1910 ieee transactions on signal processing, vol. 60,...

1910 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 4, APRIL 2012

Beam Tracking for Interference Alignment inSlowly Fading MIMO Interference Channels: APerturbations Approach Under a Linear FrameworkHeejung Yu, Student Member, IEEE, Youngchul Sung, Senior Member, IEEE, Haksoo Kim, Student Member, IEEE,

and Yong H. Lee, Senior Member, IEEE

Abstract—In this paper, the beam design for signal-space inter-ference alignment in slowly fading multiuser multiple-input mul-tiple-output (MIMO) interference channels is considered. Based ona linear formulation for interference alignment, a predictive beamtracking algorithm is proposed using matrix perturbation theory.The proposed algorithm, based on a mixture of iteration and up-date, computes interference-aligning beamforming vectors at thecurrent time by updating the previous beam vectors based on thechannel difference between the two time steps during the predic-tively updating phase, and yields significant reduction in compu-tational complexity compared with existing methods recalculatingbeams at each time step. The tracking performance of the algo-rithm is analyzed in terms of mean square error and sum rate lossbetween the predictively updating approach and the recalculatingapproach, and the impact of imperfect channel knowledge is alsoinvestigated under the state-space channel model. Numerical re-sults show that the proposed algorithm has almost the same per-formance as non-predictive methods in sum rate. Thus, the pro-posed algorithm provides a very efficient way to realize interfer-ence alignment in a realistic slowly fading MIMO channel envi-ronment.

Index Terms—Interference alignment, least squares, null spacetracking, perturbation theory, predictive algorithm, slowly fadingchannels.

I. INTRODUCTION

I NTERFERENCE alignment is one of the promisingtechniques to handle interference properly in a future inter-

ference-limited wireless environment. Interference alignmentachieves the upper bound of the degrees of freedom (DoF) inmultiuser interference channels by confining the interferencefrom all undesired transmitters into an interference subspaceat each receiver [1], and with the technique, each user can ap-proximately achieve a half of the capacity that can be achieved

Manuscript receivedMarch 03, 2011; revised August 09, 2011 andNovember22, 2011; accepted December 18, 2011. Date of publication December 23, 2011;date of current version March 06, 2012. The associate editor coordinating thereview of this manuscript and approving it for publication was Dr. Josep Vidal.This work was supported by the IT R&D program of MKE/KEIT. (2008-F-

004-02, “5G mobile communication systems based on beam division multipleaccess and relays with group cooperation”).The authors are with the Department of Electrical Engineering, KAIST,

Daejeon, 305-701, Republic of Korea (e-mail: [email protected]; [email protected]; ; [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2011.2181502

without interference. Thus, the interference alignment tech-nique is useful when wireless networks are interference-limitedwith a high signal-to-noise ratio (SNR) due to the small cellsize. One line of research in this area has focused on theDoF characterization and feasibility of exact alignment frominformation-theoretical perspectives [1]–[7]. Such researchis important since it provides the fundamental limit of thescheme and insights into developing good algorithms. Alongwith this, the other line of research is on the invention ofefficient beam design algorithms for interference alignment torealize the potential of the technique from signal processingperspectives [8]–[10]. Among several known schemes forinterference alignment, the signal-space alignment based onthe single-shot MIMO technique [1], i.e., beamforming usingmultiple antennas, is most attractive from a practical viewpoint,and this scheme is the main focus of this paper. To illustratethe necessity of efficient algorithms in this case as well as inother schemes, consider off-line beam design for interferencealignment using a look-up table. In this case, sets of beam vec-tors are calculated for different channel realizations beforehandand stored in a table, and a particular set is selected based onchannel information. Even in a simple case of 3-user 2 2MIMO channel, this scheme requires a memory space of a 72bit address even with one bit assigned for one real channelcoefficient ( , where the factor of 2 is for realand imaginary components of a channel coefficient, and andare the numbers of users and antennas, respectively). Thus,

such a scheme is impractical, and fast real-time algorithms arenecessary so that the technique can be used in practice. This isespecially true for time-varying channels in a practical wirelessenvironment since interference-aligning beam vectors shouldbe computed at every time step.

A. Our Approach and Summary of Result

In this paper, we consider the beam design problem in mul-tiuser slowly fading MIMO interference channels, where beamvectors need to be computed at every time step. The assump-tion of slow fading is suitable for realistic wireless environmentswhere the channels at two consecutive time steps are not inde-pendent and the current channel is a slightly updated versionof the previous one. One could apply a beam design methodfor interference alignment invented under the assumption of atime-invariant MIMO channel to every time step with differentchannel state information (CSI) even in this time-varying case.However, this may not be an efficient way since this approach

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YU et al.: BEAM TRACKING FOR INTERFERENCE ALIGNMENT 1911

does not exploit the time dependency of the channels betweenthe consecutive time steps. One would expect that the beam so-lution at the current time is a slightly updated version of that atthe previous time due to the channel’s time dependency. Suchupdating or recursive approaches have played a major role in in-venting efficient algorithms for important estimation problemslike the deterministic least squares and minimum mean squareerror (MMSE) estimation and have resulted in the celebratedrecursive least squares (RLS) algorithm and Kalman filtering.As evident in these two examples, updating or recursive solu-tions require some key ingredient that makes them possible, e.g.,the matrix inversion lemma for RLS and the state-space modelfor Kalman filtering. This is also true in the considered caseof beam tracking for interference alignment in slowly fadingMIMO channels.Our approach to an updating solution to this problem is based

