On the capacity of OFDM-based spatial multiplexing systems ... · fading at the BTS will be...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002 225

On the Capacity of OFDM-Based SpatialMultiplexing Systems

Helmut Bölcskei, Member, IEEE, David Gesbert, Member, IEEE, and Arogyaswami J. Paulraj, Fellow, IEEE

Abstract—This paper deals with the capacity behavior of wire-less orthogonal frequency-division multiplexing (OFDM)-basedspatial multiplexing systems in broad-band fading environmentsfor the case where the channel is unknown at the transmitter andperfectly known at the receiver. Introducing a physically moti-vated multiple-input multiple-output (MIMO) broad-band fadingchannel model, we study the influence of physical parameters suchas the amount of delay spread, cluster angle spread, and total anglespread, and system parameters such as the number of antennasand antenna spacing on ergodic capacity and outage capacity. Wefind that, in the MIMO case, unlike the single-input single-output(SISO) case, delay spread channels may provide advantages overflat fading channels not only in terms of outage capacity butalso in terms of ergodic capacity. Therefore, MIMO delay spreadchannels will in general provide both higher diversity gain andhigher multiplexing gain than MIMO flat-fading channels.

Index Terms—Broad-band fading channels, diversity gain, er-godic capacity,MIMO,multiplexing gain, OFDM, outage capacity.

I. INTRODUCTION AND OUTLINE

THEUSEOFmultipleantennasatbothendsofawireless linkhas recently been shown to have the potential of achieving

extraordinary bit rates [1]–[4]. The corresponding technology isknown as spatial multiplexing [1] or BLAST [2], [5] and allowsan increase in bit rate in a wireless radio link without additionalpower or bandwidth consumption. So far, most of the research inthis context has focused on the narrow-band flat-fading case. Ex-tensive investigations on the capacity of narrow-band flat-fading(deterministic and stochastic) multiple-input multiple-output(MIMO) channels (assuming different levels of channel stateinformation at the transmitter and the receiver) can be found in[2], [3], and [5]–[8].1) Contributions: For a broad-band MIMO fading channel

model, which is based on previous work reported in [9], [10], weprovide expressions for the ergodic capacity and the outage ca-pacity of orthogonal frequency-division multiplexing (OFDM)-

Paper approved by C. Tellambura, the Editor for Modulation and Signal De-sign of the IEEE Communications Society. Manuscript received October 12,1999; revised September 18, 2000, and April 7, 2001. This work was supportedin part by FWF under Grants J1629-TEC and J1868-TEC. This paper was pre-sented in part at the IEEE ICASSP-00, Istanbul, Turkey, June 2000.H. Bölcskei was with the Coordinated Science Laboratory and the Depart-

ment of Electrical Engineering, University of Illinois at Urbana-Champaign,Urbana, IL 61801 USA. He is now with the Communication Technology Lab-oratory, ETH Zurich, ETH Zentrum, ETF EIZZ, CH-8092 Zurich, Switzerland(e-mail: [email protected])).D. Gesbert was with Iospan Wireless Inc., San Jose, CA 95134 USA. He

is now with the Department of Informatics, University of Oslo, N-0316 OsloNorway (e-mail: [email protected]).A. J. Paulraj is with the Information Systems Laboratory, Department of Elec-

trical Engineering, Stanford University, Stanford, CA 94305-9510USA (e-mail:[email protected]).Publisher Item Identifier S 0090-6778(02)01368-5.

based spatial multiplexing systems [4], [11] considering the casewhere the channel is unknown at the transmitter and perfectlyknown at the receiver. These expressions are then used to study(analytically and numerically) the influence of propagation pa-rameterssuchasdelayspread,clusteranglespread,andtotalanglespread, and system parameters such as the number of antennasand antenna spacingon capacity.We find that, in theMIMOcase,unlike the single-input single-output (SISO) case, delay spreadchannelsmay provide an advantage over flat-fading channels notonly in terms of outage capacity but also in terms of ergodic ca-pacity. Consequently, MIMO delay spread channels provide notonly higher diversity gain than MIMO flat-fading channels butalso higher multiplexing gain.2) Relation to Previous Work: Our channel model builds on

