A limited feedback joint precoding for AF relay networks TSP10

Based on its structure, a scheme of limited feedback joint precoding using joint codebooks is then proposed, which uses a distributed codeword selection to concurrently choose two joint

A Limited Feedback Joint Precoding for

Amplify-and-Forward Relaying

Yongming Huang, Luxi Yang, Member, IEEE, Mats Bengtsson, Senior Member, IEEE, and

Björn Ottersten, Fellow, IEEE

Abstract—This paper deals with the practical precoding design

for a dual hop downlink with multiple-input multiple-output

(MIMO) amplify-and-forward relaying First, assuming that full

channel state information (CSI) of the two hop channels is

avail-able, a suboptimal dual hop joint precoding scheme, i.e., precoding

at both the base station and relay station, is investigated Based

on its structure, a scheme of limited feedback joint precoding

using joint codebooks is then proposed, which uses a distributed

codeword selection to concurrently choose two joint precoders

such that the feedback delay is considerably decreased Finally,

the joint codebook design for the limited feedback joint precoding

system is analyzed, and results reveal that independent codebook

designs at the base station and relay station using the conventional

Grassmannian subspace packing method is able to guarantee that

the overall performance of the dual hop joint precoding scheme

improves with the size of each of the two codebooks Simulation

results show that the proposed dual hop joint precoding system

using distributed codeword selection scheme exhibits a rate or

BER performance close to the one using the optimal centralized

codeword selection scheme, while having lower computational

complexity and shorter feedback delay.

Index Terms—Amplify-and-forward relaying, dual hop,

Grass-mannian codebook, joint precoding, limited feedback,

multiple-input multiple-output.

I INTRODUCTION

T HE introduction of relaying technology in cellular

net-works shows large promise to increase coverage and

system capacity at a low cost and is therefore considered in

Manuscript received November 23, 2008; accepted September 09, 2009 First

published November 06, 2009; current version published February 10, 2010 This

work was supported in part by the National Basic Research Program of China

by Grant 2007CB310603, the National Natural Science Foundation of China by

Grants 60902012 and 60672093, the National High Technology Project of China

by Grant 2007AA01Z262, Ph.D Programs Foundation of the Ministry of

Edu-cation of China under Grant 20090092120013, the European Research Council

under the European Community’s Seventh Framework Programme

(FP7/2007-2013)/ERC Grant agreement no 228044, and by the Huawei Technologies

Cor-poration The associate editor coordinating the review of this manuscript and

approving it for publication was Dr Shahram Shahbazpanahi.

Y Huang is with the School of Information Science and Engineering,

South-east University, Nanjing 210096, China He is also with the ACCESS Linnaeus

Center, KTH Signal Processing Lab, Royal Institute of Technology, SE-100 44

Stockholm, Sweden (e-mail: huangym@seu.edu.cn).

L Yang is with the School of Information Science and Engineering, Southeast

University, Nanjing 210096, China (e-mail: lxyang@seu.edu.cn).

M Bengtsson is with ACCESS Linnaeus Center, KTH Signal Processing Lab,

Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail: mats.

bengtsson@ee.kth.se).

B Ottersten is with ACCESS Linnaeus Center, KTH Signal Processing Lab,

Royal Institute of Technology, SE-100 44 Stockholm, Sweden He is also with

the securityandtrust.lu, University of Luxembourg (e-mail: bjorn.ottersten@ee.

kth.se).

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/TSP.2009.2036061

IMT-Advanced standardization work such as 3GPP LTE-Ad-vanced and IEEE 802.16m The same holds for Multiple-Input Multiple-Output (MIMO) technology [1]–[7] and its applica-tion in multiuser environments [8]–[14]

