Exploiting quantized channel norm feedback through conditional statistics in arbitrarily correlated MIMO systems

Exploiting Quantized Channel NormFeedback Through Conditional Statisticsin Arbitrarily Correlated MIMO Systems

IEEE TRANSACTIONS ON SIGNAL PROCESSINGVolume 57, Issue 10, Pages 4027-4041, October 2009.

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EMIL BJORNSON, DAVID HAMMARWALL,AND BJORN OTTERSTEN

Stockholm 2009

ACCESS Linnaeus CenterSignal Processing Lab

Royal Institute of Technology (KTH)

DOI: 10.1109/TSP.2009.2024266KTH Report: IR-EE-SB 2009:010

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009 4027

Exploiting Quantized Channel Norm FeedbackThrough Conditional Statistics in Arbitrarily

Correlated MIMO SystemsEmil Björnson, Student Member, IEEE, David Hammarwall, Member, IEEE, and Björn Ottersten, Fellow, IEEE

Abstract—In the design of narrowband multi-antenna systems,a limiting factor is the amount of channel state information (CSI)available at the transmitter. This is especially evident in multi-usersystems, where the spatial user separability determines the multi-plexing gain, but it is also important for transmission-rate adap-tation in single-user systems. To limit the feedback load, the un-known and multi-dimensional channel needs to be represented bya limited number of bits. When combined with long-term channelstatistics, the norm of the channel matrix has been shown to pro-vide substantial CSI that permits efficient user selection, linearprecoder design, and rate adaptation. Herein, we consider quan-tized feedback of the squared Frobenius norm in a Rayleigh fadingenvironment with arbitrary spatial correlation. The conditionalchannel statistics are characterized and their moments are derivedfor both identical, distinct, and sets of repeated eigenvalues. Theseresults are applied for minimum mean square error (MMSE) esti-mation of signal and interference powers in single- and multi-usersystems, for the purpose of reliable rate adaptation and resourceallocation. The problem of efficient feedback quantization is dis-cussed and an entropy-maximizing framework is developed wherethe post-user-selection distribution can be taken into account in thedesign of the quantization levels. The analytic results of this paperare directly applicable in many widely used communication tech-niques, such as space-time block codes, linear precoding, space di-vision multiple access (SDMA), and scheduling.

Index Terms—Channel gain feedback, estimation, MIMO sys-tems, norm-conditional statistics, quantization, Rayleigh fading,space division multiple access (SDMA).

I. INTRODUCTION

W IRELESS communication systems with antenna arraysat both the transmitter and receiver have the ability of

greatly improving the capacity over single-antenna systems.

Manuscript received August 16, 2008; accepted May 03, 2009. First pub-lished June 02, 2009; current version published September 16, 2009. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Zhi Tian. Parts of this work is supported in part by the FP6project Cooperative and Opportunistic Communications in Wireless Networks(COOPCOM), Project Number: FP6-033533. This work was previously pre-sented at the IEEE International Conference on Acoustics, Speech and SignalProcessing (ICASSP), Las Vegas, NV, March 30–April 4, 2008 and the IEEEInternational Symposium on Personal, Indoor and Mobile Radio Communica-tions (PIMRC), Cannes, France, September 15–18, 2008.

E. Björnson is with the Signal Processing Laboratory, ACCESS LinnaeusCenter, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden(e-mail: [email protected]).

D. Hammarwall is with Ericsson Research, SE-164 80 Stockholm, Sweden(e-mail: [email protected]).

B. Ottersten is with the Signal Processing Laboratory, ACCESS LinnaeusCenter, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden,and also with the securityandtrust.lu, University of Luxembourg, L-1359 Lux-embourg-Kirchberg, Luxembourg (e-mail: [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.2009.2024266

The potential gains have been shown for narrowband chan-nels in [1] and [2], under the assumption of independent andidentically distributed zero-mean complex Gaussian channelcoefficients between the transmit and receive antennas. Suchchannels are often referred to as uncorrelated Rayleigh fading,since there is no correlation in the spatial dimension and theenvelope of the received signal is Rayleigh distributed. Froma mathematical point of view, uncorrelated Rayleigh fadingchannels occur naturally when the antenna separation is largeand the scattering in the propagation channel is sufficiently rich.However, it has been shown experimentally that the channelcoefficients are often spatially correlated in outdoor scenarios[3], and correlation frequently occurs in indoor environmentsas well [4], [5]. This motivates the analysis of the more generalcase of Rayleigh fading where the channel coefficients arearbitrarily correlated.

Channel variations are normally characterized by small-scaleand large-scale fading [6]. The former describes changes in thesignal paths of the order of the carrier wavelength and is time-and frequency-dependent. To avoid the frequency dependencywe consider narrowband block-fading channels; that is, thechannel matrix is constant for a block of symbols and thenupdated independently from the assumed Gaussian distributionfor the next block. The large-scale fading corresponds to varia-tions in the channel statistics due to effects like shadowing bybuildings and power decay due to propagation distance. Theseeffects are typically frequency independent and slowly varyingin time. Hence, the transmitter and receiver can keep track onthe statistics by reverse-link estimation or a negligible feedbackoverhead.

In single-user multiple-input multiple-output (MIMO) sys-tems, the small-scale fading can be mitigated with using orthog-onal space-time block codes (OSTBCs) [7]–[9]. Using only sta-tistical channel state information (CSI) at the transmitter, the ca-pacity can be unexpectedly good if linear precoding takes careof the spatial correlation [9]–[12]. In practice, a small amount ofchannel gain feedback is however necessary for rate adaptationto achieve this performance. In multi-user MIMO systems thesituation is somewhat different, because the multi-user diversitygain depends on the amount of instantaneous CSI available atthe transmitter [13], [14]. This CSI can be exploited to scheduleusers for transmission on time-frequency slots and spatial direc-tions in which they experience particularly strong gains. Unfor-tunately, the amount of feedback needed to achieve full CSI isprohibitive in many realistic scenarios. Therefore, the design of

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4028 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009

limited feedback systems that capture most of the performancehas been an active research topic.

Many multi-user limited feedback systems are based on linearprecoding. Although this approach is only asymptotically op-timal in the number of users [15], the loss in performance comeswith a substantial decrease in complexity compared with non-linear precoding (e.g., optimal dirty-paper coding [16]). Oneapproach to linear precoding in space division multiple access(SDMA) is to allocate users to a set of beams based on feedbackof their achieved channel gains. These beams can either be gen-erated randomly [14] or belong to a fixed grid of beams [17].Another approach is to design and adapt the precoder matrix tostatistical user information and feedback of instantaneous CSI.This can be implemented in a zero-forcing fashion [18]–[20],where the co-user interference is made zero (for full CSI) orstatistically small and manageable (for partial CSI). Althoughthis strong zero-forcing condition is suboptimal, it provides asimple design structure and can achieve close-to-optimal per-formance if the amount of feedback is correctly scaled with thesignal-to-interference-and-noise ratio (SINR) [18]. In general,the type of approach that is most favorable depends on varioussystem parameters, such that coherence time, number of users,spatial correlation, and average SINR.