research reported in [9] and [10]. In particular, it is an extensionof the space–time channel model proposed in [9] to the case ofmultiple antennas at both the transmitter and the receiver. The ca-pacity of deterministic MIMO channels with memory and fullchannel knowledge at the transmitter and the receiver was de-rived in [12]. In [13], the capacity of deterministic two-user mul-tiaccess channels with memory is computed. Using a parametricMIMOchannelmodel inwhicheachpath isdescribedbyanangleof departure, an angle of arrival, a (complex) path gain, and apathdelay, thecapacityofthecorrespondingdeterministicMIMOdelay spread channel (full channel knowledge at the transmitterand the receiver) has been provided in [4]. Using the same para-metric channel model and defining the underlying parameters asrandom variables, a parametric MIMO fading channel model isestablished in [11], and an expression for the ergodic capacity isprovided for the caseswhere the channel is either knownat the re-ceiver only or known at both the transmitter and the receiver. Thechannel model used in [4] and [11] does not capture the effectsof spatial fading correlation, diffuse scattering, and scattering ra-diusoncapacityand is therefore fundamentallydifferent fromtheMIMO fading channel model used in this paper. Furthermore, inthe channel model used in [4] and [11], each path can only be arank-1contributor tocapacity,1 whereas inourmodel the rankde-pends on physicallymeaningful parameters such as cluster anglespread and antenna spacing. OurMIMO fading channelmodel istherefore more flexible than the one used in [4] and [11]. An in-terestingasymptotic (in thenumberofantennaelements)analysisfor both the flat-fading and the frequency-selective fading casesappears in [14].The use of OFDM in the context of spatial multiplexing has

been proposed previously in [4], [11], [15], and [16]. However,it appears that no capacity studies of OFDM-based spatial mul-tiplexing systems using the physically motivated MIMO fading1This statement will be made more precise in Section II-B.

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226 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002

channel model provided in this paper have been performed sofar. For the single-carrier narrow-band flat-fading case, the im-pact of spatial fading correlation and antenna array geometryon capacity has been studied in [7] and [8]. To the best of ourknowledge, the impact of physical parameters (delay spread,cluster angle spread, and total angle spread) and system param-eters (number of antennas and antenna spacing) on ergodic ca-pacity and outage capacity in the broad-band OFDM case hasnot been studied in the literature so far.3) Organization of the Paper: The rest of this paper is or-

ganized as follows. In Section II, we introduce our broad-bandMIMO fading channel model. In Section III, we derive expres-sions for the ergodic capacity and the outage capacity of OFDM-based spatial multiplexing systems taking into account the newchannel model. In Section IV, we study the influence of prop-agation parameters and system parameters on ergodic capacityand outage capacity. We furthermore demonstrate that, in theMIMO case, delay spread channels may provide advantage overflat-fading channels not only in terms of outage capacity but alsoin terms of ergodic capacity. In Section V, we provide numer-ical results complementing the analytical results in Section IV.Finally, Section VI provides our conclusions and some futureresearch directions.

II. BROAD-BAND MIMO FADING CHANNEL MODEL

In this section, we shall introduce a newmodel for broad-bandMIMO fading channels based on a physical description of thepropagation environment. Our channel model builds on pre-vious work reported in [9] and [10].

A. General Assumptions

1) Propagation Scenario: We assume that the subscriberunit (SU) is surrounded by local scatterers so that fading atthe SU antennas is spatially uncorrelated. The base transceiverstation (BTS), however, is sufficiently high so that it is un-obstructed and no local scattering occurs. Therefore, spatialfading at the BTS will be correlated with the exact correlationdepending on the BTS antenna spacing and the angle spreadobserved at the BTS array [17]. Our model incorporates thepower delay profile of the channel, but neglects shadowing.These assumptions on the propagation scenario are typical forcellular suburban deployments [17], where the BTS is on atower or on the roof of a building and the terminal is on thestreet level and experiences local scattering. For the sake ofsimplicity, throughout the paper, we restrict our attention to theuplink case. The results for the downlink case are similar. Inthe following, and denote the number of transmit (i.e.,SU) and receive (i.e., BTS) antennas, respectively.2) Channel: Following [9] and [10], we model the delay

spread by assuming that there are significant scatterer clusters(see Fig. 1) and that each of the paths emanating fromwithin thesame scatterer cluster experiences the same delay. In practice,local scatterers in the cluster introduce micro delay variations,which will be neglected in our model. With denoting thediscrete-time transmitted signal vector and the dis-crete-time received signal vector, respectively, we can

Fig. 1. Schematic representation of theMIMOdelay spread channel composedof multiple clustered paths. Each path cluster has a mean angle of arrival andan angle spread . The absolute antenna spacing is denoted by .

write

(1)

where the complex-valued random matrix repre-sents the th tap of the discrete-time MIMO fading channel im-pulse response. Note that in general there will be a continuumof delays. The channel model (1) is derived from the assump-tion of having resolvable paths, where with anddenoting the signal bandwidth and delay spread, respectively.

The elements of the individual are (possibly correlated) cir-cularly symmetric complex Gaussian random variables.2 Dif-ferent scatterer clusters are uncorrelated, i.e.,3

for (2)

where

with denoting the thcolumn of the matrix , and denoting the all-zero ma-trix of size . Each scatterer cluster has a meanangle of arrival at the BTS denoted as , a cluster angle spread(proportional to the scattering radius of the cluster), and a path

gain (derived from the power delay profile of the channel).3) Array Geometry: For the sake of simplicity, we assume

a uniform linear array (ULA) at both the BTS and the SU withidentical antenna elements. Most of our results, can however, beextended to nonuniform arrays. The relative antenna spacing isdenoted as , where is the absolute antenna spacingand is the wavelength of a narrow-band signal withcenter frequency .4) Fading Statistics: We assume that the (

; ) have zero mean (i.e.,2A circularly symmetric complex Gaussian random variable is a random vari-

able , where and are i.i.d.3 denotes the expectation operator and the superscript stands for conjugate

transposition.


