Introducing Space into MIMO Capacity Calculations

22 Pages·2003·0.25 MB·English

TelecommunicationSystems 24:2–4,415–436,2003
2003KluwerAcademicPublishers.ManufacturedinTheNetherlands.
Introducing Space into MIMO Capacity Calculations
∗
TONYS.POLLOCK , THUSHARAD.ABHAYAPALAandRODNEYA.KENNEDY
[email protected]
NationalICTAustralia,ResearchSchoolofInformationSciencesandEngineering,TheAustralian
NationalUniversity,Canberra,ACT0200,Australia
Abstract. The large spectral efficiencies promised for multiple-input multiple-output (MIMO) wireless
fadingchannelsarederivedundercertainconditionswhichdonotfullytakeintoaccountthespatialaspects
ofthechannel. Spatialcorrelation, duetolimitedangular spreadorinsufficientantennaspacing, signifi-
cantlyreducestheperformanceofMIMOsystems.Inthispaperweexploretheeffectsofspatiallyselective
channelsonthecapacityofMIMOsystemsviaanewcapacityexpressionwhichismoregeneralandre-
alisticthanpreviousexpressions. Byincludingspatialinformationwederiveaclosed-formexpressionfor
ergodiccapacitywhichusesthephysicsofsignalpropagationcombinedwiththestatisticsofthescattering
environment. Thisexpression gives thecapacity of a MIMO system in termsof antenna placement and
scatteringenvironmentandleadstovaluableinsightsintothefactorsdeterminingcapacityforawiderange
ofscatteringmodels.
Keywords: MIMOsystems,wireless,capacity,spatialcorrelation,multipathchannels,non-isotropicscat-
tering
1. Introduction
Multiple-Input Multiple-Output (MIMO) communication systems using multi-antenna
arrays simultaneously during transmission and reception have generated significant in-
terest in recent years. Theoretical work of [Foschini and Gans, 13; Telatar, 41] showed
the potential for significant capacity increases in wireless channels utilizing spatial di-
versity. Consider a system employing n transmit antennas and n receive antennas
T R
in a narrowband flat fading channel. It was shown in [Foschini and Gans, 13] that as
min(n ,n ) tends to infinity then the capacity of the system grows proportionally to
T R
min(n ,n ) for fixed transmit power, provided that the fading between antennas is in-
T R
dependent and identically distributed (i.i.d.) Rayleigh. This linear capacity growth has
emerged as one of the most promising solutions for overcoming the demand for higher
bitratesinwirelesscommunications. Inreality,however,thecapacityissignificantlyre-
ducedwhenthefadesarenotindependent,butcorrelatedduetoinsufficientantennasep-
arationorangularspreadofthescattererssurroundingthearrays[FoschiniandGans,13;
Shiuetal.,38].
A significant hurdle in analyzing the capacity of wireless fading MIMO systems
is the random nature of the channel. The ergodic, or mean, capacity is often used to
∗
Correspondingauthor.
416 POLLOCKETAL.
characterize the random channel capacity. However, the ergodic calculation requires
extensivesimulationswhichlimitsanalysisintothephysicalfactorsdeterminingMIMO
capacity. A closed form expression for the ergodic capacity is derived in [Telatar, 42]
for the special case of uncorrelated Rayleigh fading. However, as mentioned above,
i.i.d.Rayleighfadingmodelsanunrealistic environment notoftenseeninpractice.
Capacityresults forvarious correlation modelsusing MonteCarlosimulations are
studied in [Chizhik et al., 8; Chuah et al., 9; Driessen and Foschini, 12; Gorokhov, 19;
Pedersen et al., 31; Shiu et al., 38], and asymptotic results and bounds on the ef-
fects of correlated channels are presented in [Chen-Nee et al., 6; Gorokhov, 19;
Loyka and Kouki, 27; Pedersen et al., 31; Sengupta and Mitra, 37; Shiu et al., 38].
However, these simulations, bounds and asymptotic results have been for a limited set
of channel realizations and/or antenna configurations. For single antenna systems it
is sufficient to only consider received signal power and/or the time varying amplitude
distribution of the channel. However, for systems employing multiple antennas, con-
sideration must also be given to the angle of arrival (AOA) of the impinging signals as
well as the spatial geometry of the array. Most channel models do not include spatial
information (antenna locations and scattering environment) explicitly. Although spatial
information is represented by the correlation between channel matrix elements there is
no direct realizable physical representation, and, therefore does not easily lend itself to
insightful capacity results. In particular, of interest is the effect on channel capacity of
antenna placement, particularly in the realistic case when antenna arrays are restricted
insize,alongwithnon-isotropic scattering environments.
