mimoi

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 10, OCTOBER 2010 5219

Limited Feedback for Temporally Correlated MIMOChannels With Other Cell Interference

Salam Akoum, Student Member, IEEE, and Robert W. Heath Jr., Senior Member, IEEE

Abstract—Limited feedback improves link reliability with asmall amount of feedback from the receiver to the transmitter.In cellular systems, the performance of limited feedback will bedegraded in the presence of other cell interference, when the basestations have limited or no coordination. This paper establishesthe degradation in sum rate of users in a cellular system, dueto uncoordinated other cell interference and feedback delay. Agoodput metric is defined as the rate when the bits are successfullyreceived at the mobile station, and used to derive an upper boundon the performance of limited feedback systems with delay. Thispaper shows that the goodput gained from having delayed limitedfeedback decreases doubly exponentially as the delay increases.The analysis is extended to precoded spatial multiplexing systemswhere it is shown that the same upper bound can be used toevaluate the decay in the achievable sum rate. To reduce theeffects of interference, zero forcing interference nulling is appliedat the receiver, where it is shown that the effect of the interferenceon the achievable sum rate can be suppressed by nulling outthe interferer. Numerical results show that the decay rate of thegoodput decreases when the codebook quantization size increasesand when the Doppler shift in the channel decreases.

Index Terms—Cellular systems, feedback delay, goodput,imperfect CSI, limited feedback precoding, multiple-input–mul-tiple-output (MIMO), other-cell interference, rate outage, tempo-rally correlated channels.

I. INTRODUCTION

M ULTIPLE-INPUT–MULTIPLE-OUTPUT (MIMO)communication systems can use limited feedback of

channel state information from the receiver to the transmitterto improve the data rates and link reliability on the downlink[1]–[3]. With limited feedback, channel state information isquantized by choosing a representative element from a code-book known to both the receiver and the transmitter. Quantizedchannel state information is used at the transmitter to designintelligent transmission strategies such as precoded spatial mul-tiplexing and transmit beamforming [3], [4]. Limited feedbackconcepts have been applied to more advanced system config-urations such as MIMO-OFDM and multiuser MIMO and areproposed for current and next generation wireless systems [3].

Manuscript received December 04, 2009; accepted June 09, 2010. Date ofpublication July 01, 2010; date of current version September 15, 2010. The asso-ciate editor coordinating the review of this manuscript and approving it for pub-lication was Dr. Biao Chen. This work was supported by the Semiconductor Re-search Company (SRC) Global Research Consortium (GRC) task ID 1648.001.

The authors are with the Wireless Networking and Communications Group,Department of Electrical and Computer Engineering, The University of Texasat Austin, Austin, TX 78712 USA (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2010.2055860

Most prior work on single user limited feedback MIMO fo-cused on the block fading channel model, where the channelis assumed constant over one block and consecutive channelrealizations are assumed independent. Following this assump-tion, limited feedback was cast as vector quantization prob-lems [5]. Different methods for codebook design have been de-veloped such as line packing [4], [6], [7], and Lloyd’s algo-rithm [8]–[11]. While these approaches are optimal for block-to-block independently fading channels, they do not capture thetemporal correlation inherent in realistic wireless channels [3].Feedback methods that can track the temporal evolution of thechannel and adaptive codebook strategies are proposed to im-prove the quantization [12], [13]. In [12], an adaptive quantiza-tion strategy in which multiple codebooks are used at the trans-mitter and the receiver to adapt to a time varying distributionof the channel is proposed. In [13], a new partial channel stateinformation (CSI) acquisition algorithm that models and tracksthe variations between the dominant subspaces of channels atadjacent time instants is employed. Finite state Markov chain(FSMC) models for fading have been established and widelyused as valid channel models in wireless communication andnetworking [14]–[17]. Markov models to analyze the effect ofthe channel time evolution and consequently, the feedback delayare proposed in [18], [19]. Other temporal correlation modelsand measurement results of the wireless channel are used in[20]–[22] to evaluate the effect of the feedback delay. In [20],the authors quantize the parameters of the channel to be fed backusing adaptive delta modulation, taking into consideration thecomposite delay due to processing and propagation. The au-thors in [21] present measurement results of the performanceof limited feedback beamforming when differential quantiza-tion methods are employed. Measurement results presented in[22] show that the upper bound on the throughput gain obtainedusing a Markov model of the feedback state index, for an indoorwireless LAN setup, is accurate.

Feedback delay exists due to sources such as signal pro-cessing algorithms, propagation and channel access protocols.The effect of feedback delay on the achievable rate and bit errorrate performance of MIMO systems has been investigated inseveral scenarios [19], [23]–[26]. The feedback delay has beenfound to reduce the achievable throughput [23], [24], and tocause interference between spatial data streams [19]. Channelprediction was proposed in [25] and [27] to remedy the effectof the feedback delay. Albeit not in the context of limitedfeedback, the authors use pilot symbol assisted modulationto predict the channel based on Jakes model for temporalcorrelation of the channel. The authors in [18] and [23] derivedexpressions for the feedback bit rate, throughput gain andfeedback compression rate as a function of the feedback delay.

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5220 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 10, OCTOBER 2010

Most of the works on limited feedback MIMO consideredan ergodic metric for the achievable rate. Such a metric mightnot be appropriate to account for the slow-fading, temporallycorrelated channel. For limited feedback systems, the CSI at thetransmitter gets corrupted due to errors on the feedback channelsuch as feedback delay, quantization, as well as noise. Theuncertainty about the actual channel state causes the transmittedpackets to become corrupted whenever the transmitted rateexceeds the instantaneous mutual information of the channel,hence causing an outage [28]–[30]. The effect of packetoutage is accentuated in multicell environments, when the basestations have limited or no coordination. In the presence ofuncoordinated other cell interference, the transmitter modulatesits information at a rate that does not take into account theadded interference at the mobile station, hence increasing theprobability of outage. Thus, an analytical method that takesinto account both delay and other cell interference is importantto quantify the performance of limited feedback MIMO overslow fading channels, in multicell environments.