BOLCSKEI et al.: CAPACITY OF OFDM-BASED SPATIAL MULTIPLEXING SYSTEMS 227

pure Rayleigh fading) and that the correlation matrixis independent of , or, equivalently, the

fading statistics are the same for all transmit antennas. Definingfor ,

to be the fading correlation betweentwo BTS antenna elements spaced wavelengths apart, thecorrelation matrix can be written as

(3)

Note that we have absorbed the power delay profile of thechannel into the correlation matrices.Factoring the correlation matrix according to

, where is of size , thematrices can be written as

(4)

with the being uncorrelated matrices with i.i.d.entries. We have therefore decomposed the th tap of

the stochastic MIMO channel impulse response into the productof a deterministic matrix taking into account the spatialfading correlation at the BTS and a stochastic matrix of i.i.d.complex Gaussian random variables.Let us next assume that the angle of arrival for the th

path cluster at the BTS is Gaussian distributedaround the mean angle of arrival , i.e., the actual angle of ar-rival is given by with . The variance

is proportional to the angular spread and hence the scat-tering radius of the th path cluster. It is shown in [9] that forsmall angular spread the correlation function can be approxi-mated as

(5)

Although this approximation is accurate only for small angularspread, it does provide the correct trend for large angular spread,namely uncorrelated spatial fading. Note that in the case, the correlation matrix collapses to a rank-1 matrix andcan be written as with the array responsevector of the ULA given by

(6)

B. Differences to the Parametric Fading Channel ModelIn the parametric fading channelmodel proposed in [11], each

tap can be written as

where denotes the complex Gaussian distributed path gain,and are the random angle-of-arrival and angle-of-de-

parture, respectively, of the th path, and and theand receive and transmit array response vectors

(cf. (6)), respectively. Note that in this model every realizationof has rank 1. In our MIMO fading channel model, the rankof the matrices is controled by the fading correlation at the

BTS. If the angular spread of the th path cluster is large, willhave high rank; for decreasing angular spread the rank ofwill decrease. This follows from (4) and the fact that the corre-lation matrix loses rank if the angular spread decreases. TheMIMO fading channel model proposed in this paper is thereforemore flexible than the parametric fading channel model [11] andseems to be a more adequate description of a real-world scat-tering environment.

III. MUTUAL INFORMATION AND CAPACITY OF OFDM-BASEDSPATIAL MULTIPLEXING SYSTEMS

In this section, we derive an expression for the mutual infor-mation of OFDM-based spatial multiplexing systems. This ex-pression is then used to compute the ergodic capacity and studythe outage properties of the system.

A. OFDM-Based Spatial MultiplexingSpatial multiplexing [1], also refered to as BLAST [2], [5],

has the potential to drastically increase the capacity of wirelessradio links with no additional power or bandwidth consump-tion. The technology requires multiple antennas at both ends ofthe wireless link. The gain in terms of ergodic capacity overSISO systems resulting from the use of multiple antennas istermed multiplexing gain. The main reason for using OFDMin this context is the fact that OFDM modulation turns a fre-quency-selectiveMIMO fading channel into a set of parallel fre-quency-flat MIMO fading channels. This renders multichannelequalization particularly simple, since for each OFDM tone anarrow-band receiver can be employed [4], [11]. In OFDM-based spatial multiplexing, the (possibly coded) data streamsare first passed through OFDM modulators and then launchedfrom the individual antennas. Note that this transmission takesplace simultaneously from all transmit antennas. In the re-ceiver, the individual signals are passed through OFDM demod-ulators, separated, and then decoded. Fig. 2 shows a schematicof anOFDM-based spatial multiplexing system. Throughout thepaper, we assume that the length of the cyclic prefix (CP) in theOFDM system is greater than the length of the discrete-timebaseband channel impulse response. This assumption guaran-tees that the frequency-selective fading channel indeed decou-ples into a set of parallel frequency-flat fading channels [18].Organizing the transmitted data symbols into frequency

vectors withdenoting the data symbol transmitted from the th an-tenna on the th tone and defining

, it can be shownthat

(7)

where denotes the reconstructed data vector for the th tone,and is additive white Gaussian noise (AWGN) satisfying

(8)

where is the identity matrix of size . From (7), it canbe seen that equalization requires application of a narrow-bandreceiver for each tone .



Fig. 2. OFDM-based spatial multiplexing system (OMOD and ODEMOD denote an OFDM-modulator and demodulator, respectively).