on our previous work of linear formulation for interferencealignment, which is unlike existing bilinear formulations andwhich converts the problem of interference alignment to solvinga single linear system with dummy variables [9], [11]. Applyingmatrix perturbation theory to this linear formulation, we heredevelop a beam tracking algorithm for interference alignmentin slowly fading MIMO interference channels. Under the linearformulation, an interference-aligning beam solution is given bythe eigenvector associated with the smallest eigenvalue of themisalignment covariance matrix (which will be defined later),and the solution to a perturbed misalignment covariance matrixby channel variation can be obtained by subspace tracking.The proposed algorithm computes interference-aligning beamvectors at the current time by additively updating those ofthe previous time, and reduces computational complexitysignificantly compared with existing batch methods with whichinterference-aligning beam vectors are redesigned at every timestep when applied to time-varying MIMO interference chan-nels. In addition to the development of a tracking algorithm forinterference alignment, as a byproduct we also derive a newupdate formula for general null space tracking that is similar tothe existing result in [12] and is useful in other applications too.The tracking performance of the algorithm is analyzed in

terms of mean square error (MSE) and sum data rate. Thetracking error results from two sources: predictive beam designand imperfect channel knowledge due to channel estimation.The error by prediction is evaluated analytically with a pertur-bations approach, and the tracking error caused by imperfectchannel knowledge is examined under the state-space channelmodel and time-division duplex (TDD) transmission structure.Numerical results show that the proposed predictive algorithmtracks the time-varying beam solution well for a wide rangeof fading rate or mobile speed with significantly reducedcomplexity.

B. Related Works

The interference alignment was introduced for MIMO-channels [13], [14]. It was shown that the interference

alignment achieved the maximum DoF in -user single-inputsingle-output (SISO) time-varying interference channels basedon a vector channel model induced by time concatenation[1]. In the case of time-invariant MIMO channels (a

single-shot approach), it was shown that the maximumDoF could be achieved with a beamforming approach in the3-user case [1]. It was also shown for that exact inter-ference alignment with a signal space approach was infeasiblein time-invariant MIMO channels. Also, interference align-ment based on a signal scale using structured coding has beenproposed for deterministic channel models [15], [16]. Withasymmetric complex signaling, 1.2 DoF can be achieved for

for almost all values of channels [17]. It has been shownthat the total DoF is determined by less than in the fully con-nected constant real Gaussian interference channels dependingon the rationality and irrationality of channel coefficients [3].The total DoF of is achieved for the set of measure zero ofchannel gains. Recently, the achievability of DoF usinga new interference alignment based on the Khinchin-Groshevtheorem has been proved [4] by exploiting the properties ofrationally independent channel coefficients. Based on thisresult, interference alignment for MIMO interference channelswas proposed in [6]. In [7], the relay-aided interference align-ment including achievable DoF, feasibility, and algorithm wasdiscussed. The feasibility conditions for interference alignmentunder different frameworks were derived in [5] and [18].In addition to these information-theoretical results, there

has been research from algorithmic aspects. Gomadam et al.proposed algorithms to minimize interference and maximizethe signal-to-interference-plus-noise ratio (SINR) by solving asystem of bilinear equations iteratively [8]. A linear formulationto interference alignment using a beamforming technique fortime-invariant MIMO channels was introduced, and iterativealgorithms based on least squares were proposed in [9]. Recentadvances for interference alignment include the investigationof channel uncertainty and SINR characterization [19], [20],limited feedback in single and multiantenna channels [21],[22], interference alignment scheme based on alternating min-imization [10], and ergodic interference alignment using thestatistical properties of channels [23]. Additionally, interfer-ence alignment without channel information at transmitterswas proposed in [24].

Notations and Organization

We will make use of standard notational conventions. Vec-tors and matrices are written in boldface with matrices in cap-itals. All vectors are column vectors. For a matrix , and

indicate the transpose and Hermitian transpose of , re-spectively, and is the column vector consisting of allthe columns of . represents the column space of , i.e.,the linear subspace spanned by the columns of .means that is a matrix composed of denoting the th rowand th column element (also sometimes denoted as ), and

is the diagonal matrix of diagonal elements. and denote the pseudo inverse and trace

of matrix , respectively. We use for the 2-norm of a vector, and denote the Frobenius norm and 2-norm ofa matrix , respectively. For matrices and , and

denote the Kronecker and Hadamard products betweenthe two matrices, respectively. stands for the identity matrixof size (the subscript is omitted when unnecessary), andis a column vector with all elements as one. is an all-zero


matrix. The notation means that is complexGaussian distributed with mean vector and covariance matrix. denotes the expectation of random variable . For ascalar , is the complex conjugate of .The paper is organized as follows. The system model and

previous work on a linear framework approach to interferencealignment are described in Section II. In Section III, a predic-tive beam design algorithm for interference alignment is pro-posed based on matrix perturbation theory by exploiting ourlinear framework. In Section IV, the performance with consid-eration of channel prediction and the computational complexityof the proposed algorithm are examined. Numerical results areprovided in SectionV, followed by the conclusion in Section VI.

II. SYSTEM MODEL AND PRELIMINARIES

We consider a -user MIMO interference channelwhere the transmitter-receiver pairs exist with interferinglinks. We assume that each transmitter and receiver have andantennas, respectively, and . We also assume that

the MIMO channel is flat-fading. Due to the interference fromother transmitters, the received signal at receiver is given by

(1)

where is the time index, is the MIMOchannel matrix between transmitter and receiver ,

is an beamforming

matrix at transmitter , is thesignal vector with unit power, and

is zero-mean complex Gaussian noise vector with covariancematrix . Here, is the number of data streams at trans-mitter . It is assumed that the number of data streams at eachtransmitter is equal, i.e., . We also assumethat the MIMO channels are slowly time-varying andthat all CSI is available at each transmitter.

A. Preliminaries: Linear Framework for InterferenceAlignment [9]

In this subsection, we briefly review the previous work [9]that provides a linear framework for interference align-ment when time is fixed. Under the assumptions of

and , the interfer-ence from all undesired transmitters should be confined toan interference subspace with dimension at each receiverfor interference alignment, and the interference alignmentcondition in this case is given by

...