MIMO cellular systems are interference limited. While mul-ticell MIMO and base station cooperation techniques [31]–[33]can mitigate the effect of interference, when the base stationsshare full or partial channel state and/or data information, theyincur overhead that scales exponentially with the number of basestations. Issues such as complexity of joint processing across allthe base stations, difficulty in acquiring full CSI from all the mo-biles at each base station, and time and phase synchronizationrequirements make full coordination extremely difficult, espe-cially for large networks. Thus while base station coordinationis an attractive long term solution, in the near term, an under-standing of the impact of the interference on limited feedbackis required. Single cell limited feedback MIMO techniques areexpected to loose much of their effectiveness in the presence ofmulticell interference [34]. When each cell designs its channelstate index independently of the other cell interference, a sce-nario where every transmitter-receiver pair is trying to optimizeits own rate occurs, hence decreasing the overall sum rate ofthe limited feedback, when compared to noise limited environ-ments.

In this paper, we derive the impact of delay on the achievablesum rate of limited feedback MIMO systems in the presence ofother cell interference. To account for the packet outage, we usethe notion of goodput. We define the goodput as the number ofbits successfully transmitted to the receiver per unit of time, orin other words, the rate at the transmitter when it does not exceedthe instantaneous mutual information of the channel. We modelthe time variation of the feedback state index as a first-orderfinite state Markov chain. Assuming limited or no feedback co-ordination between adjacent base stations, and using Markovchain convergence theory, we show that the feedback delay, cou-pled with the other cell interference at the mobile station, causesthe spectral efficiency of the system to decay exponentially. Thedecay rate almost doubles when the mobile station is interfer-ence limited.

We evaluate the joint effect of delay and uncoordinated othercell interference on the achievable sum goodput of both lim-ited feedback beamforming and limited feedback precoded spa-tial multiplexing MIMO systems. We derive an upper bound onthe goodput gain for both single stream and multistream limited

feedback systems. We show that the goodput gain decays doublyexponentially with the feedback delay. To mitigate the effect ofother cell interference, while still assuming limited coordinationbetween the base stations, we also consider the net performanceimprovement through the application of zero forcing interfer-ence nulling at the receiver. Assuming one strong interferer andmultiple antennas at the receiver, we use the available degreesof freedom to apply zero forcing (ZF) nulling [35], [36]. We de-rive closed form expressions of the achievable ergodic goodputwith zero forcing nulling, and compare its performance to thatof the noise limited single cell environment.

The effect of delay on the throughput gain of a limited feed-back beamforming system was considered in [18]. The authors,however, did not consider other cell interference nor did theyaccount for the inherent packet outage. In contrast, our papertargets the performance of limited feedback systems in interfer-ence limited scenarios and derives upper bounds of the perfor-mance limits of these systems as the delay on the feedback linkincreases.

The goodput notion we consider borrows from that consid-ered in [28]–[30]. The authors in [28] jointly design the pre-coders, the rate and the quantization codebook to maximize theachievable goodput of the limited feedback system. They con-sider the noise on the feedback channel as the only driver for thepacket outage, and do not account for other cell interference.The authors in [30] use the goodput metric to design sched-uling algorithms to combat the degradation in performance dueto feedback delay in a single cell MIMO system setup. Similarly,the authors in [29] propose a greedy rate adaptation algorithmto maximize the goodput as a function of the feedback delay,using an automatic repeat request (ARQ) system to feedbackchannel state information. Practically, when the channel stateinformation obtained through limited feedback is corrupted, oneapproach is to use fast hybrid automatic repeat request (HARQ)[37], to provide closed loop channel adaptation. HARQ, how-ever, is also subject to delay and errors on the feedback channel.In this paper, we assume in our definition of goodput that theHARQ is not present, and hence packets transmitted at a ratehigher than the instantaneous rate of the channel will be lost.

This paper is organized as follows. In Section II, we de-scribe the limited feedback multicell system considered. InSection III, we present the limited feedback mechanism em-ployed. Section IV-A introduces the system goodput, andSection IV-B describes the channel state Markov chain usedfor the analysis. In Section V, we present the effect of thefeedback delay on the goodput gain. Section VI presents ZFinterference nulling at the receiver to mitigate the effect of othercell interference. Section VII extends the results in Section V toprecoded spatial multiplexing. Section VIII presents numericalresults that show the different aspects of the relation betweenthe feedback rate gain and the feedback delay. This is followedby concluding remarks in Section IX.

Bold lowercase letters are used to denote column vectors,bold uppercase letters are used to denote matrices, non boldletters are used to denote scalar values, and caligraphic letters

are used to denote sets or functions of sets. Using this notion,is the magnitude of a scalar, is the vector 2-norm,

is the conjugate transpose, is the matrix transpose, isthe scalar entry of in the th row and the th column. We use

AKOUM AND HEATH JR.: TEMPORALLY CORRELATED MIMO CHANNELS 5221

Fig. 1. A cellular system with two adjacent cells. Mobile users � and �experience other-cell interference from � and � , respectively.

to denote expectation and to denote the metric a evaluatedat the transmitter.

II. SYSTEM MODEL

We consider the modified Wyner type [38] -cell -userper cell linear array cellular model. The base stations

, with transmit antennas each, serve mobile sta-tions with receive antennas. We index the mobile usersby the same index of the base station they receive their desiredsignal from, for tractability. The users are located at the edge oftheir cells, such that each user is reachable from the two closestbase stations only. Hereafter, we limit the analysis to a two-basestation setup. The base stations have no coordination. Fig. 1 il-lustrates the cellular model for two adjacent interfering base sta-tions.