B. Mutual InformationWe start by stacking the vectors , , and according to

where and are vectors and is anvector. Note that (8) implies that the noise vector is white,i.e.,

We furthermore define the block-diagonal ma-trix

With these definitions, (7) can be rewritten as

(9)

In the following we assume that for each channel use (corre-sponding to at least one OFDM symbol) an independent real-ization of the random channel impulse response matrices isdrawn and that the channel remains constant within one channeluse. Using (9), the mutual information (in b/s/Hz) of the OFDM-based spatial multiplexing system under an average transmitterpower constraint is given by4 [19], [20]

(10)

where with5 is the covariance matrix of theGaussian input vector and is the maximum overall transmitpower. Note that mutual information is normalized by , sincedata symbols are transmitted in one OFDM symbol and that

we ignored the loss in spectral efficiency due to the presence ofthe CP. The matrix is a block-diagonal matrixgiven by

4Throughout the paper, all logarithms are to the base 2.5 stands for the trace of the matrix .

where the matrices are the covariance matricesof the Gaussian vectors , and as such determine the powerallocation across the transmit antennas and across the OFDMtones. If the channel is perfectly known at the transmitter, theoptimum power allocation is obtained by distributing the totalavailable power according to the water-filling solution [4].In OFDM-based spatial multiplexing systems, statistically in-dependent data symbols are transmitted from different antennasand different tones and the total available power is allocated uni-formly across all space–frequency subchannels [4], [11]. In thefollowing, we set, which is easily verified to result in . Using (10),

we therefore obtain

(11)

where . The quantity is the mutual infor-mation of the th MIMO OFDM subchannel. Note that, since

is random, is a random entity as well. Weshall next show that the distribution of is independent ofand hence all the have the same dis-tribution. In the following, the notation means that thedistribution of the random variable is equal to the distributionof the random variable .Proposition 1: The distribution of

is independent of and given by

for(12)

where , is an i.i.d.random matrix with entries, and . Fi-nally, denotes the th eigenvalue of .

Proof: Gaussianity of the implies Gaussianity offor . Now, using

(13)



it follows that the columns ofare uncorrelated and have the same statistics.

Denoting the first column of asand the first column of as and using (2) it followsthat

Note that the correlation matrix is independent of . We havethus shown that

with denoting an i.i.d. matrix with entries. Hence,it follows that

which can be rewritten as

Using the fact that [21], we obtain

Now, without changing the distribution, we can right-multiplyby to obtain

Using and exploiting the unitarityof , we finally get

which concludes the proof.

C. Ergodic Capacity and Outage CapacityWe shall next establish the information-theoretic value of the

results derived in Section III-B. Two scenarios are considered,the ergodic and the nonergodic case. In both cases, we assumethat the channel remains fixed within one channel use (at leastone OFDM symbol) and then changes in an independent fashionto a new realization.1) Ergodic Case: The basic assumption here is that the

transmission time is long enough to reveal the long-term er-godic properties of the fading channel. In this case, a Shannoncapacity exists and is given by with defined in(11). At rates lower than , the error probability (for a goodcode) decays exponentially with the transmission length. Theassumption here is that the fading process is ergodic, codingand interleaving are performed across OFDM symbols, andthat the number of fading blocks spanned by a codeword goes

to infinity whereas the block size (which equals the number oftones in the OFDM system multiplied by the number of OFDMsymbols spanning one channel use) remains constant (andfinite). Capacity can be achieved in principle by transmittinga codeword over a very large number of independently fadingblocks. We furthermore note that the capacity obtained for anOFDM-based spatial multiplexing system is a lower boundfor the capacity of the underlying broad-band MIMO fadingchannel.2) Nonergodic Case: In this case, we assume that a code-

word spans an arbitrary but fixed number of blocks while theblock size goes to infinity. This situation typically occurs whenstringent delay constraints are imposed, as is the case, for ex-ample, in speech transmission over wireless channels. These as-sumptions give rise to error probabilities which do not decaywith an increase of block length. A capacity in the Shannonsense does not exist since, with nonzero probability, which is in-dependent of the code length, the mutual information in (11)falls below any positive rate, as small as it may be. Thus, theconcept of capacity versus outage [22], [23] has to be invoked.Assuming that codewords extend over a single block, the outage(or failure) probability for a given rate is the probability thatfalls below that rate. In this case, capacity is viewed as a randomentity [22], [23] since it depends on the instantaneous randomchannel parameters.

IV. INFLUENCE OF CHANNEL AND SYSTEM PARAMETERS ONCAPACITY

In this section, we study the influence of channel and systemparameters on ergodic capacity and outage capacity. In partic-ular, we demonstrate that in the MIMO case, unlike the SISOcase, delay spread channels may provide advantage over flat-fading channels in terms of ergodic capacity. While the ergodiccase is to some extent amenable to analytic studies, the non-ergodic case will mainly be discussed by means of simulationresults in Section V. Analytic results seem hard to obtain in thenonergodic case. Some statements of qualitative nature on thenonergodic case will be made in this section.