Feedback of quantized gain information plays an importantrole in the design of both user-selection algorithms and linearprecoders. In [21], channel norm based user-selection wasshown to provide close-to-optimal performance asymptoti-cally in the number of transmit antennas. When consideringzero-forcing precoding and limited feedback, it was proposedin [18] that each user should feed back its normalized channelvector using a codebook and calculate a regular zero-forcingprecoder. Additional feedback of the instantaneous channelnorm is however required to estimate the SINR and performreliable rate adaptation [22]. In spatially correlated systems,the long-term statistics provide directional information andfeedback of the channel norm is sufficient to perform efficientstatistical zero-forcing [19] and estimate the instantaneousSINR that is used for rate adaptation [23]. In neither of thesepapers, channel gain quantization or multi-antenna receivers areconsidered. With multiple antennas at both sides, more degreesof freedom are available in the interference cancellation, butthe precoder and receiver combining design problem becomesconsiderably more difficult. Some of these problems wereaddressed in [20].

Herein, we analyze the impact of channel gain information onRayleigh fading MIMO systems with arbitrary spatial correla-tion. The conditional statistics and minimum mean square error(MMSE) framework derived in [23] for correlated systems withsingle-antenna users are generalized to cover general fading en-vironments, multi-antenna users, and quantized gain informa-tion. The contributions to communication are an entropy-maxi-mizing quantization framework that can be applied to gain feed-back and the derivations of closed-form estimators of the instan-taneous SINR in single- and multi-user systems, using such gainfeedback. These results can be applied to handle gain feedback

and rate adaptation in system both with and without additionalfeedback of directional channel information.

Notations

For notational convenience we use boldface (lower case)for column vectors, , and (upper case) for matrices, . With

, , and we denote the transpose, the conjugatetranspose, and the conjugate of , respectively. The Kroneckerproduct of two matrices and is denoted ,is the column vector obtained by stacking the columns of ,and is the -by- diagonal matrix with

at the main diagonal. If the th element of a matrixis , then . The distribution of circularly

symmetric complex Gaussian vectors is denoted ,with mean value and covariance matrix .

The notation is used for definitions. The squared 2-normof a vector is denoted and the squared Frobenius normof a matrix is denoted , and both are defined as the sumof the squared absolute values of all the elements. The sum ofabsolute values of all the elements in is denoted . If is aset, then the set members are denoted , where

is the cardinality of .Let . The generalized Heaviside step func-

tion is 1 if for all and , and0 otherwise. The function is 1 if , for all ,and , and 0 otherwise. Finally, denotesDirac’s delta function.

A. System Model

Consider the downlink of a communication system witha single base station equipped with an array of antennasand several mobile users, each with an array of antennas.The symbol-sampled complex baseband equivalent of thenarrowband flat-fading channel to user is represented by

. The elements of are modeled as Rayleighfading with arbitrary correlation, and thus we assume that

. The received vector ofuser at symbol slot is modeled as

(1)

where the vector of transmitted signals is denotedand the power of the system is normalized such that

is white noise with elements that are dis-tributed as .

The system model in (1) depends on three different timescales. The variations in the matrix are modeled byquasi-static block-fading; that is, the channel realization is con-stant for a block of symbols and then modeled as independentin the next block. Within a block, only the noise and thetransmitted signal are changing. The statistics change veryslowly, measured in the number of blocks, and it is thereforeassumed that the current correlation matrix is known toboth the base station and user .

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BJÖRNSON et al.: EXPLOITING QUANTIZED CHANNEL NORM FEEDBACK 4029

B. Feedback-Based Estimation of Weighted Channel Norms

To achieve reliable rate estimation and exploit the spatial andmulti-user diversity, the transmitter often needs more informa-tion than just the channel statistics. Such partial and instanta-neous CSI can be estimated at the receiver side and then fed backto the transmitter [24]. When the channel conditions changerapidly with time, the number of feedback symbols spent onachieving partial CSI not only reduces the time the informa-tion can be used at the transmitter before it is outdated but alsothe number of symbols available for data transmission on thereverse link. Hence, the feedback needs to represent some lim-ited amount of information that can be described efficiently bya small number of bits.

In a block-fading environment, the feedback system can inprinciple be described as a cyclical system that estimates andfeeds back partial CSI in the beginning of each block to im-prove the system performance during the rest of the block. Theresults herein are however not limited to this type of fading. Forsimplicity, we assume that there exists an error-free feedbackchannel from each mobile user to the base station.

The instantaneous CSI can be divided into directional in-formation and gain information, herein the latter will be con-sidered. Throughout this paper, we consider the estimation ofweighted squared Frobenius norms of the channel at the trans-mitter [20], [23], where the weights are known at the transmitterbut not necessarily at the receiver. On the contrary, the channel isonly perfectly known to the receiver and any instantaneous CSIexploited at the transmitter must be conveyed over the limitedfeedback link. The generic estimation problem that we focus onis

Estimate

given or a quantized version (2)

In this formulation, we have the weighting matrixand the effective channel , where

and are matrices known tothe receiver. In the area of communication, two interestingfeedback and estimation scenarios can be formulated in termsof the generic problem.

1) The receive combiner matrix and precoder matrix areknown to the receiver and are used as and , re-spectively. The squared norm of the effective channel

is fed back to the transmitter. This infor-mation is used to estimate the weighted squared norm

, which is either the total channel gain( ) or the gain in a certain spatial subspace.

2) Either the receive combiner matrix, the precoder matrix,or both matrices are unknown to the receiver at the timeof feedback. In these cases, the effective channel becomes

, , or , respectively, andthe squared norm is fed back. This information isused to estimate the weighted squared norm ,where may represent receive combiner and/or precodermatrices that are known to the transmitter.

The results of this paper are independent of the quantization,but a quantization framework is proposed in Section III andadapted to multi-user systems in Section IV-B.

C. Outline

In Section II, we analyze the special case of feedback ofwith a diagonal correlation matrix . Closed-form ex-

pressions of the conditional moments of the elements in arederived for both exact norm feedback and a quantized norm. Ashort overview of the applications of these results in renewaltheory is provided. In Section III, the results are generalized forcommunication purposes. A general entropy-maximizing quan-tization framework is presented and the results of Section IIare used to characterize the distribution of the effective squaredchannel norm and to derive an MMSE estimator of weightedsquared norms, given quantized norm information. Section IVshows how these results are applicable on MMSE estimationof signal/interference powers and rate adaptation in single- andmulti-user systems. Some of the results are illustrated numeri-cally in Section V and conclusions are drawn in Section VI.