Each cell employs a limited feedback beamforming system.The system, illustrated in Fig. 2, is discrete time, where con-tinuous time signals are sampled at the symbol rate , with

being the symbol duration. Consequently, each signal is rep-resented by a sequence of samples with denoting the sampleindex. Assuming perfect synchronization between the base sta-tions, matched filtering, and a narrowband channel, the th re-ceived data sample for a single user of interest in basestation can be written as

(1)

where is the received signal vector atis the small-scale fading channel between

and . and are the received powers of the desired andinterfering signal, respectively, at . rep-resents the th realization of the MIMO channel betweenand . is the desired transmit signal vector for , sub-ject to the power constraint . If transmit beam-forming is employed, and hence only one stream is trans-mitted at time , the signal , where isthe unit norm beamforming vector. If, however, precoded spa-tial multiplexing is used, , where in

is unitary is the numberof spatial multiplexing streams transmitted. is the inter-fering transmit signal vector designated for served by basestation , subject to the power constraint

Fig. 2. Base stations communicate with their mobile users through a limitedfeedback MIMO system with delay D on the feedback channel.

is , modeling the additive noise ob-served at .

The random process , (respectively, ), is as-sumed stationary, ergodic, and temporally correlated. The as-sumption that these channels are Gaussian distributed is not nec-essary for our analysis. The feedback of the precoder indices isassumed done at every sample .

We further assume that the base stations have perfect knowl-edge of the signal to noise ratio (SNR) at the receivers. The SNR(or rate) is a scalar value that can be easily quantized. Hence,perfect knowledge of this value at the transmitter is a commonassumption in the limited feedback MIMO literature [39]–[41].This implies that the beamforming vector index or the precodedspatial multiplexing index is fed back over the finite rate limitedfeedback channel, and perfect transmitted rate is available at thebase stations.

Moreover, the desired and interfering channels atand , are independent, since the base stations are geo-graphically separated. They are assumed to be perfectly knownat , thereby ignoring channel estimation error at the receiver.Knowledge of the interference channels at the receiver is usedto derive closed form expressions for the ergodic rates of thesystem.

III. CSI LIMITED FEEDBACK

In this paper, we consider a finite rate feedback link, as de-picted in Fig. 2. The mobile user first estimates the channelstate information sequence using pilot symbols sent bythe base station . Next, the CSI quantizer efficiently quan-tizes the channel sequence1 by means of a Grassmannian code-book, as outlined in [4], [7]. The quantization process dependson whether transmit beamforming or precoded spatial multi-plexing is used, as outlined in the following subsections. Thequantization index is sent to the transmitter via a limited feed-back channel.

1The quantization procedure uses the desired channel � �� CSI only. Theinterference channel at the receiver is not quantized because of the no coordi-nation assumption. This is reasonable for current cellular systems that do notuse base station coordination. Quantization based on whitened interference, forexample, results in a ping-ponging effect between the base stations, where thebase stations use precoders that depend on the delayed interference observed atthe receiver, and may result in a decrease in the achievable goodput.


A. Transmit Beamforming

For the case of transmit beamforming, the beamformingvector has rank one and the base stations senda one-dimensional stream of data to the mobile users. Thecommon criterion for designing a beamforming codebook, forthe desired channel matrix, is to maximize the SNR, which alsooptimizes the information capacity and the symbol error rate.Based on this criterion, the quantizer function at the receivermaps the channel matrix to a beamforming vector inthe codebook such that

(2)

The channel is mapped to the index if the codemaximizes the SNR metric . is then said to

be in the Voronoi cell . The feedback of the index , calledthe feedback state, is sufficient for the transmitter to choose thenecessary beamforming vector from the same codebook . Thefeedback state requires bits, where is the number ofpossible codes in the codebook.

B. Precoded Spatial Multiplexing

For the case of precoded spatial multiplexing, the precoderis a unitary matrix , with rank , where

is the number of spatial multiplexing streams. The unitaryconstraint is useful for limited codebook design, and leads tonear optimal capacity performance when the power constraintis satisfied and is variable [42]. Several different criteria areavailable to choose the optimal precoding matrix from a givencodebook [4]. We choose the criterion that maximizes the in-formation capacity. The quantizer function maps the channelmatrix to the precoding matrix that maximizes the infor-mation capacity expression,

(3)

The receiver then sends the precoding matrix index ,corresponding to the precoder , to the base station.

We assume that the feedback channel is free of error buthas a delay of D samples. The error free assumption is justifi-able as the control channels are usually protected using aggres-sive error correction coding. Given this feedback channel, thechannel state information available at the transmitter lagsbehind the actual channel state at the receiver by D samples.The delay is primarily caused by signal processing algorithmscomplexity, channel access protocols and propagation delay. Afixed delay D is assumed on the feedback channel in all basestations. Since the propagation delay has little contribution tothe total amount of delay and the other sources of delay, causedby processing at the receiver, are similar across users, differentusers in different cells experience the same amount of delay.In what follows, we study the performance of limited feedbackMIMO systems over temporally correlated channels for the caseof transmit beamforming in Section V. We extend the analysisto precoded spatial multiplexing in Section VII.

IV. THE GOODPUT EXPRESSION OVER THE

MARKOV CHANNEL MODEL

The channel state information that reaches the transmitter suf-fers from feedback delay and quantization error. Moreover, theCSI quantizer at the receiver chooses the quantization codewordto maximize the desired SNR, without taking into account theinterference from the neighboring base station. Consequently,the CSI at the transmitter does not contain any information aboutthe other cell interference affecting the mobile user. The basestation, assuming that its received CSI is accurate, modulates thetransmitted signal at a rate corresponding to its erroneous CSI,sometimes resulting in a rate outage or packet outage, when thetransmitted rate exceeds the supported rate of the channel. Inthis paper, we evaluate the effect of CSI mismatch on the achiev-able rate of the limited feedback MIMO system. We assume thatHARQ is not present, and we account for the amount of infor-mation received successfully, without errors, as a function of thefeedback delay. This is similar to the metric used in [28], [43].

