Outdoor MIMO Wireless Channels: Models and Performance Prediction

Transcription

1 Outdoor MIMO Wireless Channels: Models and Performance Prediction D. Gesbert 1),H.Bölcskei 2),D.A.Gore 2), and A. J. Paulraj 1) 1) Gigabit Wireless Inc., 3099 North First Street, San Jose, CA. Phone: (408) , Fax: (408) , (A. J. Paulraj is on part-time leave from Information Systems Laboratory, Stanford University) 2) Information Systems Laboratory, Stanford University Packard 223, 350 Serra Mall, Stanford, CA Phone: (650) , Fax: (650) , (H. Bölcskei is on leave from Institut für Nachrichtentechnik und Hochfrequenztechnik, Technische Universität Wien, Vienna, Austria) Correspondence should be addressed to D. Gesbert Abstract We present a new model for multiple-input multiple-output (MIMO) outdoor wireless fading channels which is more general and realistic than the usual i.i.d. model. The proposed model allows to investigate the behavior of channel capacity as a function of parameters such as the local scattering radius at the transmitter and the receiver, the distance between the transmit and receive arrays, and the antenna beamwidths and spacing. We show that capacity is driven by the spatial fading correlation and the condition number of the MIMO channel matrix through specific sets of propagation parameters. We use the new model to point out the existence of pin-hole channels which exhibit low fading correlation between antennas but still have poor rank properties and hence low capacity. We suggest guidelines for predicting high rank (and hence high capacity) in MIMO channels. We also show that even at long ranges high channel rank can easily be obtained under mild scattering conditions. Finally, we validate our results by simulations using ray tracing techniques. Keywords: MIMO channels, antenna arrays, smart antennas, spatial multiplexing, diversity, channel capacity, channel modeling. Submitted to IEEE Trans. Communications, July This work was supported in part by FWF-grant J1868- TEC. Part of this paper will be presented at IEEE Globecom 2000, San Francisco, CA, Nov

2 1 Introduction Both in the academic and in high-tech industry communities, the prospect of extraordinary improvement in the capacity of wireless networks has recently drawn considerable attention onto multipleinput multiple-output (MIMO) communication techniques, [1, 2]. MIMO methods make use of multielement antenna arrays at both the transmit and the receive side of a radio link and have been shown theoretically to drastically improve the capacity over more traditional single-input multiple-output (SIMO) systems (with multiple antennas being used typically at the base station only) [4, 6, 8, 20, 5]. SIMO channels in wireless networks can provide diversity gain, array gain, and interference canceling gain among other benefits. In addition to these same advantages, MIMO links can offer a so-called multiplexing gain by opening N min parallel spatial channels, where N min is the minimum of the number of transmit and receive antennas, within the same bandwidth. Under certain propagation conditions capacity gains proportional to N min can be achieved [10]. Space-time coding [16, 17] and spatial multiplexing [3, 4, 19, 8, 9] (a.k.a. BLAST ) are popular signal processing techniques making use of MIMO channels to improve the performance of wireless networks. Previous work and open problems. The literature on realistic MIMO channel models is very scarce. For the line-of-sight (LOS) case specific arrangements of the antenna arrays at the transmitter and the receiver maximizing the orthogonality between antenna signatures have been reported in [15]. A detailed treatment of array gain and capacity of MIMO channels for the case where both the transmitter and the receiver know the channel can be found in [7]. In the fading case, previous studies have mostly been confined to i.i.d. gaussian matrices, an idealistic assumption in which the entries of the channel matrix are independent complex gaussian random variables [4, 10]. The influence of spatial fading correlation on either the transmit or the receive side of a wireless MIMO radio link has been addressed in [18, 5]. In practice, however, the realization of high MIMO capacity in actual radio channels is sensitive not only to the fading correlation between individual antennas but also to the rank behavior of the channel. In the existing literature, high rank behavior has been loosely linked to the existence of a dense scattering environment. Recent successful demonstrations of MIMO technologies in indoor-to-indoor channels, where rich scattering is almost always guaranteed, corroborate this behavior [11]. However, several important questions regarding outdoor MIMO channels remain open: What is the capacity of a typical outdoor MIMO channel? 2

