Breaking Shannon's Law With MIMO

TELENORS JOURNAL: TELEKTRONIKK 1

Breaking the barriers of Shannons capacity: Anoverview of MIMO wireless systems

David Gesbert, Jabran AkhtarDepartment of Informatics, University of Oslo, Gaustadalleen 23, P.O.Box 1080 Blindern,

0316 Oslo, NorwayCorresponding author: David Gesbert, [email protected], Tel: +47 97537812

Abstract| Appearing a few years ago in a series of in-formation theory articles published by members of the Bell-Labs, multiple-input multiple-output (MIMO) systems haveevolved quickly to both become one of the most popu-lar topics among wireless communication researchers andreach a spot in todays hottest wireless technology list. Inthis overview paper, we come back on the fundamentals ofMIMO wireless systems and explain the reasons of theirsuccess, triggered mainly by the attraction of radio trans-mission capacities far greater than those available today. Wealso describe some practical transmission techniques used tosignal data over MIMO links and address channel modelingissues. The challenges and limitations posed by deployingthis technology in realistic propagation environment are dis-cussed as well.

Keywords| Spectrum eciency, wireless, MIMO, smartantennas, diversity, capacity, space-time coding.

I. Introduction

Digital communications using MIMO (multiple-inputmultiple-output), or sometimes called \volume to volume"wireless links, has emerged as one of the most promisingresearch areas in wireless communications. It also guresprominently on the list of hot technologies that may have achance to resolve the bottlenecks of trac capacity in theforthcoming high-speed broadband wireless Internet accessnetworks (UMTS1 and beyond).

MIMO systems can be dened simply. Given an arbi-trary wireless communication system, MIMO refers to alink for which the transmitting end as well as the receivingend is equipped with multiple antenna elements, as illus-trated in Fig. 1. The idea behind MIMO is that the signalson the transmit antennas on one end and that of the receiveantennas on the other end are "combined" in such a waythat the quality (Bit Error Rate) or the data rate (Bit/Sec)of the communication will be improved. MIMO systems usespace-time processing techniques in that the time dimen-sion (natural dimension of transmission signals) is com-pleted with the spatial dimension brought by the multipleantennas. MIMO systems can be viewed as an extensionof the so-called "smart antennas" [1], a popular technol-ogy for improving wireless transmission that was rst in-vented in the 70s. However, as we will see here, the un-derlying mathematical nature of MIMO environments cangive performance which goes well beyond that of conven-tional smart antennas. Perhaps the most striking propertyof MIMO systems is the ability to turn multipath propa-

1Universal Mobile Telephone Services

gation, usually a pitfall of wireless transmission, into anadvantage for increasing the users data rate, as was rstshown in groundbreaking papers by J. Foschini [2], [3].

HCHANNEL0010110 0010110codingmodulation

weighting/mapping

weighting/demappingdemodulation

decoding

Fig. 1. Diagram for a MIMO wireless transmission system. Thetransmitter and receiver are equipped with multiple antenna ele-ments. Coding, modulation and mapping of the signals onto theantennas may be realized jointly or separately.

In this paper, we attempt to explain the promise ofMIMO techniques and explain the mechanisms behind it.To highlight the specics of MIMO systems and give thenecessary intuition, we illustrate the dierence betweenMIMO and conventional smart antennas in section II. Amore theoretical (information theory) stand point is takenin part III. Practical design of MIMO solutions involveboth transmission algorithms and channel modeling tomeasure their performance. These issues are addressed insections IV and V respectively. Radio network level con-siderations to evaluate the overall benets of MIMO setupsare nally discussed in VI.

II. MIMO systems: more than Smart Antennas

In the conventional wireless terminology, smart anten-nas refer to those signal processing techniques exploitingthe data captured by multiple antenna elements located onone end of the link only, typically at the base station (BTS)where the extra cost and space are more easily aordable.The multiple signals are combined upon transmission be-fore launching into the channel or upon reception. The goalis to oer a more reliable communications link in the pres-ence of adverse propagation conditions such as multipathfading and interference. A key concept in smart antennasis that of beamforming by which one increases the averagesignal to noise ratio (SNR) through focusing energy intodesired directions. Indeed, if one estimates the response ofeach antenna element to a desired transmitted signal, onecan optimally combine the elements with weights selectedas a function of each element response. One then can maxi-mize the average desired signal level and minimize the levelof other components (noise and/or interference).