A. Conditional System Goodput

To account for the rate outage, we assume that any transmis-sion at a rate higher than the mutual information of the channelfails. In other words, if the transmitted rate , where Dis the feedback delay, exceeds the rate supported by the channel

, the transmission is declared unsuccessful. The instanta-neous system goodput is defined as

(4)

where is the indicator function, which evaluates to 1 ifthe event is true, and 0 otherwise [28]. The ergodic goodput,averaged over the set of the MIMO fading channels,

can be expressed as

(5)

In the sequel, we approximate the fading as a discrete timeMarkov process and use the Markov structural properties to de-rive closed form expressions of the ergodic goodput, as a func-tion of the feedback delay, with and without interference nullingmethods at the receiver.

B. Channel State Markov Chain

We assume that the MIMO channels are temporallycorrelated slow fading channels. Since the feedback state index

is mapped from the channel by the quantization func-tions in (2) and (3), we follow the approach in [44], and wemodel the time variation of the channel state sequenceas a first-order finite state Markov chain. The parameters ofthe Markov chain, namely the state space, the transition andthe stationary probabilities, are related to the channel statisticsand the CSI quantizer codebook. The states areone-to-one mapped with the Voronoi cells of the channel ma-trices defined as

(6)


when transmit beamforming is used, and

(7)

when precoded spatial multiplexing with the mutual informa-tion maximizing criterion is used. The probability of transitionfrom state to state is given by . The sto-chastic matrix is thus , with

(8)

The Markov chain is assumed ergodic with a stationary dis-tribution vector , whereare assumed equal, . This follows because thestationary probability of each Markov state is proportional tothe area of its corresponding Voronoi region, and the Voronoiregions for the codewords in the codebook are assumed to haveequal volume (those of the precoding codebook have equal sur-face area) [44]–[46].

For the two cell system under investigation, the random pro-cesses in different cells are assumed independent.The base stations have limited or no coordination hence the jointprobability mass function of the random processes in the set isgiven by the product of the probability mass functions of the pro-cesses in each cell individually. For the desired base station, thejoint probability mass function between andis given by

(9)

represents to the power of D. The indices are usedto denote the base station and the amount of delay in timesamples. In general, closed form expressions for the transitionprobabilities are difficult to obtain, except for the case ofsingle antennas [14], [44]. The channels corresponding to thetwo different base stations are independent, and hence the in-dividual Markov chain transition probabilities are independent.The parameters of the Markov chains corresponding tocorresponding to base station , and , correspondingto base station , are computed by Monte Carlo simulations asshown in Section VIII. The stochastic matrices correspondingto and , respectively, are assumed identical,equal to . This makes sense when both have the same channeldynamics.

V. THE EFFECT OF DELAY ON THE FEEDBACK GOODPUT GAIN

We consider the effect of fixed feedback delay on the averagesystem goodput of the MIMO interference system. The delayD on the feedback channel for the interfering as well as thedesired cell is considered fixed, caused by signal processing,propagation delay and channel access control.

The rate supported by the channel, assuming an MRC re-ceiver, in the presence of other cell interference, is expressedas

(10)

where the signal to interference noise ratio , assumingMRC combining at the mobile stations, is given by

(11)

where is theMRC combining vector. In general, the SINR distribution doesnot have a closed form expression and has to be estimated usingMonte Carlo simulations.

At the base station, the transmitter modulates its signal basedon the delayed CSI . Assuming continuous rate adaptationand Gaussian transmit signals, the instantaneous transmittedrate at depends on the delayed precoder and itscorresponding channel . Assuming perfect feedbackof the rate to the transmitter, is given by

(12)

We write the average system goodput , based on (11) and(12)

(13)

Following (9), can be obtained using the transition proba-bilities of the Markov chains of the desired and interfering chan-nels.

We distinguish between two types of interference, focused in-terference and non-focused interference. Focused interferenceoccurs when the normalized interference channel falls in thesame Voronoi cell as that of the transmit beamforming vector

, i.e., is chosen to maximize . In this case, the in-terference channel is mapped to a state that corresponds to thebeamforming vector that maximizes in the feedbackindex Markov chain. Otherwise, the interference is consideredto be non-focused. The probability of the interference beingfocused depends on the probability that the interference channelfalls in the same subspace as that of the channel and hence,on the size of the codebook used to quantize the channels.For the case of focused interference, the probability

(14)

Consequently, the ergodic goodput gain is

(15)


where the instantaneous rate is the transmitted rate given. Here, corresponds to the precoder

for the delayed channel . Thus, when, the transmitted rate

is given by

(16)

Similarly, is the instantaneous rate at the receiverwith . We obtain

as .For the case of nonfocused interference, when the codebook

index does not maximize , the joint probability massfunction of the random processes and is given by theproduct of the stationary probability of each random processindividually. Hence, is simply computed by

(17)

Finally, the ergodic goodput is written in terms of the prob-ability of focused interference.

(18)

Given a quantization size of the channel space, the proba-bility of a random channel matrix falling in a Voronoi region

is . Hence, the probability of the interference channelfalling in the Voronoi cell pertaining to the quantization

vector is . In other words, the probability of the interfer-ence being focused is .

