EfﬁcientFeedbackviaSubspace-BasedChannel ...€¦ · from MSs to BSs by using uplink resources....

Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2008, Article ID 847296, 13 pagesdoi:10.1155/2008/847296

Research ArticleEfficient Feedback via Subspace-Based ChannelQuantization for Distributed Cooperative Antenna Systemswith Temporally Correlated Channels

Jee Hyun Kim,1 Wolfgang Zirwas,1 and Martin Haardt2

1 Nokia Siemens Networks GmbH & Co. KG, St.-Martin-Strasse 76, 81541 Munich, Germany2 Communications Research Laboratory, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany

Correspondence should be addressed to Jee Hyun Kim, [email protected]

Received 15 June 2007; Revised 28 September 2007; Accepted 23 November 2007

Recommended by Ana Pérez-Neira

It is one of the biggest challenges of distributed cooperative antenna (COOPA) systems to provide base stations (BSs) with down-link channel information for transmit filtering (precoding). In this paper, we propose a novel feedback scheme via a subspace-basedchannel quantization method. The proposed scheme adopts the chordal distance as a channel quantizer criterion so as to capturechannel characteristics represented by subspaces spanned by the channel matrix. We also propose a combined feedback schemewhich is based on the hierarchical codebook construction method in an effort to reduce the feedback overhead by exploiting thetemporal correlation of the channel. The proposed methods are tested for distributed COOPA systems in terms of simulations.Simulation results show that the proposed subspace-based channel quantization method outperforms the analog pilot retransmis-sion method, and the combined feedback scheme performs as well as the permanent full-feedback scheme with a much smalleramount of uplink resources.

Copyright © 2008 Jee Hyun Kim et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Cooperative antenna (COOPA) systems have recently be-come a hot research topic, as they promise significantlyhigher spectral efficiency than conventional cellular systems[1]. COOPA systems are also termed coordinated networksystems in some references [2–5]. The gain is acquired byadopting intercell interference (ICI) cancelation schemes,for example, joint transmission/joint detection (JT/JD) algo-rithms. An improvement factor of more than 5 in spectralefficiency is observed for COOPAsystems with an antennaarrangement of 4 transmit antennas per base station and 4receive antennas per user, compared with uncoordinated cel-lular systems [2]. In COOPA systems, several adjacent basestations (BSs) are cooperating so as to support multiple mo-bile stations (MSs) which are located in the correspond-ing cooperative area (CA). Therefore, COOPA systems canbe regarded as a multiuser multiple-input multiple-output(MU-MIMO) system, in which multiple transmit antennasat the BS, which are conventionally considered to be locatedin one BS, are spread over several BSs. This distributed na-

ture, which is attributed to the fact that several geographi-cally distributed BSs are used as transmit antennas, leads tofull macro-diversity gains. Moreover, COOPA systems haveadvantageous features compared with conventional cellularsystems, for example, increased degrees of freedom, betterICI cancelation performance, the rank enhancement effect ofthe channel matrix, and so forth, [1]. In addition, JT/JD al-gorithms of COOPA systems calculate a common weightingmatrix for all BSs, cancel ICI, and allow the system to servemultiple MSs at the same time and frequency resource. Thisleads to a real-frequency reuse equal or close to 1.

COOPA systems are based on the cooperation betweenmultiple distributed BSs. This means that COOPA systemsneed a fast and efficient backbone network as well as the cen-tral unit (CU) which manages the cooperation amongst as-sociated BSs. The CU renders the overall network structuremore complex by adding one more layer in the hierarchy, andeventually increases the costs. In [6], distributed organiza-tion methods have been suggested to address this problem.

One of the main challenges of the distributed COOPAsystem is channel estimation for the downlink channel. All

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2 EURASIP Journal on Advances in Signal Processing

of the associated BSs in the CA need to know the full channelstate information to calculate the corresponding precodingweight matrix. This information is needed to be transferredfrom MSs to BSs by using uplink resources. As several BSsand several MSs are involved in COOPA systems and each BSand MS may be equipped with multiple antennas, the num-ber of channel state parameters to be fed back is expected tobe big. In an effort to reduce the amount of feedback, theanalog pilot retransmission method has been suggested andtested in [6], but the throughput of this method reaches only40% of that of the ideal case, which requires supplementaryfeedback schemes [1].

On the other hand, finite rate feedback strategies inMIMO systems have been extensively investigated recently.Beamforming codebook design methods are suggested basedon Grassmannian packing [7] and systematic unitary design[8], which guarantee substantial gains with just a small num-ber of feedback bits. A precoding matrix codebook construc-tion method, which is designed to maximize the mutual in-formation, has been developed based on vector quantization(VQ) techniques [9, 10]. These methods are designed to se-lect a beamforming vector for the MISO case or a precod-ing matrix for the MIMO case from a set of codes. They aredeveloped for point-to-point MIMO channels, in which thetransmitter serves one receiver at a time. In the single-userMIMO (SU-MIMO) case, it is known that even a small num-ber of bits per antenna can be quite beneficial [11]. In themultiuser MIMO (MU-MIMO) case, feedback rate scaling isrequired to achieve a throughput close to that with perfectfeedback information in order to compensate for the inter-ference between users [12]. The analysis in [12] is based onthe case when a user selects the precoding matrix by solelylooking at its own channel without considering the interfer-ence to other users which is caused by adopting that precod-ing matrix. Hence, a better way to handle interuser interfer-ence needs to be addressed.

In this paper, we propose a subspace-based channelquantization method which guarantees a much higher per-formance than the analog pilot retransmission method. Wealso propose an iterative codebook design algorithm whichconverges to a locally optimum codebook. Furthermore, as afeedback reduction scheme, we propose a hierarchical code-book design method. The proposed schemes can be used forcellular MU-MIMO systems as well, which involve one BS fordownlink data transmission.

Notation

Vectors and matrices are denoted by lower case bold and cap-ital bold letters, respectively. (·)T and (·)H denote transposeand Hermitian transpose, respectively. The inner product be-tween two vectors is defined as 〈u, v〉 = uH v.tr (·) denotesthe trace of a matrix. |·|, ‖·‖2, and ‖·‖F denote the magni-tude of a scalar, the two-norm of a vector or a matrix, and theFrobenius norm of a matrix, respectively. The covariance ma-trix of the vector process x is denoted by Rx = E[xxH ], whereE[·] is used for expectation. IN is the N × N identity matrixand 0M×N stands for an all-zero matrix of sizeM×N . IM×N is

defined as IM×N := [ IN0(M−N)×N ] for M > N . [A]i, j stands for the(i, j)th entry of a matrix A. |S| is the cardinality of a set S.

2. SYSTEM MODEL AND MOTIVATION FORCHANNEL QUANTIZATION METHOD

We consider a precoded MU-MIMO system in which a groupof BSs transmits data to multiple MSs simultaneously. Eachof NBS BSs and each of NMS MSs have Nt and Nr antennas,respectively. The data symbol block, s = [s1, . . . , sNtr ]T withNtr = NMSNr , is precoded by an Ntt × Ntr matrix W withNtt = NBSNt , in case that the number of data streams for eachuser ns (ns ≤ Nr) is Nr . Here, the first Nr data symbols areintended for the first user, the nextNr symbols for the seconduser, and so on. When denoting iBS/iMS as the BS/MS indexand it/ir as the transmit/receive antenna index, respectively,we can denote hi, j , where i = Nr(iMS−1)+ir , j = Nt(iBS−1)+it as the channel coefficient between the ir th receive antennaof the iMSth MS and the itth transmit antenna of the iBSth BS.The NtrNtt channel coefficients can be expressed as the Ntr ×Ntt channel matrix H with [H]i, j = hi, j . The received signalson Ntr receive antennas which are collected in the vector ycan be formulated as

y = HWs + n, (1)

where n is additive white Gaussian noise (AWGN). The sig-nal model appears to be very similar to that of the single-user MIMO case at a first glance, but the difference lies in thefact that the channel matrix in our case contains elementsbelonging to multiple BSs and multiple MSs.

