Antenna Saturation Effects on MIMO Capacity

Transcription

1 Antenna Saturation Effects on MIMO Capacity T S Pollock, T D Abhayapala, and R A Kennedy National ICT Australia Research School of Inforation Sciences and Engineering The Australian National University, ACT 2, Australia Eail: tonypollock@anueduau Abstract A theoretically derived antenna saturation point is shown to exist for MIMO systes, at which the syste suffers a capacity growth decrease fro linear to arithic with increasing antenna nubers We show this saturation point increases linearly with the radius of the region containing the receiver antennas and is independent of the nuber of antennas Using an alternative forulation of capacity for MIMO systes we derive a closed for capacity expression which uses the physics of signal propagation cobined with statistics of the scattering environent This expression gives the capacity of a MIMO syste in ters of antenna placeent and scattering environent and shows that the saturation effect is due to spatial correlation between receiver antennas I INTRODUCTION Multiple-Input Multiple-Output MIMO counication systes using ulti-antenna arrays siultaneously during transission and reception have generated significant interest in recent years Theoretical work of [1] and [2] showed the potential for significant capacity increases in wireless channels utilizing spatial diversity However, in reality the capacity is significantly reduced when the signals received by different antennas are correlated, corresponding to a reduction of the effective nuber of sub-channels between transit and receive antennas [2], [3] Previous studies have given insights and bounds into the effects of correlated channels [3] [5], however ost have been for a liited set of channel realizations In contrast, one contribution of this paper is an alternative for for capacity which overcoes current liitations, that is, with additional theory for odelling scattering environents which we refine here, we derive a odel which can be readily reconciled with a ultitude of scattering odels and allows us to derive a closed for capacity expression Using this new odel we show that there is a only a sall subset of the eigenvalues generated by a spatial correlation atrix between eleents of an array which effect capacity, regardless of the nuber of antennas in the array This result leads to the channel capacity suffering a saturation effect with increasing antenna nubers, after which additional antennas give liited capacity growth Although previous work has focused on eigenvalues as a eans to bound or siplify the tedious ergodic capacity coputation [1], [5] [7], the ephasis was on the eigenvalues of the rando channel atrix product giving little insight into environental effects on capacity In this paper we expose characteristics of the eigenvalues of an antenna spatial correlation atrix that leads to an antenna saturation effect, which has the detriental effect of reducing capacity growth fro linear to arithic with increasing antenna nubers II CAPACITY OF MULTIPLE ANTENNA SYSTEMS Consider a MIMO syste consisting of S transitting antennas with statistically independent equal power coponents each with Gaussian distributed signals, and Q receiving antennas Then for a fixed linear channel, represented by a S Q channel atrix H, with additive white Gaussian noise the channel capacity is given by [1], [2] C H = I Q + η S HH 1 where η is the average signal-to-noise ratio SNR at each receiver branch, is the deterinant operator, and the Heritian operator In the case of a rando channel odel the channel atrix is stochastic hence the capacity given by 1 is also rando In this situation the ean ergodic capacity is obtained by taking the expectation of capacity C H over all possible channel realizations, { C = E I Q + η } S HH 2 Equation 2 is often used in Monte Carlo siulations to provide capacity results for different channel atrix odels, however these siulations offer little insight and fail to provide a rigorous deonstration into factors deterining capacity Soe analytical lower and upper bounds on the ergodic capacity have been derived