To capture the effect of increasing feedback delay on theergodic system goodput, we use the notion of throughput gain,defined in [18] as the throughput with delay D minus thethroughput when the delay goes to infinity. When the delaygoes to infinity, the feedback information becomes obsoleteand thus irrelevant. The goodput gain is formally written as

(19)

where is given by

(20)

This follows from the fact that, as , the channel stateMarkov chain converges to the stationary distribution

(21)

The goodput gain allows us to analyze the effect of increasingfeedback delay on the cellular system, and draw conclusionsas to when closed loop limited feedback MIMO systems arefeasible. We derive an upper bound on the ergodic goodput gain

, based on the Markov chain convergence rate [47], [48].For this reason, we invoke [48, Theorem 2.1], to upperbound

the goodput gain in terms of the properties of the sto-chastic matrix of the channel Markov chain. Theorem 2.1 in [48]

states that for the ergodic channel state Markov chain, the fol-lowing inequality holds:

(22)

where is the second largest eigenvalue of the matrix. The matrix is defined as the time reversal of the sto-

chastic matrix .Proposition 1: For fixed feedback delay of D samples, the

feedback goodput gain can be bounded as

(23)

where , and

Proof: See Appendix.The coefficients and depend on the instantaneous rate

and the stationary probability distribution of theMarkov chain. The coefficient decreases with such that,as the codebook size increases, . This causes therate of decay of the goodput gain to approach that of thenoise limited environment [18]. In that case, the coefficient

.We make the following observations about the conclusions in

Proposition 1.1) The feedback gain decreases at least exponentially with the

feedback delay. The decreasing rate is or , dependingon the values of the coefficients and . The rate is thusdetermined by the channel coherence time and the size ofthe codebooks used.

2) The eigenvalue is a key parameter in characterizing thebehavior of the system. A larger value of indicates longerchannel coherence time and larger codebook size and viceversa.

3) The coefficients and depend on the number of precodersused to quantize the channels spaces. Both coefficients areobtained taking into account the transition probabilities forthe channel state Markov chains of channels and

. The coefficient is the prefactor of , and islargely dependent on the inverse of the size of the codebook

. It determines the effect of other cell interference in thesense that when goes to infinity, the doubly exponentialterm vanishes, and the performance of the system is thesame as that of the noise limited environment.

4) Simulation results in Section VIII show that the upperbound, derived in Proposition 1 is tight for several casesof interest.

VI. INTERFERENCE NULLING AT THE RECEIVER

The results in Section V suggest that the other cell inter-ference almost doubles the exponential rate of decrease of thegoodput gain, when compared with the rate of decrease com-puted in [18]. We propose using interference nulling techniquesat the receiver, to compensate for the rate decay. Interferencenulling techniques are suitable when the base stations have lim-ited coordination. We require the receiver to learn both its de-


sired effective channel and the effective interference channelfrom the other cell base station. We implement zero forcing (ZF)interference nulling. The performance of the zero forcing re-ceiver approaches that of the minimum mean square linear re-ceiver at high signal to noise ratio. It allows us to obtain analyt-ically tractable expressions of the distribution of the post pro-cessing signal to interference noise ratio, by applying projectionto do interference nulling.

Assuming the number of receive antennas , we rewritethe received signal at as

(24)

where is the effective desired channel atfrom . is the effective interference from .The zero forcing linear receiver effectively projects the desiredsignal onto the subspace orthogonal to the subspace ofthe interference . Hence, the resulting signal power at thereceiver is

(25)

where is the angle between and. can be viewed as a projection power loss factor

for the zero forcing receiver. And the corresponding instanta-neous rate at the receiver is expressed as

(26)

where is the signal to noise ratio at the re-ceiver. Thus the instantaneous system goodput is

(27)

The ergodic goodput follows, similar to (13),

(28)

The key to computing is to evaluate the cumulative densityfunction (cdf) of . This will allow us to express theprobability of outage as a function of the channel gains only. Wepropose the following lemma on the distribution of .

Lemma 1: The squared norm of the sine of the angle betweenthe effective desired and interference channel, in a limited feed-back beamforming system, when the channels andhave i.i.d. complex Gaussian entries with zero mean and unit

variance, can be approximated by a beta distribution with pa-rameters and 1, on the interval

(29)

where is the betafunction, defined in termed of the Gamma function , and theeffective channels are assumed Gaussian distributed.

Proof: The effective channel vectors and aretwo independent random vectors whose entries, and

can be approximated as independentlyand identically distributed complex Gaussian random variableswith mean 0 and variance 1. Consequently, is the anglebetween two independent random vectors, and isbeta distributed with parameters and 1.

When the number of receive antennasis uniformly distributed on .

Thus, a closed form expression of the complementary proba-bility of outage can be written as

(30)

Consequently, the ergodic system goodput for a delay D isgiven by

(31)

Finally, the ZF feedback goodput gain is readily written, fol-lowing (31) and (22).

Proposition 2: For fixed feedback delay of D samples, thefeedback throughput gain with ZF interference nulling of thestrongest interferer can be bounded as

(32)

where

(33)

and is the second largest eigenvalue of the matrix .The following remarks are in order.


1) Employing ZF interference nulling at the receiver bringsback the exponential rate of decrease of the ergodicgoodput to , similar to the case reported in [18], wherethe system is noise limited. The authors in [18], how-ever, derive expressions for the ergodic throughput ofthe system, without taking into account the rate outage.We reformulate the result in [18] in terms of the goodputmetric as follows:

(34)

where , and

and .2) The coefficient c depends on the effective channels at the

transmitter and the receiver, and it is smaller than the coef-ficient in the upperbound of (34). This makes sense be-cause the projection of the effective channel at the receiveronto the subspace perpendicular to that spanned by the in-terference vectors, causing an effective power loss in theSNR at the receiver.

3) In Section VIII, we run simulations to show the tightnessof the derived upper bounds. We compare the performanceof the system with and without interference nulling, withthat of the noise limited environment.

VII. PRECODED SPATIAL MULTIPLEXING

Spatial multiplexing can offer higher data rates by sendingmultiple data streams to the receiver. Limited feedbackprecoded spatial multiplexing is included in the emerging3GPP-LTE standard. The performance of this MIMO system,however, in the presence of feedback delay is yet to be analyzed.In this section, we extend the goodput analysis in Section Vto precoded spatial multiplexing systems, and we evaluate itsperformance limits in the presence of feedback delay and othercell interference.

