (56), we immediately have trDH1 (X + D2)−1D1 ≥ L i=1 |D1(i, i)|2 (X + D2)(i, i) = L i=1 |D1(i, i)|2 (diag(X) + D2) (i, i) = tr DH1 (diag(X) + D2)−1D1 (57) which proves the lemma.
Proof of (56): Let Z := (X + D2) = UΣUH be the eigendecomposition of the positive definite matrix Z. Since 1 = eT
iIei = eTi Z1/2Z−1/2ei, where eiis the ith unit standard
vector, we then have 1 =,,,eTiZ1/2Z−1/2ei,,, 2 2≤ ,, ,eT iZ1/2,,, 2 2Z −1/2e i22 (58) where the inequality in (58) follows from the submultiplicative property of the matrix norm [29]. Since eT
i Z1/222= (eiTZ1/2Z1/2ei) = Z(i, i) and
eT
iZ−1/222= (eTi Z−1/2Z−1/2ei) = Z−1(i, i), the inequality
in (58) thus leads to 1≤ Z(i, i)Z−1(i, i) or, equivalenty,
Z−1(i, i)≥ (1/Z(i, i)).
APPENDIXB
DERIVATION OF(30)AND(31)
The Lagrangian function with respect to (29) can be writ-ten as L = L i=1 1 σs−2+ ps,ipr,iσ 2 sr,iσ 2 rd,i σ2
n,rpr,iσ2rd,i+σ2n,d+ ps,i(B
−1(i, i))−1 + λs ) σs2 L i=1 ps,i− PS,T * + λr ) L i=1 pr,i σn,r2 + σ2sps,iσsr,i2 − PR,T * − L i=1 μs,ips,i− L i=1 μr,ipr,i. (59)
As mentioned, if ps,i is given, (29) is a convex optimization
problem (for pr,i). Thus, we can obtain the optimum pr,iusing
the KKT conditions [30]. The KKT optimality conditions for solving pr,i, 1≤ i ≤ L are given as follows:
−ps,iσ 2 n,dσ2sr,iσ2rd,i c(ps,i, pr,i) + λr σn,r2 + σ2sps,iσsr,i2 − μr,i= 0 (60) where c(ps,i, pr,i) = - σ−2s + ps,i B−1(i, i)−1 ×pr,iσn,r2 σrd,i2 + σ2n,d
+ ps,ipr,iσsr,i2 σ2rd,i
.2 (61) μr,i≥ 0 (62) λr≥ 0 (63) μr,ipr,i= 0 (64) λr ) L i=1 pr,i σn,r2 + σ2sps,iσsr,i2 − PR,T * = 0. (65)
pr,i= √p
s,iσn,dσsr,iσrd,i
λ1/2r (σ2n,r+σ2sps,iσsr,i2 ) 1/2 − σn,d2 σs−2+ ps,i B−1(i, i)−1 σ2 rd,i σ2 n,r
σs−2+ ps,i(B−1(i, i))−1
+ ps,iσ2sr,i
(68)
Combining (60) and (62), we have λr
σn,r2 + σs2ps,iσ2sr,i
≥ ps,iσn,d2 σ2sr,iσ2rd,i
c(ps,i, pr,i)
. (66) Substituting (60) into (64) leads to
pr,i ) λr σn,r2 + σ2sps,iσsr,i2 −ps,iσ 2 n,dσ2sr,iσ2rd,i c(ps,i, pr,i) * = 0. (67) To satisfy (67), we then have the following.
1) If λr σn,r2 + σ2sps,iσsr,i2 > ps,iσ 2 n,dσ2sr,iσ2rd,i c(ps,i, pr,i) then pr,i= 0. 2) If λr σn,r2 + σ2sps,iσsr,i2 = ps,iσ 2 n,dσ2sr,iσ2rd,i c(ps,i, pr,i)
then (68), shown at the top of the page, holds.
Considering items 1 and 2 and pr,i≥ 0, we then find the
solution of pr,ias (30), where μr= λ−1/2r is the water level that
should be chosen to satisfy the power constraint at the relay. Similarly, we can obtain the optimum ps,i for a given pr,i, as
shown in (31). The details, however, are omitted.
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Fan-Shuo Tseng (S’08) was born in Kaohsiung,
Taiwan, in November 1980. He received the B.S. degree from Fu Jen Catholic University, Taipei, Taiwan, in 2002 and the M.S. degree from National Central University, Taoyuan, Taiwan, in 2004. He is currently working toward the Ph.D. degree with the Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan.
His research interests include transceiver design in cooperative communication and statistical signal processing in digital communication.
Wen-Rong Wu (M’89) received the B.S. degree
in mechanical engineering from Tatung Institute of Technology, Taipei, Taiwan, in 1980 and the M.S. degree in mechanical engineering, the M.S. degree in electrical engineering, and the Ph.D. degree in electrical engineering from the State University of New York, Buffalo, in 1985, 1986, and 1989, respectively.
Since August 1989, he has been a faculty mem-ber with the Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan. His research interests include statistical signal processing and digital communication.
