Document7

1554 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 4, APRIL 2013

Enhanced Limited Feedback Schemes forDL MU-MIMO ZF Precoding

Wei Wang, Member, IEEE, Atsushi Harada, Member, IEEE, and Hidetoshi Kayama, Member, IEEE

Abstract—This paper proposes new limited-feedback ChannelState Information (CSI) calculation schemes for Zero Forcing(ZF)-precoded downlink Multi-User Multiple-Input Multiple-Output (MU-MIMO) systems. It is a common understanding thatthe feedback quantized by the codebook limits the performanceof MU-MIMO. In this paper, through a quasi-ZF or a quasi-Minimum Mean-Squared Error (MMSE) weight, the channelmatrix is transferred to one of the codebook vectors, basedon which, the CSI is calculated. Thus, the quantization erroris minimized. Meanwhile, the selection for the codebook vectorguarantees the maximizing of the estimated Signal to Interferenceplus Noise Ratio (SINR). We verify that the proposed schemeobtains accurate feedback information, and the predicted weightcan be the same as the optimal linear decoder as if the receiverknew all the precoder information that is fed-forward from theBS, as long as the number of antennas for each receiver equalsthat for the transmitter, and equals that for the total transmitdata streams. Compared to the commonly used Precoding MatrixIndex (PMI) based method, which uses rank-one single user(SU)-MIMO feedback, simulation results show that the proposedschemes achieve higher sum capacities in different scenarios.Moreover, since the weight can be directly used as the decoder,the feed-forward overhead is reduced.

Index Terms—MU-MIMO, broadcast channels, feedback.

I. INTRODUCTION

THE Multiple-Input Multiple-Output (MIMO) techniqueis a key technique to improve spectrum efficiency for

wireless communication systems [1-3]. In downlink Multi-User (MU)-MIMO transmission, when the transmitter knowsthe Channel State Information (CSI), a precoding techniquecan be used to reduce the inter-user interference and enhanceperformance.

Compared to non-linear Dirty Paper Coding (DPC) [4],linear Zero-Forcing Precoding (ZFP) is a more practical tech-nique, especially when the transmitter knows only a limitedamount of CSI from each user. Interestingly, linear MU-MIMO schemes can approach close to capacity limits of non-linear MU-MIMO as long as a suitable scheduling algorithmis used and the number of users is asymptotically large [5-7].

In [8-10], limited feedback ZFP was studied for the case ofa single-antenna receiver. A codebook is designed offline andstored at both the transmitter and the receiver. The receiverquantizes the channel by selecting one of the vectors from thecodebook, and feeds back the Channel Directional Index (CDI)

Manuscript received January 10, 2012; revised June 15 and July 1, 2012;accepted October 19, 2012. The associate editor coordinating the review ofthis paper and approving it for publication was A. Ghrayeb.

The authors are with DOCOMO Beijing Communications Laborato-ries Co., Ltd., Beijing, 100190 P.R. China (e-mail: {wangw, harada,kayama}@docomolabs-beijing.com.cn).

Digital Object Identifier 10.1109/TWC.2012.030413.120051

to the transmitter along with a Channel Quality Indicator(CQI). Based on the feedback information from all users, thebase station (BS) performs scheduling and transmits signalsto the selected users through ZFP technology. Due to thequantization error, ZFP is not truly ”zero-forcing” and thereceived signals are still corrupted by the residual multi-userinterference.

In future systems such as Long Term Evolution (LTE)-Advanced, multiple receiver antennas should be supported.This paper focuses on the feedback schemes for a multiple-receive-antenna case. When the maximum transmission rankper user is limited to one, the receiver should use a weightvector to transform the MIMO channel matrix into an effectiveMultiple-Input Single-Output (MISO) channel vector, andcalculate the CDI/CQI from this vector[11-13].

One commonly supported scheme is the Precoding MatrixIndex (PMI) based feedback scheme [14-16]. A similar pro-cedure to that for LTE Single User (SU)-MIMO is performedat each mobile station (MS) when calculating the feedbackinformation. More specifically, each MS selects the best PMIfrom the codebook through exhaustive search of all possiblerank-one precoding vectors, for single user rate maximization.Because only a single stream per MS is assumed, there is nointer-stream interference at the time the feedback informationis computed and thus, the linear filter is reduced to a matchedfilter and the BS assumes that the chosen precoder is matchedto the effective channel. However, the problem with the PMIbased method is that the multiuser interference caused byquantization error is neglected. This means that the CQI andrespective rate estimate of the ZF MU-MIMO at the BSbased on the PMI based channel feedback can be overlyoptimistic. This results in inaccurate precoding and unsuitableuser selection. The achieved throughput using the PMI basedmethod is much lower than the theoretical bound.