(2)

(Throughout this section, the time index is fixed and isdropped for notational simplicity.) The conditions in (2) canbe converted into a system of linear equations with dummyvariables. Consider the first equality in the first row of (2)

(3)

A necessary and sufficient condition for the equivalence ofcolumn spaces of two matrices is that a column in one matrixis represented by a linear combination of the columns of theother. Thus, the subspace equivalence in (3) can be rewrittenas the following equation:

(4)

where , are coefficients for linear com-bination. The multiple equalities in (4) can be written in a singlematrix-vector form as

(5)

where . In this manner, each row in (2) can beconverted into a linear equation. Collecting all equations gen-erated from the rows in (2) yields the following system oflinear equations with dummy variables :

(6)

where is defined as shown in (7) at the bottom of the nextpage, and

(8)

Note here that is a function of channel and dummyvariables . Once the channel and dummy variables aregiven, we can obtain the interference-aligning beam vectorfrom (6). On the other hand, the linear combination coefficients

can also be obtained easily when are given. Notethat (4) can be rewritten as

(9)

and, thus, is simply given by

(10)

where is always a tall matrix (i.e., the number of rowsof the matrix is larger than that of its columns) and, thus, itspseudoinverse almost surely exists because the number of datastreams is always less than the number of receiver antennas. [Inthe case of , the right-hand side (RHS) of (10)reduces simply to taking the inner product between two vectors.]In a similar manner, are given by

(11)

The size of is , and the existence of a non-trivial solution to (6) depends on the system parameters , ,


and . First, consider the case where exact interference align-ment is not feasible1. In this case, is necessarily a tall matrix,and a nontrivial solution to (6) does not exist for any .Thus, in this case we can find the best approximate solution tointerference alignment by resorting to a least squares approach

(12)

and the optimal solution to (6) in the least squares sense canbe found by an iterative algorithm in which (11) and (12) areused alternately at each iteration. This algorithm converges andits rate of convergence is much faster than other algorithmsfor interference alignment with similar performance [9]. Onecan view as the power of overall interference miss-alignment, and hereafter we refer to as the interference mis-alignment matrix. It is known that the solution to (12) is theeigenvector of corresponding to the smallest eigenvalue.Second, consider the case where exact interference alignment isfeasible, and, hence, a nontrivial solution solution for in (6)exists. [Note that (12) is still valid in this case.] This case canbe divided further into two subcases: one is that is still a tallmatrix (e.g., , ) and the other is that isa strictly fat matrix (e.g., , , ). In the lattercase, an interference-aligning solution can be found simply asa null vector of with an arbitrary choice of and mul-tiple linearly independent null vectors exist. In the former case,on the other hand, becomes rank-deficient only with proper

, and a nontrivial solution for in (6) can be found againby the iterative algorithm based on (11) and (12).One drawback of the iterative algorithm in [9] is that its per-

formance is degraded when due to the interference fromother data streams of the same transmitter. In the next subsec-tion, we show that this difficulty can be overcome by choosingthe beamforming vector of each stream from different basis vec-tors of the null space of .

1The feasibility analysis is not the main focus of this paper. A general feasi-bility study can be found in [5].

B. Enhanced Beamforming Vector for

In this subsection, we first discuss details about the reasonwhy the performance of the iterative algorithm in [9] is degradedwhen , and then present a method for constructing en-hanced beamforming vectors with a DoF guarantee. Here, weconsider the case that , , and . Itseems that the extension to the case of is difficult. How-ever, the considered case is a practically important case that al-lows perfect interference alignment.Theorem 1: For and , with properly

chosen , the matrix is almost surely rank-deficient withnullity of (i.e., there exists a matrix with fullcolumn rank such that ) for randomly realized .

Further, let be the partition of with

submatrices , . Then, we have

(13)

where and , , are , and vectors,respectively, and are linearly independent of each other.

Proof: See Appendix.For , the iterative algorithm in [9] selects one of the

columns of (13) as the beamforming vectors for transmitter ,sectorizes the vector into subvectors of size , and uses eachsubvector for each stream of the same transmitter. Suppose that

for a particular is chosen. Then

(14)

and , , is the beamforming vector forthe th stream in this case. Note that beamforming vectors in(14) are essentially identical. Each vector is a scaled version ofthe other, and this is the reason why the iterative algorithm in[9] does not guarantee the DoF and its performance is degradedwhen . This problem can be overcome by choosing thebeamforming vectors from different columns of in (13). Forexample, we can construct for transmitter as follows:

(15)

......

......

......

......

......

......

......

......

......

......

......

......

(7)


Now in (15), beamforming vectors for different streams are lin-early independent and can be orthonormalized by the Gram-Schmidt procedure. The orthonormalized version of (15) canalso serve as the beamforming vectors that enable perfect in-terference alignment by satisfying (6). This is becausecan be adjusted to incorporate the scaling associated with theorthonormalization. In summary, to enhance the iterative algo-rithm in [9], we here propose the construction of beamformingvectors as in (15) at each iteration and the orthonormalizationof the beamforming vectors by the Gram-Schmidt procedure atthe end of the iteration.

III. PREDICTIVE BEAMFORMING FOR

INTERFERENCE ALIGNMENT

Now consider the time-varying channel case. In this case,the beamforming vectors for interference alignment overtime-varying channels can be obtained by tracking the eigen-vectors of . This is because the solution to (12) isgiven by the eigenvector of corresponding to thesmallest eigenvalue. The basis tracking for a signal subspaceand the tracking of signal eigenvectors have been extensivelyinvestigated in the areas of temporal and spatial domainspectral analysis [26]–[30]. In these areas, efficient trackingalgorithms have been derived by exploiting the structure ofsample covariance matrices. However, the application of suchtechniques to tracking the beamforming vectors for interferencealignment does not seem feasible because beamforming forinterference alignment is not based on the sample covariancematrix but on channel estimates. In this section, we propose theuse of the perturbation theory for eigenvectors [31] for trackinginterference-aligning beam vectors. The idea is described asfollows: given and its eigenvalues andeigenvectors at time , we can predict the eigenvectors at timefor if the elements of exhibit

only small changes as compared with those of . This typeof prediction for beamforming vectors to align interferenceis possible in slowly fading channels. Next, we present twotheorems regarding the prediction of beamforming vectors forinterference alignment. In the first theorem, is assumedto have full rank and the result is obtained by directly applyingthe linear perturbation theory in [31]. The case of rank deficient

is considered in the second theorem, where a more recentresult on perturbation analysis in [32] is used.Theorem 2: Suppose that has the full rank of

. Let be the least squares solu-tion to (12), i.e., the eigenvector of corresponding to thesmallest eigenvalue at time . Then, the solution to (12) attime can be written as

(16)

where

(17)

; are the ordered eigenvalues of

, which are distinct, and are the corresponding

eigenvectors ; and

.Proof: See Appendix.