A. The Ergodic CaseThe ergodic capacity is obtained from (11) as

Now, using Proposition 1, which says that theall have the same distribution given by (12),

the ergodic capacity is obtained as

(14)

where expectation is taken with respect to . A semi-analyticresult for this expectation has been provided by Telatar in [3] forthe case where . In the general case, the evaluation of theexpectation in (14) requires the concept of zonal polynomials[21] and is significantly more complicated. We shall thereforeresort to a simple asymptotic analysis by assuming that islarge. It follows from the law of large numbers that, for fixed



as gets large, . Hence, in thelarge limit, we get

(15)

where . In the low SNR regime, i.e.,for small , it follows from (15) that in the large limit

where all the higher order terms in have been neglected. Thus,in the low SNR regime, the ergodic capacity is driven by theTrace of the sum correlation matrix . Here, comparing chan-nels on the basis of fixed energy, i.e., fixed leads to theconclusion that delay spread has no impact on ergodic capacity.In the high- SNR case, we obtain

(16)

The eigenvalue spread of the sum correlation matrixtherefore critically determines ergodic

capacity. In fact, we haveLemma 2: For , the right-hand side (RHS) in (16)

is maximized for .Proof: The proof follows easily by applying Jensen’s in-

equality to the RHS of (16).Using the developments in Section IV-A and [3, The-

orem 1], it can even be shown that, for ,maximizes the

exact (finite ) expression (14). A deviation of asa function of from a constant function will therefore resultin a loss in terms of ergodic capacity or equivalently reducedmultiplexing gain. In the following, we restrict our attention tothe high SNR case. We shall next show how the propagationand system parameters impact the eigenvalues of and henceergodic capacity. Since the individual correlation matricesare Toeplitz, the sum correlation matrix is Toeplitz as well.We can thus invoke Szegö’s theorem [24] to obtain the limiting

distribution6 of the eigenvalues of as

Using (5), we obtain

(17)with the third-order theta function given by [25]

. Although this expression yields the exacteigenvalue distribution only in the limiting case , in6Note that for the eigenvalues of are characterized by a periodic

continuous function [24]. Thus, whenever we use the term eigenvalue distribu-tion, we actually refer to this function.

(a)

(b)

Fig. 3. Limiting eigenvalue distribution of the correlation matrix for thecases of (a) high spatial fading correlation and (b) low spatial fading correlation.

the case of finite good approximations of the eigenvaluescan be obtained by sampling uniformly on the unit circle[24], which allows us to assume that the eigenvalue distributionin the finite case follows the distribution given by . We arenow able to study the impact of various propagation and systemparameters on the eigenvalue distribution of and hence theergodic capacity.1) Impact of Cluster Angle Spread and Antenna

Spacing: Let us first investigate the influence of cluster anglespread and antenna spacing on ergodic capacity. For the sakeof simplicity, take one path only and its associated correlationmatrix . The limiting eigenvalue distribution of is givenby .Now, noting that the correlation function as afunction of is essentially a modulated Gaussian function withits spread increasing for increasing antenna spacing and/orincreasing cluster angle spread and vice versa, it follows that

will be more flat in the case of large antenna spacingand/or large cluster angle spread (i.e., low spatial fading cor-relation). For small antenna spacing and/or small cluster anglespread, will be peaky. Fig. 3(a) and (b) show the limitingeigenvalue distribution of for high and low spatial fadingcorrelation, respectively. From our previous discussion, it thusfollows that the ergodic capacity will decrease for increasingconcentration of and vice versa.2) Impact of Total Angle Spread: We shall next study the

impact of total angle spread on ergodic capacity. Assume thateither the individual scatterer cluster angle spreads are smallor that antenna spacing at the BTS is small or both. Hence,the individual are peaky. Now, from (17), we can seethat the limiting distribution is obtained by adding theindividual limiting distributions . Note furthermore that

is essentially a Gaussian centered around .Now, if the total angle spread, i.e., the spread of the , is large,the sum-limiting distribution can still be flat even thoughthe individual are peaky. For a given small cluster anglespread, Fig. 4(a) and (b) show example limiting distributions



(a)

(b)

Fig. 4. Limiting eigenvalue distribution of the sum correlation matrixfor fixed cluster angle spread and for the cases of (a) small total

angle spread and (b) large total angle spread.