To address this problem, this paper proposes new limitedfeedback calculation schemes. A Quasi-ZF Weight (QZW)and a Quasi-MMSE Weight (QMW) are proposed to computethe effective channels, from which the feedback information(CDI and CQI) is obtained by using the criterion for theMinimum Quantization Error (MQE) or the Maximum Signalto Interference plus Noise Ratio (MSINR). When the numberof antennas for each receiver equals that for the transmit-ter, and equals that for the total transmit data streams, theproposed weights can change the vector of the channel toone of the codebook vectors so that the quantization errorcan be minimized. In particular, with the QZW, there is noquantization error, and the CQI is accurately estimated. Thenew feedback information calculation schemes are presented

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WANG et al.: ENHANCED LIMITED FEEDBACK SCHEMES FOR DL MU-MIMO ZF PRECODING 1555

along with simplified algorithms. The proposed weights canbe directly used as true receiver decoders without the needfor the information to be fed forward from the BS. Thus, thedownlink overhead can be reduced. This work is an extensionof the work reported in [17] and [18], where only QMW withthe MSINR criterion was presented or without derivation ofthe validity of the weights. Simulation results show that theproposed schemes achieve much better performance than thePMI based method in terms of the sum capacity in differentscenarios such as different numbers of users, different numbersof feedback bits, and different transmit antenna correlations.

The proposed method is different from that in [11], whichchooses the beamforming vectors from a transmit codebook,while our methods calculate the ZF-based precoding vectors.The proposed method is also different from that in [12,13],where the channel vector quantization (CVQ) methods areproposed based on either Euclidean distance minimizationor SINR maximization. However, the minimum Euclideandistance based CVQ method can not be used in the case whenthe number for the transmit antenna equals that for the receiveantenna. Since in that case, the Euclidean distance are alwaysthe same for different codebook entries, so the minimizationcan not be implemented. While our MQE based method canstill work in that case. Comparing with the maximum SINRbased method in [12,13] which is relatively complex due tomultiple matrix inversions, our algorithm is simplified, i.e.,only needs one matrix inversion, and we propose QZW andQMW for ZF weight and MMSE weight, respectively.

The rest of this paper is organized as follows. SectionII gives the overall description of the system. Section IIIintroduces the existing PMI based scheme. Section IV presentsthe proposed schemes. Simulation results are presented inSection V. Finally, conclusions are given in Section VI.

Throughout this paper, the superscripts (·)T , (·)∗ and (·)Hdenote transpose, conjugate and conjugate transpose of amatrix, respectively. tr(A) and det(A) represent, respectively,a trace and a determinant of matrix A. Cm×n and Cm arem-by-n and m-by-1 complex matrix and vector, respectively.The vectors are defined as column vectors, and the base ofthe logarithm is assumed to be two.

II. SYSTEM MODEL

Figure 1 shows the block diagram of the closed-loopdownlink MU-MIMO system with linear precoding and userselection. We assume one BS and K MS users. The BS hasM antennas and the k-th MS has Nk > 1 (k =1,2,. . . ,K)antennas, where K>M, and Nk ≤ M . The case that Nk = Mis discussed in this paper, and the case that Nk < M will beaddressed in another paper.

The MIMO channel of the k-th MS can be described by Nk-by-M channel matrix Hk, which is assumed to be perfectlyknown at the MS. The MS calculates a CDI and CQI, feedsthem back to the BS through a finite-rate, zero-delay anderror-free feedback channel. We assume that the maximumtransmission rank per user is limited to one.

A codebook saved at both the BS and MS is utilized toquantize the CDI. In this paper, a Discrete Fourier Transform(DFT)-based codebook is used. It consists of 2B vectors,

Fig. 1. Closed-loop downlink MU-MIMO system

where B is the number of feedback bits. Each vector has Melements. The m-th element of the i-th vector is expressed as

ci (m) = exp(− j2πim

2B

)/√M,

m = 0, ...,M − 1,

i = 0, ..., 2B − 1.

(1)

The 2B vectors compose G = 2B/M unitary matrices, andthe g-th unitary matrix is denoted as

Cg = {cg, cg+G, ..., cg+(M−1)G}, (g = 0, ..., G− 1). (2)

Since the procedure to obtain the CDI and CQI is the mainfocus of this paper, detailed introduction of the existing PMIbased scheme and the proposed schemes will be given inSections III and IV, respectively.

A. ZF Precoding and User Selection

Utilizing the CDI and CQI sent from all users, the BSselects user set Ω that includes A≤M MSs according to apredetermined criterion, then transmits signals to these MSsafter performing linear precoding.