Another powerful eect of smart antennas is called spa-tial diversity. In the presence of multipath, the received


power level is a random function of the user location and,at times, experiences fading. When using antenna arrays,the probability of losing the signal altogether vanishes ex-ponentially with the number of decorrelated antenna ele-ments. The diversity order is dened by the number ofdecorrelated spatial branches.

When multiple antennas are added at the subscribersside as well as to form a MIMO link, conventional bene-ts of smart antennas are retained since the optimizationof the transmitting and receiving antenna elements can becarried out in a larger space. But in fact MIMO links oeradvantages which go far beyond that of smart antennas [4].Multiple antennas at both the transmitter and the receivercreate a matrix channel (of size the number of receive an-tennas times the number of transmit antennas). The keyadvantage lies in the possibility of transmitting over sev-eral spatial modes of the matrix channel within the sametime-frequency slot at no additional power expenditure.

While we use information theory below to demonstratethis rigorously, the best intuition is perhaps given by asimple example of a transmission algorithm over MIMOreferred here as spatial multiplexing, which was initiallydescribed in [3], [5]. In Fig. 2, a high rate bit stream(left) is decomposed into three independent bit sequenceswhich are then transmitted simultaneously using multipleantennas. The signals are launched and naturally mixedtogether into the wireless channel as they use the samefrequency spectrum. At the receiver, after having identiedthe mixing channel matrix through training symbols, theindividual bit streams are separated and estimated. Thisoccurs in the same way as three unknowns are resolvedfrom a linear system of three equations. The separation ispossible only if the equations are independent which can beinterpreted by each antenna seeing a suciently dierentchannel. That is typically the case in the presence of richmultipath. Finally the bits are merged together to yieldthe original high rate signal.

In general though, one will dene the rank of the MIMOchannel as the number of independent equations oeredby the linear system above mentioned. It is also equal tothe algebraic rank of the channel matrix. Clearly the rankis always both less than the number of transmit antennasand less than the number of receive antennas. In turn, thenumber of independent signals that one may safely trans-mit through the MIMO system is at most equal to therank. In this example, the rank is assumed full (equal tothree) and the system shows a spectrum eciency gain ofthree. This surprising result can be demonstrated from aninformation theory standpoint.

III. Fundamental limits of wireless transmission

Todays inspiration for research and applications of wire-less MIMO systems was mostly triggered by the initialShannon capacity results obtained independently by BellLabs researchers E. Telatar [6] and J. Foschini [3], furtherdemonstrating the seminal role of information theory intelecommunications. The analysis of information theory-based channel capacity gives very useful, although idealis-

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Fig. 2. Basic spatial multiplexing (SM) scheme with 3 transmitand 3 receive antennas yielding three-fold improvement in spectraleciency.

tic, bounds on what is the maximum information transferrate one is able to realize between two points of a com-munication link modeled by a given channel. Further, theanalysis of theoretical capacity gives information on howthe channel model or the antenna setup itself may influ-ence the transmission rate. Finally it helps the system de-signer benchmark transmitter and receiver algorithm per-formance. Here we examine the capacity aspects of MIMOsystems compared with single input single output (SISO)and single input multiple output (SIMO) and multiple in-put single output (MISO) systems.

A. Shannon capacity of wireless channels

Given a single channel corrupted by an additive whiteGaussian noise (AWGN), at a level of SNR denoted by ,the capacity (rate that can be achieved with no constrainton code or signaling complexity) can be written as [7]:

C = log2(1 + ) Bit/Sec/Hz (1)

This can be interpreted by an increase of 3dB in SNR re-quired for each extra bit per second per Hertz. In practicewireless channels are time-varying and subject to randomfading. In this case we denote h the unit-power complexGaussian amplitude of the channel at the instant of obser-vation. The capacity, written as:

C = log2(1 + jhj2) Bit/Sec/Hz (2)becomes a random quantity, whose distribution can becomputed. The cumulative distribution of this \11" case(one antenna on transmit and one on receive) is shown onthe left in Fig. 3. We notice that the capacity takes, attimes, very small values, due to fading events.