For a precoding matrix with transmit streams, theachievable rate at is computed as

(35)

wheredenotes the covariance matrix of the desired signal from

is the covariance matrix of the interference from plus thenoise.

The rate at the base station is computed without taking intoaccount the presence of the noise and the interference at themobile, as in Section V

(36)

The goodput then follows as

(37)

The ergodic goodput is computed in terms of the joint prob-ability mass functions of the quantized channels . Thequantization of the channel state information according tothe capacity maximization selection criterion, as explainedin Section III, maps the channels into a first-order Markovchain model, where the different states correspond to differentVoronoi regions. Thus, the same analysis can be applied tocomputing the ergodic goodput gain for the precoded spatialmultiplexing system.

Define as the instantaneous goodput at time with, and

. The ergodic goodput gain can thus beobtained using (22) and Proposition 1.

Proposition 3: For fixed feedback delay of D samples, theergodic goodput gain is upper bounded as

(38)

where , and

The coefficients and depend on the instantaneous rate, the stationary probability distribution of the Markov

chain, as well as the codebook size . The decay rate dependson the amount of delay, the strength of the interference fromneighboring base stations, and the number of quantizationlevels. In Section VIII, we present numerical results to showthe performance of the precoded spatial multiplexing systemwith respect to delay.

VIII. SIMULATION RESULTS

In this section, we present simulation results to evaluate theperformance of limited feedback MIMO systems with delay inthe presence of other cell interference, for several scenarios.

For our simulations, we assume that the scattering environ-ment is uniform such that the channel coefficients areand the temporal correlation follows Clarke’s model and ischaracterized by the continuous zeroth-order Bessel function

, where is the maximum Doppler fre-quency and is the time separation between the samples. Thediscrete time counterpart of the continuous time autocorrelationfunction is implemented using the inverse Discrete FourierTransform method proposed in [49], such that the discrete timesamples have autocorrelation , whereis the sample separation, and is the sampling rate, and boththe continuous and the discrete correlation functions have thesame power spectrum.

To validate the Markov model quantization, we first comparethe ergodic goodput obtained without discretization to that ob-tained with discretization and averaging based on the Markovmodel stationary and transition probabilities. Fig. 3 shows theergodic goodput versus the feedback delay, for an inter-ference limited four transmit, four receive MIMO system, and acodebook size of 16. We observe from the figure that the closed


Fig. 3. The ergodic goodput versus feedback delay. The figure shows boththe nondiscretized actual ergodic goodput and that obtained using the Markovmodel for a 4� 4 MIMO system with a code size of 16.

Fig. 4. Effect of feedback delay on the ergodic sum rate gain. Comparison ofan interference limited system versus a noise limited system for a 4� 4 MIMOsystem and a codebook size of 16. The normalized Doppler shift is � � ��. The subscript SC denotes single cell, while MC denotes multicell.

form ergodic goodput obtained in (18) closely matches the ac-tual ergodic goodput, computed numerically. The slight lossfrom quantization is due to the assumptions made in deriving(18).

A. Limited Feedback Beamforming With Other CellInterference and Feedback Delay

We compare the feedback rate gain of the interference lim-ited system with that of the noise limited system, for the lim-ited feedback beamforming presented in Section V. Fig. 4 plotsboth the throughput and goodput gains of the systems versus thefeedback delay D. For the throughput gain, we use the achiev-able rate at the receiver given by

(39)

i.e., without including the rate outage. For the noise limited en-vironment, the result for the system throughput goes back to thatin [18] where the exponential rate of decrease is .

Fig. 5. The normalized feedback goodput gain and its upperbound versus feed-back delay for a Doppler shift � � � ��, a 4� 4 MIMO system with a codesize of 16. The normalized feedback goodput gain for the single cell scenario isgiven by �� .

Fig. 4 shows the effect of the feedback delay on both thethroughput gain , as well as the goodput , for afour transmit, four receive antenna system with a normalizedDoppler shift of . We observe that including therate outage at the transmitter causes a decrease in the throughputgain of the system. The decrease is constant with respect to thedelay. This can be explained by the fact that the outage does notaffect the Markov chain statistical properties, which mainly de-pend on the value of the delay. The gap between the goodputand the throughput gain can be observed for both the noise lim-ited and interference limited scenarios.

Moreover, the results in Fig. 4 show that the exponential rateof decrease of the interference limited system (one cell interfer-ence) is more pronounced than that of the noise limited system(single-cell) for the cases of small to moderate delay (up to

). The simulation results agree with the analytical resultin Proposition 1. We also observe from Fig. 4 that the rate ofdecrease in the two-cell environment increases with the delay.For , the term in the right hand side of (40) becomesmore dominant as D increases.

To evaluate the accuracy of the upper bound derived in Propo-sition 1 in predicting the performance of the system, Fig. 5 plotsthe normalized feedback rate gain and the expression in (40),as a function of feedback delay D, for a normalized Dopplershift, . As can be observed from the figure, thethroughput gain is tightly upper bounded by (40), for delay in-tervals of interest. This upper bound is different from that inthe noise limited scenario, plotted in the figure for comparison.This allows the use of the results of Proposition 1 for computinggains in achievable rate for delay values of interest.

B. ZF Nulling at the Receiver

The ZF linear receiver was proposed in Section VI to reducethe effect of other cell interference. Fig. 6(a) presents the ef-fect of employing the ZF receiver architecture on the feedbackthroughput gain. We plot the ergodic goodput gain achieved bythe noise limited (single-cell) MIMO transceiver, with that ofthe interference limited system, with and without interferencenulling. We observe that adding interference nulling brings thedecay rate of the feedback throughput gain back to , simi-larly to the noise limited environment. The effective power loss


Fig. 6. Performance of ZF interference nulling for a 4� 4 MIMO system with acodebook size of 16. The normalized Doppler frequency � � � ��. (a) Thefeedback goodput gain versus feedback delay for the noise limited, interferencelimited and interference nulling systems. (b) The ergodic goodput versus feed-back delay for the ZF interference nulling receiver and the MRC receiver.

due to the projection of the desired MIMO channel onto the in-terference subspace, causes the rate to be less than that of thenoise-limited environment.