II. ANALYSIS OF ZERO-MEAN COMPLEX GAUSSIAN VECTORS

WITH NORM INFORMATION

In this section, we consider an -dimensional vector, for , with zero-mean and indepen-

dent complex Gaussian entries—that is, . First,the distribution of the squared norm will be pre-sented. Then, expressions of the th order conditional momentand th order conditional cross-moment are derived forthe cases of either an exactly known norm or a known interval

(representing a quantization of ). These mo-ments will be used in Section IV to derive a MMSE estimatorof weighted squared norms as formulated in (2), and their cor-responding mean squared errors (MSEs).

Without loss of generality, we assume that the diagonal ele-ments, , of the positive definite correlation matrix

are ordered such that elements with identicaldistributions have adjacent indices. When analyzing , wedistinguish between three different cases, depending on the dis-tinctness of the variances (hereafter called eigenvalues):

• identical eigenvalues: , for some ;• distinct eigenvalues: , for all ;• one or more sets of repeated eigenvalues among .

While the former two cases are clearly structured and com-monly treated in literature, the third case needs further speci-fication [25]. Let the distinct values among the eigenvaluesbe ( ), with the strictly positive multi-plicities ( ). Then, we have the charac-terization

...

(3)

To simplify the notation, we gather the eigenvalue multiplic-ities in a vector and define the function

that gives the group index of from

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4030 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009

(i.e., is the integer that satisfies).

These three cases are directly applicable to systems withuncorrelated fading (identical eigenvalues), correlated fading(distinct eigenvalues), and Kronecker-structured systems (seeSection III) with correlation at either the transmitter or receiver(repeated eigenvalues with either multiplicity or ).

Next, the probability density function (pdf) of the squarednorm will be given for the three casesdescribed above. Since for all , then

and the squared norm is the sumof independent exponentially distributed variables (each withthe rate ). In the case of identical eigenvalues, the pdf isthat of a scaled -distribution (i.e., an Erlang distribution):

(4)

where is the Heaviside step function. In the case of dis-tinct eigenvalues, the pdf of is well-known in the field ofrenewal theory [25] and was derived for communications pur-poses in [23]:

(5)

In the third case, with repeated eigenvalues that satisfy (3), thepdf was derived in [25] and [26]:

(6)where

(7)with from the set of all partitions of(with ) defined as

(8)One remark is that the pdf in (6) actually becomes that in (4) if

and that in (5) if . Since the expressions withidentical and distinct eigenvalues are simpler and very useful inpractice, we will distinguish between all three cases throughoutthe paper.

A. Conditional Statistics: Known Norm Value or Interval

Next, we will consider the conditional statistics of the el-ements of when its squared norm is known exactlyor in a quantized way. The absolute value and the phase of acomplex Gaussian variable are independent [16]. Thus,

can be identically expressed as

, where the phase is uniformly dis-tributed in and for all . Observe thatinformation regarding will not provide anyknowledge of the phases. The squared magnitudes of the indi-vidual elements, , will however depend on .

In this section, we will derive closed-form expressions of theth-order conditional moment of and th order con-

ditional cross-moment of and . This will be done intwo different cases, namely when the squared normis either known exactly or when a quantization is known. Wedenote the quantized squared norm with and it represents theinformation , for some real-valued interval pa-rameters. This type of quantized information can, for example,be achieved by feedback. The conditional moments derived inthe section will be used in Section IV for MMSE estimation andMSE calculation of weighted squared norms in systems with ei-ther perfect or quantized squared norm feedback.

The following theorem gives closed-form expressions ofthe conditional moments in the case of an exactly knownsquared norm . Although the expressions are quite simplein their structure, two elementary functions and

are introduced to achieve a more convenientpresentation. These are defined and discussed in Appendix A.Observe that the mean value of an element is given by ,the quadratic mean by , and that gives thecross-correlation.

Theorem 1 (Conditional Moments With Known Norm): Let, where

has strictly positive eigenvalues and . Define. In the case of identical eigenvalues (i.e., for all

), the th order conditional moment of and th orderconditional cross-moment between and ( ) are

,

.(9)

In the case of distinct eigenvalues, the corresponding mo-ments are

,

.

(10)

Finally, if the eigenvalues are nondistinct and nonidentical,let be the eigenvalue multiplicities when the

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BJÖRNSON et al.: EXPLOITING QUANTIZED CHANNEL NORM FEEDBACK 4031

elements involved in the moments have been removed. The thorder conditional moment of and th order condi-tional cross-moment between and ( ) are

(11)

Proof: The proof is given in Appendix C.Observe that Theorem 1 only handles the case of , but

the solution in the special case of is trivial: .The theorem generalizes the previous results of [23], where ex-pressions of the first and second order moment and cross-corre-lation were derived in the special case of distinct eigenvalues.

Next, we proceed with deriving closed-form expressions ofthe same conditional moments and cross-moments as in the pre-vious theorem but in the case of quantized norm information.Once again, the expressions contain some functions that are de-fined in Appendix A.

Theorem 2 (Conditional Moments With Known NormInterval): Let , where

has strictly positive eigenvalues and. Let contain the quantized information

(where ). In the case of identical eigenvalues (i.e.,for all ), the th order conditional moment of

and th order conditional cross-moment betweenand ( ) are

(12)

where

(13)

In the case of distinct eigenvalues, the corresponding mo-ments are

(14)

where

(15)

Finally, if the eigenvalues are nondistinct and nonidentical,let be the eigenvalue multiplicities when theelements involved in the moments have been removed. The thorder conditional moment of and th order condi-tional cross-moment between and ( ) are

(16)

where

(17)

Proof: The proof is given in Appendix C.This section will be concluded by Theorem 3 that gives the

MMSE estimator of from the quantized information. Observe that the theorem completes Theorem 2 for

.Theorem 3 (Norm Estimation From Known Norm In-

terval): Let , wherehas strictly positive eigenvalues

. Let and let contain the quantized information(where ). The conditional th order

moment of , given , is

(18)

(19)

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4032 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009

and

(20)when the eigenvalues are identical (i.e., for all ), dis-tinct, or neither identical nor distinct, respectively. The variables

, , and are given in (13), (15), and (17), respec-tively.

Proof: The proof is given in Appendix C.In the remaining sections, the analytic results of Theorem 1, 2,

and 3 will be applied to problems in wireless communications.The results of this section are however general and have impor-tant applications in other areas, for example in the analysis of

-out-of- systems with exponential failure rates in renewaltheory [25], [27]. In principle, these systems consist of com-ponents and the system will keep running until of them havebreak down. The time between the th and th componentfailure is distributed as (i.e., failures may change thefailure rates of the surviving components). Thus, is thetime to system failure. The results herein can be used for MMSEestimation of the time between component failures, given theexact time of system failure or a time interval (e.g., if the func-tionality is tested only at certain occasions). Similarly, the MSEand the correlation between component failures can be calcu-lated, and the time of system failure can be MMSE estimated,given a time interval.