Interesting statistics can be extracted from the randomcapacity related with dierent practical design aspects.The average capacity Ca, average of all occurrences of C,gives information on the average data rate oered by thelink. The outage capacity Co is dened as the data ratethat can be guaranteed with a high level of certainty, for areliable service:

ProbfC Cog = 99:9::9% (3)


We will now see that MIMO systems aect Ca and Co indierent ways than conventional smart antennas do. Inparticular MIMO systems have the unique property of in-creasing signicantly both Ca and Co.

B. Multiple antennas at one end

Given a set of M antennas at the receiver (SIMO sys-tem), the channel is now composed of M distinct coe-cients h = [h0; h1::; hM1] where hi is the channel ampli-tude from the transmitter to the i-th receive antenna. Theexpression for the random capacity (2) can be generalizedto [3]:

C = log2(1 + hh) Bit/Sec/Hz (4)

where denotes the transpose conjugate. In Fig. 3, we seethe impact of multiple antennas on the capacity distribu-tion with 8 and 19 antennas respectively. Both the outagearea (bottom of the curve) and the average (middle) are im-proved. This is due to the spatial diversity which reducesfading and thanks to the higher SNR of the combined an-tennas. However going from 8 to 19 antennas does not givevery signicant improvement as spatial diversity benetsquickly level o. The increase in average capacity due toSNR improvement is also limited because the SNR is in-creasing inside the log function in (4). We also show theresults obtained in the case of multiple transmit antennasand one receive antennas,\8 1" and \19 1" when thetransmitter does not know the channel in advance (typi-cal for a frequency duplex system). In such circumstancesthe outage performance is improved but not the averagecapacity. That is because multiple transmit antennas cannot beamform blindly.

In summary, conventional multiple antenna systems aregood at improving the outage capacity performance, at-tributable to the spatial diversity eect but this eect sat-urates with the number of antennas.

C. Capacity of MIMO links

We now consider a full MIMO link as in Fig. 1 with re-spectively N transmit and M receive antennas. The chan-nel is represented by a matrix of size M N with randomindependent elements denoted by H. It was shown in [3],[6] that the capacity, still in the absence of transmit channelinformation, is derived from:

C = log2hdetIM +

NH H

i; (5)

where is the average SNR at any receiving antenna. InFig. 3, we have plotted the results for the 3 3 and the10 10 case, giving the same total of 9 and 20 antennas aspreviously. The advantage of the MIMO case is signicant,both in average and outage capacity. In fact, for a largenumber M = N of antennas the average capacity increaseslinearly with M :

Ca M log2(1 + ); (6)In general the capacity will grow proportional with the

smallest number of antennas min(N;M) outside and no

longer inside the log function. Therefore in theory and inthe case of idealized random channels, limitless capacitiescan be realized provided we can aord the cost and space ofmany antennas and RF chains. In reality the performancewill be dictated by the practical transmission algorithmsselected and by the physical channel characteristics.

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1x19

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SIMO Diversity

MISO Diversity

Fig. 3. Shannon capacity as function of number of TX RX anten-nas. The plots shows the so-called cumulative distribution of capac-ity. For each curve, the bottom and the middle give indication of theoutage performance and average data rate respectively.

IV. Data transmission over MIMO systems

A usual pitfall of information theoretic analysis is thatit does not reflect the performance achieved by actualtransmission systems, since it is an upper bound re-alized by algorithms/codes with boundless complexity.The development of algorithms with reasonable perfor-mance/complexity compromise is required to realize theMIMO gains in practice. Here we give the intuition be-hind key transmission algorithms and compare their per-formance.

A. General principles

Current transmission schemes over MIMO typically fallinto two categories: Data rate maximization or diversitymaximization schemes. The rst kind focuses on improv-ing the average capacity behavior. For example in the caseof Fig. 2, the objective is just to perform spatial multi-plexing as we send as many independent signals as we haveantennas.

More generally, however, the individual streams shouldbe encoded jointly in order to protect transmission againsterrors caused by channel fading. This leads to a secondkind of approach in which one tries also to minimize theoutage probability.

