Mobile Antenna Array

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Antenna Arrays in Mobile Communications:Gain, Diversity, and Channel Capacity

J~zrrgenBach AndersenCenter for Personkommunikation, Aalborg UniversityFr. Bajers vej 7, DK -9220 Aalborg E ast, DenmarkTel: (45)98-15 8522; Fax: (45) 98-15 1583; E-mail: [email protected]

Keywords: Antenna arrays; Land mobile radio cellular systems; antenna gain; land mobile radio diversity systems; informationrates; array antenna theory

1. Introductionre antenna arrays in mobile communications different fromA rrays in other applications? Yes, sometimes, and it is thepurpose o f this paper to explain, in a tutorial fashion, when this isthe case, and what this means for path loss in link calculations.

One thing is the classical gain of an antenna, which we have tounderstand in a new way. Another thing is the possibility for twoarrays, in a scattering environm ent, to create parallel channels, andthus, in effect, act as man y independent antennas at the same time,carrying much mo re traffic over the same bandwidth [1-31,

Le t us review the well-known free-space situation first. Con-sider two linear arrays of M an d N lements, with the assumptionthat M >N. or convenience, it is assumed that the left array of Melements is the transmitting array. The path-loss equation is givenby the classical Friis formula,

Under som e qualifying assumptions-like neglect of mutual cou-pling and element pattern -the two gains are M and N, respec-tively. A standard spacing of half a wavelength is also assumed.The underlying assumption, here, is that the other antenna lookslike a point source as seen from one antenna, and thus a plane wavefrom one specific direction is radiated (and received). If instead ofdirect line-of-sight (LOS) there was just one path, like scatteringfrom a dom inant scatterer, then the equation would still be valid asfar as the antenna gains are concemed. The m ore usual situation inmobile radio is a wide angular scattering (Figure l) , where theangular spread as seen from the mobile is often large, and wherethe scattering as seen from the base station depends on the heightand the general environment. We have h ighlighted one path out ofmany g oing from on e element at the transmitter to one element inthe receiver. The distance dependence in Equation (1) is, of course,also changed by the scatterers, but this is not our concern here. Thepoint of v iew is that the information-carrying signal is scattered inmany directions, and the general question is, how should we

This article is based on a paper presented at COMMSPHERE 99.

V

\

Y0

YN

MFigure 1. Two linear arrays of M nd N elements in a scatteringenvironment.organize the combining of elements to maximize the power trans-fer? At each antenna, we assum e that we can apply different com-plex weights at each element, so there is complete freedom at bothends to combine the a ntenna signals.

The notation is U =(U , U2, . . ,UN)T for the receiver weights,an d V =( q , V2 , . . , VM ) T for the transmitter, where T means trans-pose. Both vectors are normalized to unit length. The narrowbandchannel connecting all elements may then be described by a n M byN omplex matrix H ,where H v is the complex transmission coef-ficient from elementj, on the left side, to element i, on the rightside.

2. Antenna Gain and Diversity2.1 Transmitter Weights Fixed

For simplicity, assume that the transmitter weights are fixed,like in a beam mode where all the elements of V, are identical inmagnitude, with a uniform phase difference. This would be anobvious choice whe n the transmitter does not know the channel.

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On the receive side, the incident signal is S =H*VoAs is well known, the receiver maximum-gain weights are

and the received power is the sum of the powers from the N le-ments:

( 3 )For many scatterers and non-LOS, the power at one element willbe exponentially distributed (each transmission coefficient isRayleigh fading), and the distribution of the sum o f the powers willdepend on the correlation. If the scatterers in Figure 1 shrink to annarrow angular range, then they w ill appear as a point source, andthe fading will be spatially flat. In this case, the signals will behighly correlated:

Table la . The mean link gain for the beam modefor both arrays.

Table lb. The mean link gain for maximum-gain combiningfor both arrays.

I Transmitter IHigh SDread

If we define the array gain as the mean value of the received powerrelative to one element at each end, then the array gain in this caseis clearly N.

Table IC.The diversity order of the link formaximum-gain combining.

Receiver Low ReceiverSpread High Spread

In the other extreme, where the scatterers are spread out in alldirections, the signals will be uncorrelated but Equation (4) willstill be valid, so the array gain is also N in this case. On top of this,we get Nth order diversity gain for the uncorrelated case, where alltransmission coefficients are fading independently.. If instead of the maximum gain combining of Equation (l),we had chosen the beam mode for the receiver as well, i.e.

( 5 )

then it can be shown that the mean receiver-array gain is 1. This isnatural, since a narrow beam does not help when the energy isspread out in all directions. The above may be summarized as inTables l a- lc , for the mean link gain and diversity order.As an example, let us discuss the upper-right comers ofTables la-lc, with low angular spread (high correlation) at thetransmitter and high angular spread (low correlation) at thereceiver. In the beam mode, the transmitter array sees a point

source and has gain M , the receiver has gain 1, and the joint linkhas a gain of M . In the combining m ode, at the receiver the meangain is N as is the diversity order), so the total gain is M N . Thelower-right comers of Tables la-lc refer to the joint optimizationof the two arrays, which is the subject of the following section.