Denoting s = [s1, s2, · · · , sA]T ∈ CA,E[|sk|2

]= 1, (k =

1, ..., A, i.e., the constellation symbols are normalized tounit energy ) as the modulated symbols and denoting T =[t1, t2, · · · , tA] ∈ CM×A as the precoding matrix, transmittedsignal x ∈ CM can be expressed as

x = Ts. (3)

Under the assumption of ZF precoding, the precodingmatrix is expressed as

T = T′diag(p)1/2 = HH

(HHH

)−1

diag(p)1/2, (4)

where H = [h1, ..., hA]H , hk, (k = 1, ..., A) is the quantized

channel of user k, and hk = cCDIk . Vector p = [p1, ..., pA]includes the power loading coefficients. Under the assumptionof equal power distribution across all users, each elementpk = P/A ‖t′k‖2, (k = 1, · · · , A), where P is the total powerconstraint on the transmitted signal, and t′k is the k-th columnof T

′.

If the user selection criterion is to maximize the sum rate,the scheduler should estimate the sum rate for each possibleuser set, and then select the one having the maximum value.

The estimated sum rate of user set Ω is given by

RΩ =∑k∈Ω

log (1 + γk) , (5)

where γk =∣∣∣hH

k tk

∣∣∣2 CQIk is the estimated SINR.

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B. Receiver and Performance Measurement

At the k-th MS, soft output data sk are obtained throughmultiplying the received signal by decoding vector rk, i.e.,

sk = rHk (Hkx+ nk) = rHk

(Hk

∑i∈Ω

tisi + nk

), (6)

where nk is an additive white noise vector with elementsthat are identical independent distributed (i.i.d.) zero meancomplex Gaussian random variables with variance N0. Thechannel matrix is based on the Kronecker model [14]

Hk =R

1/2Rx,kGkR

1/2Tx,k√

tr(RRx,k)(7)

and RRx,k = E{HkH

Hk

} ∈ CNk×Nk and RTx,k =E{HT

kH∗k

} ∈ CM×M are the receiver and transmitter cor-relation matrices, respectively. Gk ∈ CNk×M has entries thatare i.i.d. zero-mean circularly symmetric complex Gaussiandistributed with unit variance. For 4 transmit antennas, thetransmitter correlation matrix can be expressed as

RTx,k =

⎛⎜⎜⎜⎜⎝

1 ρejθ ρ2ej2θ ρ3ej3θ

ρejθ 1 ρejθ ρ2ej2θ

ρ2ej2θ ρejθ 1 ρejθ

ρ3ej3θ ρ2ej2θ ρejθ 1

⎞⎟⎟⎟⎟⎠ (8)

where ρ is the correlation coefficient, and θ is the angle ofdeparture (AoD).

Due to the quantization error in the feedback informa-tion, the inter-user interference cannot be eliminated entirelythrough precoding. Therefore, the receiver either uses a ZFdetector or a MMSE detector to mitigate the residual inter-ference. The ZF detector and the MMSE detector are givenrespectively by

rZFk

H= (Hktk)

H(HkT (HkT)

H)−1

, and (9)

rMMSEk

H= (Hktk)

H

(HkT (HkT)

H+

AN0

PINk

)−1

.

(10)They are normalized to unit norm vectors.The final SINR of user k is given by

γk =P

AN0

∣∣rHk Hktk∣∣2

1 + PAN0

∑i∈Ω,i�=k

∣∣rHk Hkti∣∣2 . (11)

The final sum rate is given by

RΩ =∑k∈Ω

log (1 + γk). (12)

III. EXISTING FINITE FEEDBACK SCHEME AND PROBLEMS

In the case of PMI based ZF MU-MIMO feedback, throughexhaustive searching of all possible rank-one precoding vec-tors, each MS selects the best PMI that maximizes the assumedsum rate. Because only a single stream per MS is assumed,there is no inter-stream interference at the time when comput-ing feedback information at the MS. Thus, the linear filter is

reduced to a matched filter. When the precoder is assumed tobe the i-th codebook vector, the matched filter weight is

wik = Hkci, (13)

and the SINR becomes the Signal to Noise Ratio (SNR), i.e.,

γik =

P

N0‖Hkci‖2 . (14)

Then a codebook vector that has the maximum SNR ischosen, and the index of this code vector is fed back as aPMI, i.e.,

PMIk = arg maxi=0,...,2B−1

γik. (15)

Since the linear filter is reduced to a matched filter, the BSassumes that the effective channel direction is matched to theselected precoder. This means that although the MS feeds backa PMI, the BS sees it as a CDI.

After the best PMIk is chosen, the CQI value is determinedas

CQIk = γPMIkk , (16)

which is fed back to the BS together with PMIk.The problem with the PMI is that when the receiver uses

ZF detection (9) or MMSE detection (10), the true effectivechannel is not the same as the assumed one obtained from thematch filter-based weight (13), i.e., rkHHk �= wk

HHk. Thus,the feed back information is mismatched.

By using the mismatched CDI and CQI, precoding will beinaccurate and user selection will be inappropriate. This resultsin severe interference and degraded performance.