Note that if the level of coding is increased between thetransmit antennas, the amount of independence betweenthe signals decreases. Ultimately it is possible to code thesignals to that the eective data rate is back to that of a


single antenna system. Eectively each transmit antennathen sees a dierently encoded version of the same signal.In this case the multiple antennas are only used as a sourceof spatial diversity and not to increase data rate.

The set of schemes allowing to adjust and optimize jointencoding of multiple transmit antennas are called spacetime codes (STC). Although STC schemes were originallyrevealed in [8] in the form of convolutional codes for MISOsystems, the popularity of such techniques really took owith the discovery of the so-called space time block codes(STBC). In contrast with convolutional codes which re-quire computation-hungry trellis search algorithms at thereceiver, STBC can be decoded with much simpler linearoperators, at little loss of performance. In the interest ofspace and clarity we limit ourselves to an overview of STBCbelow. A more detailed summary of the whole area can befound in [9].

B. Maximizing diversity with space-time block codes

The eld of space-time block coding was initiated byAlamouti [10] in 1998. The objective behind this work wasto place two antennas at the transmitter side and therebyprovide an order two diversity advantage to a receiver withonly a single antenna, with no a priori channel informationat the transmitter. The very simple structure of Alam-outis method itself makes it a very attractive scheme thatis currently being considered in UMTS standards.

The strategy behind Alamoutis code is as follows. Thesymbols to be transmitted are grouped in pairs. Becausethis scheme is a pure diversity scheme and results in norate increase2, we take two symbol durations to transmita pair of symbols, such as s0 and s1. We rst transmits0 on the rst antenna while sending s1 simultaneously onthe second one. In the next time-interval s1 is sent fromthe rst antenna while s0 from the second one. In matrixnotation, this scheme can be written as:

C =1p2

s0 s1s1 s

0

: (7)

The rows in the code matrix C denote the antennas whilethe columns represent the symbol period indexes. As onecan observe the block of symbols s0 and s1 are coded acrosstime and space, giving the name space-time block code tosuch designs. The normalization factor additionally en-sures that the total amount of energy transmitted remainsat the same level as in the case of one transmitter.

The two (narrow-band) channels from the two anten-nas to the receiver can be placed in a vector format ash = [h0; h1]. The receiver collects observations over twotime frames in a vector y which can then be written asy = hC + n or equivalently as yt = H^s + n, where

H^ = 1p2

h0 h1h1 h0

, s = [s0; s1]T and n is the noise

vector.

2Diversity gains can however be used to increase the order of themodulation

Because the matrices C, H^ are orthogonal by design,the symbols can be separated/decoded in a simple mannerfrom ltering of the observed vector y. Furthermore eachsymbol comes with a diversity order of two exactly. Noticenally this happens despite the channel coecients beingunknown to the transmitter.

More recently some authors have tried to extend thework of Alamouti to more than two transmit antennas [11],[12]. It turns out however that in that case it is not possibleto design a perfectly orthogonal code, except for real valuedmodulations (e.g. PAM). In the case of a general complexsymbol constellation, full-rate orthogonal codes can not beconstructed. This has therefore led to a variety of codedesign strategies to prolong Alamoutis work where one ei-ther sacrices the data rate to preserve a simple decodingstructure or the orthogonality of the code to retain a fulldata rate [13], [14], [15]. Although transmit diversity codeshave mainly been designed with multiple transmit and sin-gle receive antenna in mind, the same ideas can easily beexpanded towards a full MIMO setup. The Alamouti codeimplemented on a system with two antennas at both trans-mitter and receiver side will for example give a four orderdiversity advantage to the user and still has a simple decod-ing algorithm. However, in a MIMO situation, one wouldnot only be interested in diversity but also in increasingthe data rate as shown below.