There are several available techniques developed fordownlink transmit filtering in MU-MIMO systems. Linearprecoding techniques (e.g., transmit matched filter (TxMF),transmit zero-forcing filter (TxZF), and transmit Wiener fil-ter (TxWF)) have an advantage in terms of computationalcomplexity [13]. Nonlinear techniques (e.g., Tomlinson-Harashima precoding (THP)) have a higher-computationalcomplexity but can usually provide a better performancethan linear techniques [14, 15]. Some linear techniques (e.g.,block diagonalization (BD) and successive minimum meansquared error precoding (SMMSE)) are developed for thecase in which there are multiple antennas at each receiver.The BD algorithm is designed to eliminate multiuser inter-ference (MUI) [16]. BD outperforms the TxZF and asymp-totically approaches the sum capacity of the channel at highSNR. SMMSE performs better than some nonlinear tech-niques (e.g., successive optimization (SO) THP and MMSETHP) with a relatively low-computational complexity [17].In our case, we adopt the TxZF which completely suppressesthe interference at the receiver [13] as follows:

{W, g} = arg min{W,g}

|g|2tr(Rn)

s.t.: gHW = INtr , tr(

WRsWH) = Ptx

(2)

where Ptx, Rn, and Rs are the maximum transmit power, thecovariance matrix of the noise, and the covariance matrix ofthe data symbol, respectively. The TxZF strategy, while gen-erally suboptimal, is known to achieve the same asymptotic

Jee Hyun Kim et al. 3

sum capacity as that of dirty paper coding (DPC) which is theoptimal (channel capacity achieving) method, as the numberof users goes to infinity [18]. The transmit precoding matrixW which satisfies the design criteria (2) takes the followingform:

W = g−1HH(HHH)−1,

where g =

√√√√ tr

((HHH

)−1Rs)

Ptr.

(3)

The challenge here is that BSs should know the downlinkchannel matrix H so as to construct the precoding matrix W.The analog pilot retransmission method has been proposedas a way of transferring channel state information to the BS[6]. As shown in [6], the analog pilot retransmission methodis vulnerable to noise enhancement effects and this weaknessof the analog method brings about a significant performancedegradation, even though it is efficient in terms of requiredresources. As a way of combating noise, a digital method canbe used instead of the analog method. A digital method im-plies that MSs measure the downlink channel and encode thisinformation into a digital code and send it back to the BSs af-ter performing appropriate digital signal processing (modu-lation, spreading, repetition, or channel coding, etc.) to guar-antee robust data transmission.

As it is explained in the previous section, most of the fi-nite rate feedback strategies in MIMO systems are designedfor the single user case, focusing on the selection and con-struction of the precoding matrix codebook. If this strategyis directly applied to the multiuser case, the performance willbe degraded since the user is supposed to select the precodingmatrix which is suitable in terms of its criterion (e.g., maxi-mizing the mutual information or SNR), which may cause asevere interference to other users. Here, we propose to quan-tize the channel from the MS side instead of quantizing theprecoding matrix. Both methods are similar from the signalprocessing perspective in the sense that both schemes com-press the information in a matrix, while the channel quanti-zation method is better positioned to cope with interuser in-terferences. The BSs, after receiving feedback messages fromthe MSs, can now build a precoding matrix with interuserinterferences taken into account, since the transmitters havethe whole channel state information, albeit it is not perfectdue to the limited feedback.

One way of quantizing the channel matrix is to view thechannel matrix as a set of complex matrices, and to quan-tize every individual matrix by looking up a predefined code-book. As explained above, the overall channel matrix is anNMSNr ×NBSNt matrix, and is composed of the channel ma-trices for each user, which are of size Nr × NBSNt . Equation(4) depicts this relationship as follows:

H = [H1, . . . , H j , . . . , HNMS]T

, j : user index. (4)

Here, H j is the transpose of the channel matrix for user j,which is an NBSNt ×Nr matrix. If we allocate nCB bits for thecodebook, we need nCBNMS bits in total for every subcarrier.This method is suitable for the limited feedback in terms of

required feedback bits, and the conventional vector quantiza-tion (VQ) method can be applied with some modifications.

The system model is depicted in Figure 1. The NBS BSsneed overall downlink channel state information H for thecalculation of the precoding matrix W so as to form multi-ple spatial beams which enable independent and decoupleddata streams for NMS users. The individual user j estimatesits portion of the channel H j and quantizes it by finding thebest candidate from the predefined set of codes Ci. The in-dex of the chosen code i j is sent back to the BSs through thelimited feedback channel. The BSs reconstruct the channelmatrix Ĥ by looking up the codebook, which is shared bytransmitters and receivers. This reconstructed channel ma-trix is used for the calculation of the precoding matrix W.We should note that in this case all of the NBS-associated BSshave the same channel matrix, as long as the feedback mes-sages are received without errors. In case of the analog pilotretransmission method, the individual BS has its own versionof the channel matrix, which is in general different from eachother due to the nature of the analog transmission scheme,and this entails a significant performance degradation [1, 6].The principles of the analog pilot retransmission method canbe found in [1, Section 10.3.3.1].

3. SUBSPACE-BASED CHANNELQUANTIZATION METHOD

As proposed in the previous section, MS j is supposed toquantize its channel matrix H j . We view H j not just as acomplex matrix but as a subspace which is spanned by itscolumns. We perform a singular value decomposition (SVD)to extract the unitary matrix U j which includes the basis vec-

tors U(S)j spanning the column space of H j (H j : Ntt ×Nr , U j :Ntt × Ntt, U(S)j : Ntt × Nr). Here, the superscripts (S) and (0)are used to denote a basis for the signal subspace and the nullspace, respectively,

H j = U jΣ jVHj , U j =[

U(S)j U(0)j

]. (5)

The channel quantizer uses the chordal distance as a distancemetric, since we should measure the distance between sub-spaces. There are other subspace distance metrics [19], butthe chordal distance is the one which leads us to an analyticsolution when designing the codebook [20]. The chordal dis-tance is defined as

dc(

Ti, T j) = 1√

2

∥∥∥TiTHi − T jTHj

∥∥∥F

(6)

for matrices Ti, T j which have orthonormal columns.The quantized version of the column space basis vectors

U(S)j is chosen to be the code which has the minimum chordaldistance from it. Thus, the subspace quantization process canbe written as

Û(S)j = Q(

U(S)j)= arg min

Ci∈Cdc(

U(S)j , Ci)

, (7)

where C is the codebook of size N (N= 2nCB ) which hasthe code Ci ∈ CNtt×Nr as its elements. Here, Ci has unitary


H

BS1

W

BS NBS

MS1

MS NMS

Feedback{i1, . . . , iNMS

}

Ci1 = Q(H1)

CiNMS = Q(HNMS )

Ĥ =[

Ci1 · · · CiNMS]T

H =

⎡

⎢⎢⎢⎣

H1 · · · HNMS

⎤

⎥⎥⎥⎦

T

...

...

...

...

...

...

Figure 1: COOPA system downlink with NBS cooperating base stations and NMS mobile stations.

columns (CHi Ci = INr ), hence the code represents only thecolumn space of H j . No channel magnitude information isfed back to the transmitter, since extensive simulation resultsshow that extra magnitude information does not improve thesystem performance, compared with the case in which onlythe code index is provided to the transmitters when the linkstrengths (large-scale fading due to path loss and shadowing)are assumed to be provided at the BSs. In the case that chan-nel magnitude information is to be fed back, the quantizedversion of the channel at the transmitter which takes this intoaccount can be formulated as

Ĥ j = Û(S)j Σ(S)j , (8)

where Σ(S)j ∈ RNr×Nr+ is a diagonal matrix which is composedof Nr ×Nr elements in the upper left-hand corner of Σ j . Thediagonal elements of Σ(S)j constitute the channel magnitudeinformation, which can be regarded as a refinement of thelink strengths which are already available at the BSs. Thischannel quantization model (8) can provide a better view ofthe channel, as it considers not only the channel directionalinformation, but also the channel magnitude information(link strength refinement information). However, the sim-ulation results show that this extra information does not en-hance the system performance in terms of SINR, comparedwith the case in which only the channel directional informa-tion is provided. Some of the simulation results can be foundin Section 6. It is known that the performance can be im-proved by providing the transmitter with the channel quality

information (e.g., SINR) in addition to the directional in-formation (the column space basis vectors), when a multi-antenna downlink system carrying more users than transmitantennas is considered [21]. In our case, the multiuser di-versity gain is not considered at the moment, so we focuson the directional information of the channel. The bottomline is that the MS needs to quantize the column space of H jonly. The channel magnitude information contained in Σ jis, therefore, not required. In this case, the MS is supposedto send only nCB bits of feedback. The channel quantizationformula can be simplified as

Ĥdj = Û(S)j = arg minCi∈C

dc(

U(S)j , Ci). (9)

The superscript d implies that the channel is quantized interms of the direction, with its magnitude information ig-nored.

The subspace-based channel quantization method worksas follows. MS j finds the code Ci which provides the mini-

mum chordal distance with U(S)j . Then, it sends back an nCBbit code index to all associated BSs. The reconstructed down-link channel matrix at the BSs is as follows:

Ĥ =[

Ĥd1 , . . . , Ĥdj , . . . , Ĥ

dNMS

]T, j : user index. (10)

Finally, the BSs calculate the TxZF precoding matrix W byusing the reconstructed channel matrix Ĥ as follows:

W = g−1ĤH(ĤĤH)−1, (11)


where g is the normalization factor imposed by the transmitpower constraint (3). (Actually, it is not a true zero-forcingprecoding matrix in the strict sense, since the channel mag-nitude information is not considered. In this paper, we usethe term TxZF interchangeably for this particular case, as-suming that readers are not to be confused.)