IV. PROPOSED FINITE FEEDBACK SCHEMES

To calculate the CDI and CQI, each MS must computeeffective channel vector hk from Hk. The effective channelis given by multiplying Hk

H with weight vector wk, i.e.,hk = Hk

Hwk. Different from rk which is used for truesignal decoding, wk is temporally used for the CDI and CQIcalculation. In this section, three new schemes are proposedto obtain the CDI and CQI. These schemes better match thetrue effective channel and consequently, the performance isimproved.

A. QZW - Based Feedback Scheme

In this sub-section, a QZW is proposed to compute theeffective channel and obtain the CDI and CQI.

Let us first consider that if wk = rZFk , the feedback

information would match the true channel. However, the ZFweight (9) includes transmit matrix T, which cannot be knownwhen calculating the feedback information. Then the weightcannot be predicted exactly. However, the effective channelneeds to be quantized to one of the codebook vectors. Ifthe effective channel has the same direction as one of thecodebook vectors, then there will be no quantization error. Inthat case, the effective channel vector should be

wCDIk

HHk = cHCDI . (17)

If we assume the effective channel is ci,(i = 0, ..., 2B − 1),then the weight becomes

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wik

H= cHi H†

k, (18)

where H†k is the pseudo-inverse of Hk . When M = Nk,

H†k = H−1

k .Compared to the ZF decoder in (9), it seems that there is an

assumption in calculating the weight in which T is assumedto be one of the unitary codebook matrices, i.e.,Cg, and theprecoding vector of user k is one code vector, ci, in Cg . Thismeans that M selected users are assumed whose quantizedchannels are orthogonal to each other. Then the QZW shouldbe normalized as

wik

H= wi

k

H/∥∥∥wi

k

H∥∥∥ = cHi H†

k/∥∥∥cHi H†

k

∥∥∥ . (19)

When M = Nk, since H−1k Hk = I, hi

k can be expressedas

hik

H= wi

k

HHk = cHi H−1

k Hk/∥∥cHi H−1

k

∥∥ = cHi /∥∥cHi H−1

k

∥∥ .(20)

Thus, the approximate lower bound of the expected SINRwith the assumed ci can be estimated as [10]

γik =

∥∥hik

∥∥2 cos2 θikMN0

P +∥∥hi

k

∥∥2 sin2 θik . (21)

where θik is the angle between the effective channel hik and

ci . Then cos2 θik =∣∣∣hi

kHci

∣∣∣2 / ∥∥hik

∥∥2.

Since‖ci‖2 = 1, cos2 θik =∣∣cHi ci

∣∣2 / ‖ci‖2 = 1 andsin2 θik = 0, γi

k can be simplified to:

γik =

P

MN0

∥∥hik

∥∥2 =P

MN0

∥∥cHi H−1k

∥∥2 . (22)

Then CDIk is selected as the index of one codebook vectorthat can maximize the SINR, i.e.,

CDIk = arg maxi=0,...,2B−1

∣∣γik

∣∣= arg min

i=0,...,2B−1

∥∥cHi H−1k

∥∥ . (23)

The corresponding CQIk is given by replacing i with CDIkin (22).

CQIk =P

MN0

∥∥cHCDIkH−1

k

∥∥2 . (24)

It should be noted that when M = Nk = A, this assumedweight is exactly the same as the true ZF decoder, i.e.,wCDIk

k = rZFk . The verification is shown hereafter.

Proof:Since

cHCDIkH−1

k

= cHCDIktkt

Hk (TH)−1(T)−1(Hk)

−1

= cHCDIktkt

Hk (HkTTH)−1

= cHCDIktkt

Hk HH

k (HHk )−1(HkTTH)−1

= |cHCDIktk|(Hktk)

H(HkTTHHHk )−1

= |cHCDIktk|(Hktk)

H(HkT(HkT)H)−1,

thus,

wCDIkk

H=

cHCDIkH−1

k

‖cHCDIkH−1

k ‖

=(Hktk)

H(HkT(HkT)H)−1

‖(Hktk)H(HkT(HkT)H)−1‖= rZF

k

H. (25)

In the following, we will also show that when M = Nk =A, by using the proposed scheme, the multiuser interferencecan be totally eliminated and the SINR can be estimatedaccurately at the BS.

After zero-forcing precoding (4), the interference in thedirections of the quantized channel for other users is forcedto zero, i.e., hH

k ti∈Ω,i�=k = cHCDIkti∈Ω,i�=k = 0. When the

decoder is rHk = wCDIkk

H= cHCDIk

H−1k /

∥∥cHCDIkH−1

k

∥∥ , thereceived interference becomes

rHk Hk

∑i∈Ω,i�=k

tisi =cHCDIk

H−1k

‖cHCDIkH−1

k ‖Hk

∑i∈Ω,i�=k

tisi

=cHCDIk

‖cHCDIkH−1

k ‖∑

i∈Ω,i�=k

tisi

= 0. (26)

This means that the multiuser interference can be com-pletely canceled.