C. Spatial Multiplexing

Spatial multiplexing, or V-BLAST (Vertical Bell LabsLayered Space-Time) [3], [16] can be regarded as a spe-cial class of space-time block codes where streams of inde-pendent data are transmitted over dierent antennas, thusmaximizing the average data rate over the MIMO system.One may generalize the example given in II in the followingway: Assuming a block of independent data C is transmit-ted over the NM MIMO system, the receiver will obtainY = HC + N. In order to perform symbol detection, thereceiver must un-mix the channel, in one of various pos-sible ways. Zero-forcing techniques use a straight matrixinversion, a simple approach which can also result in poorresults when the matrix H becomes very ill-conditionedin certain random fading events. The optimum decodingmethod on the other hand is known as maximum likelihood(ML) where the receiver compares all possible combina-tions of symbols which could have been transmitted withwhat is observed:

C^ = arg minC^jjY HC^jj (8)

The complexity of ML decoding is high, and even pro-hibitive when many antennas or high order modulations areused. Enhanced variants of this like sphere decoding [17]have recently been proposed. Another popular decodingstrategy proposed along side V-BLAST is known as nullingand cancelling which gives a reasonable tradeo betweencomplexity and performance. The matrix inversion processin nulling and cancelling is performed in layers where oneestimates a symbol, subtracts this symbol estimate from Yand continues the decoding successively [3].


Straight spatial multiplexing allows for full independentusage of the antennas, however it gives limited diversitybenet and is not always the best transmission scheme fora given BER target. Coding the symbols within a blockcan result in additional coding and diversity gain whichcan help improve the performance, even though the datarate is kept at the same level. It is also possible to sacricesome data rate for more diversity. Methods to design suchcodes start from a general structure where one often as-sumes that a weighted linear combination of symbols maybe transmitted from any given antenna at any given time.The weights themselves are selected in dierent fashions byusing analytical tools or optimizing various cost functions[11], [18], [19], [20].

In what follows we compare four transmission strategiesover a 2 2 MIMO system with ideally uncorrelated ele-ments. All schemes result in the same spectrum eciencybut oer dierent BER performance.

Figure 4 shows such a plot where the BER of various ap-proaches are compared: The Alamouti code in (7), spatialmultiplexing (SM) with zero forcing (ZF) and with max-imum likelihood decoding (ML), and a combined STBCspatial multiplexing scheme [20]. A 4-QAM constellationis used for the symbols except for the Alamouti code whichis simulated under 16-QAM to keep the data rate at thesame level. It can be seen from the gure that spatialmultiplexing with zero-forcing returns rather poor results,while the curves for other coding methods are more or lesscloser to each other. Coding schemes, such as Alamoutiand the block code give better results than what can beachieved with spatial multiplexing alone for the case of twoantennas. The Alamouti curve has the best slope at highSNR because it focuses entirely on diversity (order four).At lower SNR, the scheme combining spatial multiplexingwith some block coding is the best one.

It is important to note that as the number of antennasincreases, the diversity eect will give diminishing returns.While the data rate gain of spatial multiplexing remainslinear with the number of antennas. Therefore, for a largernumber of antennas it is expected that more weight hasto be put on spatial multiplexing and less on space timecoding. Interestingly, having a larger number of antennasdoes not need to result in a larger number of RF chains. Byusing antenna selection techniques (see for example [21]) itis possible to retain the benets of a large MIMO arraywith just a subset of antennas being active at the sametime.

V. channel modeling

Channel modeling has always been an important area inwireless communications and this area of research is partic-ularly critical in the case of MIMO systems. In particular,as we have seen earlier, the promise of high MIMO ca-pacities largely relies on decorrelation properties betweenantennas as well as the full-rankness of the MIMO chan-nel matrix. The performance of MIMO algorithms such asthose above can vary enormously depending on the realiza-tion or not of such properties. In particular spatial mul-

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Bit E

rror R

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Alamouti Linear (16QAM)SM ZF (4QAM)SM ML (4QAM)STBC ML (4QAM)

Fig. 4. Bit Error Rate (BER) comparisons for various transmissiontechniques over MIMO. All schemes results use the same transmissionrate.

tiplexing becomes completely inecient if the channel hasrank one. The nal aim of channel modeling is thereforeto get and understanding of, by the means of convertingmeasurement data into tractable formulas, what perfor-mance can be reasonably expected from MIMO systems inpractical propagation situations. The other role of channelmodels is to provide with the necessary tools to analyzethe impact of selected antenna or propagation parameters(spacing, frequency, antenna height etc.) onto the capacityto influence the system design in the best way. Finally,models are used to try out transmit and receive processingalgorithms with more realistic simulation scenarios thanthose normally assumed in the literature.