for a three-path channel with a total angle spread of 22.5degrees and a total angle spread of 90 degrees, respectively.We can clearly see the impact of total angle spread on thelimiting eigenvalue distribution and hence on ergodiccapacity. Large total angle spread renders flat and there-fore increases ergodic capacity, whereas small total anglespread makes peaky and hence reduces ergodic capacity.This impact can further be illustrated by studying the extremecase , i.e., small cluster angle spread (or equivalentlylarge distance between BTS and SU). In this case, the sumcorrelation matrix is given by

with defined in (6). Take the simple example ,, and . In this case, for and , we

get and for and we have. For , the ergodic capacities obtained by

Monte Carlo evaluation of (14) are b/s/Hz in the caseof small total angle spread and b/s/Hz in the case oflarge total angle spread.3) Ergodic Capacity in the SISO and in the MIMO Case: It

is well known that in the SISO case delay spread channels do notoffer advantage over flat-fading channels in terms of ergodic ca-pacity[22],[23].Thiscaneasilybeseenfrom(14)bynotingthat intheSISOcase andhenceergodiccapacityis only a function of the total energy in the channel. In theMIMOcase, the situation is in general different. Fix , and take aflat-fading scenario with small cluster angle spread where

has rank 1. In this case, the matrix has rank 1 withprobabilityoneandhenceonlyonespatialdatapipecanbeopenedup, or equivalently there is no multiplexing gain. Now, comparethis scenario to a delay-spread scenariowhere and eachof the has rank 1 but the sum-corre-lation matrix has full rank. For this to happen, a sufficientlylarge total anglespread isnecessary.Clearly, in thiscase, spa-

(a)

(b)

Fig. 5. Example histograms of the mutual information in b/s/Hz in the (a)flat-fading case and (b) the high delay-spread case.

tial data pipes can be opened up and we will get a higher ergodiccapacity because the rank of is higher than in the flat-fadingcase. We note that in the case where all the correlation matricessatisfy this effect does notoccur. However, since this scenario corresponds to fully uncorre-lated spatial fading it is very unlikely.We can therefore concludethat inpracticeMIMOdelayspreadchannelsofferadvantageoverMIMOflat-fading channels in termsof ergodic capacity.Wecau-tion the reader that this conclusion is a result of the assumptionthatdelayedpathsincreasethetotalanglespread.Thisassumptionhasbeenverifiedbymeasurement foroutdoorMIMObroad-bandchannels in the 2.5-GHz band [26].

B. The Nonergodic CaseIn [22], it has been demonstrated that SISO delay-spread

channels offer significant advantage over flat-fading channelsin terms of outage probabilities or outage capacity. The outageproperties are determined by the number of diversity degreesof freedom in the channel. In our case, we have both spatialdiversity and frequency diversity available. We can thereforeexpect that both diversity sources will contribute to the outagecharacteristics of the system. Assuming that a codeword spansone block, we recall that the outage probability for a given rateis the probability that falls below that

rate. The distribution of is hard to compute analytically7 . We7In this context, we would like to point out an error in [27] and simplifications

of some of the results reported in [27], in which the defined in [27, eq.(9)] should readand [27, eq. (11)] should be replaced by

. Furthermore, it follows fromProposition 1 in this paper that [27eq. (11)] can be simplified to yield

with defined inProposition 1 in this paper. It can also be shown that the in [27]are independent of and equal the defined in Proposition 1 in thispaper, and that the in [27] are independent of and equal rankwith . With this, [27, eq. (12)] can be simplified toyield . Furthermore, [27, eq. (13)]should be replaced byand the last equation in [27] has to be replaced by

.



therefore resort to numerical studies presented in Section V andmake a few qualitative statements below.The individual all have the same distribution. The correla-

tion between the , however, strongly depends on the amountof delay spread in the system. In order to establish the value ofspace–frequency diversity in terms of outage properties, let usconsider the two limiting cases of flat-fading (i.e., no frequencydiversity) and high delay spread. In the high delay spread case,we assume that the correlation between the is small, whichcorresponds to the assumption of a high number of indepen-dently fading taps in the channel. The mean of is indepen-dent of the correlation between the and is given by (14).The variance of , however, and hence the outage propertiesdepend significantly on the amount of space–frequency diver-sity. Denote the variance of as (recall that the distribu-tion of is independent of ). In the flat-fading case, we have

, whereas in the high delay-spread case (underthe idealistic assumption of full decorrelation of the ) we ob-tain . Fig. 5(a) and (b) illustrate examplehistograms of for a 64-tone OFDM system in the flat-fadingcase and in the high delay-spread case, respectively. It can beseen that in the high delay-spread case the distribution is signif-icantly more concentrated around the mean. Take a rate of say7.5 b/s/Hz. For this rate, clearly from Fig. 5 the outage proba-bility will be much lower for the high delay-spread case than forthe flat-fading case.Since the rank of the individual correlation matrices

determines the number of spatial degrees offreedom in each path of the MIMO channel, it is to be expectedthat the rank of the individual correlation matrices and not therank of the sum correlation matrix determines the outageproperties. This can be illustrated by assuming a simple examplewhere all the have rank 1 but are such that the sum correla-tion matrix has full rank. In this case, it readily follows from (4)that the number of degrees of freedom in each path is andhence the total number of degrees of freedom in the channel is

, irrespectively of the rank of the sum correlation matrix. In the case where the individual correlation matrices are

full rank, the sum correlation matrix is also full rank,8 but thenumber of degrees of freedom in the channel will be ,and hence significantly better outage properties than in the fullycorrelated case can be expected. These statements will be cor-roborated by means of simulation results in Section V. We con-clude by noting that, while the multiplexing gain is determinedby the rank of the sum correlation matrix , the diversity gainwill be governed by the rank of the individual correlation ma-trices .