2.2 Transmit-Receive Gain fo rWide Angular Spreads

The SVD (singular-value decomposition) [4] is an attractivetechnique for solving the joint optimization of the two sides, thetransmit side and the receive side. In the following, it is assumed

TransmitterLow SpreadTransmitterHigh Spread

that the complex matrix (the channel matrix) is known at both thetransmitter and receiver. This is not so strange as it sounds, i.e., ina TD D (time-division-duplex) case, where the channel is reciprocalbecause the frequency is the same in both directions, the channelwill be known at the transmitter as well, unless the channelchanges too rapidly.

An SVD expansion is a description of H as given by

where D is a diagonal matrix of real, non-negative singular values,the square roots of the eigenvalues of G = H ' H , a Hermitianmatrix. The colum ns of the orthogonal matrices U and V are thecorresponding singular vectors. Since G may be written as

it follows that the columns of V are eigenvectors o f GThe SVD is particularly useful for interpretation in theantenna context. Writing Equation (7 ) differently,

for one particular eigenvalue, it is noted that VI is the transmitweight factor for excitation of the singular value & A receive

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Mean gain of (N,N)array2018161412

8 108642n I I I I I I I I I" 1 2 3 4 5 6 7 8 9N

Figure 2. The gain relative to one element of ( N ,N ) arrays in acorrelated situation ( p =0), and in an uncorrelated case( p=1 ) . The upper bound equals 4 N , and is the asymptoticupper bound for the gain for N tending to infinity.

weight factor of U; , conjugate match, gives the receive voltage,and the square o f that gives the received power:

(9)

Thus, the eigenvalues correspond to the power gains, and all weneed to do is to extract the largest eigenvalue with correspondingV and U vectors, and maximum gain is achieved for that par-ticular channel matrix.

Recent results [5], onceming the distribution of the eigen-values of a random Hermitian matrix, can give some insight intothe maximum gain and how it varies with M an d N . In the asymp-totic limit when M an d N are large, it may be shown that the largesteigenvalue is bounded above by

This indicates that the joint link gain can no longer be separatedinto a transmitter-antenna gain and a receiver-antenna gain. ForM =N , the gain equals 4N, which is much less than the N 2available when the spreading is small. As an explanation, comparethe ( M , N ) case with the (1,N) case. In the latter case, we have Ndegrees of freedom (elements of the weight vector) and N ifferentsignals, which matches. In the former case, we have M +Ndegrees of freedom but M N different signals, a clear mismatch forlarge M an d N , and we m ust accept a reduced gain. Although thegain is reduced, it is still greater than M an d N , and the diversityorder is truly large, namely MN. This follows from the fact thatthere are M N different signals, and the probability of having allpaths fading at the same time is vanishing small. Thus, the fadinghas practically disappeared for reasonable values of M an d N .

The mean gains for p =0 (uncorrelated signals) are shownin Figure 2, together with the upper bound and the gain for the cor-related, free-space case, N 2 . Fo r N =10 , the true mean gain isjust 1 dB below the upper bound. Thus, the price to pay for therandom scattering is a diminishment of the gain from N 2 to 4N forN large. For a partly correlated case, we can expect the gain to liebetween the p =0 and the p =1 cases.

2.3 Implications for Path LossEquation (1) implies that the received power decreases as the

square of the carrier frequency for frequency-independent gainslike dipole, or other small handset, antennas. If instead the antennaapertures are introduced in the free-space case (or in the case ofsmall angular spread s), the situation reverses, as is well known. LetAI =M - , A2 =N- hen, the path-loss equation for the cor-related case reads

R2 R24 r 4n

& = 4 - AI '42R 2 R 2 '

and for the uncorrelated case,

Pr =e 4 n R 2This has the interesting result that the frequency dependence hasdisappeared for the uncorrelated case. The assumptions behindEquation (12) are the sam e as in Equation (10).

The equations imply that we can only gain from going tohigher carrier frequencies for given areas of antenna arrays, whenatmospheric absorption and diffraction are ignored. When thespreading is small, the joint gain may be very high, and when thespreading is large, the worst situation is a constant power: it doesnot decrease as in Equation (1). The true benefit would be thatthere is more bandwidth available at the higher microwave fre-quencies.

"

YM N

r 0

Figure 3. N different signals are distributed over the M anten-nas, with the resulting N parallel channels.