In contrast, the contribution of this paper is an expression for MIMO capacity
whichovercomes theselimitations, thatis,withadditional theory formodelling scatter-
ing environments which werefine here, wederive amodel which can be readily recon-
ciled with a multitude of scattering distributions and antenna configurations and allows
ustoderiveaclosedformexpression fortheMIMOcapacity.
In this paper, we exploit the convergence of ergodic capacity as the number of
transmitantennasisincreasedtoderiveaclosed-formchannelcapacityexpressionwhich
depends on the correlation between the receive antennas. To model the correlation we
generalize the one-ring model [Lee, 24; Shiu et al., 38], to develop a closed form ex-
pression which depends on the antenna spacing and placement along with the mean
angle-of-arrival (AOA) and angular spread for a wide range of common scattering dis-
tributions. Thisallowsforacapacity expression whichcanbeevaluated foranygeneral
spatialscenario,givingsignificantinsightsintothecapacityofsuchsystems,withoutthe
needformultiplecorrelation modelsorextensivesimulations.
Theremainderofthispaperisorganizedasfollows. Insection2,wederiveaclosed
form expression for the ergodic capacity of a MIMO system and discuss limitations to
capacity scaling. In section 3, we develop a generalized 3D multipath channel model
which includes antenna locations and environmental factors to characterize the MIMO
wireless channel. Using this model, a closed-form spatial correlation expression for
generalscattering environments isderived,andwesummarizesomecommonscattering
distributions. MIMOcapacityisthenanalyzedinsection4foravarietyofphysicallyre-
INTRODUCINGSPACEINTOMIMOCAPACITYCALCULATIONS 417
alisticspatialscenarios,andspatiallimitationstocapacitygrowtharediscussed. Finally,
wemakesomeconcluding remarksinsection5.
2. Convergenceofergodiccapacity
Consider a MIMO system consisting of n transmit antenna and n receive antennas.
T R
Whenthetransmittedsignalvectoriscomposedofstatisticallyindependentequalpower
components, eachwithaGaussiandistribution, theergodicchannelcapacitywasshown
tobe[Telatar,41;FoschiniandGans,13],
(cid:1) (cid:2) (cid:2)(cid:3)
(cid:2) (cid:2)
Cerg = EH log(cid:2)(cid:2)InR + nη HH†(cid:2)(cid:2) , (1)
T
where H is the n ×n random flat fading channel matrix, assumed known at the re-
R T
ceiver, and normalized such that E{|h |2} = 1, where h is the channel gain from the
rt rt
tthtransmittertotherthreceiver. Notethatthelogarithmisbase2andgivesthecapacity
inbits-per-second-per-Hertz (bps/Hz).
Let H = [h h ... h ], where h is the n ×1 complex zero-mean Gaussian
1 2 nT t R
vector of channel gains corresponding to the tth transmit antenna, then the correlation
matrix at the receiver is defined as R (cid:1) E{h h†}, where the expectation is over all
rx t t
transmitters andchannel realizations, then
 
ρ ρ ... ρ
 11 12 1nR 
 ρ ρ 
Rrx =  ...21 22 ... ... , (2)
ρ ... ρ
nR1 nRnR
with elements Rrx|rr(cid:10) = ρrr(cid:10) corresponding to the spatial correlation between two sen-
sorsr andr(cid:10) atthereceiver.
Considerthesituation wherethetransmitarrayhasuncorrelated transmitbranches
corresponding toindependenth vectors. Thiscanoccurwhenthetransmitantennasare
t
sufficiently separated foragiven angular spread ofthe scatterers surrounding thetrans-
mit array. For example, using the correlation expression developed in section 3, a two
dimensional isotropic scattering environment with antennas separated by 0.35λ gives
correlation of0.1. However,formorerealistic scattering environments, whenthetrans-
mitter is usually mounted high above the scatterers, the angular spread is considerably
less. For angular spread of 3◦ the antennas must be separated by around 9λ to achieve
thesamelevelofcorrelation. Withsufficienttransmitantennaspacingthevectorsh are
t
independent andthesamplecorrelation matrix,definedas
1 (cid:11)nT
R(cid:10) (cid:1) h h†, (3)