V. SIMULATION RESULTS

In every simulation example, 1000 independent Monte Carloruns were performed. Unless specified otherwise, the powerdelay profile was taken to be exponential, the number of tonesin the OFDM system was , the CP length was 64, andthe relative antenna spacing was set to . For the sakeof simplicity, we assume uniform tap spacing in all simulation

8This follows from application of [28, Lemma 4.1] to andnoting that is nonnegative.

(a)

(b)

Fig. 6. Ergodic capacity (in b/s/Hz) as a function of SNR for various values ofand (a) small cluster angle spread and (b) large cluster angle spread.

examples. Finally, the SNR was defined as SNR.

A. Simulation Results1) Simulation Example 1: In the first simulation example,

we study the impact of delay spread on ergodic capacity cor-roborating the statement that in the MIMO case delay spreadmay provide advantage over the flat-fading case in terms of er-godic capacity (provided that the total angle spread is large).The number of antennas was . In order to makethe comparison fair, we normalize the energy in the channel bysetting for all cases. The cluster angle spread wasassumed to be . In the flat-fadingcase, the mean angle of arrival was set to . In thedelay-spread case, we assumed a total angle spread of 90 de-grees. Fig. 6(a) shows the ergodic capacity (in b/s/Hz) as a func-tion of SNR for different values of . We can see that ergodiccapacity indeed increases for increasing delay spread. We canfurthermore observe that increasing the number of resolvable



Fig. 7. Outage probability for , 5, and 16 at an SNR of 10 dB.

Fig. 8. Outage probability for various values of at an SNR of 10 dB.

taps beyond 4 does not further increase ergodic capacity. Thereason for this is that the number of transmit and receive an-tennas was set to 4 and hence the maximum rank of the sumcorrelation matrix is 4. Fig. 6(b) shows the ergodic capacityfor the same parameters as above except for the cluster anglespreadwhichwas increased to .In this case, the rank of the individual correlation matrices ishigher than 1 and the improvement in terms of ergodic capacityresulting from the presence of multiple taps is less pronounced.We emphasize that, as already stated in Section IV-A, this re-sult is a consequence of the assumption that delayed paths tendto increase the total angle spread.2) Simulation Example 2: In the second simulation ex-

ample, we investigate the impact of delay spread on the outageproperties of the system. Again, for fixed , Fig. 7shows the outage probability for , 5 and 16 and anSNR of 10 dB. Here, we assumed that there is no spatialfading correlation. It is clearly seen that the outage probabilitydecreases significantly with increasing delay spread.3) Simulation Example 3: In the last simulation example,

we investigate the impact of spatial fading correlation on outage

probability. For , and ,0.5, and 0.7, Fig. 8 shows the outage probability as a functionof rate for an SNR of 10 dB. In all three simulations, the meanangles of arrival were chosen such that the sum correlation ma-trix was full rank. Again, in all cases . It can beseen that, even though the sum correlation matrix has fullrank, the outage probability depends critically on the individualcluster angle spreads and hence the rank of the individual cor-relation matrices .

VI. CONCLUSION

Based on a physically motivated model for broad-bandMIMO fading channels, we derived expressions for the ergodiccapacity and for outage capacity of OFDM-based spatial mul-tiplexing systems for the case where the channel is unknown atthe transmitter and perfectly known at the receiver. We studiedthe influence of propagation parameters and system parameterson ergodic capacity and outage probability and demonstratedthe beneficial impact of delay spread and angle spread oncapacity. Specifically, we showed that, in the MIMO case asopposed to the SISO case, delay-spread channels may provideadvantage over flat-fading channels not only in terms of outagecapacity but also in terms of ergodic capacity (provided theassumption that delayed paths tend to increase the total anglespread is true). We furthermore found that, while the multi-plexing gain is governed by the rank of the sum correlationmatrix , the diversity gain seems to be governed by the rankof the individual correlation matrices .Directions for further work include the analysis of the case

where there is scattering at both the transmitter and the receiver.A question of particular importance seems to be the analysisof the influence of scattering radii and distance between BTSand SU on capacity. Furthermore, a detailed study of the influ-ence of different antenna geometries on the capacity of OFDM-based spatial multiplexing systems appears to be of interest.This problem has been studied to some extent in [7] for thenarrow-band frequency-flat fading case.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their constructive criticism which helped to significantly im-prove the quality of the paper and the exposition. They wouldfurthermore like to thank G. Wunder and R. W. Heath, Jr., fortheir detailed comments on the paper.