eg, see [2], [3], ore recently, [4] gave an upper bound on ergodic capacity based on the correlation between each channel atrix eleent In contrast, we use a capacity forula based on the spatial correlation of the antennas at the receiver and the statistics of the scattering environent, and thus reove any explicit dependencies on the rando channel atrix Due to subtle differences between our capacity result and that of the classical forulation we provide a detailed derivation To study the effects of correlation on capacity we restrict ourselves to receive correlation only and assue no transit correlation, a valid assuption given the less geoetrical size restrictions for base-station arrays A Alternative MIMO Capacity Derivation Let x = [x 1, x 2,, x S ] T be the vector of sybols sent by the S transitting antennas, n = [n 1, n 2,, n Q ] T be the zero ean additive white gaussian noise vector, and

2 y = [y 1, y 2,, y Q ] T be the vector of received sybols, where T denotes the vector transpose, then y = r + n 3 where r is the vector of Q noiseless sybols received after propagation of S sybols x through a flat fading channel The channel capacity of a MIMO channel with total transit power restricted to P is defined as [8] C = ax Ix; y 4 px:trv x P where px is the transitter statistical distribution, V x = E { xx } is the covariance atrix of the transitted sybols x, with E{ } the expectation operator, and Ix; y is the utual inforation of the transitted and received sybols Assuing the receiver noise is independent fro the transitted sybols, the utual inforation is given by Ix; y = Hy Hn 5 with H the differential entropy of a continuous rando variable, hence axiizing capacity is achieved by axiizing the entropy Hy The entropy of a coplex gaussian vector x has the following inequality [1] Hx πev x 6 with base 2 and entropy is expressed in bits Equality in 6 is achieved if and only if x is a circularly syetric coplex Gaussian It is easy to show if x is a circularly syetric coplex Gaussian then so is y, thus, assuing optial gaussian distribution for the transit vector x, the utual inforation becoes Ix; y = V y V n where V y = E{yy }, and V n = E{nn } are the received and noise covariance atrices, respectively Let V r = E{rr } be the covariance atrix of the noiseless received sybols, then, assuing the receiver noise is independent fro the transitted signals, the received covariance atrix V y becoes V y = V r + V n 7 then, assuing uncorrelated noise in each receiver branch, the noise covariance atrix V n becoes σ 2 I Q, with noise variance σ 2, and the utual inforation can be written as, Ix; y = I Q + 1 σ 2 V r 8 In the situation where the transitter has no knowledge about the channel, it is optial to uniforly distribute the power across all the transit antennas [1], thus fro 4 and 8 the MIMO channel capacity is given by C = I Q + 1 σ 2 V r 9 The capacity in 9 is the Shannon capacity for the noiseless received sybol correlation atrix V r which is defined by the statistical scattering environent, antenna nubers and placeent In this paper we use a receiver antenna spatial correlation approach which gives the capacity without the explicit need for the use of a rando channel atrix Here the scattering environent and antenna placeent is captured by the noiseless received sybols correlation atrix V r and is utilized to give an analytical capacity forula B Spatial Correlation Matrix Approach Define the noralized spatial correlation between the coplex envelopes of the received signal at two antennas p and q as ρ pq E{r pr q} σ 2 s 1 where σs 2 is the average signal power received at any receive antenna, assuing noralized channel gains We can then write 1 σ 2 V r = ηr Q 11 with the Q Q spatial correlation atrix R Q ρ 11 ρ 1Q ρ Q1 ρ QQ 12 where each ρ pq depends on antenna separation and the power distribution of the scattering environent For a two diensional propagation environent [9] showed that ρ pq = i n β n J n k z p z q e inφ pq 13 n= where z p is the location of the pth point, φ pq is the angle of the vector connecting z p and z q, k = 2π/λ is the wavenuber with λ the wavelength, and J n is the nth