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3. Spectral Eff iciency of Parallel Channels

An illustrative case is shown in Figure 4, for a basic meanSN R of 10 dB for one antenna. The spectral efficiency hasincreased in the mean from 2.9 to 18.8 b/s/Hz, when using 12-

The joint gain of the link corresponded to the largest eigen-value of G. The first question is, how many eigenvalues are there?For the completely correlated case, there is only one, but for theuncorrelated case, there are m in (M , N ) d istinct eigenvalues withcorresponding pairs of V and U vectors. Since we have beenassuming M >N , e have N igenvalues. This tells us that it ispossible to send multiple sets of data over the same physical chan-nel for the same band width, since the weight vectors correspondingto separate eigenvalues are orthogonal [l-3, 6-71, In this way, thespectral efficiency can be greatly increased. Physically, the manydifferent paths in the environment create the possibility o f multiplechannels, as illustrated in Figure 3: All we have to do is todiagonalize the channel matrix, as was done in the previous sec-tion.

The seemingly complicated, random environment in mobileand personal communications gives rise to some new possibilitiesin the antenna area. The received fields come from many differentdirections, and fo r a noise-limited system, it is important to absorball the energy. For those situations where the channel transfersfrom all elements to all elements are known, it is possible tomaximize the total transfer power by jointly adjusting the antennaweights in the arrays. The resulting joint antenna gain depends onthe angular spread of the environment as seen from the two anten-nas: In on e extreme of high correlation, we get the usual free-spacegain, while in the other extreme, we get a smaller gain, due to thelack of degrees of freedom. If we introduce antenna apertures inthe link budget instead of directivities, it is interesting to observethat we obtain a link gain that is independent of carrier frequency.This seems to indicate that it should be worthwhile to go to higherfrequencies to get the high er bandwidths available.

The situation is easily described by Shannons information Th e other possibility in a wide-scattering situation is to applydifferent information signals to the various antennas, and to utilizethe inherently high spectral efficiencies of the channel. This maybe done by effectively creating a number of parallel orthogonalchannels, which all have high gain and high-order diversity, espe-number of receive antennas.

measure:C =log2 (1 +P / c ) b/s/Hz, (13)

where p/ is the signal-to-noise ratio, S N R , for one channel, For cially when the number Of transmit antennas is higher than theN arallel chann els, the capacities ad d:

NC = C l o g 2( l+l i4/ f f ) ,I

(14) 5. Referenceswhere 4 is the power put into channel i, and 1, s the gain of thatchannel. The total power, P , as the sum of all the separate powers,is assumed constant, to make comparisons fair. The problem ofassigning powers to the individual channels may be solved by thewater-filling scheme [8], which makes sure that the channelswith the highest gains get most of the power. If P is very small,only the largest gain gets any power, and we are in the situation of

1. G . J. Fosch ini, Layered Space -Time Architecture for WirelessCommu nication in a Fading Environment W hen Using Multi-Ele-ment Antennas, Bell Labs Technical Journal, Autumn 1996, pp.41-59.2. G. G. Raleigh, J. M. Cioffi, Spatio-Temporal Coding for Wire-less Communication, IEEE Transactions on Communications,the one-eig envalu e case of the previous section. COM-46, 3, March 1998, pp. 357-366.

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3. J. H. inters, On the Capacity of Radio Communication Sys-tems with Diversity in a RayIeigh Fading Environment, IEEEJournal on Selected Areas in Communications,5 , 5, June 1987, pp.871-878.4. L. L. Scharf, Statistical Signal Processing, Reading, Pennsylva-nia, Addison-Wesley, 1991.5 . U. Haagerup, S. Thorbjmsson, Random Matrices with Com-plex Gaussian Entries, Centre for Mathematical Physics and Sto-chastics, Aarhus University, Denmark, Research Report No. 14,September 1998.6. V. Tarokh, N. Seshadri, A. R. Calderbank, Space-Time Codesfor High Data Rate Wireless Communication: Performance Crite-rion and Code Construction, IEEE Transactions on InformationTheory, IT-4 4,2, March 1998, pp. 744-765.7 . I. E. Telatar, Capacity o f Multi-antenna Gaussian Channels,AT&T Bell Laboratories technical note, 1996.8. R. G. Gallager, Information Theory and Reliable Communica-tion, New York, Wiley, 1968.

Introduc ing Feature Article Author

Dr. Jergen Bach Andersen is a professor at Aalborg Un i-versity, Denmark, where he heads a research center, the Center forPersonkommunikation, dealing with modem wireless systems. Dr.Andersen is a graduate from the Technical University of Denmark(1961),where he also obtained th e DrTechn degree, in 1971, at theElectro magn etics Institute. Sin ce 19 73, he has been in Aalborg,interrupted by visiting positions in New Zealand, Austria, and theUnited States. He has published widely on various aspects ofantenn as and propagation for wireless communications, and alsohas an inter est in the bioelectromagnetic issues of radio waves, inwhich capacity he ha s been an advisor for the European Union. Heis a former Vice President of U R S I , the International Union ofRadio S cience, and is a Fellow of the IEEE. He serves on the edito-rial boards o f several journals, and is presently an Associate E ditorfor the IEEE Transactions on Antennas and Propagation. .:e

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