3 What are the key propagation parameters governing capacity? Under what conditions do we get a full rank MIMO channel (and hence high capacity)? What is a simple analytical model describing the capacity behavior of outdoor MIMO wireless channels well? Contributions and organization of the paper. In this paper, we address the above mentioned questions and other problems related to the prediction of the capacity of outdoor MIMO channels. We start with theoretical model concepts and illustrate their capacity behavior. We then suggest a simple classification of MIMO channels and devise a MIMO channel model which seems to be general enough to encompass all cases of practical relevance (Sec. 2). The channel model used in [18, 5] does not encompass important propagation scenarios such as the uncorrelated-low-rank (ULR) and correlated-low-rank (CLR) channel models presented in Sec. 2. The key difference between the new model presented in this paper and previous models such as those in [18, 5] is that in the new model the impact of spatial fading correlation and channel rank are decoupled although not fully independent, which allows for example to describe MIMO channels with uncorrelated spatial fading at the transmitter and the receiver but reduced channel rank (and hence low capacity). This situation typically occurs when the distance between transmitter and receiver is large. Furthermore, our model allows description of MIMO channels with spatial fading correlation at both the transmitter and the receiver. First we consider the case of deterministic MIMO channels in a green-field (i.e. non-scattering) environment, and we derive a simple condition guaranteeing high rank (and hence high capacity) behavior (Sec. 3). Then we turn to scattering situations and introduce a stochastic channel model describing the capacity behavior as a function of the wavelength, the scattering radii at the transmitter and the receiver, the distance between transmit and receive arrays, antenna beamwidths, and antenna spacing. The impact of each of these parameters on capacity is interpreted and studied. Our model suggests that full MIMO capacity gain can be achieved for very realistic values of scattering radii, antenna spacing and range. It shows, in contrast to usual intuition, that large antenna spacing has only limited impact on capacity under fairly general conditions. We use the new model to predict high and low rank behavior of MIMO channels with potentially uncorrelated antenna fading on the transmit and receive sides. This leads to the concept of a pin-hole channel where spatial fading is uncorrelated and yet the channel has low rank and hence low capacity. We show that this situation 3

4 typically occurs for very large distances between transmitter and receiver. In the 1 1 case (i.e. one transmit and one receive antenna), the pin-hole channel yields capacities worse than the traditional Rayleigh fading channel (Sec. 4). Our results are validated by comparing the capacity obtained from the new stochastic channel model with a ray tracing-based channel simulation where each scatterer and path is simulated. We find a good match between the two models over a wide range of situations (Sec. 5). Finally, we conclude in Sec Capacity of MIMO Channels and Model Classification In this section, we briefly review the capacity formula for MIMO channels and we present a classification of MIMO channels. Throughout the paper, we restrict our discussion to the frequency-flat fading case and we assume that the transmitter has no channel knowledge whereas the receiver has perfect channel knowledge. 2.1 Capacity of MIMO channels In what follows, we assume M receive and N transmit antennas. The capacity in bits/sec/hz of a MIMO channel under an average transmitter power constraint is given by 1 [4] C =log 2 [det (I M + ρ )] N HH, (1) where H is the M N channel matrix, I M denotes the identity matrix of size M, and ρ is the average signal-to-noise ratio (SNR) at each receiver branch. The elements of H are complex gaussian with zero mean and unit variance, i.e., [H] m,n CN(0, 1) for m =1, 2,..., M, n =1, 2,..., N. Notethat since H is random C will be random as well. Assuming a piece-wise constant fading model and coding over many independent fading intervals 2, E H {C} will be the Shannon capacity of the random MIMO channel [6]. 2.2 Model classification Let us next introduce the following MIMO fading channel models: Uncorrelated high rank (UHR, a.k.a. i.i.d.) model: The elements of H are i.i.d. CN(0, 1). 1 The superscript stands for Hermitian transpose. 2 E H stands for the expectation over all channel realizations. 4