order bessel function of the first kind The coefficients β n characterize any possible scattering environent and are given by β n = 2π Pϕe inϕ dϕ 14 with Pϕ the average power density distribution over ϕ the angle to the scatters For essentially all coon choices of P ϕ: von-mises, gaussian, truncated gaussian, unifor, piecewise constant, polynoial, Laplacian, Fourier series expansion, etc, there is a closed for expression for the β n [9] Therefore we have a closed for representation for the spatial correlation 13 and, as we will see next, for the capacity of a MIMO channel The capacity in 9 can now be expressed as C = I Q + ηr Q 15 which is the capacity for the MIMO syste given the scattering environent, described by the average power density distribution Pϕ, and antenna placeent The capacity given

3 by 15 is axiized when there is no correlation between the receive antennas, ie, R Q = I Q, giving, C ax = Q 1 + η 16 Therefore, in the idealistic situation of zero correlation between receiver antennas we see the best capacity growth achievable is linear in the nuber of antennas This result agrees with the traditional capacity forulation [1], [2] which is widely used to advocate the use of MIMO systes However, as we shall show in the following section, the linear capacity growth result does not hold for the ore realistic situation where the antennas are restricted to a region in space III CAPACITY OF A UNIFORM CIRCULAR ARRAY IN A 2D ISOTROPIC DIFFUSE FIELD Consider a unifor circular array UCA with radius r and Q receiver eleents Denote the set {d l }, l =,, Q 1 as the distance between any eleent and the other eleents in the array in a clockwise or anticlockwise direction, with d = being the distance between the eleent and itself, then d l 2r sinπl/q 17 For the special case of scattering over all angles in the plane we have a 2D isotropic diffuse field often referred to as a rich scattering environent at the receiver and the noralized average power reduces to Pϕ = 1/2π, ϕ [, 2π, giving the spatial correlation between any two points in the plane as ρ pq = J k z p z q 18 Define the spatial correlation between any eleent on the UCA and its lth neighbour as ρ l J k d l 19 then due to UCA syetry, ρ l = ρ Q l, and the correlation atrix becoes a Q Q syetric circulant atrix, [ ] R Q = Circ ρ, ρ 1,, ρ, ρ 2,, ρ 2, ρ where and are the ceiling and floor operators respectively, and x 1 x 2 x N x N x 1 x N 2 Circ [x 1, x 2,, x N ] 21 x 2 x 3 x 1 defines the circulant atrix The deterinant in the capacity forula 15 can be expanded by the product of eigenvalues of the arguent, giving, C = = 1 + ηλ 22 where λ σr Q are the Q eigenvalues of the circulant spatial correlation atrix R Q Therefore we see that the capacity is governed by the eigenvalues of the spatial correlation atrix, and as such their properties dictate the behavior of the capacity given differing scattering environents, antenna nubers and placeent It is iportant to note that the eigenvalues in 22 belong to the spatial correlation atrix R Q and not the rando channel atrix product HH, as analyzed in other work for exaple see [1], [5] [7] Unlike previous work where the eigenvalues of the channel atrix product are used siply as a eans to bound or siplify the tedious ergodic capacity coputation, in the following we expose characteristics of the spatial correlation atrix eigenvalues that lead to valuable insights into the channel capacity A Eigenvalues of Spatial Correlation Matrix R Q The eigenvalues of the syetric circulant atrix R Q are given by a siple closed for expression [1] λ = l= ρ l e i2πl/q 23 For a UCA in a 2D isotropic diffuse field the correlation coefficients are real and syetric, hence 23 represents the Discrete Cosine Transfor DCT of the spatial correlation coefficients; λ = l= ρ l cos 2πl/Q 24 Since R Q is a positive-seidefinite Heritian atrix and with the properties of the DCT it is easy to show the following: λ R 25 λ 26 λ Q = λ, > 27 that is, all the eigenvalues of R Q are real, non-negative and syetric Theore 1 Maxiu Eigenvalue Threshold: For a UCA of radius r in a 2D isotropic diffuse field define the Maxiu Eigenvalue Threshold: M πer/λ 28 then, for {M + 1, Q M 1} the eigenvalues λ are vanishingly sall Before proving Theore 1 we clarify its significance with the following interpretation: For any UCA in a 2D isotropic diffuse field there is a finite set of non-zero spatial correlation atrix eigenvalues, where the set size increases linearly with the radius of the array and is independent of the nuber of antennas Proof: Substitution of 19 and 17 into 24 gives λ = l= J 2kr sin πl/q cos 2πl/Q 29 letting ξ = πl/q and assuing a large nuber of antennas, we can approxiate 29 with the integral λ Q π π J 2kr sin ξ cos 2ξ dξ 3