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Helmut Bölcskei (M’98) was born in Mödling, Aus-tria, on May 29, 1970. He received the Dipl.-Ing. andDr. techn. degrees in electrical engineering/commu-nications from Vienna University of Technology, Vi-enna, Austria, in 1994 and 1997, respectively.From 1994 to 2001, he was with the Institute of

Communications and Radio-Frequency Engineering,Vienna University of Technology. From March 2001to January 2002, he was an Assistant Professor ofElectrical Engineering at the University of Illinoisat Urbana-Champaign. Since February 2002, he

has been an Assistant Professor of Communication Theory at ETH Zürich,Switzerland. From February to May 1996, he was a Visiting Researcher at

Philips Research Laboratories Eindhoven, The Netherlands. From Februaryto March 1998, he visited the Signal and Image Processing Department atENST Paris, France. From February 1999 to February 2001, he was an ErwinSchrödinger Fellow of the Austrian National Science Foundation (FWF)performing research in the Smart Antennas Research Group in the InformationSystems Laboratory, Department of Electrical Engineering, Stanford Univer-sity, Stanford, CA. From 1999 to 2001, he was a consultant for IospanWireless,Inc. (formerly Gigabit Wireless Inc.), San Jose, CA. His research interestsinclude communication and information theory and statistical signal processingwith special emphasis on wireless communications, multi-input multi-output(MIMO) antenna systems, space-time coding, orthogonal frequency divisionmultiplexing (OFDM), and wireless networks.Dr. Bölcskei received a 2001 IEEE Signal Processing Society Young Author

paper awards and serves as an Associate Editor for the IEEE TRANSACTIONS ONSIGNAL PROCESSING.

David Gesbert (S’96–M’99) was born in France in1969. He received theM.Sc. degree in electrical engi-neering from National Institute for Telecommunica-tions (INT), Evry, France, in 1993 and the Ph.D. de-gree from Ecole Nationale Superieure des Telecom-munications, Paris, France, in 1997.From 1993 to 1997, he was with France Telecom

Research, where he was involved in the developmentand study of receiver algorithms for digital radiocommunications systems, with emphasis on blindsignal detection. From April 1997 to October 1998,

he was a Post-Doctoral Fellow in the Smart Antennas Research Group, Infor-mation Systems Laboratory, Stanford University, Stanford, CA. In October1998, he took part in the founding engineering team of Iospan Wireless, Inc.,formerly Gigabit Wireless Inc., San Jose, CA, a startup company promotinghigh-speed wireless data networks using smart antennas. In January 2001,he joined the Signal Processing Group, Department of Informatics at theUniversity of Oslo, Norway, as an Adjunct Associate Professor in parallel to hisactivities at Iospan. His research interests are in the area of signal processingfor digital communications, blind array processing, multi-input multi-output(MIMO) systems, multi-user communications, and adaptive wireless networks.

Arogyaswami J. Paulraj (SM’85–F’91) receivedthe Ph.D. degree from the Naval Engineering Collegeand Indian Institute of Technology, Bangalore, in1973.He has been a Professor at the Department

of Electrical Engineering, Stanford University,Stanford, CA, since 1993, where he supervisesthe Smart Antennas Research Group. This groupconsists of approximately 12 researchers workingon applications of space–time signal processing forwireless communications networks. His research

group has developed many key fundamentals of this new field and helpedshape a worldwide research and development focus onto this technology. Hisnonacademic positions included Head, Sonar Division, Naval OceanographicLaboratory, Cochin, India; Director, Center for Artificial Intelligence andRobotics, Bangalore; Director, Center for Development of Advanced Com-puting; Chief Scientist, Bharat Electronics, Bangalore, and Chief TechnicalOfficer and Founder, Iospan Wireless, Inc., San Jose, CA. He has also heldvisiting appointments at Indian Institute of Technology, Delhi, LoughboroughUniversity of Technology, and Stanford University. He sits on several boardsof directors and advisory boards for U.S. and Indian companies/venture part-nerships. His research has spanned several disciplines, emphasizing estimationtheory, sensor signal processing, parallel computer architectures/algorithmsand space–time wireless communications. His engineering experience includeddevelopment of sonar systems, massively parallel computers, and more recentlybroad-band wireless systems. He is the author of over 250 research papers andholds eight patents.Dr. Paulraj has won several awards for his engineering and research con-

tributions. These include two President of India Medals, the CNS Medal, theJain Medal, the Distinguished Service Medal, the Most Distinguished ServiceMedal, the VASVIK Medal, and the IEEE Best Paper Award (Joint), amongstothers. He is a member of the Indian National Academy of Engineering.


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