It is worth noticing that the channel quantization crite-rion (9) can be expressed as

ĥ j = arg maxci∈C

∣∣〈v j , ci〉∣∣, where v j =

h j∥∥h j

∥∥

2

, (12)

for the MISO case where the MS is equipped with one an-tenna. In this case, the task of quantizing the channel boilsdown to that of quantizing the channel vector, instead of thechannel matrix. It basically selects the code of which the di-rection is closely aligned with the direction of the channel.Here, the channel to be quantized, the directional informa-tion of the channel, and the corresponding code are all vec-tors of the same size (h j , v j , c j ∈ CNtt×1). For the proof of thisformula, please refer to the appendix.

4. CODEBOOK CONSTRUCTION BASED ONMODIFIED LBG VQ ALGORITHM

The Grassmannian subspace packing is optimal in terms ofquantization for the uncorrelated Rayleigh fading channel[7]. The Grassmannian space G(m,n) is the set of all n-dimensional subspaces of the space Cm, and the Grassman-nian subspace packing problem is the problem of findingthe best packing of N n-dimensional subspaces in Cm. Thebest packing means that N points in G(m,n) are maximallyspaced such that the minimal distance between any two ofthe subspaces is as large as possible.

In our case, the Linde, Buzo, and Gray (LBG) vectorquantization (VQ) algorithm [22] is used to construct thecodebook C. The LBG VQ algorithm is an iterative algorithmbased on the Lloyd’s algorithm which is known to provide analternative systematic approach for the Grassmannian sub-space packing problem [20]. We in this paper acquire thecodebook through the iterative algorithm described in [20].The main difference of the proposed method is attributedto the fact that the codebooks in [20] are precoder code-books, while the codebooks to be constructed here are chan-nel quantizer codebooks. The proposed algorithm aims atfinding a tradeoff between good quantization properties andthe Grassmannian subspace packing requirements by adopt-ing the minimum chordal distance of the codebook as a de-cision criterion for iterations.

4.1. Design issue

The LBG-VQ-based codebook C design problem can bestated as follows. For a given source vector, a given distortionmeasure, a given codebook evaluation measure, and giventhe size of the codebook, find a codebook and a partitionwhich result in maximizing the minimum chordal distanceof the codebook. (The partition of the space is defined as theset of all encoding regions.) In other words, we want to find

maximally spaced N points in G(Ntt,Nr) with given channelrealization samples.

Suppose that we have a training sequence T to capturethe statistical properties of the column space basis vectorsU(S)j of size Ntt ×Nr :

T = {X1, X2, . . . , XM}

, (13)

where Xm ∈ CNtt×Nr is a sample of U(S)j which can be obtainedby taking an SVD of the channel matrix H j . The codebookcan be represented as follows:

C = {C1, C2, . . . , CN}. (14)

The individual code is of the same size as a training matrix(Cn ∈ CNtt×Nr ). Let Rn be the encoding region associatedwith the code Cn and let

P = {R1, R2, . . . , RN}

(15)

denote the partition of the space. (The encoding region iscalled a Voronoi cell in some publications.) If the source ma-trix Xm belongs to the encoding region Rn, then it is quan-tized to Cn as follows:

Q(

Xm) = Cn, if Xm ∈Rn. (16)

Our aim is to find a codebook of which the minimumchordal distance is maximized. There are several subspacedistance metrics, for example, the Fubini-Study distance, theprojection two-norm distance, and the chordal distance met-rics. It has been shown that the chordal distance is the onlydistance measure which makes the iterative algorithm feasi-ble [20]. The minimum chordal distance of the codebook isgiven by

dc,min(C) := mindc(

Ci, C j), for Ci, C j ∈ C,∀i /= j. (17)

The design problem can be stated as follows. Given T andN ,find C and P such that dc,min(C) is maximized:

Copt = arg maxC

dc,min(C). (18)

4.2. Optimality criteria

C and P must satisfy the following two criteria so as to bea solution to the above-mentioned design problem [22]. Weshould note that the chordal distance is used as a distancemetric.

(i) Nearest neighbor condition:

Rn ={

X : dc(

X, Cn)< dc

(X, Cn′

), ∀n′ /=n

}. (19)

This condition says that any channel sample X, which iscloser to the code Cn than any other codes in the chordal dis-tance sense, should be assigned to the encoding region Rn,and be represented by Cn.

(ii) Centroid condition:

Cn = URINtt×Nr , (20)


Table 1: The minimum codebook distances dc,min(C).

(Ntt ,Nr) nCB mLBG VQ Grassmann

(2, 1) 3 0.3895 0.3820

(3, 1)3 0.5706 0.5429

4 0.4882 0.4167

where UR is an eigenvector matrix of the sample covariancematrix R which is defined as

R := 1NRn

∑

Xm∈RnXmXHm , where NRn =

∣∣Rn

∣∣, (21)

provided that eigenvalues in the eigenvalue matrix ΣR of R =URΣRUHR are sorted in the descending order. This conditionmeans that the code Cn of the encoding region Rn should bethe principal eigenvectors of the sample covariance matrix R,meaning theNr eigenvectors of R corresponding toNr largesteigenvalues. The centroid condition is designed to minimizethe average distortion in the encoding region Rn, when C

optn

represents Rn [20].

4.3. Modified LBG VQ algorithm

The modified LBG VQ (mLBG VQ) design algorithm is an it-erative algorithm which finds the solution satisfying the twooptimality criteria in Section 4.2. The algorithm requires aninitial codebook C(0). C(0) is obtained by the splitting of aninitial code, which is the centroid of the entire training se-quence, into two codes. The iterative algorithm runs withthese two codes as the initial codebook. The final two codesare split into four and the same process is repeated untilthe desired number of codes, which leads to the minimumchordal distance, is obtained.

The minimum distances of the codebooks are collectedin Table 1. A training sequence of the length 50,000 is usedfor obtaining the codebook. It shows that the codebooks ac-quired by the modified LBG VQ algorithm have better dis-tance properties than the Grassmannian codebooks listed in[23].

5. COMBINED CODEBOOK: HIERARCHICALCODEBOOK DESIGN METHOD

In this section, we propose a hierarchical codebook designmethod, which exploits the temporal correlation of the chan-nel, as a way of reducing the feedback overhead. It is knownthat the wireless channel does not change radically within thecoherence time Tc. Accordingly, the code index would notchange so often during this time period, since the code in-dex can be considered as a channel state indicator. On theother hand, we can easily draw the conclusion that the code-book index transition rate over time is dependent on the sizeof the codebook, which decides the resolution of the chan-nel quantizer. It means that for a given channel, a bigger size

Ric

(a) Coarse encoding region

Ric ,i f

(b) Fine encoding region

Coarsecodebook index

Fine code index

ic

i f

τc

τ f

t

(c)

Figure 2: Coarse/fine encoding regions and feedback time frame.

codebook (fine codebook) has a higher capability of differen-tiating encoding regions than a smaller size codebook (coarsecodebook). The period during which the coarse codebookprovides the same code index (let us call this a nontransi-tion period) can be composed of several shorter nontransi-tion periods when the fine codebook is used to quantize thechannel.

Thus, if we are able to design a codebook hierarchicallyso that a codebook has several layers, say two layers, one ofwhich represents coarse encoding regions and another pro-vides fine encoding regions, we can achieve the performanceof a fine channel quantizer by using much smaller feedbackresources. This can be achieved by organizing the coarse/finecodebook feedback periods in a smart way to take advantageof nontransition periods of the coarse/fine codebook.

The operation scenario of the combined codebook is asfollows. As a preparation, we need to design the combinedcodebook which has a two-layer structure, namely, an nc bitcoarse codebook and an n f bit fine codebook. Correspond-ing feedback periods should be decided, based on the statisti-cal properties of the nontransition time. The nt = nc +n f bitcombined codebook, as a whole, is designed to be composedof 2nc groups of the fine codes, and each fine codebook groupconsists of 2n f fine codes. The feedback operation works asfollows. For every coarse feedback period τc of the nc bitcoarse codebook, the MS sends the coarse codebook index ic(nc bit) back to the BSs to indicate the fine codebook groupindex to which the subsequent fine code indices belong. Atthe same time, the MS sends the fine code index i f (n f bit)to indicate the fine code index of the chosen fine codebookgroup. This subsequent feedback is done for every fine feed-back period τ f of the nt bit fine codebook. Since that, nc bitfeedback is sent back for every τc and only n f bit extra feed-back is needed for every τ f , we can save uplink resource whencompared to the case of sending back nt bit feedback for ev-ery time. Interested readers can consult Figure 2 for betterunderstanding.