5 Uncorrelated low rank (ULR) (or pin-hole ) model: H = g rx gtx, where g rx and g tx are independent receive and transmit fading vectors with i.i.d. complex-valued components g rx CN(0, I M ), g tx CN(0, I N ). In this model every realization of H has rank 1 and therefore although diversity is present capacity must be expected to be less than in the ULR model since there is no multiplexing gain. Intuitively, in this case the diversity order is equal to min(m,n). Correlated low rank (CLR) model: H = g rx gtxu rx u tx where g rx CN(0, 1) and g tx CN(0, 1) are independent scalar variables and u rx and u tx are fixed deterministic vectors of size M 1 and N 1, respectively, and with unit modulus entries. This model yields no diversity or multiplexing gain whatsoever, just receive array gain. We also define the following single-antenna models to which we extend the low rank and high rank concepts: 1 1 HR, defined by the UHR model with M = N = 1, also known as Rayleigh fading channel. 1 1 LR, defined by the ULR or CLR model with M = N = 1 (double Rayleigh channel). Note that the low rank models (ULR, CLR, 1 1 LR) above do not use the traditional normal distribution for the entries of H but instead the product of two gaussian variables. This type of distribution will be shown later to occur in important practical situations. The above models exhibit very different capacity behavior. The capacity cumulative distribution function (c.d.f.) of the corresponding capacities is depicted in Figs. 1 and 2 for ρ = 10dB. Fig. 1 clearly shows the impact of rank loss on capacity. The loss in the 3 3 ULR case is due to the fact that there is only one spatial data pipe. However, in this case, much of the diversity gain is preserved because the antennas still fade in an uncorrelated manner. Antenna correlation causes additional loss in capacity, which can be seen from the c.d.f. of the 3 3 CLR channel in Fig. 1. From Fig. 2 it follows that the 1 1 LR model yields less capacity than the 1 1 HR model (Rayleigh) in most of the capacity region. This is due to the intuitive fact that a double Rayleigh channel will fade twice as often as a standard Rayleigh channel. In Section 4, we comment on the practical relevance of the model categories introduced above. 5

6 3 Green-field MIMO Channels In this section, we derive conditions guaranteeing a high rank MIMO channel in a green field (or LOS) environment. We suggest in particular that rank properties are governed by simple geometrical propagation parameters. We concentrate on the ideal non-scattering non-fading case (i.e., we consider a deterministic channel). The results below are thus applicable in flat/rural wireless deployments. We shall see later in the paper that our findings suggest guidelines which have broader applicability. Considering the N transmitter, M receiver setup described in Fig. 3, we assume boresight propagation from the transmit array to the receive array. In addition, we assume the signal radiated by the l-th transmit antenna to impinge as a plane wave on the receive array at an angle of θ l. This assumption is justified when the antenna aperture is much smaller than the range R. Finally, using the same assumption the effect of path loss can be ignored. Denoting H =[h 1, h 2,..., h N ]withh l denoting the signature of the l-th transmit antenna, we find h l = [ 1,e 2πj sin(θ l)d r/λ,..., e ] 2πj(M 1) sin(θ l)d r/λ,whereλis the wavelength, and d t and d r is the antenna spacing at the transmitter and the receiver, respectively. The common phase shift due to the distance R between transmitter and receiver has no impact on capacity and is ignored here. Clearly, when the θ l (l =1, 2,..., N) (all other parameters being fixed) approach zero we find that H approaches the all one matrix and therefore has rank 1. In practice, this happens for large range R. As the range decreases, linear independence between the signatures starts to build up. We choose to use the full orthogonality between the signatures of adjacent pairs of transmit antennas as a criterion for the receiver to be able to separate the transmit signatures well which implies high capacity. This condition reads h l, h l+1 = M 1 m=0 e 2πj[sin(θ l+1) sin(θ l )]m dr λ =0. (2) For practical values of R, d t, d r, orthogonality will occur for small θ l. We can therefore assume that sin θ l (l 1)d t /R (l =1, 2,..., N). Condition (2) can therefore be rewritten as M 1 m=0 e j2πm d t dr λr =0, (3) which implies d t d r R λ M. (4) In a pure LOS situation orthogonality will be achieved for very small values of the range R. For example at a frequency of 2GHz with M =3,amaximumofR = 20m is acceptable for 1m antenna spacing. Therefore a non-line-of-sight (NLOS) situation (with scatterers) is required to build up rank 6