rx n t t
T t=1
418 POLLOCKETAL.
converges toR forlargenumbersoftransmitantennas (n → ∞). Observing thatthe
rx T
channel matrixproductcanbeexpressed as
(cid:11)nT
HH† = h h†, (4)
t t
t=1
then for alarge number of sufficiently separated transmit antennas the ergodic capacity
converges tothecapacity C,
lim C = C (cid:1) log|I +ηR |. (5)
nT→∞ erg nR rx
The convergence error of the ergodic capacity to the expression (5) is shown in fig-
ure 1for increasing numbers of uncorrelated transmit antennas and various numbers of
receiver antennas. It is important to observe that the ergodic capacity approaches the
capacity C for finite numbers of transmit antennas, with faster convergence for smaller
numbers of receive antennas. Therefore, the capacity expression will be accurate for
many practical fixed wireless scenarios, where the receiver has a small number of an-
tennas whilstthebasestation islessrestricted ingeometrical sizeandisabletoprovide
a sufficient number of uncorrelated transmit branches. Therefore, provided the channel
does not contain keyholes1 and the transmit antennas are sufficient in number and sep-
Figure1.ConvergenceerrorofergodiccapacityCerg(1)toboundC(5)withincreasingnumberoftransmit
antennasforvariousnumbersofreceiveantennasandSNR10dB.
1Ithasbeenshowntheoreticallythatsomechannelsmayexhibitlowcapacityeventhoughthereisnospatial
correlationatthetransmitterorreceiver[Chizhiketal.,7;Gesbertetal.,15].However,noobservationsof
INTRODUCINGSPACEINTOMIMOCAPACITYCALCULATIONS 419
aration, the capacity C provides agood estimate ofthe ergodic capacity C . Notethat
erg
a bound similar to (5) has also been derived in [Abdi and Kaveh, 2] to demonstrate a
space–time cross-correlation modelforavon-Mises scattering environment.
2.1. Capacityscalinglimits
Thecapacitygivenby(5)ismaximizedwhenthereisnocorrelationbetweenthereceive
antennas, i.e.,R = I ,giving
rx nR
C = n log(1+η). (6)
max R
Therefore, in the idealistic situation of zero correlation at both transmitter and receiver
arrays (corresponding to the i.i.d. case) we see the maximum capacity scaling is linear
in the number of receive antennas. In this case, the system achieves the equivalent of
n independent nonfading subchannels, each with SNR η. This result agrees with the
R
traditional capacity formulation [Telatar, 42] which is widely used to advocate the use
ofMIMOsystems.
Conversely,wheneachpairofantennaelementsatthereceiverarefullycorrelated,
thecorrelation matrixbecomesthen ×n matrixofones,R = 1 ,andthecapacity
R R rx nR
oftheMIMOsystemwillbeminimizedto
C = log(1+n η). (7)
min R
Here the logarithmic capacity growth with increasing receiver antennas is due to an
effective increase inthe average SNRofthe single antenna case, due tothe assumption
of independent noise ateach receiver, and iswidely known asareceiver diversity array
gaineffect.
Thecapacity (5) provides an expression for the ergodic capacity without the need
forextensivesimulations. Incontrasttocurrentsimulationstudiespresentedintheliter-
ature, whicharedifficulttorelatetophysicalfactorsofthesystem,inthenextsection it
isshownthatforessentially allcommonscattering distributions, andanyarrayconfigu-
rations, itispossible tocomputeaclosedformcapacity expression.
3. Receiverspatialcorrelation forgeneraldistributionsofFarfieldscatterers
3.1. Channelmodel
The one-ring model was initially proposed in [Lee, 24] to model fixed wireless com-
munications systemswherethebasestation iselevatedandnotobstructed bylocalscat-
tering, whilst the user is uniformly surrounded by scatterers. In [Shiu et al., 38] this
model was extended to include multiple transmit and receive antennas, and the capac-
ity wasstudied for various transmit angular spreads and receive antenna geometries. In
keyhole(pinhole)degeneratechanneleffectsfrompracticalmeasurementshaveappearedintheliterature.
Therefore,inthispaperwewillassumenondegeneratechannelsonly.