5.1. Hierarchical codebook construction

The main design problem of hierarchical codebook construc-tion is to divide a coarse encoding region into equally proba-ble fine encoding regions. Here, the term “equally probable”means that the probability that a channel sample falling intoa certain encoding region is the same for all candidate en-coding regions. Equally probable encoding regions allow usto fix the feedback period for given channel-dependent con-straints.

The modified LBG VQ (mLBG VQ) algorithm, whichis used for codebook construction, generates the codebookwhich pertains this property. The resulting codes are maxi-mally spaced codes of which an individual code is designedto provide the minimum mean squared chordal distancebetween the code and the channel samples in that encod-ing region. This criterion places finer encoding regions indensely populated areas, and the resulting encoding regionsare asymptotically equally probable. This is shown to be truein [24] as well, for codes generated by the Lloyd’s algorithm-based codebook construction method.

The design problem of hierarchical codebook construc-tion with an nc bit coarse and an nt bit fine codebook is todivide the given channel space which is a subspace of CNtt×Nr

into 2nc equally probable coarse encoding regions and to di-vide each coarse encoding region into 2n f equally probablefine encoding regions. In the end, we want to have 2nc groupsof codebooks each of which is composed of 2n f codes. Thiscan be solved as follows. We first perform the mLBG VQ algo-rithm to get Nc= 2nc coarse codes and corresponding coarseencoding regions. These encoding regions are supposed tobe equally probable (Pn = 1/Nc,∀n ∈ {1, . . . ,Nc}). Then,we perform the mLBG VQ for channel samples which be-long to each coarse encoding region, individually. As a resultof Nc parallel codebook generation processes for each coarseencoding region, we can acquire Ncb= 2nt= 2nc+n f fine codeswith corresponding equally probable fine encoding regions(Pn = 1/N f ,∀n ∈ {1, . . . ,Nf }). Each coarse encoding regionconsists of Nf= 2n f fine encoding regions.

The overall codebook C can be regarded as a set of code-books Ccic , where ic is the codebook index. Here, we differ-entiate between the terms code and codebook, in such a waythat a code indicates an individual code, whereas a codebookindicates a set of codes. Thus ic indicates not an individualcode index, but a codebook index to which a fine code be-longs. It means that the codebook Ccic is constructed basedon the icth coarse encoding region Ric . The elements of C

cic

are fine codes. The overall codebook C and the icth fine code-book Ccic can be expressed as

C = {Cc1, Cc2, . . . , CcNc} = {C1, C2, . . . , CNcb

}, (22)

Ccic ={

Cic1 , Cic2 , . . . , C

ici f , . . . , C

icN f

}, for ic ∈

{1, 2, . . . ,Nc

},

(23)

where fine codes are arranged in such an order that the con-dition Cici f = C(ic−1)Nf +i f ∈ CNtt×Nr is satisfied and i f is thefine code index within the coarse encoding region Ric . It be-comes clear at this point that the resulting codebook has a

hierarchical structure. This is the reason why it is termed ahierarchical codebook.

For example, Figure 2 shows the case with nc = 3 andn f = 2. The partition of the channel sample space consists ofNc = 8 coarse encoding regions, the icth of which is denotedas Ric , as a result of the mLBG VQ procedure. Each individ-ual coarse encoding region is again decomposed into Nf = 4fine encoding regions, the i f th of which is Ric ,i f in case thatit is based on Ric . In the end, we get Ncb = 32 fine codesassociated with the corresponding fine encoding regions.

5.2. Operation scenario

The operation scenario of the hierarchical codebook deploy-ment, which is also termed a combined codebook in this arti-cle, is as follows.

(i) Coarse feedback: feedback of the codebook index ic.For every coarse feedback period τc, the MS sends thenc bit codebook index ic back to the associated BSsso as to indicate the chosen codebook. Based on thechannel information observed for the time period τc,the MS quantizes the channel matrix and finds the bestcode in terms of the chordal distance. The index of thechosen code C(ic−1)Nf +i f ∈ CNtt×Nr can be decomposedinto two parts, for example, the codebook index partic and the code index part i f . The coarse feedback in-volves sending back ic.

(ii) Fine feedback: feedback of the code index i f .For every fine feedback period τ f within τc, the MSsends the n f bit code index i f back to the associatedBSs so as to indicate the chosen code. It means thatthe MS performs the channel quantization for everyτ f , but the scope of candidate codes is restricted withinthe codebook Ccic . This can save a lot of computationalburden for the MS, since the number of candidatecodes isNf instead ofNcNf . The fine feedback involvessending back i f .

The BSs collect the coarse and fine feedback messages,and combine this information to find the chosen code, whichis one of Ncb= 2nc+n f fine codes which are predefined andshared by both BSs and MSs.

There are several points to be worth our attention.

(1) The feedback periods τc and τ f have a significant ef-fect on the system performance. It is a challenging taskto find an analytical solution for calculating an opti-mum feedback period. The optimization problem issupposed to maximize the performance or to mini-mize the performance degradation compared with theideal case (τc, τ f are equal to the shortest possible feed-back period), and it is a function not only of the di-mension of the channel matrix to quantize, the speedof the MS, and the carrier frequency, but also of thenumber of feedback bits nc,n f which decide the reso-lution of the channel quantizer.

(2) Once found, τc and τ f can have fixed values forgiven channel-related parameters (the channel matrixdimension, the carrier frequency, and the speed of


BS1MS1

Sector

BS3

BS2

MS2

Cell

Cooperative area

BS1

h11

h21

MS1

h12

h13

BS3

h23

MS2

BS2

h22

Figure 3: CA topology based on 3-sector-cell system.

the MS) and the number of feedback bits, and are stillable to guarantee the target performance. Therefore,we do not need a feedback period of variable lengthfor given circumstances, which makes the system de-sign problem easy. If parameters other than the mobilespeed remain same, we only need to scale the feedbackperiod with respect to the mobile speed.

(3) Within the coarse feedback period τc, the resultingcodebook has a limited scope, since it selects the bestcode from the chosen codebook only. If the actualchannel realization falls into a different coarse encod-ing region, the quantization error would be significant.

The BS may benefit from saving the coarse codebookCc = {Cc1, Cc2, . . . , CcNc} into memory, just in case that MSsare temporarily disabled to send fine feedback messages toBSs. In this case, the BSs are accessible to coarse feedback in-dices only. However, the BSs can still reconstruct the chan-nel if the coarse codebook is available at the BSs. The BS issupposed to have two codebooks, one of which is the coarsecodebook of size Nc, and another is the combined codebookof size Ncb. Only the combined codebook needs to be savedon the MS side.

5.3. Further comments

The feedback overhead reduction of the combined codebookcan be also achieved by constructing a single fine codebookfollowed by an adequate codeword assignment. For example,we can accomplish the same effect by assigning codewords toquantization regions in such a way that quantization regionsclose in chordal distance are assigned to codewords close, say,in Hamming distance. In addition, another alternative way toexploit temporal correlations to reduce the amount of feed-back is by differential encoding, that is, by transmitting onlythe difference between the actual index and the previous in-dex. If the channel has slightly changed, only the least signif-icant bits will be transmitted. The most significant bits willbe transmitted only when the channel has experienced anabrupt change.

On the other hand, the hierarchical codebook designmethod can be further improved by endowing tracking ca-pability. A subspace tracking codebook can be defined as asubset of the entire codebook which consists of neighboringcodewords of the currently chosen codeword. As this smallsize neighboring codebook is able to change its elementsadaptively to the current status, it is capable of tracking asubspace, which leads to a further reduction of the feedbackoverhead.

6. NUMERICAL RESULTS

In this section, we present numerical results. First, simu-lations have been performed for the 2BSs-2MSs and 3BSs-2MSs cases to evaluate the performance of the proposedchannel quantizer and the codebook construction method.Two (three) BSs are cooperating to transmit data signal fortwo MSs through the same resources at the same time. BothBSs and MSs have a single antenna, so it yields 2×2 and 2×3overall channel matrices, respectively. We employ the trans-mit zero-forcing filter as an example ofbeamforming schemeto prove the quality of the proposed quantization method.The extended 3GPP spatial channel model (SCM) is used forthe simulations; and the proposed methods are tested for anurban macro channel with a mobile speed of 10 m/s. (TheMATLAB code provided in [25] supports a channel matrixgeneration function for links between multiple BSs and mul-tiple MSs.) The system performance is evaluated in terms ofthe received SINR at the MS. Simulations are performed for30,000 channel realizations and the cumulative distributionfunction (CDF) at one MS is obtained. OFDMA is assumedas the data transmission scheme and we focus on one subcar-rier. The transmit power at the BS is set to be 10 W and it isequally allocated to 1201 subcarriers.

The cooperative area (CA) topology is shown in Figure 3.As in the conventional cellular topology, one cell that is com-posed of three sectors and the hexagonal area, which is com-posed of three sectors which are served by three BSs, forms aCA. Two MSs in the CA are served by three BSs simultane-ously. In case of the 2BSs-2MSs case, two BSs which maintain


the strongest two links with MSs are chosen for downlinktransmission. The cell radius is 600 m and MSs are equallydistributed in the CA for every drop.