III. NORM FEEDBACK AND MMSE ESTIMATION OF WEIGHTED

SQUARED CHANNEL NORMS

In this section, we return to the generic estimation problemin (2) and the system model in (1). Thus, the effectivechannel used for norm feedback is , where

and are arbitrary matricesknown at the receiver. In this section, we will first develop ageneral entropy-maximizing quantization framework. Then,the results of Section II will be used to derive the distributionof the squared norm of the effective channel,which is necessary to apply the quantization framework tothe problem at hand. Finally, we solve the estimation problemin (2) by deriving the MMSE estimator, and its MSE, of theweighted squared norm , conditioned on exactor quantized feedback of . As described in Section I-B,the weighting matrix can represent receive combiningand precoding matrices. The applications of this section onuser-selection, link-adaptation, and linear precoding will beconsidered in Section IV. The user index will be dropped inthis section for brevity.

The results herein are derived for a general positive semi-definite correlation matrix , but we will also give the corre-sponding expressions in the special case of Kronecker-struc-tured correlation. In this widely used model, the transmit andreceive side correlation are separable as ,where and are the positive

semi-definite transmit and receive correlation matrices, respec-tively. As a result, the matrix can in this case be decomposedas

(21)

where the elements of are independent and identically dis-tributed (i.i.d.) as . The eigenvalues of become theproducts of any two eigenvalues of and , respectively.Depending on the amount of spatial correlation at the trans-mitter and receiver, the eigenvalues of are either identical(e.g., if ), distinct (e.g., if distinct eigenvaluesat both sides), or consist repeated eigenvalues (e.g., when oneof the sides is spatially uncorrelated with either or

). Eigenvalues that are measured in practice are natu-rally distinct, but clustering of those that are close-to-equal maybe necessary to achieve numerical stability. Recall that thesethree cases correspond to those in Section II.

A. General Entropy-Maximizing Quantization Framework

Next, we will present a general framework for quantizationof a stochastic variable , with the cumulative distri-bution function (cdf) , for the purpose of finite rate feed-back. This variable may represent the weighted squared norm ofa communication system, but the results are valid for any con-tinuous cdf that fulfills , for , and , for

.With quantization, we mean the process of dividing a

continuous range of values into a finite number of intervals.Herein, we consider -bits quantization of the rangeof , which means that the range is divided into disjointintervals , . In our context, the purposeof the quantization is feedback and storage of the variableusing bits. Note that each interval, , should be seen as arepresentative for all values of the original variable that lies inthe interval. The actual value in the interval that best representsthe quantized information, , will change depending on theapplication (e.g., estimation of or some function of it). Whendesigning the quantization, we need to choose the decisionboundaries , for , so that some design criteriais fulfilled. There is no over-all optimal criteria, but from aninformation-theoretical perspective it makes sense to maximizethe entropy of the quantization and thereby the average amountof channel information that is fed back.

Lemma 1 (Entropy-Maximizing Quantization): Let bea stochastic variable with a continuous cdf , that fulfills

, for , and , for . Assume thatthe sample space, , of is quantized into disjointintervals ( ), where the th interval is with

and . The interval boundaries that maximizesthe entropy of are given by

(22)

This quantization will make the outcome of equally probablein all the quantization intervals.

Let denote the index such that the out-come . The quantization maximizes the mutual

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BJÖRNSON et al.: EXPLOITING QUANTIZED CHANNEL NORM FEEDBACK 4033

information between and , for any invertible function.

Proof: The lemma follows from a division of the cdf ofinto disjoint intervals of equal probability, and from the

observation that and contain the same information.The inverses of cdfs can in general not be given in closed

form, but since the function is bijective and nondecreasing thequantization boundaries in the lemma can be calculated effi-ciently using line search.

An important result of Lemma 1 is that even if we are inter-ested in some function of (e.g., the capacity if representsthe SNR), the entropy-maximizing quantization is still that of(22). Next, we will derive the distribution of the squared normand apply this quantization framework.

B. Distribution and Feedback of the Squared Channel Norm

Consider feedback of the squared norm of theeffective channel. Although we have assumed full CSI at thereceiver [24], it is unreasonable to feedback the positive real-valued squared norm with unlimited accuracy in a fading en-vironment (if it still should provide information on the currentchannel conditions at the time of reception). Hence, we willquantize the squared norm so it can be described by a finitebit sequence. In order to apply the entropy-maximizing quan-tization framework in Lemma 1 we need to derive the cdf of ,which is given by the following corollary.

Corollary 1 (Distribution of the General SquaredNorm): Let the channel be distributed as

. Let andbe arbitrary matrices such that the effective channelhas the distribution , where

. If the nonzero eigenvaluesof are denoted (for ), then the pdfof is given by (4), (5), or (6), in the casesof identical, distinct, or nonidentical nondistinct eigenvalues,respectively. The corresponding cdf, , is given by (13),(15), and (17), respectively, using and .

Proof: The proof is given in Appendix C.In the Kronecker-structured case, , the ef-

fective channel inherits this property: , whereand . The nonzero

eigenvalues of are given as the product of any two nonzeroeigenvalues of and , respectively.

To summarize, the distribution of the squared norm, , of theeffective channel is given by Corollary 1. Using Lemma 1, thisdistribution can be used to calculate the entropy-maximizingquantization of .

C. MMSE Estimation of Weighted Squared Channel Norms

Next, we assume that the receiver has fed back informa-tion regarding the squared norm of the effectivechannel and the transmitter wants to estimate the weightedsquared norm . This corresponds to the genericestimation problem in (2). Using the conditional momentsand cross-moments derived in Theorem 1 and 2, we willsolve this problem by deriving the MMSE estimator of

and its corresponding MSE. The followingcorollary extends results of [20], [23] by deriving the first

two conditional moments of the weighted squared norm forarbitrary eigenvalue structure of the effective channel. Observethat the first moment, , is the MMSE es-timator, while the corresponding MSE can be calculated as

.Corollary 2 (MMSE Estimation of Weighted Squared

Norms): Let the effective channel be dis-tributed as . Let be theeigenvalue decomposition of the correlation matrix, where

is positive semi-definite. If theweighting matrix is independent of and if

contains information regarding the squared norm , then

(23)

where , , ,, , and

.(24)

For all such that , we have that. If represents the

exact value of or the quantized information, then the remaining conditional moments of

(24) are given by Theorem 1 and 2 (by removing all zero-valuedeigenvalues), respectively.

Proof: The proof is given in Appendix C.In the Kronecker-structured case, , let

and be the eigen-value decompositions of the effective transmit and receive cor-relation, respectively. Then, we have and

. Thus, the nonzero eigenvalues of are theproducts of any two nonzero eigenvalues of and , re-spectively. If the weighting matrix is also Kronecker-structured,

, then the weighted squared norm can be ex-pressed as .