Fig. 6(b) compares the ergodic goodput achieved usingZF interference nulling to that obtained in (18). The figureshows that the ergodic goodput achieved using a ZF receiveris higher than that achieved using a linear MRC receiver. Thisfollows because the MRC receiver maximizes the signal tonoise ratio, without taking the interference into account, whilethe ZF receiver suppresses the interference.

Fig. 7 compares the normalized feedback goodput gain, and its approximation with the closed form upper-

bound in Proposition 2, for different values of the feedbackdelay D. The exact and closed form expression for theare shown in solid and dashed lines, respectively. The approx-imate solution simulated based on the probability ofoutage is also shown in the figure. One can observe that theclosed form expression closely matches the exact .

Fig. 7. The feedback goodput gain and its approximations versus feedbackdelay for the ZF interference cancellation receiver for a 4� 4 MIMO systemwith a code size of 16.

C. Precoded Spatial Multiplexing With Feedback Delay andOther Cell Interference

We investigate the performance of the precoded spatial mul-tiplexing system in the presence of other cell interference inFig. 8(a). We consider a four transmit, four receive antennaMIMO system, with two spatial multiplexing streamsand a codebook size of 16. Fig. 8(a) plots the throughput gainfor the precoded multistream system for both the interferencelimited and the noise limited environments, for a Grassman-nian codebook of size 16.2 We observe from the figure that thegoodput gain of the precoded multistream decreases exponen-tially with the feedback delay. This exponential decrease is morepronounced in the presence of other cell interference, as pre-dicted in Proposition 3. The figure also shows the upperboundobtained in (3) on the precoded spatial multiplexing system. Theupper bound predicts the rate of decay of the goodput gain withthe feedback delay.

Fig. 8(b) plots the throughput gain of the precoded spatialmultiplexing system and that of transmit beamforming, for thesame codebook. The behavior of both systems vis-a-vis othercell interference and delay is the same. However, the throughputgain achieved by the precoded system is higher than that oftransmit beamforming, due to the multiple transmit streams.

D. Effect of Doppler Frequency and Codebook Size

The performance of the two-cell system in the presence ofdelay with varying codebook sizes is shown in Fig. 9(a) for a twotransmit, two receive MIMO system and codebook sizes 4, 8,and 16. One can clearly see that the rate of decay of the goodputgain increases as the codebook size increases, it approaches thatof the single cell system for larger codebook sizes. This suggestsa tradeoff between the feedback rate and the goodput gain ininterference limited scenarios.

In Fig. 9(b), we plot the feedback goodput gain versus thefeedback delay for different Doppler frequencies normalized bythe sampling period

. We consider a four transmit, four receiveMIMO system with a Grassmannian codebook size of 16.

2The codebooks used in these simulations are available at http://cobweb.ecn.purdue.edu/~djlove/grass.html


Fig. 8. Performance of precoded spatial multiplexing in an interference limitedenvironment for a 4� 4 MIMO system with � � �, codebook size of 16 andnormalized Doppler frequency � � � ��. (a) Effect of feedback delayon the ergodic sum goodput gain for a precoded spatial multiplexing system.(b) Comparison of the effect of feedback delay on the ergodic goodput gain fora precoded spatial multiplexing system and a transmit beamforming system.

Clearly, the exponential rate of decrease of the feedback gainis sensitive to the Doppler shift. As the Doppler shift, or thevelocity of the mobile is increased, the feedback gain is de-creased, resulting in a steeper decrease rate with respect to thefeedback delay. This can be explained as follows. Over a fixeddelay, higher Doppler causes larger channel variation, whichaccelerates the decrease of the feedback capacity gain with thefeedback delay and hence results in a steeper curve slope.

E. 3GPP-LTE Design Example

In this subsection, we present a design example thatdemonstrates the application of the results in this paper todesigning limited feedback system for 3GPP LTE-Advancedstandard [50]. The standard proposes using limited feedbackbeamforming and precoded spatial multiplexing over orthog-onal frequency division multiple access (OFDMA). For thedownlink OFDMA, the available frequency bandwidth is par-titioned into frequency slots, called frequency subbands, andassigned to different users based on a given scheduling algo-rithm. Each subband consists of several orthogonal frequencydivision multiplexing (OFDM) symbols. Limited feedback isperformed on every OFDM symbol to increase its data rate.

Fig. 9. Effect of feedback delay on the feedback goodput gain of the beam-forming system with limited feedback. (a) With varying codebook sizes, for a2� 2 MIMO system with and a normalized Doppler shift of � � � ��.(b) With varying Doppler shifts, for a 4� 4 MIMO system with a codebooksize of 16.

Channel estimation is performed using pilot symbols located atthe center of the subbands. One CSI feedback link is requiredfor every subband, and feedback is performed every two sub-frames.

The design specifications are summarized in Table I. Thetransmission bandwidth is 10 MHz at a carrier frequency of2 GHz. Downlink transmission is organized into radio frameswith a frame duration of 10 ms. We consider the frame struc-ture applicable to frequency division duplex (FDD), in which10 subframes, of duration 1 ms each, are available for down-link transmission [50]. At each scheduled user equipment (UE),the channel state information is fed back every 2 subframes,or 2 ms. The design of the precoder for 4 transmit antennas isbased on the Householder transformation [51]. The choice ofthe codebook is based on its reduced computational complexity.The Householder precoding codebook satisfies a constant mod-ulus where all the transmit antennas keep the same power level,regardless of which precoding matrix is used to maximize thepower amplifier efficiency. Moreover, the nested property of thecodewords permits the precoding matrix in a lower rank (such aswith transmit beamforming) to be a submatrix of a higher rankprecoding matrix (when sending multiple streams). The code-book also features a constrained alphabet that avoids the needfor matrix multiplication [52].