As for the combination of MIMO and relaying technology, most previous studies focus on the information theoretic limits for multi-antenna relay channels with different protocols Capacity bounds of relaying channels in a single MIMO relay network have been developed in [15], where a regenerative MIMO relay is considered For the multiple MIMO relay net-work, an asymptotical quantitative capacity result is presented

in [16], where distributive diversity is achieved through coop-eration among all the nonregenerative relays available in the network This paper focuses on practical signalling design for

a dual hop transmission with MIMO relay Although the use of regenerative relays employing decode-and-forward (DF) shows advantages over nonregenerative relays using amplify-and-for-ward (AF) in many scenarios, it requires much higher delay tolerance and may cause security problems, thus here we concentrate on the AF MIMO relaying strategy For dual hop transmission with a single MIMO AF relay station, the optimal linear transceiver design at the relay-destination link has been developed [17], [18], assuming that the channel state informa-tion (CSI) of both the source-relay and relay-destinainforma-tion links

is available at the relay station It is revealed that such a dual hop transmission can be transformed into several simultaneous data streams transmitted over orthogonal subchannels In the case of multiple AF relay stations, a relay selection scheme is presented in [19] to exploit the additional diversity offered by the multiple relay stations available in the network, where the preferred relay station is chosen as a function of CSI to imple-ment a dual hop transmission Moreover, assuming that the CSI

of all the links is available, a quasi-optimal joint design of linear transceivers at both the source-relay and the relay-destination links is developed in [20] and [21], which achieves very good performance while requiring high computational complexity Note that the above dual hop transmit schemes all require full CSI of both two hop channels and are unfortunately infeasible

in practical frequency division duplex (FDD) systems, though they provide considerable performance gains To overcome this problem, a limited feedback beamforming scheme for MIMO

AF relaying was proposed in [22], which employs Grassman-nian codebook to reduce the feedback overhead It can even

be extended to the case where the second order statistics of channel vectors are used instead of the limited instantaneous channel knowledge However, this scheme is only limited in the beamforming case and its extension to the precoding case (mul-tiple simultaneous data streams) is nontrivial, which usually re-sults in a rate performance loss especially when all the nodes 1053-587X/$26.00 © 2010 IEEE

are equipped with multiple antennas, due to the fact that the

multiplexing gain offered by MIMO channels can not be fully

exploited In this paper we aim to design a practical dual hop

transmit scheme which can fully exploit the multiplexing gains

offered by multiple antennas More specifically, we propose a

limited feedback joint precoding scheme using the criterion of

optimizing the system rate or the BER performance, where the

reduction of both feedback overhead and feedback delay will be

fully considered The main contributions are listed as follows:

1) We first present a CSI based suboptimal joint precoding

scheme for a dual hop downlink with AF, where the overall

dual hop MIMO channels can be effectively transformed

into several orthogonal subchannels by using the optimal

pairing between the eigenmodes of the dual MIMO

chan-nels Based on this, we then propose a codebook based

lim-ited feedback joint precoding scheme, where a distributed

codeword selection (CS) scheme is further proposed based

on the newly derived bounds for the capacity and the mean

square error (MSE) sum of a dual hop MIMO

transmis-sion with a linear minimum mean square error (MMSE)

receiver, such that the feedback burden and feedback delay

are both greatly reduced

2) Furthermore, we investigate the codebook design for the

proposed limited feedback joint precoding scheme, and

disclose that if the conventional method of Grassmannian

subspace packing is separately employed to construct the

codebooks at the base station and relay station, the overall

performance of the dual hop transmit scheme can be

guar-anteed to improve with the size of each of the two

code-books

The rest of this paper is organized as follows In the next

section we introduce the system model for the dual hop joint

precoding In Section III we investigate the expression of the

optimal joint precoders based on full CSI, and provide a

sub-optimal joint precoding scheme which can reduce to a limited

feedback scheme In Section IV we first present a codebook

based joint precoding system using a centralized codeword

se-lection scheme, and then propose a distributed codeword

selec-tion scheme to reduce computaselec-tional complexity and feedback

delay In Section V we analyze the design criterion of the joint

codebooks used in the dual hop precoding system Simulation

results are presented in Section VI and conclusions are drawn

in Section VII

II SYSTEMMODEL

We consider a dual hop downlink model which consists of a

base station and a relay station transmitting through two time

slots We assume that the base station is equipped with

an-tennas, the relay station is equipped with antennas and the user

terminal is equipped with antennas As depicted in Fig 1,

during the first slot, the base station employs linear precoding

to transmit simultaneous data streams, i.e., a data vector

, to the relay station Without loss of generality, we assume

, with denoting the expectation operator

The received baseband signal at the relay station is written as

(1)

Fig 1 The signal model for the dual hop joint precoding system.