420 POLLOCKETAL.
(cid:10)
Figure2.ScatteringmodelforaflatfadingMIMOsystem.gt(ψ)representstheeffectiverandomcomplex
(cid:10)
gain of the scatterers for a transmitted signal xt arriving at the receiver array from direction ψ via any
numberpathsthroughthescatteringenvironment.Thespheresurroundingthereceiveantennascontainsno
scatterersandisassumedlargeenoughthatanyscatterersarefarfieldtoallreceiveantennaslocatedwithin.
this paper we model the multipath propagation and fading correlation using a general-
ized3Done-ringmodelthatallowsformoregeneralscatteringenvironmentsatboththe
transmit and receive arrays. This model allows for greater insights into the spatial fac-
tors determining channel capacity, in particular, closed-form capacity expressions can
becomputedforawiderangeofscattering environments andantennageometries.
Consider the narrowband transmission of n statistically independent uniform
T
power signals {x ,x ,...,x }through ageneral 3D flatfading scattering environment
1 2 nT
with scatterers assumed distributed in the farfield from the receiver antennas, as shown
(cid:10)
infigure2,thentheincomingsignalfromdirection ψ atthereceiverisgivenby
(cid:12) (cid:13) (cid:11)nT (cid:12) (cid:13)
(cid:17) ψ(cid:10) = x g ψ(cid:10) , (8)
t t
t=1
(cid:10)
where g (ψ) is the effective random complex gain of the scatters for the transmitted
t
(cid:10)
signal x from the tth transmit antenna arriving at the receiver array from direction ψ.
t
Since the scatterers are assumed farfield to the receiver antennas, signals impinging on
the receiver array will be plane waves, therefore, the received signal at the rth sensor
INTRODUCINGSPACEINTOMIMOCAPACITYCALCULATIONS 421
located aty isgivenby
r
(cid:14) (cid:14)
(cid:12) (cid:13) (cid:12) (cid:13) (cid:11)nT (cid:12) (cid:13) (cid:12) (cid:13)
z = (cid:17) ψ(cid:10) e−ikyr·ψ(cid:10)d(cid:19) ψ(cid:10) = x g ψ(cid:10) e−ikyr·ψ(cid:10)d(cid:19) ψ(cid:10) , (9)
r t t
(cid:19) t=1 (cid:19)
where k = 2π/λ is the wavenumber with λ the wavelength, and d(cid:19)(ψ(cid:10)) is a surface
elementoftheunitsphere(cid:19). Therefore,thechannelgainh fromthetthantennatothe
rt
rthreceiverisgivenby
(cid:14)
(cid:12) (cid:13) (cid:12) (cid:13)
h = g ψ(cid:10) e−ikyr·ψ(cid:10)d(cid:19) ψ(cid:10) , (10)
rt t
(cid:19)
(cid:15)
withnormalization E{|g (ψ(cid:10))|2}d(cid:19)(ψ(cid:10)) = 1.
(cid:19) t
3.2. Correlationofthereceivedcomplexenvelopes
Define the normalized spatial correlation function between the complex envelopes of
tworeceivedsignals r andr(cid:10) locatedatpositions yr andyr(cid:10),respectively, as
ρrr(cid:10) = √E{zErz{zr}rEzr{(cid:10)}zr(cid:10)zr(cid:10)}, (11)
wherexdenotesthecomplexconjugateofx,andz denotesthenoiselessreceivedsignal
r
atreceiverr.
From(9)thecovariance betweensignals atsensorsr andr(cid:10) isgivenby
(cid:16) (cid:17)
(cid:14) (cid:14)
E{zrzr(cid:10)}=E (cid:11)nT xt gt(cid:12)ψ(cid:10)(cid:13)e−ikyr·ψ(cid:10)d(cid:19)(cid:12)ψ(cid:10)(cid:13)(cid:11)nT xt(cid:10) gt(cid:10)(cid:12)ψ(cid:10)(cid:10)(cid:13)eikyr(cid:10)·ψ(cid:10)(cid:10)d(cid:19)(cid:12)ψ(cid:10)(cid:10)(cid:13)
t=1 (cid:19) t(cid:10)=1 (cid:19)
(cid:14)
=σT2 (cid:11)nT E(cid:18)gt(cid:12)ψ(cid:10)(cid:13)gt(cid:12)ψ(cid:10)(cid:10)(cid:13)(cid:19)e−ik(yr·ψ(cid:10)−yr(cid:10)·ψ(cid:10)(cid:10))d(cid:19)(cid:12)ψ(cid:10)(cid:13)d(cid:19)(cid:12)ψ(cid:10)(cid:10)(cid:13), (12)
(cid:19) t=1
where σ2 = E{|x |2} ∀t is the transmit power for each antenna, and it is assumed the
T t
transmitted symbols are independent across antennas and independent of the scattering
environment. Assumingazero-meanuncorrelatedscatteringenvironment,thescattering
channel is characterized by the second-order statistics of the scattering gain function
(cid:10)
g (ψ),givenby
t
(cid:18) (cid:12) (cid:13) (cid:12) (cid:13)(cid:19) (cid:12) (cid:13) (cid:12) (cid:13)
E g ψ(cid:10) g ψ(cid:10)(cid:10) = G ψ(cid:10) δ ψ(cid:10)−ψ(cid:10)(cid:10) , (13)
t t t
whereG (ψ(cid:10)) = E{|g (ψ(cid:10))|2},then(12)simplifiesto
t t
(cid:14)
(cid:11)nT (cid:12) (cid:13) (cid:12) (cid:13)
E{zrzr(cid:10)} = σT2 Gt ψ(cid:10) e−ik(yr−yr(cid:10))·ψ(cid:10)d(cid:19) ψ(cid:10) . (14)