A. Theoretical models

The original papers on MIMO capacity used an idealis-tic channel matrix model consisting of perfectly uncorre-lated (i.i.d.) random Gaussian elements. This correspondsto a rich multipath environment, yielding maximum exci-tation of all channel modes. It is also possible to deneother types of theoretical models for the channel matrixH which are not as ideal. In particular we emphasize theseparate roles played by antenna correlation (on transmitor on receive) and the rank of the channel matrix. If fullycorrelated antennas will lead to a low rank channel, theconverse is not true in general.

Let us next consider the following MIMO theoreticalmodel classication, starting from Foschinis ideal i.i.d.model, and interpret the performance. In each case belowwe consider a frequency-flat channel. In the case of broad-band, frequency selective, channels, a dierent frequency-flat channel can be dened at each frequency. Uncorrelated High Rank (UHR, a.k.a. i.i.d.) model: Theelements of H are i.i.d. complex Gaussian. Correlated Low Rank (CLR) model: H = grxgtxurxutxwhere grx and gtx are independent Gaussian coecients(receive and transmit fading) and urx and utx are xed


deterministic vectors of size M 1 and N 1, respectively,and with unit modulus entries. This model is obtainedwhen antennas are placed to close to each other or thereis too little angular spread at both the transmitter andthe receiver. This case yields no diversity nor multiplexinggain whatsoever, just receive array/beamforming gain. Wemay also imagine the case of uncorrelated antennas at thetransmitter and decorrelated at the receiver or vice versa. Uncorrelated Low Rank (ULR) (or \pin-hole" [22])model: H = grx gtx, where grx and gtx are independentreceive and transmit fading vectors with i.i.d. complex-valued components. In this model every realization of Hhas rank 1 despite uncorrelated transmit and receive an-tennas. Therefore although diversity is present capacitymust be expected to be less than in the UHR model sincethere is no multiplexing gain. Intuitively, in this case thediversity order is equal to min(M;N).

B. Heuristic models

In practice of course, the complexity of radio propaga-tion is such that MIMO channels will not fall completelyin either of the theoretical cases described above. Antennacorrelation and matrix rank are influenced by as many pa-rameters as the antenna spacing, the antenna height, thepresence and disposition of local and remote scatterers, thedegree of line of sight and more. Fig. 5 depicts a generalsetting for MIMO propagation. The goal of heuristic mod-els is to display a wide range of MIMO channel behaviorsthrough the use of as few relevant parameters as possiblewith as much realism as possible.

A good model shall give us answers to the following prob-lems: What is the typical capacity of an outdoor or indoorMIMO channel? What are the key parameters governingcapacity? Under what simple conditions do we get a fullrank channel? If possible the model parameters should becontrollable (such as antenna spacing) or measurable (suchas angular spread of multipath [23], [24]) which is not al-ways easy to achieve.

The literature on these problems is still very scarce. Forthe line-of-sight (LOS) case it has only been shown howvery specic arrangements of the antenna arrays at thetransmitter and the receiver can maximize the orthogo-nality between antenna signatures and produce maximumcapacity as reported in [25]. But, in a general situationwith fading, which is the true promising case, this work isnot applicable.

In the presence of fading, the rst step in increasing themodels realism consists in taking into account the correla-tion of antennas at either the transmit or receive side. Thecorrelation can be modeled to be inversely proportionalto the angular spread of the arriving/departing multipath.The experience suggests that higher correlation at the BTSside can be expected because the BTS antenna is usuallyhigher above the clutter, causing reduced angular spread.In contrast the subscribers antenna will be buried in theclutter (if installed at street level) and will experience moremultipath angle spread, hence less correlation for the samespacing. The way MIMO models can take correlation into

LOCAL RX SCATTERERS

LOCAL TX SCATTERERS

REMOTE SCATTERERS

Fig. 5. MIMO channel propagation. The complicated dispositionof the scatterers in the environment will determine the number ofexcitable modes in the MIMO channel.

account is similar to how usual smart antenna channel mod-els do it. The channel matrix is pre- (or post-) multipliedby a correlation matrix controlling the antenna correlationas function of the path angles, the spacing and the wave-length. For example, for a MIMO channel with correlatedreceive antennas, we have:

H = R1=2r;drH0 (9)

where H0 is an ideal i.i.d. MIMO channel matrix andRr;dr is the M M correlation matrix. r is the receiveangle spread and dr is the receive antenna spacing. Dier-ent assumptions on the statistics of the paths directionsof arrival (DOA) will yield dierent expressions for Rr;dr[26], [27], [28]. For uniformly distributed DOAs, we nd[27], [26]

[Rr;dr ]m;k =1S

i=S12Xi=S12

e2j(km)dr cos(2 +r;i) (10)

where S (assumed odd) is the number of paths with corre-sponding DOAs r;i. For \large" values of the angle spreadand/or antenna spacing, Rr;dr will converge to the iden-tity matrix, which gives uncorrelated fading. For \small"values of r; dr, the correlation matrix becomes rank de-cient (eventually rank one) causing fully correlated fading.The impact of the correlation on the capacity was analyzedin several papers, including [29]. Note that it is possible togeneralize this model to include correlation on both sidesby using two distinct correlation matrices:

H = R1=2r;drH0R1=2t;dt

(11)

B.1 Impact of scattering radius

One limitation of simple models like the one in (11) isthat it implies that rank loss of H can only come from


rank loss in Rr;dr or in Rt;dt , i.e. a high correlationbetween the antennas. However as suggested by the theo-retical model "ULR" above, it may not always be so. Inpractice such a situation can arise where there is signicantlocal scattering around both the BTS and the subscribersantenna and still only a low rank is realized by the chan-nel matrix. That may happen because the energy trav-els through a narrow "pipe" , i.e. if the scattering radiusaround the transmitter and receiver is small compared tothe traveling distance. This is depicted in Fig. 6. Thissituation is referred to a pinhole or keyhole channel in theliterature [22], [30].

scatter ring

scatter ring

Narrow pipe

Users area

BTS Area

dddddddd

Fig. 6. An example of pin-hole realization. Reflections around theBTS and subscribers cause uncorrelated fading, yet the scatter ringsare too small for the rank to build up.

In order to describe the pinhole situation and more, so-called double scattering models are developed that take intothe account the impact of the scattering radius at the trans-mitter and at the receiver. The model is based on a sim-plied version of Fig. 5 shown in Fig. 7 where only localscatterers contributing to the total aperture of the antennaas seen by the other end are considered. The model can bewritten as [22]:

H =1pS

R1=2r;drH0;rR1=2S;2Dr=S

H0;tR1=2t;dt

: (12)

where the presence of two (instead of one) i.i.d. randommatrices H0;t and H0;r account for the double scatteringeect. The matrix RS;2Dr=S dictates the correlation be-tween scattering elements, considered as virtual receive an-tennas with virtual aperture 2Dr. When the virtual aper-ture is small, either on transmit or receive, the rank of theoverall MIMO channel will fall regardless of whether theactual antennas are correlated or not.

N TXsM RXs

Dt

Dr

dr

dt

R

Or sO O t

Fig. 7. Double scattering MIMO channel model

C. Broadband channels

In broadband applications the channel experiences fre-quency selective fading. In this case the channel model canbe written as H(f) where a new MIMO matrix is obtainedat each frequency/subband. This type of model is of inter-est in the case of orthogonal frequency division multiplex-ing (OFDM) modulation with MIMO. It was shown thatthe MIMO capacity actually benets from the frequencyselectivity because the additional paths that contribute tothe selectivity will also contribute to a greater overall an-gular spread and therefore improve the average rank of theMIMO channel across frequencies [31].

D. Measured channels

In order to validate the models as well as to foster the ac-ceptance of MIMO systems into wireless standards, a num-ber of MIMO measurement campaign have been launchedin the last two years, mainly led by Lucent and ATT Labsand by various smaller institutions or companies such asIospan wireless in California. More recently Telenor FoUput together its own MIMO measurement capability.