Then the final SINR (11) becomes

γk =P

MN0

∣∣rHk Hktk∣∣2

=P

MN0

∣∣cHCDIkH−1k Hktk

∣∣2 / ∥∥cHCDIkH−1k

∥∥2=

P

MN0

∣∣cHCDIktk∣∣2 / ∥∥cHCDIkH

−1k

∥∥2=∣∣∣hH

k tk

∣∣∣2 CQIk. (27)

Therefore, the SINR can be accurately estimated by

simply multiplying CQIk (24) by∣∣∣hH

k tk

∣∣∣2 , i.e., γk =∣∣∣hHk tk

∣∣∣2 CQIk. = γk.Since the estimated SINR is accurate, the sum rate can also

be accurately estimated. Then suitable users can be selected,and the precoding can be true zero forcing. This means that theMS can expect an exact receiver weight and there is no need tofeed-forward the precoding information to the MS. Thus, theproposed scheme can reduce the downlink signaling overhead.

With the proposed scheme, the receiver can directly usewCDIk

k as the decoder, i.e., rZFk = wCDIk

k . If the channel haschanged, the receiver can use CDIk to recalculate wCDIk

k ,and then use it as the decoder, as if the receiver knew all thereal precoders, i.e., tk and T. When M = Nk > A, it canalso be verified that

wCDIkk ≈ rZF

k .

So this weight is called a QZW.

B. QMW - Based Feedback Scheme

Similar to the QZW, in this sub-section, a QMW is proposedto compute the effective channel and obtain the CDI and CQI.

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The QMW with ci(i = 0, ..., 2B − 1) is given by

wik

H= cHi HH

k

(HkH

Hk +

MN0

PINk

)−1

. (28)

Then this weight is normalized by∥∥wi

k

∥∥ to obtain the unitnorm.

The effective channel is given by

hik

H= wi

k

HHk. (29)

With this weight, the corresponding SINR is estimated asthe approximate lower bound of the expected SINR, i.e.,

γik =

∥∥hik

∥∥2 cos2 θikMN0

P +∥∥hi

k

∥∥2 sin2 θik=

|wikHHkci|2

MN0

P + ‖ wikHHk ‖2 −|wi

kHHkci|2

. (30)

By substituting (28) into (30) we obtain

γik = cHi HH

k (HkHHk −Hkcic

Hi HH

k +MN0

PINk

)−1Hkci.

(31)Then CDIk is chosen as the index of one code vector that

achieves the maximum γik, i.e.,


γik . (32)

It is noted that to select the CDI, the above algorithm mustcalculate γi

k 2B times, which includes 2B matrix inversions,that causes a high computational load to be placed on theMS, as also noticed in [12,13]. In the following, a simplifiedmethod using only one matrix inversion is presented, i.e.,

CDIk

= argmaxi=0,...,2B−1

|cHi HHk (HkH

Hk −Hkcic

Hi HH

k + MN0

P INk)−1

Hkci|= argmax

i=0,...,2B−1

|cHi HHk (HkH

Hk + MN0

P INk)−1Hkci|.

(33)Thus, only one matrix inversion and O

(2B)

matrix/vectormultiplications need to be calculated.

Finally, CQIk is given by replacing i with CDIk in (31),i.e.,

CQIk

= cHCDIkHH

k (HkHHk −HkcCDIkc

HCDIk

HHk + MN0

P INk)−1

HkcCDIk .

(34)This scheme is called as the MSINR–QMW feedback

scheme.We note that angle θik between the effective channel and

the quantized channel has the largest impact on γik. Since θik

is proportional to the quantization error, we propose anotheralgorithm that aims to minimize the quantization error.

For each code vector ci, the corresponding cos θik is calcu-lated as

cos θik = wik

HHkci/

∥∥∥wik

HHk

∥∥∥ . (35)

To minimize the quantization error, CDIk is chosen as theindex of one code vector that achieves the minimum θik, orequally the largest

∣∣cos θik∣∣ , i.e.,


| cos θik|. (36)

This scheme is called as the MQE–QMW feedback scheme.The verification of the similarity between the QMW and

true MMSE decoder is given in the Appendix.

C. Impact on Downlink Signaling Overhead Reduction

In the existing schemes, the BS should inform each selecteduser of the precoding vectors through dedicated downlinkcontrol signaling in order to allow the users to calculateZF detector rZF

k (9) or MMSE decoder rMMSEk (10). The

procedure is shown in Fig. 2(a), where both a commonreference signal and precoded reference signal are needed.

In the proposed schemes, since the MS can use the selectedCDI to recalculate the receiver decoder according to thecurrent channel information, the BS does not need to feed-forward the precoder information to each MS. Then only acommon reference signal is needed, as shown in Fig. 2(b).Thus the downlink overhead can be reduced.