4 for = {, 1,, /2 } Using the identity [11, p32] J 2 nz = 1 π π then the eigenvalues can be expressed as J 2z sin ψ cos2nψdψ 31 λ QJ 2 kr 32 which is asyptotically equal to 29 with the antenna nuber Using the the following bound [12, p362] on the bessel functions for n J n z z n 2 n Γn + 1 the eigenvalues are then upper-bounded by 33 C/2λ 2 λ Q 34 Γ + 1 with C = 2πr the circuference of the circular array Since Gaa function Γ+1 increases faster than the exponential C/2λ then 34 will rapidly approach for soe > for which Γ + 1 > C/2λ Using a relaxed Stirling lower bound 1 for Γ + 1, we wish to find for which C > 35 e 2λ The inequality in 35 is clearly satisfied when > Ce/2λ, asserting that ust be an integer and fro the definition of C we see that the eigenvalues vanish for > πer/λ, thus giving the axiu eigenvalue threshold in 28 Given the syetric nature of the eigenvalues in 27 then for any nuber of antennas, Q 2M + 1, there is a finite set of 2M + 1 non-zero eigenvalues, λ = {λ, λ 1,, λ M, λ Q M,, λ Q 2, λ } 36 whose nuber of eleents grows only with the radius of the array, and is independent on the nuber of antennas Fig 1 shows the eigenvalues of the spatial correlation atrix R Q for various UCA radii in a 2D isotropic diffuse field Shown as a solid black line, it can be seen that the theoretical axiu eigenvalue threshold derived in Theore 1 gives a good indication of the axiu non-vanishing eigenvalue for each radius B Capacity Growth Liits: Antenna Saturation Due to the dependence of the capacity forula 22 on the eigenvalues of the spatial correlation atrix we see that Theore 1 has significant iplications on the capacity growth with increasing antenna nubers In this section we show that this fixed set size of eigenvalues, regardless of the nuber of antennas, leads to an antenna saturation effect on MIMO capacity Theore 2 Antenna Saturation Point: For a UCA of radius r in a 2D isotropic diffuse field define a saturation point 1 Γz + 1 > 2πz z z e z > z z e z, z > Fig 1 The eigenvalues of the spatial correlation atrix for various UCA radii in a 2D isotropic diffuse scattering field The dark solid line represents the theoretical Maxiu Eigenvalue Threshold derived in Theore 1, and clearly shows the boundary between the significant and vanishing eigenvalues of the spatial correlation atrix for each array radius as the iniu nuber of antennas that generate a full set of non-zero eigenvalues λ ; 2M then, for any Q the channel capacity is governed by C 1 + ηλ + 2 M 1 + ηλ 38 =1 where η = Q/ η is the scaled average SNR ratio at each receiver branch Before giving a proof of Theore 2 we give the following interpretation: For a MIMO syste with a UCA in a 2D isotropic diffuse field there exists a saturation point in the nuber of antennas, which is dependent only on the radius of the array, after which the addition of ore antennas gives diinishing arithic capacity gains Proof: Denoting λ Q σr Q as the eigenvalues generated fro Q antennas on a UCA of radius r, then the capacity 22 can be expressed as = 1 + ηλ Q 39 which, when using 27 and assuing an odd nuber of antennas can be written as 2 C = 1 + ηλ Q /2 + 2 =1 1 + ηλ Q 4 Consider the UCA placed in a 2D isotropic diffuse field, then as a direct result of Theore 1 for Q 2M + 1 the channel 2 fro Theore 1 the case of even Q gives identical results, however to siplify notation we assue an odd nuber of antennas