We use the first and second terms in the RHS of (16) for pre-dicting . Specifically, let denote the prediction of .Then,

(18)

The third term in the RHS of (16) represents the dominant pre-diction error and is denoted as .Corollary 1: The dominant prediction error in (16),

, is upper bounded as

(19)

Proof: is the norm of the third term in theRHS of (16). By using the subadditivity and submultiplicity

(20)

Here, the last inequality holds becauseand. The RHS of (20)

is less than or equal to the RHS of (19) because.

The upper bound in (19) is given by a function of

and .

Theorem 3: Suppose that is rank-deficient with rank. Let be a matrix whose columns form the null space


of . Then, the perturbed null space for at timeis given by

(21)For an arbitrary null vector of , we have

(22)

where is defined in (17); ; areordered eigenvalues of that are not necessarily distinct

, and are the correspondingeigenvectors.

Proof: See Appendix.Again the first and second terms of RHS of (22) are used for

prediction, and the third term represents the dominant predictionerror. The prediction of is given by

(23)

Corollary 2: The dominant prediction error in (22), denotedby , is upper bounded as

(24)

This inequality can be proved in a manner similar to the case ofCorollary 1. In (19) and (24), the impact of and onprediction error is separated. As expected, theMSE of beam pre-diction increases as progresses, and the increase rate is givenby in both cases. Interestingly, the impact of atthe reference time is seen by the spectral gap of , i.e., thedifference between the two distinct smallest eigenvalues; whenthe spectral gap is large, the tracking performance is good.The perturbation form (22) of the null vector in Theorem 3

is very similar to the result of Theorem 2. Note that the resultobtained by substituting into (16)is identical to (22) in the rank-deficient case with a nullity ofone, i.e., . In this case, the zero eigenvalue has amultiplicity of one and, thus, Theorem 2 holds. When the nul-lity is larger than one, as in the example of ,

or , , and , however, the as-sumption for the derivation of Theorem 2 [12] is broken, andthe general result of Theorem 3 should be applied. Thus, thenewly derived update formula in Theorem 3 is a generalizedversion for the null vector update in a rank-deficient case with

arbitrary nullity. Theorems 2 and 3 provide a basis for a predic-tive beam solution to interference alignment in slowly fadingchannels. Once the full eigen-decomposition of is per-formed at time , the beam solutions at successive time steps

are obtained by these formulaswhere is the prediction depth. In the proposed algorithm, thefull eigen-decomposition is performed once in every timesteps and linear prediction is performed in between two fulleigen-decompositions. During the prediction period, we onlyneed to compute the channel difference matrix and perform ma-trix additions and multiplications. The value of prediction depthdepends on the fading rate (or mobile speed) and allowable

prediction error. In typical slow fading, the value of can belarge (which will be seen in Section V), and the reduction ofcomputational complexity can be significant as compared withthe method performing full eigen-decomposition at each time.The proposed algorithm is summarized below.1) The Proposed Algorithm: At for some integer

and (eigen-decomposition phase)1 Initialize for and for .2 Construct with and available ; up-date by solving (11) and (12) iteratively until it con-verges. (A proper normalization step can be added here.)

3 Reconstruct with obtained in Step 2.4 Perform the eigen-decomposition of

.5 Set to the eigenvector associated with the smallesteigenvalue.for (prediction phase)

6 Construct using obtained in Step 2 and.

7 Compute , andupdate the beam vectors using either (18) or (23) to obtain

depending on the rank of . (The rank is obtainedin Step 4.)end

Since for in Step 1, updatingin Step 2 requires only a few iterations to converge. Some addi-tional remarks of interest are as follows:• As mentioned in Section II-B, an orthonormalization stepcan be added to equalize power among the streams andusers without disturbing the interference subspace. Duringthe prediction phase, an orthonormalization step can beinserted after obtaining in Step 7.

• We can consider one modification of the proposed algo-rithm to incorporate signal power as well as the minimiza-tion of interference, which was considered in [9]. In thismodification, is usedinstead of , where is a weightingfactor and .In this case, almost surely has full rank, and The-orem 2 needs to be applied.

• In the proposed algorithm, we assume that the variationof is small enough to be negligible. Without thisassumption, a step to update with (11) should beincluded after Step. 7. However, we do not updatein the prediction phase because exclusion of this step does


Fig. 1. Complexity comparison (a) , , . (b) , , .

not cause performance loss much in terms of the sumthroughput.

• In the feasible multiple-stream case , the eigen-vectors associated with the smallest eigenvalues ofare obtained in the eigen-decomposition phase, and eachof the eigenvectors is updated according to (23) in the pre-diction phase. Then, a beamforming matrix with linearlyindependent columns can be obtained with these eigen-vectors, as explained in Section II-B.

IV. PERFORMANCE ANALYSIS AND APPLICATION

In this section, we examine the overall performance by con-sidering both the beam prediction and channel prediction withan application to TDD downlink. To implement the proposedpredictive beam design, channel state information is required atthe transmitters, and channel prediction should be performed inthe case of TDD operation. Therefore, the prediction error re-sults from two sources: one is the predictive beam computationitself and the other is imperfect channel knowledge due to theuse of a channel predictor or estimator to obtain the channel in-formation.