The transmit zero-forcing filter formula follows (3),based on downlink channel information which is either per-fect channel (pCh), or is provided by a downlink channelestimation method which is shared by the BSs through aprompt, error free backbone network (centralized CA de-noted by cCA), or is acquired by the analog pilot retrans-mission method (distributed CA denoted by dCA), or is cap-tured and reconstructed by looking up an n bit codebook(n bit channel quantization referred to as nbCQ). The cCAcase assumes that the system employs a time division duplex(TDD) scheme and the backbone network connecting asso-ciated BSs is delay free and error free. The downlink chan-nel state information can be acquired by estimating uplinkchannel by using uplink-downlink channel reciprocity, whenthere exists a direct link between a BS and an MS. In simula-tions, the uplink channel is assumed to be estimated by usingthe uplink pilot signal. The nondirect link channel informa-tion can be provided by BSs with direct links through promptdata communication over the backbone network. In the ana-log pilot retransmission method, the MS sends the receivedpilot which pertains to the downlink channel state informa-tion to all associated BSs over the uplink channel [1, 6]. Inthis case, two pilots are required in the uplink. One is forconveying the received pilot directly to the BSs (analog pilotretransmission), and the other is for estimating the uplinkchannel itself, which is necessary to compensate the retrans-mitted pilot for the uplink channel influence so as to acquirethe downlink channel information. Therefore, these two pi-lots should be adjacent in time and frequency. Since the esti-mated version of the channel state information is used to peeloff distortions caused by the uplink channel from the retrans-mitted pilot, this method is vulnerable to noise enhancementeffects.

The BSs are assumed to be aware of the large-scale fadingof the channel; and the channel quantization process (9) isbased on true channel information. (Some readers may finda direct comparison between cCA and nbCQ inadequate inthe sense that cCA is based on the realistic channel estimationmethod while nbCQ is based on the ideal channel knowledge.However, the performance of the cCA case is provided here asa mere reference for the mapping of the performance of theproposed method in relation to an alternative method.) Thecodebooks are acquired by the modified LBG VQ algorithm.The feedback link is error free and delay free.

First, two proposals concerning the channel quantiza-tion model are evaluated. One model (Ĥdj , equation (9))adopts the channel directional information only, and theother (Ĥ j , equation (8)) takes the channel magnitude in-formation into account, as well as the channel directionalinformation. Figure 4 shows the CDF of the SINR for the3BSs-2MSs case. The channel directional information-basedmodel (4bCDI, 5bCDI) performs closely to or in the lowSINR region even slightly better than the model which com-bines the directional and magnitude information (4bCDMI,5bCDMI). We assume that the BSs have access to the linkstrengths (large-scale fading due to path loss and shadowing)

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0 5 10 15 20 25 30

SINR (dB)

4bCDI5bCDI

4bCDMI5bCDMI

SINR@50% cdf4bCDI: 13.8 dB5bCDI: 15.3 dB4bCDMI: 13.8 dB5bCDMI: 15.2 dB

cdf of SINR for MS1, 3BS-2MS case (ZF),Rcell = 600 m, Urban Macro10 m/s

Figure 4: Performance comparison of channel quantization models(4bCDI: 4-bit-channel directional information-based quantization;5bCDI: 5-bit CDI; 4bCDMI: 4-bit channel directional/magnitudeinformation-based quantization; 5bCDMI: 5-bit CDMI).

for both cases, and the BSs have perfect knowledge of Σ(S)jfor the latter case. Simulation results indicate that the extrachannel magnitude information does not improve the SINRperformance in these cases. (We should be careful in inter-preting the simulation results. We have simulated relativelylow-bit (4 and 5 bits) quantization cases. In this case, theprecision of the channel directional information is more rele-vant to the system performance than the channel magnitudeinformation. The channel magnitude information can playan important role for higher-bit quantization cases, wherethe accuracy of the channel directional information is suffi-ciently high that only the magnitude information can helpimprove performance. In this paper, we focus on the lim-ited feedback case in which the channel directional informa-tion matters most.) Please note that in this paper the chan-nel quantizer (CQ) or the subspace-based CQ refers to thechannel directional information-based model, unless other-wise mentioned.

Figure 5 shows the CDF of the SINR for the 2BSs-2MSscase. At 50% outage SINR, the 3-bit channel quantizer(3bCQ) shows 7.8 dB gain over the analog pilot retransmis-sion case (dCA) and it is only 0.1 dB away from the central-ized CA (cCA). The channel matrix at MS j, H j ( j = 1, 2), isin this case a 2× 1 complex vector and this is represented bya codebook of size 23 = 8. Compared with the channel quan-tization method, the resource efficient dCA case requires 3-pilot tones per MS in case of FDD. Therefore, the proposedscheme performs much better than the pilot retransmissionmethod without requiring extra resources. Figure 6 dealswith simulation results of the 3BSs-2MSs case. The 3bCQ,4bCQ, and 5bCQ cases have 3.2 dB, 5.0 dB, and 6.5 dB gainsover the dCA case, respectively. In this case, the proposed


10−2

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−20 −10 0 10 20 30 40SINR (dB)

dCAcCA

pCh

3bCQ

SINR@50% cdfdCA: 7.4 dBcCA: 15.3 dBpCh: 23.7 dB3bCQ: 15.2 dB


Figure 5: 2BSs-2MSs case simulation results (dCA: distributed CA;cCA: centralized CA; pCh: perfect channel; 3bCQ: 3-bit CQ).

10−2

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−20 −10 0 10 20 30 40SINR (dB)

dCAcCApCh

3bCQ4bCQ5bCQ

SINR@50% cdfdCA: 8.8 dBcCA: 16.9 dBpCh: 27.2 dB3bCQ: 12 dB4bCQ: 13.8 dB5bCQ: 15.3 dB


Figure 6: 3BSs-2MSs case simulation results (dCA: distributed CA;cCA: centralized CA; pCh: perfect channel; 3bCQ: 3-bit CQ; 4bCQ:4 bit CQ; 5bCQ: 5-bit CQ).

method still has a lot of room for improvement even thoughthe expected gain over the conventional method is not in-significant. The gap between the proposed method and theideal case can be reduced by adopting a smart schedulingstrategy like user grouping which selects users with orthogo-nal channel signatures so as to reduce interferences betweendifferent users [18].

The proposed method is to quantize the channel matrixbased on the chordal distance, and the LBG VQ algorithmis modified as such. Conventional VQ methods use the Eu-clidean distance instead. Is the subspace-based method betterthan the conventional method? A performance comparisonresult is shown in Figure 7. The Euclidean distance-based CQ(nbeCQ) adopts the Euclidean distance as a distance metricfor channel quantization. The simulation results show thatthe subspace-based CQ has a substantial gain over the Eu-clidean distance-based CQ. At 50% outage SINR, the 4bCQand 5bCQ outperform the 4beCQ and 5beCQ by 2.9 dB and3.0 dB, respectively.

There exists another CQ method which exploits Givensrotations [26]. This method allows us to represent the col-umn space basis vectors U(S) ∈ Ct×n of the channel by(2t − 1)n− n2 real numbers. t = 3,n = 1 holds for the 3BSs-2MSs case, and it requires 4 real number parameters (φ1,2,φ1,3, θ1,1, θ1,2) for channel matrix construction. The per-formance comparison result between the proposed methodand the Givens-rotation-based channel matrix decomposi-tion method is shown in Figure 8. The Givens-rotation-basedmethod with n-bit feedback is denoted by nbGR. 4bGR allo-cates 1 bit for each parameter, and 5bGR assigns 2, 2, 1, and 0bit(s) for φ1,2, φ1,3, θ1,1, and θ1,2, respectively. (In this case, thevalue of θ1,2 is predefined and fixed.) At 50% outage SINR,the 4bCQ case outperforms the 4bGR case by 2.5 dB, whilethe 5bCQ case shows comparable performance to the 5bGRcase of which the computational complexity at MS is higherthan that of the 5bCQ case.

The performance of the combined codebook is shownin Figure 9. Simulations have been performed for the 2BSs-2MSs case. The combined codebooks are acquired by themodified LBG VQ algorithm and the hierarchical codebookconstruction method as described in Section 5. The com-bined codebook of the nc-bit coarse and n f -bit fine feed-backs with corresponding feedback periods τc and τ f is de-noted by nc + n f bCQ ([τc], [τ f ]), where the unit of feed-back period is the number of OFDM symbols: [τ] = m,when τ = m·Ts (Ts = 71.37 μs). At 50% outage SINR, the5-bit codebook case (5bCQ), which is generated by the hier-archical codebook construction method, shows 11.8 dB and3.9 dB gains over the analog pilot retransmission case (dCA)and the centralized CA case (cCA), respectively. In this case,the 5-bit feedback is being sent back for every symbol. The 3+ 2-bit combined codebook with the empirically found opti-mum feedback period pair ([τc], [τ f ])=(10,5), (3 + 2bCQ1)is less than 0.1 dB away from the performance of the 5bCQcase, even though some degradation is observed in the lowSINR region. In terms of the required resources, the 5bCQcase requires 5 bits/symbol for feedback, while the 3 + 2bCQ1case needs only 0.7 bit/symbol. Thus, the 3 + 2-bit combinedcodebook can achieve the performance of the 5-bit codebookwith negligible degradation by using just 14% of the feedbackresource.