V. SIMULATION RESULTS

The sum rate performance of the proposed methods iscompared to that for the PMI based method based on computersimulations. The BS is assumed to have M = 4 transmit anten-nas, and each MS is assumed to have N = 4 receive antennas.If not noted specifically otherwise, it is assumed that thecorrelation matrixes RRx,k and RTx,k are unit matrixes, andthere is no delay in the feedback procedure. Each MS sendsback a B-bit CDI and non-quantized CQI. The calculation ofthe sum rate is given by (12). For each point, 5000 MonteCarlo simulations are performed and the average performanceis shown. Greedy scheduling based on the maximum of thesum rate is used.

Figure 3 shows the sum rate for various SNRs ( defined asthe average per user ) of the 3 proposed weight calculationmethods (MSINR-QMW, MQE-QMW, and QZW) and theexisting PMI based method. Ten users are assumed and fourbits are used to feed back the CDI information. For eachmethod, not only the results with feed-forwarding but alsothose without feed-forwarding are shown. For comparison,the upper bound is also shown with the assumption thatthe BS knows all channel information to calculate iterativelyoptimal precoders and decoders, and that the BS can feed-forward the decoders to the MS without any error. It can beseen that when with feed-forwarding, MSINR-QMW exhibitsthe highest performance. Compared to it, MQE-QMW isapproximately 0.5 bps/Hz lower at all SNRs. The reason forthis is that MQE is not meant to maximize the sum rate, but tominimize the quantization error. For QZW, the gap comparedto MSINR-QMW is relatively large in a low SNR range, i.e., atthe SNR of 5 dB, the gap is 1.2 bps/Hz, but the gap decreasesto less than 0.2 bps/Hz when the SNR is 30 dB. This is becausethe QZW amplifies the noise when the SNR is low, and QMWbecomes asymptotically QZW at high SNR. MSINR-QMW

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(a) With feed-forward

(b) Without feed-forward

Fig. 2. Procedure of DL MU-MIMO

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

45

Average SNR (dB)

Spect

rum

effic

iency

(bps/

Hz)

MSINR−QMW with feedforwardMSINR−QMW w/o feedforwardMQE−QMW with feedforwardMQE−QMW w/o feedforwardQZW with feedforwardQZW w/o feedforwardPMI with feedforwardPMI w/o feedforwardUpper bound

Fig. 3. Sum rate vs. average SNR, M = N = 4, K = 10 and B = 4.

and MQE-QMW both outperform the PMI based method atall SNRs, and QZW outperforms the PMI based method whenthe SNR is larger than 10dB. At the spectrum efficiency of25 bps/Hz, the MSINR-QMW and QZW both outperformthe PMI based method by approximately 3.2 dB (12.8%).For the three proposed methods, the difference between theresults with and without feed-forwarding is less than 1%when the SNR is higher than 15 dB. It is noted that thethree proposed methods without feed-forwarding outperform

the PMI based method with feed-forwarding when the SNR islarger than 12dB. The performance of the PMI-based methodwithout feed-forwarding is very poor and is almost infeasible.Since the performance of the three proposed methods arevery similar, in the following simulations, we use MSINR-QMW as a representative of the proposed methods to make acomparison with the PMI based method.

Figure 4 shows the sum rate of the MSINR-QMW and thePMI based method versus the number of users. Here, theSNR = 10 dB and B = 4 bits. For the PMI based method,the performance with feed-forwarding is shown. This meansthat the BS should send the precoding vector indexes orthe precoded reference signals to each selected MS for thedecoding vector calculation. For the MSINR-QMW scheme,the performance without feed-forwarding is shown, i.e., the BSdoes not need to send the precoding information and the MSsimply uses wCDIk

k as the decoding vector. An increase in theperformance gain is observed with the increase in the numberof users. When there are 10 users, MSINR-QMW outperformsthe PMI based method by 4.1 bps/Hz. When there are 50users, MSINR-QMW outperforms the PMI based method by5.1 bps/Hz.

Figure 5 shows the sum rate of the MSINR-QMW and thePMI based method versus the number of feedback bits. In thiscase, the SNR = 20 dB and K= 10 users. The figure showsthat with the increase in the number of feedback bits, boththe performance of the MSINR-QMW and that for the PMIbased method increase, while the MSINR-QMW without anyfeed-forwarding always outperforms the PMI based with feed-forwarding. When the the number of feedback bits is 3, thegain is about 3.7 bps/Hz, and when that number is 4, 6 or 8,the gain becomes always about 4 bps/Hz.