A. Channel Model, Frame Structure, and Application

To investigate the impact of channel dynamics and estima-tion on the prediction performance, we need a time-varyingchannel model. In this paper, we use the state-space channelmodel with zero-mean Gaussian channel gain, i.e., the first-order Gauss-Markov channel model, which has widely beenused as a good model for Rayleigh-fading time-varying chan-nels [33]–[35]. The channel gain between antenna of trans-mitter and antenna of receiver , i.e., the th element of

, is given by

(25)

where , is the correlation co-efficient, and is the plant noise ofthe channel process. The parameter depends on the Dopplerspread and symbol duration [34], [35]. Note that depends

on indices and of transmitter and receiver but not on an-tenna indices and of the same transmitter-receiver pair sincethe fading rate is determined by the relative speed between thetransmitter and receiver. For simplicity, we here assume thateach element in the MIMO link evolves independently and thestatistics of channel process are identical for all and .We assume that the proposed interference alignment is ap-

plied to TDD downlink transmission with one base station con-troller (BSC) and pairs of base station (BS) and mobile sta-tion (MS). It is assumed that the TDD frame size is

, and and are the numbers of uplink and downlinksymbols, respectively. To compute interference-aligning beamvectors at the BSC, channel matrices correspondingto all MIMO links between BSs and MSs are required2. Toobtain CSI at BSs, pilot symbols can be embedded in the uplink,and the length of uplink transmission can be set to be longenough to guarantee that the pilot sequence of one uplink an-tenna is orthogonal to those from all others. Then, the obtainedchannel information can be sent to the BSC for the computa-tion of interference-aligning beam vectors. Under this scenario,the channel can be estimated at the transmitter directly due tochannel reciprocity. For the period of uplink pilot symbols, thereceived signal at antenna of transmitter is given by

(26)

where denotes the known pilot symbol from antenna of

receiver in the uplink, due to the channel reci-

procity, and is the additive white Gaussiannoise at time . Equations (25) and (26) form a state-spacechannel model to which Kalman filtering can be applied to ob-tain an optimal channel estimate. The optimal estimation underthe state-space model was thoroughly studied by Dong et al.[33]. The optimal estimation is given by Kalman filtering duringthe uplink pilot period and Kalman prediction during the down-link data phase

(27)

2For interference alignment, desired links from each transmitter to a cor-responding receiver are not required. To incorporate signal power in the beamdesign, the desired links are also required.


where is the channel estimate at time . Closed-formexpressions for the steady-state channel estimationMSE of pilotand data phases are known and available in [33].

B. Sum Rate Loss

In Section III, the beam prediction performance is analyzedand the dominant error term is obtained under the assumption ofperfect channel knowledge. In practical systems like the TDDtransmission explained in the previous subsection, the MSE ofbeam prediction should further include the channel estimationerror. The expected MSE of beam prediction averaged overchannel realizations can be used to assess the behavior of beamdesign MSE and corresponding sum rate loss as functions offading rate and prediction depth. Both and

in (19) and (24) are composed of two compo-nents: one depends on the eigenvalues of at reference timeand the other relies on the channel difference matrix

between two time steps, and . The expected eigenvalues ofand, thus, those of in (19) and (24) depend only on

time , and they are dominantly affected by the size of , i.e.,the numbers of users, transmit streams, and antennas. Under theconsidered channel model in Section IV-A, ,

, where

and is in the form of (7) with estimated channelsinstead of true channels , can be obtained explicitly, andis given in the following theorem.Theorem 4: Under the assumption of the Gauss-Markov

channel model and for all and , the upper bound ofthe expected 2-norm of is given by

(28)where is the maximum of the fourth moment ofviewed as a function of channel random variables under the as-sumption of its existence.

Proof: See Appendix.

Note that the overall beam design MSE in this case can beupperbounded using (28). It is seen from Theorem 4 that thetracking error decreases with the fading coefficient and in-creases with other parameters including , , , etc. In partic-ular, with fixed , and , the MSE of tracking errorduring the prediction period is given by with someconstant ; that is, the tracking error does not increase withoutbound, but saturates as time elapses.Now we analyze the sum rate loss caused by both predic-

tive beam design and channel estimation under the interferencealignment condition in the case of a single data stream. (Similaranalysis for multiple streams for one user is possible, but it doesnot provide more insight.) We assume that a zero-forcing (ZF)receive beam vector obtained based on the predicted channelis used at the receiver, and the zero-forced received signal isgiven by (29) at the bottom of the page, where

is the channel prediction (or estimation) error,denotes the error between beam correctly matched to

the predicted channel and predictively obtained beam usingthe predicted channel , and .In the last equality, we use the fact that byinterference alignment for the exact beam based on the pre-dicted channel. In (29), the first term stands for the desiredsignal component, the second and third terms are the interfer-ence from channel and beam vector predictions, respectively.The loss of the expected sum rate can then be defined as (30) atthe bottom of the page, where

and . An upper bound of the expectedrate loss due to channel prediction (or estimation) and beam pre-diction is obtained and given in the following theorem.Theorem 5: Assuming independent and identically dis-

tributed (i.i.d.) channel coefficients with zero mean andvariance for all elements of all MIMO channel matrices, wehave an upper bound of the expected sum rate loss, given by

(31)

(29)

(30)


Fig. 2. MSE of beam vector prediction under different mobile speeds (a) , , , . (b) , , .

TABLE ICOMPLEXITY OF THE PROPOSED ALGORITHM

where and are the MSEs of channel estimation and beamprediction, respectively.