To summarize the section, the entropy-maximizing quantiza-tion of an arbitrary nonnegative random variable was given inLemma 1. The distribution of the squared normof the efficient channel was derived in Corollary 1 and this dis-tribution is sufficient to calculate the entropy-maximizing quan-tization of . Finally, the MMSE estimator of weighted squarednorms with the structure , and their correspondingMSEs, was derived in Corollary 2 when feedback of either theexact value or a quantization of is available.

IV. APPLICATIONS IN SINGLE- AND MULTI-USER SYSTEMS

In both single-user and multi-user systems, there is a need offeeding back a limited number of bits to shape the transmissionto the spatial properties of the multi-antenna channel, adapt thesymbol constellations to current conditions, and to perform ef-ficient user-selection (in the multi-user case). There is a tight

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4034 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009

connection between these goals and the SINR; we want to se-lect users for transmission in spatial directions that permit hightransmissions rates and the Shannon capacity, which gives anupper bound on the achievable rate, is a function of the SINR[2]. Hence, it is important to choose a feedback parameter thatprovides a reliable way of estimating the SINR, and to quantizethis parameter efficiently by maximizing the amount of infor-mation per bit.

In this section, we consider norm based feedback for the pur-pose of rate adaptation and MMSE estimation of signal/inter-ference powers. It will be shown that the results of Section IIIfit naturally into both single-user systems with OSTBCs andmulti-user SDMA systems with beamforming.

A. Orthogonal Space-Time Block Codes With Precoding

We consider linear precoded OSTBCs, which should be re-garded as the general type of OSTBCs that can exploit the spatialproperties of the channel to improve the performance [9]–[12].Recall the system model in (1) and assume that there is onlyone active user, so the user index can be dropped. Assume thata OSTBC is used to transmit symbols over symbols slots(i.e., the coding rate is ). Letbe the vector of data symbols, where we have normalized suchthat for all . These symbols are coded in amatrix that fulfills the orthogonality property

[8]. In addition, we use an arbitrary pre-coding matrix that projects the code into spatialdirections and is known to both the transmitter and the receiver[9]. The transmitted signals over consecutive symbol slots isthus and the correspondingsystem model is

(25)

where ,contains i.i.d. noise samples with , and the data sym-bols are present in the entries of . From [11], [28], it isknown that OSTBCs provide the possibility of decomposing(25) into independent and virtual single-antenna systems as

(26)

where . The corresponding SNR and maximumrate per source symbol are

(27)The exact SNR and rate values are known at the receiver, whilethe transmitter only knows the statistics. The SNR can be es-timated at the transmitter as the average , but theestimation error will typically be large if no instantaneous CSIfeedback is available. More robust performance is achieved bysimply feeding back a quantized version of to improvethe estimation.

The effective channel is . The entropy-maximizingquantization of is given by Lemma 1, with the cdfof given by Corollary 1 (with and ). The quan-tization boundaries are functions of the precoder and the channelstatistics, and need only to be updated at the relatively slow rate

that these are changing. Given the quantized feedback informa-tion of , the MMSE estimator (and the correspondingMSE) of the SNR is given by Corollary 2 (with ).

When estimation is used to choose an appropriate transmis-sion rate, it might be necessary to include a fade-margin toachieve a target frame error rate, denoted . Observe thatpackages sent in outage should not be considered lost since theinformation in them can still be utilized using, for example, hy-brid ARQ. To control the error rate, we propose to include afade-margin parameter that is designed such that

, where the SNR estimate is deter-mined as

(28)whereand contains the quantized information. Hence, the SNR estimate can be calculated directly, using

Corollary 2. MMSE estimation of the maximum ratecan be treated in a similar manner [29].

To summarize, the framework in Lemma 1 can be used for en-tropy-maximizing quantization of the channel gain with linearprecoded OSTBCs. Using Corollary 1, the SNR can be esti-mated either in the MMSE sense or in an outage-robust way asproposed in (28).

B. Beamforming for SDMA

Next, we consider a downlink multi-user SDMA system withbeamforming transmission. The problem of efficient precodingand receive combining will be discussed, but the main focus willbe on adapting the quantization framework of Section III-A tosystems with user-selection and on developing a robust SINRestimation framework with feedback of norm based channel in-formation.

Assume that users have been scheduled for transmissionand let the transmit beamforming vector and the data symbol in-tended for user be denoted and , respectively.Without loss of generality, we assume that .Using the system model in (1), the transmitted signal is

(29)

where is the precoding matrix andis the vector of all transmitted sym-

bols. Linear combining is assumed at the receiver side; that is,each user uses a receive beamforming vector , with

, to achieve a scalar received signal .In principle, the purpose of the precoding matrix

is to transmit simultaneous data streams with an ac-ceptably low co-user interference, while the linear com-bining at each receiver is used to further reduce boththe inter- and intra-cell interference. With the notation

, the SINR (whenaveraging over the noise and transmitted symbols) of user is

(30)

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BJÖRNSON et al.: EXPLOITING QUANTIZED CHANNEL NORM FEEDBACK 4035

In order to optimize the system performance, we want to choosethe beamforming vectors to maximize the sum rate of the se-lected users, possibly under some fairness condition. The op-timal user-selection and beamforming scheme is very difficultto obtain in practice since base stations and users have asym-metric information; herein, the base station knows the channelstatistics and some quantized feedback from each user, whileeach user knows its own channel perfectly but has limited infor-mation regarding the co-users. The main difficulty lies in the de-sign of the limited feedback; it should reflect the channel prop-erties when an SINR maximizing receive beamformer has beenapplied. Such a receive beamformer can in general not be de-signed until the user-selection and precoder design is finished,which is a stage when the transmitter truly needs instantaneouschannel information. To resolve the receive beamformer ambi-guity, for the sake of feedback design, we propose a two-stepapproach.

• Stage 1, Feedback and Transmitter Design: A rea-sonable, but suboptimal, virtual receive beamformer isassumed which is derived such that the efficient channel

has statistical properties which may bederived at both the receiver and the transmitter (changes ona slow basis). The squared norm is quantizedand fed back. Using this feedback information the trans-mitter selects users and design its precoder, assuming thatall receivers uses their as receive beamformers.Additional directional feedback might be necessary if thespatial correlation is weak.

• Stage 2, Data Transmission and Receiver Design: Thebase station transmits data using the selected precoding.The receivers are free to select more beneficial receivebeamformers if they desire, which could potentially in-crease their SINRs. These receive beamformers may forexample be functions of the own channel matrix, , andsome overhead or measurement of the interference. TheSINRs estimated by the transmitter will then act as slightlypessimistic estimates.

Next, we will describe the first stage in greater detail. User se-lection and precoding design was thoroughly analyzed in [19]with similar prerequisites. Hence, our focus will be on feedbackdesign and estimation of the SINR for a given precoder ma-trix and set of users. First, the entropy-maximizing frameworkof Section III-A will be adapted to multi-user systems. Then,the design of virtual receive beamformers will be discussed. Fi-nally, observe that the signal and interference powers in (30) areweighted squared norms and therefore we will show how Corol-lary 2 can be used to estimate these from quantized feedback of

. The user indexes will be droppedfor brevity.