Figure 6 shows the performance of MSINR-QMW andthe PMI based method versus the SNR when the correlationbetween the transmitter antennas is high, i.e., the correlationcoefficient in RTx,k is set to 0.92, and the AoD is randomlyspread within 360 degrees. K = 10 users is assumed andthere are 4 feedback bits. Compared to figure 3, figure6 shows that the high transmitter correlation yields higherperformance for both methods at low SNRs, i.e., when SNRis smaller than 20dB, and yields lower performance at highSNRs. That is because at low SNR, the number of the totaltransmit streams (A) is relatively smaller, then high correlationantennas generates beams which concentrate more energy foreach user. While at high SNRs, when the A becomes larger, thehigh correlation brings more interference among each beams,thus the performance becomes lower. The advantage of theMSINR-QMW method over the PMI based method still exists,i.e., when the SNR = 20 dB, the gain is approximately 4.5bps/Hz, and the higher the SNR, the larger the gain.

VI. CONCLUSION

Novel schemes are proposed for ZF-precoded downlinkMU-MIMO systems to calculate the codebook-based feedbackinformation. Under the assumption that the maximum trans-mission rank per user is limited to one, a Quasi-MMSE Weightand a Quasi-ZF Weight are proposed for each user to transformthe MIMO channel into an effective MISO channel. Then

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10 15 20 25 30 35 40 45 5019

20

21

22

23

24

25

26

27

28

number of users (K)

Spect

rum

effic

iency

(bps/

Hz)

MSINR−QMW w/o feedforwardPMI with feedforward

Fig. 4. Sum rate vs. the number of users K,M = N = 4, SNR = 20 dBand B = 4.

2 3 4 5 6 7 818

19

20

21

22

23

24

Number of feedback bits (B)

Spe

ctru

m e

ffici

ency

(bp

s/H

z)

MSINR−QMW w/o feedforwardPMI with feedforward

Fig. 5. Sum rate vs. the number of CDI feedback bits B, M = N = 4,SNR = 20 dB and K = 10.

the maximum-SINR criterion or the minimum-quantization-error criterion is used to choose a suitable CDI and calculatethe corresponding CQI. Compared to the existing PMI basedmethod, which uses the rank-one SU-MIMO feedback, theproposed schemes achieve more accurate feedback informa-tion to match the true channel, and reduce the feed-forwardoverhead. Simulations verified that the proposed methodsachieve much higher sum rates than that for the PMI basedmethod, e.g., for a 4 by 4 antenna configuration with the CDIfeedback of 4 bits and with 10 users, at the spectrum efficiencyof 25 bps/Hz, the gain is approximately 12.8%.

APPENDIX APROOF OF THE QMW, I.E., WHEN M=N=A,

wCDIkk ≈ rMMSE

k .

Proof:Since

Fig. 6. Sum rate vs. average SNR ,M = N = 4, K = 10, and B = 4,high Tx correlation

cHCDIk

HHk

(HkH

Hk +

MN0

PINk

)−1

= cHCDIktkt

Hk

(TTH

)−1HH

k

(HkH

Hk +

MN0

PINk

)−1

= cHCDIk

tktHk

((HkH

Hk +

MN0

PINk

)(H

Hk

)−1TT

H

)−1

= cHCDIktkt

Hk

(HkTTH +

MN0

PINk

(HH

k

)−1TTH

)−1

= cHCDIktkt

Hk HH

k

(HkTTHHH

k +MN0

PINk

(HH

k

)−1TTHHH

k

)−1

≈∣∣∣cHCDIk

tk

∣∣∣ (Hktk)H

(HkTTHHH

k +MN0

PINk

)−1

, (37)

thus,

wCDIkk =

cHCDIkHH

k

(HkH

Hk + MN0

P INk

)−1∥∥∥cHCDIkHH

k

(HkHH

k + MN0

P INk

)−1∥∥∥

≈(Hktk)

H(HkTTHHH

k + MN0

P INk

)−1

∥∥∥∥(Hktk)H(HkTTHHH

k + MN0

P INk

)−1∥∥∥∥

= rMMSEk .

(38)

REFERENCES

[1] G. Caire and S. Shamai, “On the achievable throughput of a multi antennaGaussian broadcast channel,” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp.1691–1706, 2003.

[2] S. Vishwanath, N. Jindal, and A. Goldsmith. “Duality, achievable rates,and sum-rate capacity of Gaussian MIMO broadcast channels.” IEEETrans. Inf. Theory, vol. 49, no. 10, pp. 2658–2668, Oct. 2003.

[3] H. Weingarten, Y. Steinberg, and S. S. Shamai, “The capacity regionof the Gaussian multiple-input multiple-output broadcast channel,” IEEETrans. Inf. Theory, vol. 52, no. 9, pp. 3936–3964, 2006.

[4] M. Costa, “Writing on dirty paper,” IEEE Trans. Inf. Theory, vol. 29, no.3, pp. 439–441, 1983.

[5] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methodsfor downlink spatial multiplexing in multiuser MIMO channels,” IEEETrans. Signal Process., vol. 52, no. 2, pp. 461–471, 2004.

[6] T. Yoo and A. Goldsmith, “On the optimality of multi antenna broadcastscheduling using zero-forcing beamforming,” IEEE J. Sel. Areas Com-mun., vol. 24, no. 3, pp. 528–541, 2006.