5 capacity given by 4 is well approxiated using the set of 2M + 1 non-vanishing eigenvalues, that is, C 1 + ηλ Q + 2 M =1 1 + ηλ Q 41 Given two UCAs of equal radius r with antenna nubers Q 1, Q 2 2M +1, and spatial correlation atrix eigenvalues λ Q1 and λ Q2 respectively, then fro 32 we have the following relationship between eigenvalues for systes with different nubers of receive antennas, λ Q 1 Q 1 λq2 42 Q 2 with the approxiation asyptotically equal with the nuber of antennas Define 2M +1 as the iniu nuber of antennas which generate the full set of non-zero eigenvalues, then letting Q 1 = and Q 2 = Q we have λ Q Q λ 43 where λ are the eigenvalues of the spatial correlation atrix R QM Thus the eigenvalues for any UCA of radius r with nuber of antennas Q are siply scaled versions of the eigenvalues generated by an array with antennas Substituting 43 into 41 gives C 1 + ηqλ M ηqλ 44 =1 which behaves arithically with Q since the eigenvalues are constant for all Q, hence the capacity gain is reduced to arithic growth once the antenna nuber reaches the saturation point given by Let η = Q/ η be the scaled average SNR at each antenna, then the capacity 44 becoes 38, thus we see that the effect of any additional antennas above the saturation point is just an increase in the average SNR, or in other words, a noise-averaging effect due to the assuption of independent noise at each antenna It can be observed fro Fig 2 that the capacity does indeed increase approxiately linearly up until the theoretical saturation point defined in Theore 2, after which the capacity reduces to arithic increase with antenna nuber This result has significant iplications for practical MIMO systes, we have shown that there is a saturation point in antenna nubers after which there are diinishing returns in capacity for additional antennas, therefore the saturation point gives the optial nuber of antennas that achieve axiu capacity with iniu cost Further to the UCA case, epirical studies have shown that there are only ever 2M + 1 significant eigenvalues generated by arbitrarily placed antennas within a circular region of radius r We believe the saturation effect seen in UCAs also holds for any antenna configuration within a circular region and we are currently developing theoretical results to support this Capacity bps/hz C ax r=7λ r=5λ r=3λ r=1λ Nuber of Antennas Fig 2 Capacity of MIMO systes for various antenna nubers of a UCA with radii r = 1, 3, 5, and 7 wavelengths in a 2D isotropic diffuse scattering field, along with the theoretical axiu capacity As indicated by the dashed lines for each radii, the Antenna Saturation Point theoretically derived in Theore 2 gives a good indication where the MIMO syste saturates and hence increasing antenna nubers gives only arginal capacity gain REFERENCES [1] IE Telatar, Capacity of ulti-antenna gaussian channels, Tech Rep, ATT Bell Labs, 1995 [2] GJ Foschini and MJ Gans, On liits of wireless counications in a fading environent when using ultiple antennas, Wireless Personal Counications, vol 6, no 3, pp 311, 1998 [3] Da-Shan Shiu, GJ Foschini, MJ Gans, and JM Kahn, Fading correlation and its effect on the capacity of ultieleent antenna systes, IEEE Trans Coun, vol 48, no 3, pp , 2 [4] S Loyka and A Kouki, New copound upper bound on MIMO channel capacity, IEEE Coun Lett, vol 6, no 3, pp 96 98, 22 [5] C Chen-Nee, DNC Tse, JM Kahn, and RA Valenzuela, Capacity scaling in MIMO wireless systes under correlated fading, IEEE Trans Infor Theory, vol 48, no 3, pp , 22 [6] TL Marzetta and BM Hochwald, Capacity of a obile ultipleantenna counication link in rayleigh flat fading, IEEE Trans Infor Theory, vol 45, no 1, pp , 1999 [7] L Hanlen and M Fu, MIMO wireless counication systes: Capacity liits for sparse scattering, in AusCTW 2, Canberra, Australia, 22, pp [8] T Cover and J Thoas, Eleents of Inforation Theory, Wiley, New York, 1991 [9] PD Teal, TD Abhayapala, and RA Kennedy, Spatial correlation for general distributions of scatterers, in ICASSP 2, Florida, 22, pp [1] PJ Davis, Circulant Matrices, Wiley, New York, 1979 [11] GN Watson, Theory of Bessel Functions, Cabridge University Press, 2nd edition, 1958 [12] M Abraowitz and IA Stegun, Handbook of Matheatical Functions, Dover Publications, Inc, New York, 1972