Proof: See Appendix.Theorem 5 is general in the sense that the Gauss-Markov

channel assumption is not required, and it is valid for gen-eral channel estimator and beam design. In the case of theGauss-Markov channel model, the channel estimation MSEis known in [33], and an upperbound of beam prediction MSEis given by (19), (24) and Theorem 4. As expected, there is

no loss if the channel estimation and beam tracking is perfect,i.e., . It is interesting to see that the impact bybeam design error is scaled by the channel variance . Thedominance of either term depends on the number of users andthe variance of channel process.

C. Complexity

Here, we examine the computational complexity of theproposed algorithm and compare it with simple redesigningmethods applying existing algorithms at every time step. Forthe redesigning methods, we consider the iterative interferencealignment (IIA) algorithm proposed in [8], which has lesscomplexity than the MAX-SINR algorithm in [8] and which isoptimal at high SNR [36], and the iterative LS (ILS) approach[9] described in Section II-A. Here, we also considered themodified IIA algorithm to fit in slowly fading channels byinitializing the transmitter and receiver beam vectors by thevalue of the previous time to reduce the number of iterationsto a minimum. In this way, the modified IIA requires only afew iterations to converge and to yield the same performance

Fig. 3. MSE of beam vector prediction with two different channel realizationswhen , , , and . (marker: numerical MSE, line:analytical MSE).

as the proposed algorithm3. As a complexity measure, we usethe number of required complex multiplications during onedata period with symbols. We mainly consider the case of

and . The IIA algorithm requiresin

each iteration, where the first and second terms are the num-bers of multiplications for computation of the interferencecovariance and extreme eigenvectors using the iterativeQR method with iterations, respectively. Under the as-sumption that the numbers of iterations required to obtain aconvergent solution with an arbitrary initialization and smartinitialization are and two, respectively, the total numbers ofmultiplications of the redesigning and modified IIA for one dataperiod are given by and ,

3For , , and , the modified IIA with one or twoiterations for beam update with this smart initialization has a sum rate loss ofabout 1 or , respectively, at 35 dB SNR under mobile speed. Theproposed algorithm shows less than rate loss under the same condition,as shown in Fig. 4(b), so we used two as the iteration number for the IIAmethod.


Fig. 4. (a) Sum rate losses with channel estimation and (or) beam prediction at different and . (b) Upper bounds of sum rate loss obtained with (31) andanalytical MSEs.

respectively. On the other hand, the redesigning ILS algorithmrequires

operations, where is the number of iterationsfor ILS with an arbitrary initialization [9]. The first three termsare the complexity of obtaining , and the fourth and fifthterms are the complexity to obtain extreme eigenvectors usingthe iterative QR method and to update , respectively. Byexploiting the structure of , we further reduce the complexityof the ILS algorithm than the result in [9]. Now, consider theproposed predictive algorithm. The number of multiplicationsin each step is shown in Table I. Here, we assume that thefull eigen-decomposition of is done by the Householdermethod to reduce the symmetric matrix to tri-diagonal formand the following QL algorithm to obtain the eigenvalues andeigenvectors. These two steps require andoperations [37], and we simply assume that all these operationsare complex multiplications. For , ,and , , , we plotted the complexity inFig. 1. We used , , and to assurethat both algorithms terminate with the same interferenceleakage of . As seen in Fig. 1, the proposed algorithm re-duces the complexity significantly compared with redesigningapproaches and shows less complexity than the best modifiedIIA method in slowly fading channels (and even the modifiedIIA method with only one iteration per step—the slope of theproposed algorithm is smaller). As increases, the complexitygain over the other methods increases linearly with respect to(w.r.t.) since the proposed method requires fewer operationsin each update step than all the other methods. This is becauseiterative operations like the iterative QR method to obtain theextreme eigenvector(s) are required to update the beam vectorin the other three algorithms. However, the proposed predictivebeam design does not require any iterative operation during theupdate phase.

Fig. 5. Sum rate losses with channel estimation and beam prediction at dif-ferent mobile speed and .

V. NUMERICAL RESULTS

To assess the performance of the proposed algorithm, weprovide numerical results in this section. We assume 1.0 GHzcarrier frequency and a symbol duration of , which isthe length of one OFDM symbol in 3GPP long-term evolu-tion (LTE). We used the channel and transmission model inSection IV-A with andfor all simulations here. The fading coefficient of the Gauss-Markov channel model is determined by the zero-order Besselfunction of the first kind , where and arethe maximum Doppler frequency and symbol duration, respec-tively.First, we validate the error analysis of the proposed algorithm

based on the perturbations approach in Fig. 3 to Fig. 5. Since thesolution to interference alignment is not unique, the interferencemisalignment is used as the measure of beam designMSE. When , , , and , the analyt-ical MSEs match the numerical ones well for a wide range ofmobile speed, as shown in Fig. 2(a), although the up-


dating step is not included in the prediction phase of the pro-posed algorithm. In the case of , , and

, on the other hand, the updating step is includedin the prediction phase of the proposed algorithm since inter-ference alignment is feasible only with proper in thiscase. The numerical results again match the analysis well in thiscase, as shown in Fig. 2(b). Fig. 3 shows the MSE results ac-cording to mobile speed with two different channel realizationsfor , , , and : one has a smallminimum non-zero eigenvalue of , and the other has alarge minimum nonzero eigenvalue of . As expected byCorollaries 1 and 2, the beam prediction MSE is smaller whenthe minimum nonzero eigenvalue of is larger, i.e., the spec-tral gap of is larger. Fig. 4(a) shows the sum rate loss withthe predictive beam design by the proposed algorithm and/orchannel estimation by using the Kalman predictor at 35 dB SNRwhen , , and , ,