where denotes the precoding matrix at the base station, without loss of generality, we assume

with being the trace operator, denotes the first hop channel matrix between the base station and the relay station, denotes the total transmit power at the base station and denotes a white Gaussian noise vector with zero mean and variance

Keeping in mind that a multiuser downlink can be trans-formed into several single-user downlinks by employing multiple access techniques such as TDMA and OFDMA, here

we concentrate on the single-user dual hop downlink More-over, we focus on relay deployments intended for coverage expansion, where the direct link between the base station and the user terminal can be neglected due to path loss or severe shadowing To succeed a downlink communication between the base station and the user terminal, during the second slot the relay station will forward its received signal using a linear precoding matrix that has to be designed With the transmit power constraint at the relay station, should satisfy that

(2) The received baseband signal at the user terminal during this time slot is written as

(3) where denotes the second hop channel matrix be-tween the relay station and the user terminal, and denotes a white Gaussian noise vector with zero mean and variance Note that in the above system model we can normalize the vari-ances of both and , and have the effects of large scale fading incorporated into the noise variances of and The key point of the above dual hop joint precoding system lies in the design of two precoders and , which commonly re-quires channel information feedback in FDD systems

Also, the number of simultaneous data streams should be carefully determined It is well known that a MIMO channel with transmit antennas and receive antennas can be transformed into a maximum of orthogonal sub-channels via singular value decomposition (SVD) The simul-taneous transmission of data streams over or-thogonal subchannels can fully utilize the multiplexing gain and

is thereby capacity-approaching, while the scheme of always transmitting a single data stream in general cannot achieve the

potential rate offered by MIMO channels, due to the fact that the

multiplexing gain cannot be fully exploited in this case This

re-sult can be easily extended to the dual hop MIMO transmission

Considering that the overall performance of the dual hop

down-link is dominated by the worse one of the two hops, it is

reason-able to choose the number of simultaneous data streams in our

system equal to if possible, instead of always

using a single data stream regardless of antenna configuration,

such that the overall rate performance can be optimized

III JOINTPRECODINGWITHFULLCSI

This section concentrates on the design of two joint precoders

assuming that full channel state information of the two hops is

available In difference to the previous related work which aims

at the optimal performance by using an iterative approach, we

are more interested in the suboptimal scheme which has a simple

structure and can provide some insight on the design of a limited

feedback joint precoding scheme

We consider an MMSE receiver at the user terminal, as shown

in [17], [18], the MSE matrix for the dual hop joint precoding

can be written as (4), shown at the top of the next page

(4)

The sum rate achieved by an MMSE receiver is upper bounded

by the instantaneous capacity , which can be expressed as [17],

[23]

(5)

where denotes the th diagonal element of ,

de-notes of the determinant of , the factor 0.5 is due to the two

channel uses which are needed by a dual hop downlink, and will

be omitted henceforth for convenience Obviously, the equality

in (5) holds when is diagonal, which means that the capacity

is achieved by an MMSE receiver in this case Therefore, the

design of and should first satisfy the condition that the

MSE matrix is diagonalized [19] Let the SVD of and

be

(6)

where , , , and are unitary matrices, and are diagonal matrices with their elements being the singular values of and , respectively Obvi-ously, the ordering of the singular values in and (and the corresponding ordering of the singular vectors in , , 2) influences the specific decomposition expressions Here we first assume an arbitrary ordering and leave its opti-mization to be solved later By substituting (6) in (4) the MSE matrix can be rewritten as

(7) The diagonalization of can be obtained by

(8) (9) where denotes the submatrix formed by the first columns

of , and are two diagonal matrices with nonnegative

respectively We partition the matrices , , and as

(10)

By substituting (8)–(10) in (7), the MSE matrix can be simplified as

(11)

respectively, the achieved sum rate can be easily derived as

(12)

It is shown that with the above joint precoding, the overall dual hop channel can be transformed into orthogonal subchannels, with their channel gains each represented by the product of a pair of eigenmodes and , while the diagonal matrices and can be viewed as the power allocation for the joint precoding Since does not influence the sum rate, it should

be set to zero to avoid wasting power The resulting precoding matrix at the relay station is

(13) where and denote the submatrices formed by the first columns of and , respectively Aiming to maximize the sum rate of the dual hop transmission, we need to optimize the

power allocation matrices and by solving the following

optimization problem:

(14)

where the two constraints are obtained from the power

con-straints at the base station and the relay station Specifically, the

first constrain is obtained by substituting (8) in

, while the second constraint is obtained by substituting (8)

and (9) in (2) We would like to note here that this

optimiza-tion should be done over the optimal ordering of the singular

values at the SVD of and , since different ordering will

give different values of Defining new notations

, by replacing the notations

in (14) with the newly defined notations, the above optimization

problem can be simplified as

(15)

It is clear from the above steps that the ordering of singular

values at the SVD of and influences both the sum rate

and the specific expression of the optimal joint precoders Thus,

the joint optimal ordering of singular values at the SVD of

and needs to be addressed It is seen from (12) that only

singular values of each hop, i.e., eigenmodes of each hop,

affect the sum rate Therefore, the problem reduces to the

op-timal selection of active eigenmodes from each hop followed

by the optimal pairing of active eigenmodes between the two

hops Since the sum rate expressed in (12) monotonically

in-creases with both the eigenmode and the eigenmode ,

the scheme of selecting the largest eigenmodes from each

hop will give a maximum sum rate Moreover, it is found from

(15) that the eigenmode pairing problem is equivalent to the

sub-channel pairing problem of the dual hop MIMO-OFDM systems

in [20] The results in [20] showed that it is optimal to pair the

active eigenmodes of the first hop ordered in

with the active eigenmodes of the second hop

optimal joint precoders should be given by the SVD of and

both having its singular values arranged in a nonincreasing

order For notation simplicity, henceforth the SVD expressions

of and refer to a nonincreasing ordering of singular

values

It should be noted that although (8) and (13) provide a simple

expression for the optimal joint precoders, the closed-form

solution for the included power allocation matrices and

are difficult to obtain Hammerström et al [20] showed that

the optimization problem in (15) cannot be exactly solved but its quasi-optimal solution can be obtained using an iterative method, and the optimal power allocation schemes at both the base station and relay station are similar to the waterfilling scheme in point-to-point MIMO systems Since it is well known that an uniform power allocation (UPA) in general only suffers from slight performance loss compared to the optimal waterfilling scheme, while having lower cost and reduced feedback burden in FDD systems, we will use UPA to form

a suboptimal joint precoding scheme Next we will show that such a UPA based dual hop joint precoding scheme can reduce

to a practical limited feedback joint precoding scheme

IV LIMITEDFEEDBACKPRECODING

By employing UPA, it is seen from (8), (13) that the joint precoders with full channel knowledge of and can be simplified as

(16) (17) where is a common scaling to fulfill the transmit power con-straint at the relay station Since it is reasonable to assume that

is available at the relay station and available at the user terminal, the above joint precoding solution requires the feedback of to the base station and to the relay sta-tion In order to reduce the feedback burden, we use two code-books to quantize and , such that, similar to the precoding for point-to-point MIMO systems, only the indices of the pre-ferred codewords are required to be fed back to the base sta-tion and relay stasta-tion, respectively However, the extension of point-to-point precoding to a dual hop transmission is nontrivial and the following problems need to be addressed

1) Though the optimal and depend on and , respectively, the codebook based choice of the precoder at the base station or the relay station is in general a function

of both and In practical FDD systems, however, only the user terminal may know the channel of both two hops without feedback If both two precoders are selected

by the user, it will suffer from a severe feedback delay due to the fact that the communication between the base station and the user terminal has to be forwarded by the relay station Therefore, the precoder selection and feed-back scheme should be carefully designed to reduce the feedback delay

2) The criterion for precoding codebook design has been widely studied in point-to-point MIMO communication systems However, it is an open problem whether these developed codebook design criteria can be directly em-ployed in the dual hop joint precoding systems

In order to address the first problem, we first present a cen-tralized codeword selection scheme which provides the optimal performance but a high feedback delay Then, we propose a sub-optimal distributed codeword selection scheme where feedback delay and complexity are both greatly reduced