7 and achieve high MIMO capacity, as intuitively expected. Note that the orthogonality condition (4) depends on the number of receive antennas M only. This is so since we are seeking separability of only the two closest transmit antennas. Clearly, linear independence of adjacent transmit antenna signatures is a necessary (but not sufficient) condition for the global channel matrix H to have full rank. We show later in the paper how the guideline condition (4) extends nicely to scattering channels. 4 Distributed Scattering MIMO Model We now turn to the case of NLOS channels, where fading is induced by the presence of scatterers. The purposes of this section are to develop a stochastic channel model that captures separately the diversity andrank properties as suggested by Figs. 1 and 2. to suggest how the guideline offered by (4) for LOS channels can be extended to fading channels upon appropriate redefinition of d t and d r. In the following, for the sake of simplicity, we consider the effect of near-field scatterers only, i.e., the scatterers which are either in the vicinity (typically a few tens to hundreds of meters away) of the transmitter or the receiver. We ignore remote scatterers assuming that the path loss will tend to limit their contribution to the total channel energy. Finally, we consider a frequency-flat fading channel. 4.1 SIMO Fading Correlation Model We consider a linear array of M omni-directional receive antennas with spacing d r. A number of distributed scatterers act as ideal reflectors (i.e. perfect omnidirectional scatterers) of a signal which eventually impinges on the receive array. The plane-wave directions of arrival (DOAs) of these signals span an angular spread of θ r radians (see Fig. 4). Several distributions can be considered for the DOAs, including uniform, Gaussian, Laplacian etc. [14, 12, 13]. The addition of different planewaves causes space-selective fading at the receive antennas. It is well known that the resulting fading correlation is governed by the angle spread, the antenna spacing and the wavelength. The receive 7

8 array response vector h cannowbemodeledas h CN(0, R θr,d r ) or equivalently h = R 1/2 θ r,d r g with g CN(0, I M ), (5) where R θr,dr is the M M correlation matrix. Different assumptions on the statistics of the DOAs will yield different expressions for R θr,dr [14, 12, 13]. For uniformly distributed DOAs, we find [12, 14] [R θr,d r ] m,k = 1 S i= S 1 2 i= S 1 2 e 2πj(k m)dr cos( π 2 +θ r,i) (6) where S (odd) is the number of scatterers with corresponding DOAs θ r,i. For large values of the angle spread and/or antenna spacing, R θr,dr will converge to the identity matrix, which gives uncorrelated fading. For small values of θ r,d r, the correlation matrix becomes rank deficient (eventually rank one) causing (fully) correlated fading. For the sake of simplicity, we furthermore assume the mean DOA to be orthogonal to the array (bore-sight). Some comments on this model are now in order. Impact of directional antennas: If directional antennas are used instead of omni-antennas the effective angle spread seen by the array can be obtained by intersecting the scattering angle spread with the main lobe of the antennas. In what follows the directionality of antennas is accounted for by selecting the effective angle spread properly. Spatial fading correlation at the transmitter: The model provided in (5) can readily be applied to an array of transmit antennas with corresponding antenna spacing and signal departure angle spread. 4.2 MIMO Correlated Fading Model We consider the NLOS propagation scenario depicted in Fig. 5. The propagation path between the two arrays is obstructed on both sides of the link by a set of significant near-field scatterers (such as buildings and large objects) refered to as transmit or receive scatterers. Scatterers are modeled as omni-directional ideal reflectors. The extent of the scatterers from the horizontal axis is denoted as D t and D r, respectively. When omni-directional antennas are used D t and D r correspond to the transmit and receive scattering radius, respectively. On the receive side, the signal reflected by the scatterers onto the antennas impinge on the array with an angular spread denoted by θ r,whereθ r is function of the position of the array with respect to the scatterers. Similarly on the transmit side 8