1) Post-User-Selection Quantization: The entropy-maxi-mizing quantization framework in Lemma 1 can be used tocalculate an efficient quantization of the squared norm .In multi-user systems, user-selection can however change thestatistics of the norm. If the scheduler takes its decisions basedon, for example, the instantaneous sum rate, then users thatexperience strong channel norms are more likely to be selected.

Hence, the post-user-selection cdf of the squared norm willbe the result of a transformation from the pre-scheduling cdf,

, that shifts the probability mass towards larger values.The feedback information can be used both in the process of

selecting users and in subsequent precoding design for the se-lected users. As discussed in [30], less CSI is required to chooseappropriate users than to design a precoder that guarantees highand robust throughput. Thus, it makes more sense to maximizethe post-user-selection entropy, than the pre-user-selection en-tropy as was done in Lemma 1.

The post-user-selection distribution depends strongly on thetype of selection criterion, and is often difficult to derive an-alytically. In [31], the distribution was derived in a single-an-tenna system with known co-channel statistics, but the latterassumption is unreasonable in most multi-user scenarios. Ob-serve that the post-user-selection cdf can be written as ,for some transformation function . Using this notation, thefollowing theorem gives the entropy-maximizing post-user-se-lection quantization.

Theorem 4 (Entropy-Maximizing Post-User-Selection Quan-tization): Let have the continuous pre-user-selection cdf

, which fulfills the properties in Lemma 1. Let thepost-user-selection cdf be denoted for some contin-uous transformation function , whichwill be increasing and bijective on if the probability ofselecting a user increases with its value .

If the sample space, , of is quantized into disjointintervals ( ), where the th interval is with

and , then the entropy-maximizing post-user-selection quantization is given by

(31)

Proof: The theorem follows directly from Lemma 1.To illustrate the usefulness of the notation with a transfor-

mation function , we consider the following scheduler forwhich can be derived in closed form.

Definition 1 (Greatest Quality Probability Scheduler): Con-sider a scheduler that selects users out of . Let the channelquality of user be measured by and let its cdf be

, for all users . Then, the Greatest QualityProbability (GQP) scheduler selects those users that have thelargest cdf values of their current realization of .

The proposed scheduler selects users based on the cdf valuesof their current channel quality (i.e., the percentage of realiza-tions with worse performance). The quality, , may representthe squared norm, or some other suitable measure. An importantproperty of the proposed scheduler is that it provides fairnessin terms of selecting users with identical probability, because

for all users . The spatial separability be-tween users is however ignored, but this is of minor importancewhen the number of transmit antennas grows [21]. When theusers have identical statistics and represents the SNR, thenthe GQP scheduler coincides with maximum throughput sched-

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4036 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009

uling [13] (i.e., the users with the highest rates are selected). Forthe proposed scheduler, the transformation function becomes

(32)

This is shown by observing that the cdf values, , areidentically distributed among all users and that a selected userhas any of the th largest with equal probability. It isworth noting that the selection scheme in Definition 1 becomesidealized when quantization is introduced; the exact values of

are unknown and have to be estimated based on theavailable feedback information. The point is however that thetransformation function can be determined explicitly forcertain schedulers. In general, the function depends on all usersand will therefore be unavailable at the receivers. It can how-ever be approximated in various ways. In Section V, it will beillustrated numerically that even a simple parametrization as

, for some parameter , can signifi-cantly improve the performance. Thus, the gain of post-user-se-lection quantization can be exploited by simple means.

2) Design of Virtual Receive Beamformers: The virtualreceive beamformer should be designed such thatthe statistics of the effective channel can bederived deterministically at both the receiver and transmitter.At first sight, this assumption seems to lead to the conclusionthat needs be independent of the realization . Thisrequirement can however be relaxed, since the effective channelwill be deterministic in eigendirections with eigenvalues thatbecome zero. Thus, the system can be designed such thatthe transmitter knows that always will cancel out thechannel in some predefined eigendirections (e.g., such that areexpected to contain much interference).

As an example, the following virtual receive beamformer wasproposed in [20] for Kronecker-structured systems with, but can be generalized for arbitrary receiver side correlation.

Let the eigenvalue decomposition of the transmit side correla-tion matrix be partitioned as

(33)

where and contain eigen-vectors, and the eigenvalues are ordered in some (predefined)arbitrary way. If , then there exist a receive beam-former that will completely cancel out the power inthe eigensubspace such that the experienced channel

has the distribution , with

. To achieve this, the re-ceive beamformer should be chosen arbitrarily in the null spaceof (i.e., ). Using this virtual receivebeamformer, the transmitter knows that the experienced channelwill have the correlation matrix .

In practice, the virtual receive beamformer can be designedin various ways depending on the environment. The design canalso be relaxed such that the effective channel only becomesapproximately Gaussian; the important thing is that the first andsecond order statistics are approximately known at the trans-mitter.

3) Estimation of the SINR: Finally, we consider estimationof the SINR in (30) at the transmitter (e.g., for the purpose ofuser-selection and rate adaptation). Apart from the channel sta-tistics, the transmitter has received quantized feedback of ,the squared norm of the effective channel with the virtual receivebeamformer. The unknown quantities in the SINR expressionare the signal and interference powers, which both are weightedsquared norms: ,where the weighting matrix contains one or several transmitbeamformers. These beamformers are either directly known totransmitter or they should be selected in the precoder design tomaximize the (weighted) sum rate. In any way, the SINR canbe estimated as a function of the transmit beamformers.

Similar to [19], [20], [23], we propose to use the pessimisticSINR estimator in (34), at the bottom of the page. In this esti-mator, and . The MSEsare calculated as

and represents either exact norm information orthe quantized feedback information . The designparameter in (34) can be used to achieve a target frame errorrate, . This adaptive fade-margin is similar to the one inSection IV-A and is an essential control-feature in most systems,including those with advanced error control.

If the virtual receive beamformer, , is designed asdescribed in the previous section, the signal and interferencepowers (and their MSEs) in (34) can be MMSE estimated usingCorollary 2. If only approximately fulfills the require-ments and/or an improved receive beamformer is used in theactual data transmission, then the SINR estimate in (34) willnot be the ideal one. The performance loss is however limited inmany practical systems, as illustrated in [23]. The explanationis that small estimation errors have limited consequences sincethe adaptive fade-margin in (34) is used to adapt the SINRestimate to control the error rate.

(34)

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BJÖRNSON et al.: EXPLOITING QUANTIZED CHANNEL NORM FEEDBACK 4037

To summarize the section, we have considered a multi-usersystem with beamforming at both the base station and the re-ceiving users. The entropy-maximizing quantization frameworkin Lemma 1 has been extended to take the effect of user-selec-tion into account. A virtual receive beamformer was proposedto overcome the receive beamformer ambiguity in the feedbackdesign. Finally, it was shown how Corollary 2 can be used to per-form robust SINR estimation in systems with norm feedback.