TABLE ILTE DESIGN SPECIFICATIONS

The size of the codebook used for quantizing feedback CSI is16. The mobility of the UE is up to 30 km/h. The Doppler shiftfor the maximum speed of 30 km/h is of Hz. Normal-izing this shift by the subframe time of 1 ms, we get a normalizedDoppler shift of . The control delay is of 4 ms.The roundtrip ARQ delay is of 4 ms. Based on these require-ments, we compute the goodput gain of the limited feedbackprecoding systems, using the LTE Householder codebooks,3 forrealistic delay values of 4 and 6 ms.

The second largest eigenvalue used in Proposition 1 is firstcomputed by using the stochastic matrix of the feedback statemarkov chain, based on the codebook size and the normalizedDoppler shift to be . The maximum feedbackgoodput gain for delay free CSI feedback is computed as 2.453bps/Hz. By using Proposition 1 for the LTE codebook, with

stream, the normalized goodput gain is computed at(4, 6) ms, respectively, as (0.4708, 0.2904), this implies thatthe goodput gain at these values is (1.1549, 0.7124) bps persubcarrier, and (83.1528, 51.2928) bps per subband.

IX. CONCLUSION

In this paper, we analyzed the effect of the feedback delay onlimited feedback systems in the presence of uncoordinated othercell interference. We showed that the feedback delay causes thebase station to transmit at a rate higher than the instantaneousmutual information of the channel, resulting in a rate outage. Byonly considering the successful transmissions over the channel,or the goodput, we showed that the other cell interference causesthe exponential rate of decay of the ergodic feedback gain todouble, when compared to the noise-limited single cell sce-nario, especially at low to moderate feedback delay values. Theanalysis was carried out for both transmit beamforming andprecoded spatial multiplexing systems. Numerical results con-firmed that a smaller time correlation and a lower codebook sizeleads to a faster reduction of the goodput gain with the feedbackdelay.

We then implemented interference nulling, assuming thatthe receiver has perfect knowledge of the interference channel,while the base station is oblivious to the presence of any in-terference. We used the zero forcing receiver architecture, andwe showed that the effect of the other cell interference can besimply mitigated, by bringing the decay rate of the throughputgain back to that of the noise limited case, at the expense of arate loss due to the projection power loss of the ZF receiver.

This paper opens up several issues for future investigation.First, the results in this work focus on the feedback delay asthe major bottleneck in the performance, these results can be

3The codebooks used in these simulations are available at http://users.ece.utexas.edu/~inoue/codebook/index.html

extended to include other nuisance sources on the channel,such as errors in estimating the CSI at the receiver, and noiseon the feedback channel. Future work should also include ex-tending the results to multiuser MIMO scenarios and multipleaccess (MAC) channels. More importantly, this paper sets thegrounds for implementing a joint rate adaptation and codebookdesign technique that takes the rate outage caused by the othercell interference as the major constraint, and optimizes thesystem goodput accordingly.

APPENDIX APROOF OF PROPOSITION (1)

Proposition 1: For fixed feedback delay of D samples, thefeedback goodput gain can be bounded as

(40)

where , and

Proof: We define

and we expand it in terms of as follows:

(41)

Consequently, is written as

(42)


Using (22), can be upperbounded as follows:

where .Similarly upper bounding the other terms yields expressions

in

(43)

where , and

with .Finally, can be upperbounded by

(44)

where , and

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Salam Akoum (S’06) received the B.E. degree incomputer and communications engineering and aminor in mathematics (Honors) from the AmericanUniversity of Beirut, Lebanon, in 2006. She receivedthe M.S.E.E. degree from the University of Utah,Salt Lake City, in 2008.

She is a Ph.D. student at the University of Texasat Austin. She joined the Wireless Networking andCommunication Group (WNCG) in fall 2008, towork under the supervision of Prof. Robert W.Heath, Jr. in the Wireless Systems and Innovation

Lab (WSIL) group. Her current research interests include management ofuncoordinated interference in wireless mobile networks, heterogeneous cellularsystems, and various aspects of MIMO communication.

Robert W. Heath, Jr. (S’96-M’01-SM’06) receivedthe B.S. and M.S. degrees from the University of Vir-ginia, Charlottesville, in 1996 and 1997, respectively,and the Ph.D. degree from Stanford University, Stan-ford, CA, in 2002, all in electrical engineering.

From 1998 to 2001, he was a Senior Member ofthe Technical Staff then a Senior Consultant withIospan Wireless Inc., San Jose, CA, where he workedon the design and implementation of the physical andlink layers of the first commercial MIMO-OFDMcommunication system. In 2003, he founded MIMO

Wireless Inc., a consulting company dedicated to the advancement of MIMOtechnology. Since January 2002, he has been with the Department of Electricaland Computer Engineering, The University of Texas at Austin, where heis currently an Associate Professor and Associate Director of the WirelessNetworking and Communications Group. His research interests include severalaspects of MIMO communication: limited feedback techniques, multihopnetworking, multiuser MIMO, antenna design, and scheduling algorithms, aswell as 60-GHz communication techniques and multimedia signal processing.

Dr. Heath has been an Editor for the IEEE TRANSACTIONS ON

COMMUNICATIONS and an Associate Editor for the IEEE TRANSACTIONS

ON VEHICULAR TECHNOLOGY. He is a member of the Signal Processing forCommunications Technical Committee in the IEEE Signal Processing Societyand is the Vice Chair of the IEEE COMSOC Communications TechnicalTheory Workshop, is a general organizer for the 2009 CAMSAP Conference,and was a Technical Co-Chair for the 2010 IEEE International Symposiumon Information Theory. He is the recipient of the David and Doris LybargerEndowed Faculty Fellowship in Engineering. He is a licensed Amateur RadioOperator and is a registered Professional Engineer in Texas.

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