9 we define an angular spread θ t. In general, using directional antennas instead of omni-antennas will tend to decrease the effective values of D t or D r as well as the angular spreads. The scatterers are assumed to be located sufficiently far from the antennas for the plane-wave assumption to hold. We furthermore assume that D t,d r R (local scattering condition) Signal at the Receive Scatterers We assume S scatterers on both sides, where S is an arbitrary, large enough number for random fading to occur (typically S > 10 is sufficient). The exact distribution of the scatterers is irrelevant here. Every transmit scatterer captures the radio signal and re-radiates it in the form of a plane wave towards the receive scatterers. The receive scatterers are viewed as an array of S virtual antennas with average spacing 2D r /S, and as such experience an angle spread defined by tan(θ S /2) = D t /R. We denote the vector signal originating from the n-th transmit antenna and captured by the S receive scatterers as y n =[y 1,n, y 2,n,..., y S,n ] T. Approximating the receive scatterers as a uniform array of sensors and using the correlation model of (5), we find y n CN(0, R θs,2d r/s) or equivalently y n = R 1/2 θ S,2D r/s g n with g n CN(0, I S ). (7) For uncorrelated transmit antennas, the S N channel matrix describing the propagation between the N transmit antennas and the S scatterers Y =[y 1, y 2,..., y N ] simply writes Y = R 1/2 θ S,2D r/s G t, (8) where G t =[g 1, g 2,..., g N ]isans N i.i.d. Rayleigh fading matrix. However, there is generally correlation between the transmit antennas because of finite angle spread and insufficient antenna spacing. Therefore, a more appropriate model becomes Y = R 1/2 θ S,2D r/s G t R 1/2 θ t,d t, (9) where R 1/2 θ t,d t is the N N matrix controlling the transmit antenna correlation as suggested in the transmit form of model (5) The MIMO Model Like the transmit scatterers, the receiver scatterers are assumed here to ideally reradiate the captured energy. As shown in figure (5), a set of plane waves, with total angle spread θ r, impinge on the receive 9

10 array. Denoting the distance between the s-th scatterer and the m-th receive antenna as d s,m,the vector of received signals from the n-th transmit antenna can be written as e 2πjd 1,1/λ... e 2πjd S,1/λ z n = : : y n. e 2πjd 1,M /λ... e 2πjd S,M /λ (10) } {{ } Φ Collecting all receive and transmit antennas according to Z =[z 1, z 2,..., z N ], we obtain Z = ΦY, (11) where Φ is the M S matrix shown in (10). The problem with the expression in (11) is the explicit use of deterministic phase shifts in the matrix Φ which makes the model inconvenient. The simple equivalence result below allows us to get rid of this inconvenience and obtain a new and entirely stochastic MIMO model. Lemma. For S, Z = ΦY has the same p.d.f. as R 1/2 θ r,d r G r Y where G r is an i.i.d. Rayleigh fading matrix of size M S. Proof. See the appendix. 2 After proper power normalization 3 and replacing Y by (9), we obtain the following simple MIMO transfer function H = 1 S R 1/2 θ r,d r G r R 1/2 θ S,2D r/s G tr 1/2 θ t,d t. (12) 4.3 Interpretation & The Pin-Hole Channel The model suggested in (12) lends itself to several useful interpretations, explaining the effect of propagation parameters on the capacity behavior of the channel. We make the following remarks: The model in (12) is symmetric in structure, which was to be expected from the scenario considered. The spatial fading correlation between the transmit antennas, and therefore the transmit diversity gain, is governed by the deterministic matrix R 1/2 θ t,d t and hence implicitly by the local transmit angle spread, the transmit antenna beamwidth and spacing. On the receive side, the fading correlation is similarly controled by the receive angle spread and antenna spacing through R 1/2 θ r,d r. 3 We use a normalization to fix the channel energy regardless of how many scatterers are considered. 10