V. NUMERICAL EXAMPLES

This section will illustrate how the analytic results of thepaper can be used to improve the performance of MIMO com-munication systems. Two numerical examples will be given,corresponding to the single-user case in Section IV-A and themulti-user case Section IV-B, respectively. In the single-usercase, we consider transmission from a four-antenna transmitterto a two-antenna receiver, using OSTBCs ( ) and thelinear precoder in [10] that adapts the coding to the channelstatistics. The average SNR (defined as ) is 10 dBand we assume the Kronecker channel model in (21) with un-correlated receive antennas ( ). The transmit correlationfollows the exponential model of [32], which models a uniformlinear array (ULA) with the correlation between adjacent an-tennas as a parameter. The instantaneous SNR of this system is

and is quantized and fed back using the entropy-max-imizing framework in Lemma 1.

The average throughput over realizations for differentnumbers of feedback bits is shown in Fig. 1 with varying antennacorrelation (absolute value of the coefficient in [32]) and withan outage probability of 5%. Observe that a logarithmic scalehas been used on the -axis. The corresponding throughputswith half a wavelength antenna separation and different angularspreads (standard deviation of Gaussian distributed scatterers,as seen from the transmitter) are given as a reference to showthat many measured systems in fact have antenna correlationsaround 0.9 (cf. [3]). From Fig. 1, it is clear that just a few bitsof norm feedback are sufficient to achieve performance close tothat of full CSI; 52% of the feedback gain is achieved with onebit of feedback, while three bits gives 84% and five bits 95%.The amount of correlation has little impact on the percentage offeedback gain. Finally, observe that the performance of this pre-coded system increases with the transmit antenna correlation, asexpected from [12].

In the multi-user case, we consider downlink zero-forcingSDMA communication from a transmitter with an eight-antennauniform circular array (UCA) to 20 users, each equipped withfour uncorrelated receive antennas. The angular spread is 10 de-grees and the transmit antenna separation is half a wavelength.The users are uniformly distributed in the areaof a circular cell of radius . The average SNR is 10 dB at thecell boundary and the power decay is proportional to . Thescheduling is performed using the greedy user selection [19],[33] and with proportional fairness as scheduling criterion [34].The performance is measured in terms of the cdf of the averagecell throughput over different scenarios. Each of the consid-ered scenarios represent a unique random constellation of mo-biles with fixed statistics, while the average cell throughput iscalculated over 150 scheduling decisions.

Fig. 1. The average throughput as a function of the absolute value of the cor-relation between adjacent antennas at the transmitter. The performance is givenfor the cases with exact SNR/norm feedback, with quantized feedback using 1,3, or 5 bits (increasing performance), and without feedback. The performancewith different amounts of angular spread is marked with circles as a reference.Observe the logarithmic scale of the �-axis.

Fig. 2. The cumulative distribution functions (cdfs) of the average cellthroughput over scenarios with 20 uniformly distributed users in a circular cell.The performance of zero-forcing with full CSI is compared with generalizedzero-forcing with gain feedback [19], directional-quantized zero-forcing [18],and multi-user opportunistic beamforming [14]. The GZF uses 3 bits of gainfeedback, while the latter two schemes uses 3 bits of directional feedback andperfect gain feedback.

In Fig. 2, the performance of zero-forcing (ZF) precodingwith full CSI is compared with 1) directional-quantized ZF [18]with a Grassmannian codebook [35]; 2) multi-user opportunisticbeamforming [14]; and 3) the generalized zero-forcing (GZF)scheme in [19]. The GZF uses the receive antennas to suppressthe interference sensitive subspace of [19] and uses 3 bits ofnorm feedback (with post-user-selection quantization and thetransformation function in Theorem 4 approximated as

). The outage probability is 5%. The quan-tized ZF and opportunistic beamforming schemes are based on3 bits directional information and perfect gain feedback. It ishowever seen in Fig. 2 that the GZF scheme outperforms theother partial CSI schemes, although it is based on a consider-ably smaller feedback load. Observe that the framework derivedherein can be applied to handle quantized gain feedback in thetwo competing schemes.

Finally, in Fig. 3 the performance of the GZF scheme isshown for different numbers of feedback bits and with both pre-and post-user-selection quantization. With one bit of feedback,71–73% of the feedback gain is achieved, depending on thetype of quantization. The corresponding interval is 90–92%for three bits and 97–98% for five bits. It is clear that a fewbits of feedback are sufficient to achieve most of the feedbackgain, and that the benefit of considering the post-scheduling

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4038 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009

Fig. 3. The cumulative distribution functions (cdfs) of the average cellthroughput over scenarios with 20 uniformly distributed users in a circularcell. The performance of the generalized zero-forcing scheme [19] is shownfor different types of norm feedback: no feedback, pre- or post-schedulingquantization with 1, 3, or 5 bits, and perfect feedback (increasing performance).Zero-forcing precoding with full CSI is given as a reference.

distribution in the quantization is nonnegligible. The lack ofinstantaneous directional information will however make thescheme suboptimal, even for perfect norm feedback.

VI. CONCLUSION

For arbitrarily correlated zero-mean complex Gaussian ma-trices, closed-form expressions for the conditional matrix dis-tribution and moments of individual elements have been de-rived when the squared Frobenius norm of the matrix is eitherknown exactly or known to lie in a specific quantization in-terval. In addition, MMSE estimators (and their resulting MSEs)of weighted squared norms have been derived, given quantizednorm information. This mathematical contribution has clear ap-plications in renewal theory, but herein the main focus has beenon the applications in wireless communication systems with sta-tistical CSI and limited feedback. In these systems, the signaland interference powers are weighted squared norms. An en-tropy-maximizing framework was proposed for feedback quan-tization and it has been shown how feedback of quantized norminformation enables robust estimation of the SINR in an MMSEbased framework. The usefulness of the results were exempli-fied in single-user systems with linearly precoded OSTBCs andin multi-user SDMA systems with beamforming and quantiza-tion that takes the post-user-selection distribution into account.

APPENDIX ASOME USEFUL ELEMENTARY FUNCTIONS

Throughout the paper, a few nonstandard elementary func-tions have been used extensively. Specifically, they appear inthe derivation of Theorem 1, 2, 3, and Lemma 4. This appendixwill first define the functions and then provide integral expres-sions that have these functions as solutions.

Definition 2: For nonnegative integers wedefine the five functions in (35)–(39). These are shown at thebottom of the next page.

The next two lemmas show how the functions in Definition 2appear as the solutions to certain integrals.

Lemma 2: Let and be two nonnegative integers andlet be a nonzero real-valued scalar. Then,

(40)

(41)

where is a real-valued constant and is some arbitrary con-stant.

Proof: First, assume that and observe that (40)holds for , since . Then bythe principle of induction, we have that

where we integrated by parts and used (40) for and. Then, for , the expression in (40) follows by the Bi-

nomial series expansion .The expression in (41) follows by the same kind of Binomialseries expansion and pure integration.