A Centralized Codeword Selection

We employ precoding according to (16) and (17) and assume that two codebooks for and have been designed and de-noted as and , respectively In order to maximize the

ca-pacity expressed in (5), the codeword selection for and

can be written as

(18) Alternatively, considering that the minimization of the trace of

MSE matrix means to some degree the optimization of the error

rate performance of an MMSE receiver, an MSE-trace selection

scheme aiming to minimize the error rate may be employed and

is expressed as

(19) Obviously, the codeword selection either from the sum rate

or the error rate perspective is a function of both and ,

which requires the selection operator to know full CSI of both

two hops, and thereby is called a centralized codeword

selec-tion scheme Due to the fact that each calculaselec-tion of the

objec-tive function includes one or two matrix inversions, this

cen-tralized selection scheme requires a high computational

com-plexity Moreover, since full knowledge of the two hop channels

may only be available at the user terminal without feedback in

practical FDD systems, the codeword selection for and

should be both conducted by the user Unfortunately, the

feed-back of selection result for from the user terminal to the base

station has to be forwarded by the relay station, which results in

a high delay

B Distributed Codeword Selection

In order to reduce the feedback latency, we propose a

dis-tributed codeword selection scheme where the codeword

selec-tions for and can be concurrently conducted by the relay

station and the user terminal, respectively Since in practical

systems only can be available at the relay station without

feedback, while only can be easily available at the user

ter-minal ( should be fed forward by the relay station if the user

terminal needs), the distributed codeword selection for and should be merely based on and , respectively, such that the feedback overhead and feedback delay can be consider-ably reduced To this end, a new objective function, either from the capacity or the error rate perspective, should be designed

In this section we will derive bounds for the capacity and the MSE-trace, and then use them as the objective functions

By replacing with its SVD expression, the MSE matrix

in (4) can be simplified as

(20) Based on this, the capacity of the dual hop transmission can be lower bounded by

(21)

and are the eigenvalues of the Hermitian matrix arranged in a nonincreasing order For a proof, refer to Appendix A

Note that this capacity lower bound increases with both and , namely, the lower bound increases if

is increased, for any value of , or increases if

is increased, for any value of Since and merely depend on and respectively, the following distributed codeword selection scheme for and , will maximize the lower bound of the capacity

(22)

In order that the proposed distributed codeword selection scheme can minimize the error rate of the dual hop transmis-sion, we derive two upper bounds for the MSE trace Both decrease with two decoupled functions of and , and can

be utilized as the codeword selection criteria Based on (20),

upper bounds of the MSE trace can be expressed as

(23)

(24)

See Appendix B for proofs Obviously, minimization of the

upper bound in (23) is equivalent to the maximization of the

lower bound in (21) Thus, the distributed codeword selection

scheme of (22) also works from the perspective of minimizing

the error rate In addition, since the upper bound in (24) is

formed by a sum of two functions of and , an alternative

distributed codeword selection scheme, to optimize the error

rate performance, is given as

(25)

C Distributed Beamforming Selection

In general, the proposed distributed codeword selection

schemes for the joint precoding system are able to reduce both the

overall feedback delay and the computational complexity, while

they may suffer from a performance loss compared to the

central-ized selection scheme, due to the fact that the employed selection

objective functions are not the exact capacity or the MSE trace,

but their bounds However, our following brief analysis shows

that the proposed distributed selection scheme in the special case

of beamforming (it happens when ) will

suffer from no performance loss as compared with the centralized

one, which is consistent with the result found in [22], though

different analyzing methods are used

For the beamforming case, the MSE matrix in (20) reduces

into a scalar and can be written as

(26)

scalars, their eigenvalues are equal to themselves Also, it

follows from (2), (16), and (17) that

(27) Substituting (27) in (26), yields

(28)

that the MSE is minimized when both and

are maximized, which means that the pro-posed distributed codeword selection schemes are optimal from

the perspective of both the capacity and the error rate

V CODEBOOKDESIGNCRITERIA

We have derived codeword selection schemes for the dual hop joint precoding system, and it is important that the codebook pair of and are designed specifically for the chosen

se-lection schemes Love et al [5] have shown that the criterion of

maximizing the minimum Grassmannian subspace distance be-tween any pair of codewords is quasi-optimal for point-to-point precoding systems In dual hop precoding systems using the proposed distributed codeword selection scheme, our following analysis shows that a separate design for and using the conventional Grassmannian subspace packing method is able to guarantee that the overall performance increases with the size

of each of the two codebooks

To define a notion of an optimal codebook, we need a distortion measure with which to measure the average distor-tion It is seen from (21), (23), and (24) that when the term

is maximized, the lower bound of capacity will be maximized, and the upper bound of MSE trace will be minimized as well Thus, we utilize this term as a performance metric and define the following error difference:

(29) which is nonnegative for any choices of and , since the first term is the performance metric obtained by the optimal precoders of and Furthermore, we will design our codebook pair to minimize the average distortion

(30) where denotes the expectation with respect to and

If we define the minimum distances of the codebook pair , as

(31)

namely, the so-called projection two-norm distance between two subspaces is employed, the average distortion can be upper bounded as

(32)

where and denote the sizes of the codebooks and ,

respectively For a proof, refer to Appendix C Similar to the

, we always have that the average dis-tortion is decreased with both and Thus, we can design

the codebook pair and separately, with each codebook

constructed to maximize the minimum projection two-norm

dis-tance between any pair of codewords

VI SIMULATIONRESULTS Monte Carlo simulations are performed to illustrate the

performance of the proposed dual hop joint precoding system

with distributed and centralized codeword selection schemes

A block fading flat MIMO channel model is used throughout

the simulations The two hop channel matrices and

are both assumed to have entries independently and

identi-cally distributed with , with the large scale factors of

channels incorporated into the effective noise variances The

antenna configurations are focused on ,

Grassmannian codebook provided in [24] is employed in our

simulations, and we use the same codebook at the base station

and relay station, with its size shown in figures in terms of

the number of feedback bits The average SNR at the relay

station and the user terminal are defined as and ,

respectively For comparison, some optimal or suboptimal

dual hop precoding systems based on full channel state

infor-mation are also simulated, where the hereinafter mentioned

joint optimal scheme denotes the precoding system in (8) and

(13), the suboptimal scheme denotes the precoding system in

(16) and (17) with uniform power allocation, and the relay

side optimal scheme denotes the system in [17], [18], where

only the precoding matrix at the relay station is optimized

based on full CSI, and its rate performance is calculated as the

information theoretic instantaneous capacity of an equivalent

open-loop MIMO system Note that in the case of , the

joint optimal precoding can not be analytically solved since the

objective function in (15) is not concave with respect to

Here we use the alternating optimization method presented in

[20] to find the global or local optimum and repeat it with 50

randomly generated starting vectors, using the maximum one

in comparison

A Dual Hop Joint Beamforming

This section focuses on the configurations

only equipped with single antenna, a joint beamforming, i.e.,

, should be employed As disclosed in Section IV-C, in

this case the proposed distributed codeword selection scheme

will not result in any performance loss as compared with the

centralized codeword selection scheme, and it reduces to the

same scheme as the one presented in [22] Fig 2 shows that the

proposed dual hop joint beamforming scheme using distributed

CS exhibits slight rate loss as compared with the full CSI

based joint optimal beamforming scheme, especially for the

case of Fig 3 illustrates the cumulative

distribution function of the rate achieved by the dual hop joint

beamforming, the results also show a slight gap between the

proposed limited feedback joint beamforming scheme and the

Fig 2 The rate of the dual hop joint beamforming system with (M; L; N) = (4; 4; 1) and (M; L; N) = (2; 2; 1), 15 dB SNR at the relay station.

Fig 3 The cumulative distribution functions of the rate achieved by the pro-posed dual hop joint beamforming system, with (M; L; N) = (4; 4; 1) and (M; L; N) = (2; 2; 1), 15 dB SNR at both the receiver and relay station.

joint optimal scheme Fig 4 illustrates the BER performance

of the proposed dual hop joint beamforming using QPSK modulation Similar results are also observed

B Dual Hop Joint Precoding

This section focuses on the configurations and Fig 5 shows the sum rate of the dual hop joint precoding system using two different codeword selection schemes It is seen that the performances of the dual hop joint precoding schemes using distributed and centralized CS both increase with the codebook size Compared with the centralized

CS, the distributed CS suffers from a slight rate loss This

is because the distributed CS is based on a bound but not an exact rate metric However, the distributed CS has a shorter feedback delay and requires much lower computational com-plexity Also, it is reasonable to see that even the scheme using centralized CS has a gap from the full CSI based suboptimal scheme, due to the quantization of the optimal joint precoders

Fig 4 The BER of the proposed dual hop joint beamforming system with

(M; L; N) = (4; 4; 1) and (M; L; N) = (2; 2; 1), 15 dB SNR at the relay

station.