(cid:19) t=1
422 POLLOCKETAL.
Substitution of(14)into(11)givesthecorrelation betweenthetworeceiversensors as
(cid:14)
(cid:12) (cid:13) (cid:12) (cid:13)
ρrr(cid:10) = P ψ(cid:10) e−ik(yr−yr(cid:10))·ψ(cid:10)d(cid:19) ψ(cid:10) , (15)
(cid:19)
whereP(ψ(cid:10))isthenormalized averagepowerreceivedfromdirection ψ(cid:10),definedby
(cid:20)
P(cid:12)ψ(cid:10)(cid:13)(cid:1) (cid:15) (cid:20) nt=T1Gt(ψ(cid:10)) . (16)
nT G (ψ(cid:10))d(cid:19)(ψ(cid:10))
(cid:19) t=1 t
In two dimensional scattering environments the power distribution (16) is commonly
known as the power azimuth spectrum (PAS) [Kermoal et al., 23], or power azimuth
distribution (PAD)[Gesbertetal.,18].
To highlight the factors which effect spatial correlation the Jacobi-Anger plane
waveexpansion isemployed[ColtonandKress,10],
(cid:11)∞ (cid:12) (cid:13)
eiky·ψ(cid:10) = im(2m+1)j k(cid:18)y(cid:18) P (cosξ), (17)
m m
m=0
where ξ = (cid:19) (y,ψ(cid:10)) denotes the angle between y and ψ(cid:10), j (·) are the spherical Bessel
m
functions of the first kind, and P (·) are the Legendre polynomials of degree m. To
m
further separate the effects of the scattering and the sensor positioning, consider the
identity [Arfken,4,p. 694]
(cid:11)m (cid:12) (cid:13)
P (cosξ) = 4π Yn((cid:10)y)Yn ψ(cid:10) , (18)
m 2m+1 m m
n=−m
where(cid:10)y = y/(cid:18)y(cid:18)andYn(·)representsphericalharmonics[ColtonandKress,10,p.25],
m
thenthespatialcorrelation (15)canbeexpressedas
(cid:21) (cid:22)
(cid:11)∞ (cid:11)m (cid:12) (cid:13) y −y
ρrr(cid:10) = 4π (−i)mαmn jm k(cid:18)yr −yr(cid:10)(cid:18) Ymn (cid:18)yr(cid:10)−yr(cid:18) , (19)
m=0n=−m r r(cid:10)
wherecoefficients αn characterize thescattering environment,
m
(cid:14)
(cid:12) (cid:13) (cid:12) (cid:13) (cid:12) (cid:13)
αn = P ψ(cid:10) Yn ψ(cid:10) d(cid:19) ψ(cid:10) , (20)
m m
(cid:19)
and are independent of the antenna positions. The spatial correlation (19) is now com-
posedofasummationofterms,whereeachtermhasindependent factorscharacterizing
the scattering environment and the antenna locations. Therefore, unlike previous mod-
els,wheretheantennalocationsandthescatteringenvironmentarecoupled,theseparate
effectsofantennageometryandscatteringenvironmentonthechannelcapacitycannow
bestudied.
INTRODUCINGSPACEINTOMIMOCAPACITYCALCULATIONS 423
Figure3. Capacityof2Dand3Disotropicscatteringenvironmentsforfixedlengthaperture(1λ)uniform
linear(ULA)anduniformcircular(UCA)arraysforincreasingnumberofreceiveantennas. Insert:spatial
correlationbetweentwoantennasagainstspatialseparationforthe2Dand3Disotropicscatteringenviron-
ments.