We have tested the 3 + 2bCQ case with various subopti-mum feedback periods, for example, (10,10) and (20,5), andeach case is denoted by the subscripts 2 and 3, respectively.None of these cases outperforms the optimum case (10, 5),and performance degradations are 0.7 dB and 0.8 dB for the


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−10 −5 0 5 10 15 20 25 30SINR (dB)

dCAcCA4bCQ

5bCQ4beCQ5beCQ

SINR@50% cdf4bCQ: 13.8 dB5bCQ: 15.3 dB4beCQ: 10.9 dB5beCQ: 12.3 dB


Figure 7: Performance comparison of the Euclidean distance-basedCQ and the subspace-based CQ (dCA: distributed CA; cCA: cen-tralized CA; 4bCQ: 4 bit subspace-based CQ; 5bCQ: 5-bit subspace-based CQ; 4beCQ: 4-bit Euclidean distance-based CQ; 5beCQ: 5-bitEuclidean distance-based CQ).

longer fine feedback period case and the longer coarse feed-back period case, respectively. The following points shouldbe noted with respect to simulation results.

(i) The optimum feedback period pair guaranteesthe target performance

In this case, ([τc], [τ f ]) = (10, 5) is the optimum feedbackperiod pair, and any other case with longer period results inperformance degradation. The task of finding an optimumfeedback period pair in an analytical way is an interestingtopic for future research.

(ii) The coarse feedback period is decisivein the performance

Even though both τc and τ f are important in deciding theperformance, τc has a more profound impact than τ f , sincethe coarse codebook is associated with the bigger encodingregion which entails a bigger error if the feedback infor-mation is outdated. Both the (10,10) and (20,5) cases showdegradations in performance. The degradation of the (20,5)case with a longer coarse feedback period is 0.1 dB whichis bigger than the (10,10) case with a longer fine feedbackperiod, even though the former requires 0.55 bit/symbolfeedback overhead while the latter needs 0.5 bit/symboloverhead. In short, the (20,5) case performs worse thanthe (10,10) case despite its higher feedback overhead, andthis performance degradation comes from the suboptimumcoarse feedback period.

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0 5 10 15 20 25 30

SINR (dB)

dCAcCA4bCQ

5bCQ4bGR5bGR

SINR@50% cdf4bCQ: 13.8 dB5bCQ: 15.3 dB4bGR: 11.3 dB5bGR: 15.3 dB


Figure 8: Performance comparison of the Givens rotation-basedCQ and the subspace-based CQ (dCA: distributed CA; cCA: cen-tralized CA; 4bCQ: 4-bit subspace-based CA; 5bCQ: 5-bit subspace-based CQ; 4bGR: 4-bit Givens-rotation-based CQ; 5bGR: 5-bitGivens-rotation-based CQ).

7. CONCLUSIONS

In this paper, we have investigated precoded MU-MIMOsystems with limited feedback. The subspace-based chan-nel quantization method is proposed as a way of providingBSs with downlink channel state information in the pres-ence of interuser interference, which is applicable to the dis-tributed COOPA systems as well as MU-MIMO systems. Thesubspace-based channel quantizer improves the system per-formance significantly, compared to the analog pilot retrans-mission method with relatively small feedback overhead. Wealso developed an efficient codebook construction algorithmbased on well-known LBG VQ by adopting the chordal dis-tance and modifying the optimality criteria accordingly. Thecodebooks generated by the proposed algorithm have betterdistance properties than Grassmannian codebooks that arecurrently available.

We have also proposed a feedback overhead reductionscheme which makes use of the temporal correlation of thechannel. It constructs the codebook with a hierarchical struc-ture so that the feedback index can be divided into two parts,that is, a coarse feedback which points to the codebook in-dex and a fine feedback which indicates the code index.The resulting codebook is termed the combined codebook.The simulation results suggest that we can save a significantamount of feedback resources while maintaining the sameperformance level as the case with fully loaded feedback. Thedecision of the optimum feedback periods is an open issuefor future research.


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−10 −5 0 5 10 15 20 25 30SINR (dB)

dCAcCApCh5bCQ

3 + 2bCQ1(10, 5)3 + 2bCQ2(10, 10)3 + 2bCQ3(20, 5)

50% outage SINRdCA: 7.4 dBcCA: 15.3 dBpCh: 23.7 dB5bCQ: 19.2 dB3 + 2bCQ1: 19.1 dB3 + 2bCQ2: 18.4 dB3 + 2bCQ3: 18.3 dB


Figure 9: Combined codebook simulation results for (nc + nf b) =(3 + 2b) case (dCA: distributed CA; cCA: centralized CA; pCh: per-fect channel; 5bCQ: 5-bit CQ; 3 + 2bCQi ([τc], [τ f ]): 3-bit coarseand 2-bit fine feedbacks with a feedback period pair ([τc], [τ f ]) ∈{(10, 5), (10, 10), (20, 5)}).

APPENDIX

PROOF OF EQUATION (12)

First, we provide the formula which explains how the chordaldistance is related with the inner product, when it is used fortwo unit norm vectors vi, v j , where vHi vi = vHj v j = 1;

d2c(

vi, v j) =

(1√2

∥∥vivHi − v jvHj

∥∥F

)2

= 12tr((

vivHi − v jvHj)(

vivHi − v jvHj)H)

= 12tr(

vHj(

vivHi − v jvHj)(

vivHi − v jvHj)H

v j)

= 12

(1− vHj vivHi v j)

= 12

(1− ∣∣〈vi, v j

〉∣∣2),

(A.1)

where 〈vi, v j〉 = vHi v j . The following property is used fromthe first to the second line: ‖A‖F =

√tr (AAH ); and from

the third to the fourth line, the tr (·) operation is omittedsince its argument has a scalar value. From (A.1), the decisioncriterion in terms of the chordal distance can be formulatedas follows:

arg mindc(

vi, v j) = arg max ∣∣〈vi, v j

〉∣∣. (A.2)

On the other hand, the column space basis vector uS ∈CNtt×1 of the channel vector h ∈ CNtt×1 can be found as fol-lows (the user index j is omitted for brevity):

h = UhΣhVHh = vhσhuS, (A.3)

where vh and σh have scalar values and uS ∈ CNtt×1. Therightmost form is an “economy size” version of the SVD,where vh is in effect the same as Vh (size: 1 × 1), and σh isthe only nonzero singular value in Σh. The followings holds:σh = ‖h‖2 and vh ∈ {+1,−1}, since h is a vector. Therefore,uS can be expressed in terms of the directional vector of thechannel as follows:

uS = vh h‖h‖2= vhv, (A.4)

where v = h/‖h‖2 is the directional vector of the channel.Since vh decides the sign only, the following criterion holds:

ĥ = arg minci∈C

dc(

uS, ci) = arg max

ci∈C

∣∣〈v, ci

〉∣∣. (A.5)

Therefore, the chordal distance-based channel quantizationcriterion (9) can be simplified to the inner product-based cri-terion (12).

ACKNOWLEDGMENT

Part of this work was presented at the International ITG/IEEEWorkshop on Smart Antennas (WSA’07), Vienna, Austria,February 2007.

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[21] T. Yoo, N. Jindal, and A. Goldsmith, “Finite-rate feedbackMIMO broadcast channels with a large number of users,” inProceedings of the IEEE International Symposium on Informa-tion Theory (ISIT ’06), pp. 1214–1218, Seattle, Wash, USA, July2006.

[22] Y. Linde, A. Buzo, and R. M. Gray, “An algorithm for vectorquantizer design,” IEEE Transactions on Communications, pp.702–710, 1980.

[23] D. J. Love, “Personal webpage on Grassmannian subspacepacking,” http://cobweb.ecn.purdue.edu/djlove/grass.html.

[24] K. Huang, B. Mondal, R. W. Heath Jr., and J. G. Andrews,“Limited feedback for temporally-correlated channels: feed-back rate and delay,” to appear in IEEE Transactions on Infor-mation Theory, http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0606022.

[25] D. S. Baum, J. Salo, M. Milojevic, P. Kyösti, and J. Hansen,“MATLAB implementation of the interim channel modelfor Beyond-3G systems (SCME),” http://www.tkk.fi/Units/Radio/scm/.