Fig 5 The rate of the dual hop joint precoding system with (M; L; N) =

(4; 4; 2) and 15 dB SNR at the relay station.

Moreover, the results reveal that the proposed joint precoding

scheme with distributed CS shows obvious advantage over the

beamforming scheme presented in [22] in terms of the rate

performance, especially in medium-to-high SNR regions This

is due to the fact that our proposed precoding scheme employs

multiple simultaneous data streams and thus can fully exploit

the multiplexing gain offered by the dual hop MIMO channels

Fig 6 shows the cumulative distribution function of the rate

achieved by the dual hop joint precoding system Similar results

are seen as in Fig 5

Interestingly, it is also found from Fig 5 and Fig 6 that the

full CSI based suboptimal scheme with UPA shows slight

per-formance loss as compared with the joint optimal scheme, only

in the range of medium-to-high SNRs And, the relay side

op-timal scheme shows the worst performance among three full CSI

based schemes, especially in high SNR region This is due to

the fact that the precoder at the base station is not optimized It

should also be noted that, though it seems from the curves that

the relay side optimal scheme outperforms the proposed scheme

Fig 6 The cumulative distribution functions of the rate achieved by the pro-posed dual hop joint precoding system, with (M; L; N) = (4; 4; 2), 15 dB SNR at both the receiver and relay station.

Fig 7 The BER of the proposed dual hop joint precoding system with (M; L; N) = (4; 4; 2) and 15 dB SNR at the relay station.

in most of the SNR region, this is a result of unfair compar-ison, where the performance of the relay side optimal scheme

is calculated as the instantaneous capacity, but not the sum rate achieved by an MMSE receiver

Fig 7 shows the BER performance of the dual hop joint pre-coding scheme using QPSK and MMSE receiver Both the pro-posed two distributed CS schemes, i.e., (22) and (25), are simu-lated It is seen that the BER performance of these two schemes (denoted as distributed CS #1 and #2) are very close, and they both increase with the codebook size Compared with the cen-tralized CS scheme, a loss of less than 2 dB is observed in the proposed two distributed CS schemes

VII CONCLUSION

In this paper we have presented a limited feedback joint precoding for the dual hop downlink with amplify-and-forward relaying The proposed scheme employs a distributed codeword selection and thus has lower computational complexity and feedback delay Also, we have analyzed the joint codebook

design for the joint precoding system, and revealed that a

separate codebook design for the base station and the relay

station using Grassmannian subspace packing method can

guarantee that the overall performance of the proposed scheme

improves with the size of each of the two codebooks Finally,

computer simulations have confirmed the advantage of the

proposed scheme in terms of the tradeoff between performance

and complexity, as compared with the limited feedback joint

precoding with a centralized codeword selection

APPENDIXA

PROOF OF(21)

We first present the following matrix inequalities [25]: Given

two positive semidefinite Hermitian matrices and

with eigenvalues and arranged in

nonin-creasing order, respectively, we have

(33) Since the matrix determinant equals the product of the

eigen-values, the capacity of the dual hop transmission with precoders

and can be rewritten as

(34)

By applying the inequality in (33), this yields

(35)

Assuming that the relay station transmit signal with full power,

it is derived from (2) that

(36)

Thus, we further have

(37)

This concludes the proof

APPENDIXB

PROOF OF(23)AND(24)

We first prove the first upper bound of the MSE trace in (23)

(38)

where

and the inequality in (a) comes from the lower bound of , which has been derived in Appendix A

Similar to the above derivation, the second upper bound of the MSE trace in (24) can be obtained as follows:

(39)

This concludes the proof

APPENDIXC

PROOF OF(32) Before the proof of (32), we first give the following

in-equality Given arbitrary nonnegative variables , , and

, we have [22, Lemma 1]

(40) With that, the average distortion can now be upper bounded as

shown in (41) at the top of the page, where the inequality is a

result of direct use of (40) Based on the results in [5, eq 29,

30], the two terms in the right-hand side (RHS) can be further

upper bounded as

(42)

(43) Thus, the upper bound of the average distortion can modified as

(44) This concludes the proof

ACKNOWLEDGMENT The authors would like to thank all the anonymous reviewers and the editor for their valuable comments that have helped to improve the quality of this paper

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