Sample of analysis for UMTS type scenarios can be foundin [32], [33], [34]. Measurements conducted at 2.5GHz forbroadband wireless access applications can be found in [35].So far the results reported largely conrm the high level ofdormant capacity of MIMO arrays, at least in urban or sub-urban environments. Indoor scenarios lead to even betterresults due to a very rich multipath structure. Eigenvaluesanalysis reveal that a large number of the modes of MIMOchannels can be exploited to transmit data. Which par-ticular combination of spatial multiplexing and space timecoding will lead to the best performance complexity trade-o over such channels remains however an area of activeresearch.

VI. System Level Issues

A. Optimum use of multiple antennas

Multiple antenna techniques are not new in commercialwireless networks. Spatial diversity system, using two orthree antenna elements, co- or cross-polarized, have been inuse since the early stages of mobile network deployments.More recently, beamforming-type BTS products equippedwith ve to ten or more antennas have been oered on themarket. These products are using diversity to improve thelink budget and the beamforming capability to extend thecell range or help in load balancing.

Beyond the information theory aspects addressed earlier,there are signicant network-level dierences between thebeamforming approach and the MIMO approach to usingmultiple antennas.

While beamforming systems tend to use a larger numberof closely spaced antennas, MIMO will operate with typi-cally fewer antennas (although the only true constraint isat the subscriber side rather than at the BTS side). Fur-thermore the MIMO antennas will use as much space as canbe aorded to try and realize decorrelation between the el-ements while the directional-based beamforming operation


imposes stringent limits on spacing. Also most MIMO algo-rithms focus on diversity or data rate maximization ratherthan just increasing the average SNR at the receiver or re-ducing interference. Finally beamforming systems thrive innear line of sight environments because the beams can bemore easily optimized to match one or two multipaths thana hundred of them. In contrast, MIMO systems turn richmultipath into an advantage and lose their multiplexingbenets in line of sight cases.

Because of these dierences, the optimal way of usingmultiple antenna systems, at least at the BTS, is likelyto depend on the situation. The search for compromis-ing solutions, in which the degrees of freedom oered bythe multiple antennas are best used at each location, isan active area of work. A key to this problem resides inadaptive techniques, which through the tracking of envi-ronment/propagation characteristics are able to pick theright solution at all times.

B. MIMO in broadband Internet access

One unfavorable aspect of MIMO systems, when com-pared with traditional smart antennas, lies in the increasedcost and size of the subscribers equipment. Although asensible design can extract signicant gains with just twoor three antennas at the users side, it may already provetoo much for simple mobile phone devices. Instead wire-less LAN modems, PDAs and other high speed wirelessInternet access, xed or mobile, devices constitute the realopportunity for MIMO because of less stringent size andalgorithmic complexity limitations. In Fig. 8 we show thedata rates achieved by a xed broadband wireless accesssystem with 23 MIMO. The realized users data rates arecolor coded from 0 to 13.5 Mb/s in a 2MHz RF channel3,function of the users location. The access point is locatedin the middle of an idealized hexagonal cell. Detailed as-sumptions can be found in [36]. The gure illustrates theadvantages over a system with just one transmit antennaand one or two receive antennas. Current studies demon-strating the system level advantages of MIMO in wirelessInternet access focus mainly on performance. While verypromising, the evaluation of overall benets of MIMO sys-tems taking into account deployment and cost constraintsis still in progress.

VII. Conclusions

This paper reviews the major features of MIMO linksfor use in future wireless networks. Information theory re-veals the great capacity gains which can be realized fromMIMO. Whether we achieve this fully or at least partiallyin practice depends on a sensible design of transmit and re-ceive signal processing algorithms. More progress in chan-nel modeling will also be needed. In particular upcomingperformance measurements in specic deployment condi-tions will be key to evaluate precisely the overall benetsof MIMO systems in real-world wireless systems scenariossuch as UMTS.

3A user gets zero if the link quality does not satisfy the target BER.

1 1 1 2 2 3

0.0Mbps 1.1Mbps 2.2Mbps 3.4Mbps 4.5Mbps 5.6Mbps 6.7Mbps 9.0Mbps 11.2Mbps13.5Mbps

Fig. 8. User rates in 2MHz FDD channels in a xed wireless accesssystem. The plots shows the relative gains between various numberof antennas at transmiter receiver (SISO, SIMO, MIMO).

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Breaking Shannon's Law With MIMO

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