[7] A. Wiesel, Y. Eldar, and S. Shamai, “Zero-forcing precoding and gener-alized inverses,” IEEE Trans. Signal Process., vol. 56, no. 9, pp. 4409–4418, 2008.

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[8] N. Jindal, “MIMO broadcast channels with finite rate feedback,” IEEETrans. Inf. Theory, vol. 52, no. 11, pp. 5045–5060, 2006.

[9] P. Ding, D. Love, and M. Zoltowski, “Multiple antenna broadcastchannels with shape feedback and limited feedback,” IEEE Trans. SignalProcess., vol. 55, no. 7, pp. 3417–3428, July 2007.

[10] T. Yoo, N. Jindal, and A. Goldsmith, “Multi antenna broadcast channelswith limited feedback and user selection,” IEEE J. Sel. Areas Commun.,vol. 25, no. 72, pp. 1478–1491, 2007.

[11] J. Schreck, P. Jung, and G. Wunder. “Approximation of multiuser ratesin MIMO OFDM downlink systems,” in 2009 OFDM Workshop.

[12] M. Trivellato, F. Boccardi, and H. Huang, “On transceiver design andchannel quantization for downlink multi-user MIMO systems with limitedfeedback,” IEEE J. Sel. Areas Commun., vol. 26, no. 8, pp. 1494–1504,2008.

[13] G. Dietl, O. Labreche, and W. Utschick, “Channel vector quantizationfor multiuser MIMO systems aiming at maximum sum rate,” in Proc.2009 IEEE Globecom.

[14] K. Kusume, K. Khashaba, G. Dietl, and W. Utschick, “Hybridsingle/multi-user MIMO transmission based on implicit channel feed-back,” in Proc. 2011 IEEE ICC.

[15] 3GPP R1-113293 NTT DOCOMO, “Capacity enhancement of DLMU-MIMO with increased feedback bits for single point transmissionscenarios,” Oct. 2011.

[16] B. Clerckx, G. Kim, J. Choi, and Y. J. Hong, “Explicit vs. implicitfeedback for SU and MU-MIMO,” in Proc. 2010 IEEE Globecom.

[17] W. Wang, Z. Zhang, and H. Kayama, “Limited feedback zero-forcingprecoding scheme for MIMO broadcast channels,” in Proc. 2008 IEEEPIMRC.

[18] W. Wang, Z. Zhang, and H. Kayama, “Quasi-MMSE/ZF based limitedfeedback scheme for MU-MIMO precoding,” in Proc. 2010 IEEE Globe-com.

Wei Wang received the Ph.D. degrees in Elec-tronic and Information Engineering from BeijingUniversity of Aeronautics and Astronautics, Chinain 2002. Since 2002, she has been an algorithmengineer in VIA Telecom Inc. working on researchand development of GSM/GPRS and CDMA. In2006, she joined DOCOMO Beijing Labs, where sheis an advanced researcher in the Innovative RadioTransmission Laboratory, engaged in the researchof the physical layer technologies for 4G mobilecommunications. Her main research interests include

MIMO precoding and decoding, multiuser scheduling, space-time/frequencyprocessing. She received the Outstanding Paper Award in 2001 from ChineseInstitute of Electronics.

Atsushi Harada received his B.E. and M.E. degreesfrom Kyushu University, Fukuoka, Japan in 1996and 1998, respectively. In 1998, he joined NTTMobile Communications Network, Inc. (now NTTDOCOMO, Inc.). Since joining the company, hehas been engaged in the research and developmentof adaptive antenna array for DS-CDMA and L2protocol design for mobile communications. Since2010, he has been the director of the Innovative Ra-dio Transmission Laboratory in DOCOMO BeijingCommunications Labs. Co., Ltd. P.R. China. He was

a recipient of the IEICE Young Engineer Award in 2004 and the 2004 ActiveResearch Award in Radio Communication Systems.

Hidetoshi Kayama received the B.E., M.E. andPh.D. degrees from Kyoto University in 1987, 1989and 2004, respectively. He has been with NTTRadio Systems Laboratories since 1989, workingon research of packet radio access protocols forcellular and wireless LAN. Since 1998, he waswith NTT DoCoMo, Inc., where he engaged in thedevelopment of the PHS 64kbps data communica-tion and research for QoS and MAC protocol for4G mobile communications. Since 2004, he was adirector of Lab in DOCOMO Beijing Labs. From

2010 to 2012, he was a leader of a research project about cognitive radioorganized by MIA, Japan. Now he is the President and CEO of DOCOMOBeijing Communications Laboratories Co., Ltd., China. He received theScholarship Encouragement Award in 1995 from IEICE, and Best PaperAward of ICT2002, APCC2008 and 2012, ICCTA2009 respectively.

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