[26] J. C. Roh and B. D. Rao, “An efficient feedback method forMIMO systems with slowly time-varying channels,” in Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC ’04), vol. 2, pp. 760–764, Atlanta, Ga, USA,March 2004.

http://cobweb.ecn.purdue.edu/˜djlove/grass.htmlhttp://www.citebase.org/abstract?id=oai:arXiv.org:cs/0606022http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0606022http://www.tkk.fi/Units/Radio/scm/http://www.tkk.fi/Units/Radio/scm/

EURASIP Journal on Embedded Systems

Special Issue on

Selected Papers from SLA++P 2007 Model-DrivenHigh-Level Programming of Embedded Systems

Call for Papers

Model-based high-level programming of embedded systemshas become a reality in the automotive and avionics indus-tries. These industries place high demands on the efficiencyand maintainabiliy of the design process as well as on theperformance and functional correctness of embedded com-ponents. These goals are hard to reconcile in the face of theincreasing complexity of embedded applications and targetarchitecures that we see today. Research efforts towards meet-ing these goals have brought about a variety of high-level en-gineering design languages, tools, and methodologies. Theirstrength resides in clean behavioral models with strong se-mantical foundations providing a rigorous way to go froma high-level description to provable, that is, mathematicallycertifiable, executable code.

Undebatably, the most successful representatives of thistrend of putting logic and mathematics behind design au-tomation in embedded systems (known as Mike Fourman’s“Lambda” programme) are synchronous languages. Firmlygrounded in clean mathematical semantics, they have beenreceiving increasing attention in industry ever since theyemerged in the 1980s. Lustre, Esterel, Signal are now widelyand successfully used to program real-time and safety criticalapplications, from nuclear power plant management layer toAirbus air flight control systems. Their recent successes in theautomatic control industry highlight the benefits of formalverification and automatic code generation from high-levelmodels.

Model-based programming is making its way in otherfields of software engineering, too, often involving cyclic syn-chronous paradigms. Strong interest is emerging in compo-nent programming for large-scale embedded systems, in thelink between simulation tools and compiler tools, in lan-guages for describing the system and its environment, inte-grated tools for both compilation and simulation of moregeneral models of communication and coordination, and soon. The impact of such unifying methodologies will depend,among other things, on the extent to which it will be possi-ble to maintain the high degree of predictability and verifi-ability of system behavior that is the strength of the classicsynchronous world.

Topics of interest for this special issue cover, among others,the following:

• Synchronous programming formalisms• Novel language paradigms blending synchrony with

asynchrony and nondeterminism, discrete with con-tinuous control

• Techniques for component abstraction and refinement• New models of communication and coordination for

embedded systems• Model-based compilation and simulation techniques• Specification, verification, and model-based testing• Case-studies, industrial and teaching experiences

Submission to this special issue limited to the participants ofthe SLA++P conference who have been invited to submit tothis issue.

Authors should follow the EURASIP Journal on Embed-ded Systems manuscript format described at the journalsite http://www.hindawi.com/journals/es/. Prospective au-thors should submit an electronic copy of their completemanuscript through the journal Manuscript Tracking Sys-tem at http://mts.hindawi.com/, according to the followingtimetable:

Manuscript Due March 1, 2008

First Round of Reviews June 1, 2008

Publication Date September 1, 2008

Guest Editors

Florence Maraninchi, VERIMAG Laboratory, 38610Gieres, France; [email protected]

Michael Mendler, University of Bamberg, 96045 Bamberg,Germany; [email protected]

Marc Pouzet, Laboratoire de Recherche en Informatique(LRI), Université Paris-Sud 11, 91405 Orsay Cedex, France;[email protected]

Hindawi Publishing Corporationhttp://www.hindawi.com

http://www.hindawi.com/journals/es/http://mts.hindawi.com/mailto:[email protected]:[email protected]:[email protected]

EURASIP Journal on Bioinformatics and Systems Biology

Special Issue on

Network Structure and Biological Function:Reconstruction, Modelling, and Statistical Approaches

Call for Papers

We are particularly interested in contributions, which eluci-date the relationship between structure or dynamics of bi-ological networks and biological function. This relationshipmay be observed on different scales, for example, on a globalscale, or on the level of subnetworks or motifs.

Several levels exist on which to relate biological function tonetwork structure. Given molecular biological interactions,networks may be analysed with respect to their structural anddynamical patterns, which are associated with phenotypes ofinterest. On the other hand, experimental profiles (e.g., timeseries, disturbations) can be used to reverse engineer networkstructures based on a model of the underlying functional net-work.

Is it possible to detect the interesting features with the cur-rent methods? And how is our picture of the relationsshipbetween network structure and biological function affectedby the choice of methods?

Perspectives both from simulation approaches as wellas the evaluation of experimental data and combinationsthereof are welcome and will be integrated within this spe-cial issue.

Authors should follow the EURASIP Journal on Bioin-formatics and Systems Biology manuscript format describedat the journal site http://www.hindawi.com/journals/bsb/.Prospective authors should submit an electronic copy of theircomplete manuscript through the journal Manuscript Track-ing System at http://mts.hindawi.com/, according to the fol-lowing timetable:

Manuscript Due June 1, 2008

First Round of Reviews September 1, 2008

Publication Date December 1, 2008

Guest Editors

J. Selbig, Bioinformatics Chair, Institute for Biochemistryand Biology, University of Potsdam, Germany;[email protected]

M. Steinfath, Institute for Biochemistry and Biology,University of Potsdam, Germany;[email protected]

D. Repsilber, AG Biomathematics & Bioinformatics,Genetics and Biometry and Genetics Section, ResearchInstitute for the Biology of Farm Animals, Dummerstorf,Germany; [email protected]


http://www.hindawi.com/journals/bsb/http://mts.hindawi.com/

International Journal of Digital Multimedia Broadcasting

Special Issue on

Personalization of Mobile Multimedia Broadcasting

Call for Papers

In recent years, the widespread adoption of multimedia com-puting, the deployment of mobile and broadband networks,and the growing availability of cheap yet powerful mobilehave converged to gradually increase the range and complex-ity of mobile multimedia content delivery services for devicessuch as PDAs and cell phones. Basic multimedia applicationsare already available for current generation devices, and morecomplex broadcasting services are under development or ex-pected to be launched soon, among which mobile and inter-active television (ITV). Among the many challenging issuesopened by these developments is the problem of personaliza-tion of such services: adaptation of the content to the techni-cal environment of the users (device and network type) andto their individual preferences, providing personalized assis-tance for selecting and locating interesting programes amongan overwhelming number of proposed services.

This special issue is intended to foster state-of-the-art re-search contributions to all research areas either directly ap-plying or contributing to solving the issues related to digitalmultimedia broadcasting personalization. Topics of interestinclude (but are not limited to):

• Mobile TV• Mobile multimedia broadcasting personalization• Interactive broadcasting services/interactive television• Personalization and multimedia home platform

(MHP)• Multimedia content adaptation for personalization• User behavior and usage modelling• Standards for modelling and processing (MPEG-21,

CC/PP, etc.)• Personalization issues in DVB-H, DMB, MediaFLO,

CMMB, MBMS, and other systems• Mobile web initiative• Personalized multimedia and location-based services• Security and digital rights management• Applications for personalized mobile multimedia

broadcasting with cost-effective implementation

Authors should follow the International Journal of Digi-tal Multimedia Broadcasting manuscript format describedat the journal site http://www.hindawi.com/journals/ijdmb/.Prospective authors should submit an electronic copy of theircomplete manuscript through the journal Manuscript Track-ing System at http://mts.hindawi.com/ according to the fol-lowing timetable:

Manuscript Due March 1, 2008

First Round of Reviews June 1, 2008

Publication Date September 1, 2008

Guest Editors

Harald Kosch, University of Passau, 94030 Passau,Germany; [email protected]

Jörg Heuer, Siemens AG, 80333 Munich, Germany;[email protected]

Günther Hölbling, University of Passau, 94030 Passau,Germany; [email protected]

László Böszörményi, University Klagenfurt, 9020Klagenfurt, Austria; [email protected]

David Coquil, University of Passau, 94030 Passau,Germany; [email protected]


http://www.hindawi.com/journals/ijdmb/http://mts.hindawi.com/mailto:[email protected]:[email protected]:[email protected]:[email protected]

EURASIP Journal on Wireless Communications and Networking

Special Issue on

Fairness in Radio Resource Management forWireless Networks

Call for Papers

Radio resource management (RRM) techniques (such as ad-mission control, scheduling, sub-carrier allocation, channelassignment, power allocation, and rate control) are essentialfor maximizing the resource utilization and providing qual-ity of service (QoS) in wireless networks.