Lemma 3: Let be strictly positive scalars. Thefunctions introduced in Definition 2 satisfy

Proof: The results for andfollow from Lemma 2, when the lower and upper bound are 0and , respectively. Similarly, the results for ,

, and are achieved from thelemma when the lower and upper bounds are and , respec-tively.

APPENDIX BJOINT CONDITIONAL DISTRIBUTIONS

Consider ,as defined in Section II. Joint conditional pdfs of sets of

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BJÖRNSON et al.: EXPLOITING QUANTIZED CHANNEL NORM FEEDBACK 4039

(with known ) are used in the proofs of several the-orems. Since for any choice of , notethat the joint conditional pdf of can be factorized as

(42)

where and .While the joint conditional pdf is a function ofcomplex-valued variables, the expression in (42) has separatedit into an -dimensional uniform phase distribution and the

-dimensional conditional distribution .The following lemma derives closed-form expressions for thispdf in the three cases of identical, distinct, and neither identicalnor distinct eigenvalues.

Lemma 4: Let , wherehas strictly positive eigenvalues , and

define . Let be a nonempty set with distinct indexesfrom and with cardinality . If theeigenvalues are identical (i.e., for all ), then thejoint conditional pdf of , when

is known, is

(43)where . If the eigenvalues are dis-tinct, then the joint conditional pdf is

(44)

Finally, if the eigenvalues are nondistinct and nonidentical,then assume that they are ordered such that the characterization

(35)

(36)

(37)

,

(38)

(39)

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4040 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009

in (3) is fulfilled. Let be the eigenvalue mul-tiplicities when the elements in have been removed. Then,the joint conditional pdf is

(45)

Proof: The expression for the joint conditional distributionwas proved in [23], in the case of distinct eigen-

values, using induction. Herein, all three cases will be provedusing a somewhat shorter approach based on the law of totalprobability. Let be the vector with all elements that remainwhen those with indexes in have been removed. By using thelaw of total probability to condition on and then Bayes’formula and that , wehave that

(46)

Then, the theorem follows from observing that, and that and

are given in (4), (5), and (6) for the three different cases.

APPENDIX CCOLLECTION OF PROOFS

Proof of Theorem 1: By definition, the conditional thorder moment is

where the conditional distribution is given in Lemma 4. In allthree eigenvalue cases, the integral can be solved using Lemma3, by observing that the terms that depend on form an inte-gral that equals the function , from Definition 2, fordifferent values of , , , and .

The th order cross-moment is defined as

The joint conditional distribution isgiven by Lemma 4 for . In this case, the double integralcan be determined (using Lemma 3) by observing that the termsthat depend on and form a double integral that equals thefunction for different values of , , , ,, and . In the special case of , the joint conditional

distribution becomes degenerate, since . Using

the second equality in (42), the cross-moment can be expressedas

which is solved by a similar identification.Proof of Theorem 2: Using the law of total probability, the

conditional moment can be expressed as

(47)

where represents the exact value of . Observe that theconditional moment is given by Theorem 1, andthat the pdf of the norm is given in (4), (5), and (6) for the threedifferent cases of eigenvalue structure. The integral in the nu-merator of (47) can be solved directly (using Lemma 3), whilethe integral in the denominator can be solved by straightforwardintegration using Lemma 2. The conditional cross-moments canbe derived by the same approach.

Proof of Theorem 3: The theorem follows from observingthat

(48)

The numerator is given by Lemma 3 and the denominator wascalculated in the proof of Theorem 2.

Proof of Corollary 1: The expressionis obtained by using the rule, for general matrices , , and . The

distribution of the squared norm of the effective channel isachieved by using that the Frobenius norm and that the distri-bution of a complex Gaussian vector is invariant under unitarymatrix transformations. Since zero-valued eigenvalues have noimpact on the norm, the distribution of is equivalent to that of

, where and is a diagonal matrix with allnonzero (i.e., strictly positive) eigenvalues of .

Proof of Corollary 2: Let and observethat . The corollary follows from straightfor-ward and tedious expansion of

and . The expecta-tion is evaluated using Theorem 1 and 2.

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Emil Björnson (S’07) was born in Malmö, Sweden,in 1983. He received the M.S. degree in engineeringmathematics from Lund University, Lund, Sweden,in 2007. He is currently working towards the Ph.D.degree in telecommunications at the Signal Pro-cessing Laboratory, Royal Institute of Technology(KTH), Stockholm, Sweden.

His research interests include wireless commu-nications, resource allocation, estimation theory,stochastic signal processing, and mathematicaloptimization.

David Hammarwall (S’03–M’07) was born inStockholm, Sweden, in 1977. He received the M.S.degree (with highest hons.) in electrical engineeringfrom the Royal Institute of Technology (KTH),Stockholm, Sweden, in 2003. In 2001-02 he pursuedM.S.-level studies at the Department of ElectricalEngineering, Stanford University, Stanford, CA(as part of the KTH degree). He was awarded an”Excellent Graduate Student Position,” from thepresident’s office at KTH, and received the Ph.D.degree in telecommunications from the same univer-

sity, in 2007.He has since joined Ericsson Research, Stockholm, Sweden. His research in-

terests include wireless communications, resource optimization, beamforming,and scheduling.

Björn Ottersten (S’87–M’89–SM’99–F’04) wasborn in Stockholm, Sweden, in 1961. He receivedthe M.S. degree in electrical engineering and appliedphysics from Linköping University, Linköping,Sweden, in 1986 and the Ph.D. degree in electricalengineering from Stanford University, Stanford, CA,in 1989.

He has held research positions at the Department ofElectrical Engineering, Linköping University; the In-formation Systems Laboratory, Stanford University;and the Katholieke Universiteit Leuven, Leuven, Bel-

gium. During 1996–1997, he was Director of Research at ArrayComm Inc.,San Jose, CA, a start-up company based on Ottersten’s patented technology. In1991, he was appointed Professor of Signal Processing at the Royal Institute ofTechnology (KTH), Stockholm, Sweden. From 2004 to 2008, he was Dean ofthe School of Electrical Engineering at KTH, and from 1992 to 2004 he washead of the Department for Signals, Sensors, and Systems at KTH. He is alsoDirector of security and trust at the University of Luxembourg. His researchinterests include wireless communications, stochastic signal processing, sensorarray processing, and time-series analysis.

Dr. Ottersten has coauthored papers that received an IEEE Signal ProcessingSociety Best Paper Award in 1993, 2001, and 2006. He has served as Asso-ciate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and on theEditorial Board of the IEEE Signal Processing Magazine. He is currently Ed-itor-in-Chief of the EURASIP Signal Processing Journal and a member of theEditorial Board of the EURASIP Journal of Advances Signal Processing. He isa Fellow of EURASIP. He is a first recipient of the European Research Counciladvanced research grant.

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