3.2.1.Isotropicscattering environment
For the special case of isotropic scattering (omnidirectional diffuse fields) the power
distributionisgivenbytheconstants,P(ψ(cid:10)) = 1/2π2andP(ψ) = 1/2π,forthe3Dand
2D scattering environments, respectively.2 In this case, the summation in (19) reduces
toasingletermandthe3Dand2Dspatialcorrelation canbeshowntobe:
(cid:12) (cid:13)
ρr3rD(cid:10) =j0(cid:12)k(cid:18)yr −yr(cid:10)(cid:18)(cid:13), (21)
ρr2rD(cid:10) =J0 k(cid:18)yr −yr(cid:10)(cid:18) , (22)
which agree with the classical results [Cook et al., 11; Jakes, 20], where J (·) are the
n
Besselfunctions ofordern,andbydefinition j (·) = sinc(·). Withtheseanalytic forms
0
for thespatial correlation wecan compute thecapacity (5). Figure 3showsthe theoret-
ical capacity for increasing antenna numbers for a fixed aperture uniform linear (ULA)
and circular (UCA) arrays in 2D and 3D isotropic scattering with SNR of 10 dB. It is
clear from comparison of the 2D and 3D capacities that any elevation spread has little
effect on the capacity of an array in the horizontal plane. This can be seen in the insert
2The2Dscatteringenvironmentisaspecialcaseofthe3Dcasewhenthesignalsarrivefromtheazimuthal
planeonly.
424 POLLOCKETAL.
offigure3wherethespatialcorrelationforincreasingspatialseparationisshownforthe
2D and 3D isotropic scattering environments, here the twofunctions J (·) and j (·)are
0 0
qualitatively similar, particularly for low spatial separation. Therefore, without loss of
generality, wefocusonscatteringenvironmentswherethereisnegligiblepowerarriving
fromelevationangles.
3.3. Two-dimensional scattering environment
Consider a2D scattering environment where the signals arrive only from the azimuthal
plane, then y = ((cid:18)y(cid:18),θ ) and ψ(cid:10) = (1,ψ) inpolar coordinates, and (17) can beshown
y
toreduceto,
(cid:12) (cid:13) (cid:11)∞ (cid:12) (cid:13) (cid:11)∞
eiky·ψ(cid:10) = J k(cid:18)y(cid:18) +2 imJ k(cid:18)y(cid:18) cos(mξ) = J (k(cid:18)y(cid:18))e−im(θy−π/2)eimψ.
0 m m
m=1 m=−∞
(23)
Substitution of(23)into(15)givesthespatialcorrelation fora2Denvironment as
(cid:11)∞ (cid:12) (cid:13)
ρrr(cid:10) = αmJm k(cid:18)yr −yr(cid:10)(cid:18) eimθrr(cid:10), (24)
m=−∞
where θrr(cid:10) is the angle of the vector connecting yr and yr(cid:10). The coefficients αm char-
acterize any possible 2D scattering environment surrounding the receiver and are given
by
(cid:14)
2π
α = P(ψ)e−imψ dψ, (25)
m
0
where P(ψ) is the average angular power distribution, or power azimuth distribution
(PAD).
Bessel functions J (x), |n| > 0 exhibit spatially high pass behavior, that is, for
n
fixed order n, J (x) starts small and becomes significant for arguments x ≈ O(n).
n
Therefore, to compute the spatial correlation for points closely located in space, only
afewtermsinthesum(24)needtobeevaluatedinordertogiveaverygoodapproxima-
tion[Jones etal.,21]. Thus,closed-form solutions forthecorrelation canbefound pro-
vided closed-form expressions existforthescattering coefficients (25). Assummarized
next,formanycommonscatteringdistributions, thereexistsclosed-formexpressionsfor
thescatteringcoefficientsα andthusthecapacity canbecomputedforawiderangeof
m
realistic scattering environments.
3.4. Non-isotropic scattering environments
The spatial correlation as a function of receive antenna separation depends on the scat-
tering distribution surrounding the receiver. One of the most commonly used dis-
tributions is the isotropic scattering model, where the power is assumed to be uni-
form over all AOA [Jakes, 20]. However, as discussed in [Braun and Dersch, 5;