, . The rate loss increases with the symbol timesince both and are increasing functions of in the formof , as seen in Theorem 4 and [33]. So, it is seen thatthe increment of the rate loss monotonically decreases as in-creases. The rate loss for , , and is smallerthan that for and since the former casehas smaller MSE of beam prediction, as predicted by Theorem5. To verify Theorem 5, the upper bounds of sum rate loss withchannel estimation and/or beam prediction MSEs are evaluatedwhen and . To obtain the upper bounds,we use (31). For the upper bound only with beam predictionerror, we set . The upper bound only with channel es-timation error is evaluated with . The upper bound ofsum rate loss is not tight with the numerical result, but the be-havior of each term is well predicted by Theorem 5; the rela-tive contributions of predictive beam design and channel esti-mation errors to sum rate loss are similar in Fig. 4(a) and (b).Note also that the sum rate loss by channel estimation error isdominant compared with that by beam prediction. The rate lossw.r.t. different mobile speeds is also shown in Fig. 4. Under theslow-fading assumption , the fading coefficient canbe approximated as

. Then, can be approximated as

, where andare the mobile speed and the wavelength, respectively, since

. Thus, behaves like ,and behaves like at high SNR according to Theorems4 and 5, as seen in Fig. 5.Next, we compare the sum rate performance of the predictive

beam design algorithm with that of others. We considered threedifferent mobile speeds of 3, 10, and 30 km/h and four differentcases: two feasible cases with tall (or square) (, , and , , ), a strictly fatwith one more transmit antenna, i.e., , , ,

and , and finally an infeasible case ofand . First, Fig. 6 shows the average sum rate w.r.t.average link SNR at 30 km/h of six different approaches in-cluding the recalculating ILS redesigning transmit beams everytime step, the proposed method, their modifications incorpo-rating signal power where the weighting factor is set by the

inverse of SNR as in [9], the modified IIA, and the MAX-SINRalgorithm. To see the relative performance only by the differentbeam design methods, we here use perfect CSI. In the singlestream cases (a), (b), and (c), the proposed algorithm performsas well as the others, except the MAX-SINR algorithm. Also,in these cases the proposed method, redesigning ILS and modi-fied IIA algorithms yield almost the same performance whereasthe proposed and redesigning ILS algorithms modified to in-clude signal power perform in-between the MAX-SINR andthe other three algorithms. Hence, the modified approach withsignal power consideration is a reasonable tradeoff between thesum rate and complexity. In the infeasible case (c), the sum ratesaturates after the linear increase w.r.t. SNR due to interferenceleakage, as expected. In the multistream case (d) with ,the proposed algorithm performs worse than the IIA methodsince theMSE of the predictive approach increases with . Next,Fig. 7 shows the average sum rate including channel predic-tion using the Kalman filter for the case with ,

, and . The rate loss due to imperfect channelinformation and beam update increases as the mobile speed in-creases, as expected, and the loss occurs mainly at high SNR. Itis seen that the rate loss caused by channel prediction is largerthan that by predictive beam design and the redesigning and pre-dictively updating approaches using imperfect CSI show nearlythe same performance due to the dominance of channel pre-diction error. Thus, at high SNR, the predicted channel qualityshould be also improved to avoid the rate loss in systems op-erating under interference alignment. As mentioned previously,we used the first-order Gauss-Markov channel model for simpleanalysis in this paper. By adopting a higher-order channel modeland corresponding Kalman filtering, the quality of channel pre-diction can be improved.

VI. CONCLUSION

We have proposed a predictive algorithm for the beam designfor interference alignment in slowly fading MIMO interferencechannels based on matrix perturbation theory and newly de-rived null space update formulas. The proposed algorithmmixeseigen-decomposition and prediction steps properly to minimizecomplexity while predicting interference-aligning beams effec-tively in slowly fading channels. We have analyzed the perfor-mance of the proposed algorithm to gain insights into the fac-tors affecting its prediction performance and have provided nu-merical results to validate the proposed algorithm. The algo-rithm provides an efficient way to track interference-aligningbeams with comparable sum rate performance with less com-plexity than the previous redesigning approaches with time-varying channel information.

APPENDIX

Proof of Theorem 1 [25]: It is sufficient to show that a fullcolumn-rank matrix (with columns) and exist suchthat . ( cannot be greater than by the known


Fig. 6. Average sum throughput with perfect CSI and mobile speed (a) , , . (b) , , , . (c), , . (d) , , .

feasibility result [1].) By partitioning into ,

can be rewritten as

(32)

(33)

(34)

For randomly realized channel , the matrices are almostsurely invertible. From (33) and (34), we express andin terms of and use them in (32). Then, (32) is expressedassince for invertible matricesand , and . Defining

and , weobtain

(35)

Note that (35) is an equation for subspace invariance withas the invariant subspace of operator . We

denote the sets of eigenvalues and eigenvectors of byand , respectively, and those of

by and , respectively. Sincethe set of complex matrices that are not diagonaliz-able has measure zero in the space of , the randomlygenerated matrix is almost surely diagonalizable and itseigenvectors are linearly independent. (Also,

.) It is always possible to design sothat has eigenvalues such that for somepermutation . For such an , has a unit eigenvalue

with geometric multiplicity , and the correspondingeigenvectors are given by . Theseeigenvectors are linearly independent because of the linear in-dependence of . Let be a matrix consistingof the eigenvectors as its columnsin the form of (13). Then, satisfies (35). The expressionsfor and can be derived in a similar manner. Finally, thematrix constructed as such has rank since the submatrix

of has rank .Proof of Theorem 2: Define for

. Then, . We denote the


Fig. 7. Average sum throughput with perfect/predicted CSI and different mo-bile speeds (a) , , , . (b) , ,

, . (c) , , , .

smallest eigenvalue of by and the correspondingeigenvector by . Then

(36)

Assuming that both and are differentiable w.r.t. ,the result in (16) can be obtained by deriving the second-orderTaylor expansion of , which is an extension of the first-order Taylor expansion in [12, p. 346 ]. By differentiating (36)w.r.t. , we have

(37)

Differentiating the above equation w.r.t. again and settingyields

(38)

Since the eigenvectors of form a basis inin the full rank case, and are represented as

(39)

and

(40)

respectively. From [12, p. 346],

. Hence

for

for .(41)

where . Thus, (38) is rewritten as

(42)

Since and , (42) becomes

(43)

Multiplying to (43) from the left yields

(44)


where for . To evaluate , we setin (37) and multiply the equation by . Then

(45)

where . Since is a symmetric matrix,, and (45) reduces to

From (39), (40), and (44), the Taylor expansion for isgiven by

(46)

The desired result in (16) is obtained from (46) becauseimplies .