In many cases, the performance metrics (e.g., overallthroughput) can be optimized if opportunistic algorithmsare employed. However, opportunistic RRM techniques al-ways favor advantaged users who have good channel condi-tions and/or low interference levels. The problem becomeseven worse when the wireless terminals have low mobilitysince the channel conditions become slowly varying (or evenstatic), which might lead to long-term unfairness. The prob-lem of fair resource allocation is more challenging in multi-hop wireless networks (e.g., mesh and multihop cellular net-works).

The performance fairness can be introduced as one ofthe QoS requirements (e.g., as a condition on the minimumthroughput per user). Fair RRM schemes might penalize ad-vantaged users; hence, there should be a tradeoff between theoverall system performance and the fairness requirements.

We are soliciting high-quality unpublished research pa-pers addressing the problem of fairness of RRM techniquesin wireless communication systems. Topics include (but arenot limited to):

• Fairness of scheduling schemes in wireless networks• Tradeoff between maximizing the overall throughput

and achieving throughput fairness• RRM fairness: problem definition and solution tech-

niques• Fairness performance in emerging wireless systems

(WiMAX, ad hoc networks, mesh networks, etc.)• Cross-layer RRM design with fairness• Short-term and long-term fairness requirements• Adaptive RRM to support fairness• Fairness in cooperative wireless communications• Issues and approaches for achieving fairness in multi-

hop wireless networks

• RRM framework and QoS architecture• Complexity and scalability issues• Experimental and implementation results and issues• Fairness in multiple-antenna transmission/reception

systems• Fairness in end-to-end QoS provisioning

Authors should follow the EURASIP Journal on Wire-less Communications and Networking manuscript for-mat described at the journal site http://www.hindawi.com/journals/wcn/. Prospective authors should submit an elec-tronic copy of their complete manuscript through the jour-nal Manuscript Tracking System at http://mts.hindawi.com/,according to the following timetable:

Manuscript Due July 1, 2008

First Round of Reviews October 1, 2008

Publication Date January 1, 2009

Guest Editors

Mohamed Hossam Ahmed, Faculty of Engineering andApplied Science, Memorial University of Newfoundland,St. John’s, NF, Canada A1B 3V6; [email protected]

Alagan Anpalagan, Department of Electrical andComputer Engineering, Ryerson University, Toronto, ON,Canada M5G 2C5; [email protected]

Kwang-Cheng Chen, Department of ElectricalEngineering, National Taiwan University, Taipei, Taiwan;[email protected]

Zhu Han, Department of Electrical and ComputerEngineering, College of Engineering, Boise State UniversityBoise, Idaho, USA; [email protected]

Ekram Hossain, Department of Electrical and ComputerEngineering, University of Manitoba, Winnipeg, MB,Canada R3T 2N2; [email protected]


http://www.hindawi.com/journals/wcn/http://www.hindawi.com/journals/wcn/http://mts.hindawi.com/

EURASIP Journal on Wireless Communications and Networking

Special Issue on

Wireless Access in Vehicular Environments

Call for PapersWireless access in vehicular environments (WAVE) technol-ogy comes into sight as a state-of-the-art solution to con-temporary vehicular communications, which is anticipatedto be widely applied in the near future to radically im-prove the transportation environment in aspects of safety,intelligent management, and data exchange services. WAVEsystems will fundamentally smooth the progress of intelli-gent transportation systems (ITSs) by providing them withhigh-performance physical platforms. WAVE systems willbuild upon the IEEE 802.11p standard, which is still activeand expected to be ratified in April, 2009. Meanwhile, theVHF/UHF (700 MHz) band vehicular communication sys-tems are attracting increasingly attention recently.

The fast varying and harsh vehicular environment bringsabout several fresh research topics on the study of WAVE sys-tems and future vehicular communication systems, which in-clude physical layer challenges associated with mobile chan-nels, capacity evaluation, novel network configuration, effec-tive media access control (MAC) protocols, and robust rout-ing and congestion control schemes.

The objective of this special section is to gather and cir-culate recent progresses in this fast developing area of WAVEand future vehicular communication systems spanning fromtheoretical analysis to testbed setup, and from physical/MAClayers’ enabling technology to network protocol. These re-search and implementation activities will be considerablyhelpful to the design of WAVE and future vehicular commu-nications systems by removing major technical barriers andpresenting theoretical guidance. This special issue will cover,but not limited to, the following main topics:

• 700 MHz/5.8 GHz vehicle-to-vehicle (V2V)/vehicle-to-infrastructure (V2I) single-input single-output(SISO)/MIMO channel measurement and modeling,channel spatial and temporal characteristics explo-ration

• Doppler shift study, evaluation and estimate, time andfrequency synchronizations, channel estimate and pre-diction

• Utilization of MIMO, space-time coding, smart an-tenna, adaptive modulation and coding

• Performance study and capacity analysis of V2V andV2I communications operating over both 5.8GHz and700Mz

• Software radio, cognitive radio, and dynamic spectrumaccess technologies applied to WAVE and future vehic-ular communication systems

• Mesh network and other novel network configurationsfor vehicular networks

• Efficient MAC protocols development• Routing algorithms and congestion control schemes

for both real-time traffic warning message broadcast-ing and high-speed data exchange

• Cross-layer design and optimization• Testbed or prototype activities

Authors should follow the EURASIP Journal on Wire-less Communications and Networking manuscript for-mat described at the journal site http://www.hindawi.com/journals/wcn/. Prospective authors should submit an elec-tronic copy of their complete manuscript through theEURASIP Journal on Wireless Communications and Net-working’s Manuscript Tracking System at http://mts.hindawi.com/, according to the following timetable.

Manuscript Due April 1, 2008

First Round of Reviews July 1, 2008

Publication Date October 1, 2008

Guest Editors

Weidong Xiang, University of Michigan, Dearborn, USA;[email protected]

Javier Gozalvez, University Miguel Hernández, Spain;[email protected]

Zhisheng Niu, Tsinghua University, China;[email protected]

Onur Altintas, Toyota InfoTechnology Center, Co. Ltd,Tokyo, Japan; [email protected]

Eylem Ekici, Ohio State University, USA; [email protected]


http://www.hindawi.com/journals/wcn/http://www.hindawi.com/journals/wcn/http://mts.hindawi.com/http://mts.hindawi.com/mailto:[email protected]

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2007

Signal ProcessingResearch Letters in

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ISSN: 1687-6911; e-ISSN: 1687-692X; doi:10.1155/RLSP



Research Letters in Signal Processing is devoted to very fast publication of short, high-quality manuscripts in the broad field of signal processing. Manuscripts should not exceed 4 pages in their final published form. Average time from submission to publication shall be around 60 days.

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Editorial BoardTyseer Aboulnasr, Canada

T. Adali, USA

S. S. Agaian, USA

Tamal Bose, USA

Jonathon Chambers, UK

Liang-Gee Chen, Taiwan

P. Dan Dan Cristea, Romania

Karen Egiazarian, Finland

Fary Z. Ghassemlooy, UK

Ling Guan, Canada

M. Haardt, Germany

Peter Handel, Sweden

Alfred Hanssen, Norway

Andreas Jakobsson, Sweden

Jiri Jan, Czech Republic

Soren Holdt Jensen, Denmark

Stefan Kaiser, Germany

C. Chung Ko, Singapore

S. Maw Kuo, USA

Miguel Angel Lagunas, Spain

James Lam, Hong Kong

Mark Liao, Taiwan

Stephen Marshall, UK

Stephen McLaughlin, UK

Antonio Napolitano, Italy

C. Richard, France

M. Rupp, Austria

William Allan Sandham, UK

Ravi Sankar, USA

John J. Shynk, USA

A. Spanias, USA

Yannis Stylianou, Greece

Jarmo Henrik Takala, Finland

S. Theodoridis, Greece

Luc Vandendorpe, Belgium

Jar-Ferr Kevin Yang, Taiwan

INTRODUCTIONNotation

SYSTEM MODEL AND MOTIVATION FORCHANNEL QUANTIZATION METHODSUBSPACE-BASED CHANNELQUANTIZATION METHODCODEBOOK CONSTRUCTION BASED ONMODIFIED LBG VQ ALGORITHMDesign issueOptimality criteriaModified LBG VQ algorithm

COMBINED CODEBOOK: HIERARCHICALCODEBOOK DESIGN METHODHierarchical codebook constructionOperation scenarioFurther comments

NUMERICAL RESULTS(i) The optimum feedback period pair guarantees the target performance(ii) The coarse feedback period is decisivein the performance

CONCLUSIONSAPPENDIXPROOF OF EQUATION (12)ACKNOWLEDGMENTREFERENCES1Call for Papers [-4pt]Guest Editors1Call for Papers0ptGuest Editors1Call for Papers4ptGuest Editors1Call for Papers4ptGuest Editors1Call for Papers-4ptGuest Editors

EfﬁcientFeedbackviaSubspace-BasedChannel ...€¦ · from MSs to BSs by using uplink resources....

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