Theorem A.1 (Avrachenkov and Haviv, 2003): Supposethat is a rank-preserving perturbation of ; i.e., and

have the same rank and .Then, there exists a holomorphic family of eigenvectors,

, corresponding to thezero eigenvalue and satisfying the normalization condition

, where such that and

. The coefficients of the power series for can becalculated recursively by the formula

(47)

Proof of Theorem 3: To prove Theorem 3, we need theabove result regarding the perturbation of a null space in [32].Theorem 3 can be proved by considering the first-order ap-

proximation of and the second-order approximation ofin Theorem A.1. Specifically, we writewhere and , and con-

sider the eigen-decompositions of and

and

where ; is a submatrix of the eigenvectormatrix of corresponding to the zero eigenvalues;

and are submatrices of the eigenvector matrix of ,which are perturbed versions of and , respectively.

is a rank-preserving perturbation of because therank of is almost surely identical to that of . There-fore, can be written as

(48)

where and are obtained from (47)

[The following relations are used in (47): ,and for .] Since

, and are rewritten as

From (48), we have

(49)

The interference aligning beam vectors and are givenby linear combinations of the columns of and , respec-tively; i.e., and for some constantvector . Multiplying to (49) from the right, we have

(50)

where the second equality holds because is a

scalar and .


Proof of Theorem 4: Using the definition ofand , we have

(51)

where is defined below. In the second equality, we use thefollowing relations in (52), shown at the bottom of the page.The inequality holds since

. Under the assumption of for all and , we have

(53)

By taking expectation w.r.t. channel distribution, we have

(54)

In , two different kinds of nonzero submatrices exist.Some submatrices are Kronecker products of a channel matrixand an identity matrix, and the others are those of a channelmatrix and a linear combining matrix . The numbers ofelements corresponding to the former and later cases are

......

......

......

......

......

......

......

......

......

(52)

(55)


and , respectively. Addi-tionally, we use

where is the maximum of the fourth moment of .The first inequality is the Cauchy-Schwartz inequality, and thesecond inequality holds because and,Proof of Theorem 5: The sum rate loss is given by

(55) at the bottom of the previous page, whereand [38].

(a) is by and . (b) holds because , ,, and are isotropically distributed unit vectors and are

independent of [38]. (c) is by Jensen’s inequality and theconcavity of logarithm. Applying Jensen’s inequality again tothe last equation, we have

(56)

and

(57)

since and are unit-norm vectors and forany matrix .

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Heejung Yu (S’07) received the B.S. degree in radioscience and engineering from the Korea University,Seoul, in 1999, and the M.S. and Ph.D. degrees inelectrical engineering from the Korea Advanced In-stitute of Science and Technology (KAIST), Daejeon,in 2001 and 2011, respectively.Since 2001, he has been with the Electronics

and Telecommunications Research Institute (ETRI),Korea. He has participated in the IEEE 802.11 stan-dardization, where he made technical contributionsfrom 2003. His areas of interest include statistical

signal processing and communication theory.Dr. Yu is the recipient of the Bronze Prize in the 17th Humantech Paper Con-

test and the Best Paper Award in the 21st Joint Conference on Communicationsand Information (JCCI) in 2011.

Youngchul Sung (S’92–M’93–SM’09) receivedB.S. and M.S. degrees from Seoul National Uni-versity, Seoul, Korea, in electronics engineering in1993 and 1995, respectively, and the Ph.D. degreein electrical and computer engineering from CornellUniversity, Ithaca NY, in 2005.He is an Associate Professor in the Department of

Electrical Engineering, Korea Advanced Institute ofScience and Technology (KAIST), Daejeon. From2005 until 2007, he was a senior engineer in theCorporate R&D Center, Qualcomm, Inc., San Diego,

CA, and also worked as a senior research engineer with LG Electronics, Ltd.,

Seoul, from 1995 to 2000. His research interests include signal processing forcommunications, statistical signal processing, and asymptotic statistics withapplications to wireless communications and related areas.Dr. Sung is an associate member of the IEEE SPS SPCOM TC, a member

of Signal and Information Processing Theory and Methods (SIPTM) TC ofAsia-Pacific Signal and Information Processing Association (APSIPA), a TPCmember of Globecom 2011/2010/2009, ICC 2011, MILCOM 2010, DCOSS2010, WiOpt 2009, and its sponsorship chair, APSIPA 2010/2009, IEEE SAM2008.

Haksoo Kim (S’06) received the B.S. and M.S. de-grees in electrical engineering for Korea AdvancedInstitute of Science and Technology (KAIST), Dae-jeon, in 2006 and 2008, respectively.He is currently working toward the Ph.D. degree

in the Department of Electrical Engineering, KAIST.His research interests include statistical signal pro-cessing and communication theory.

Yong H. Lee (S’81–M’84–SM’98) was born inSeoul, Korea, on July 12, 1955. He received theB.S. and M.S. degrees in electrical engineeringfrom Seoul National University, in 1978 and 1980,respectively, and the Ph.D. degree in electricalengineering from the University of Pennsylvania,Philadelphia, in 1984.From 1984 to 1988, he was an Assistant Professor

with the Department of Electrical and ComputerEngineering, State University of New York, Buffalo.Since 1989, he has been with the Department of

Electrical Engineering, Korea Advanced Institute of Science and Technology(KAIST), where he is currently a Professor and the Provost of KAIST. Hisresearch activities are in the area of communication signal processing, whichincludes interference management, resource allocation, synchronization, andestimation for OFDM/MIMO systems with relays.

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