11 The rank of the MIMO channel model, and therefore our ability to apply spatial multiplexing (BLAST-like) techniques [4, 3], is primarily controled through R 1/2 θ S,2D r/s. The model in (12) shows that it is well possible to have uncorrelated fading at both sides, and yet have a rank deficient MIMO channel with reduced capacity. Such a channel is dubbed a pin-hole because scattering (fading) energy travels through a very thin air pipe, preventing the rank to build up. In practice, this occurs when the product D t D r is small compared to the range R, making θ S small, and causing the rank of R 1/2 θ S,2D r/s to drop. This extends the analysis carried out in the green-field case with the formula provided in (4) and is confirmed by our simulations (see Sec. 5). Note that D t, D r play a role analogous to d t, d r in the green field case. Also this suggests that additional scatterers that would lie in between the transmission path and not contribute to increased scattering angle spread will not contribute to capacity. Conversely, remote scatterers with significant impact on the total channel energy if they exist will increase the effective value of θ s and quickly help build up additional capacity. Eq. (12) suggests that in the scattering case the rank behavior of the MIMO channel is mainly governed by the scattering radii and by the range. Scatterers can be viewed as virtual antenna arrays with very large spacing and aperture. Unlike the usual intuition, the physical antenna spacing has limited impact on the capacity provided antennas remain uncorrelated, which occurs at λ/2 spacing for reasonably high local angle spread/antenna beamwidth. Note that if scattering is absent at one end of the link, the relevant parameter on that particular end driving the MIMO rank becomes the antenna spacing, which then must be huge in order to achieve high rank. Note that using cross-polarized antennas removes the need for scatterers because power loss between two orthogonal polarizations tends to make the channel matrix orthogonal as well and hence yields high capacity. When either the transmit or the receive antennas are fully correlated due to small local angle spread, the rank of the MIMO channel also drops. In this situation, the diversity and multiplexing gains vanish, preserving only the receive array gain. Note that there is no transmit array gain since we assumed that the channel is unknown in the transmitter. From the remarks above it follows that antenna correlation causes rank loss but the converse is not true. Our model is therefore much more general than the previously reported models. The new model contains not one but the product of two random Rayleigh distributed matrices. 11

12 This is in contrast with the traditional Rayleigh MIMO model of [4, 10]. Depending on the rank of R 1/2 θ S,2D r/s, the resulting MIMO fading statistics ranges smoothly from Gaussian to product of two independent Gaussians. In the high rank region, R 1/2 θ S,2D r/s becomes the identity matrix. Using the central limit theorem, the product G r G t approaches a single Rayleigh distributed matrix, which justifies the traditional model in that particular case. In the low rank (i.e. rank one) region, R 1/2 θ S,2D r/s is the all one matrix. The MIMO channel becomes R 1/2 θ r,d r g rx gtx R1/2 θ t,d t, an outer-product with independent transmit and receive Rayleigh fading vectors. In this case we have no multiplexing gain, but there is still diversity gain with the exact amount depending on the transmit and receive fading correlation. In practice depending on local angle spread and antenna spacing, the model will range smoothly from the CLR to UHR models. In the 1 1 case, meaningful high rank and low rank models can still be defined, according to the rank taken by R 1/2 θ S,2D r/s. The high rank model is the traditional Rayleigh channel. The low rank model has double Rayleigh distribution with mostly less capacity than Rayleigh as was shown earlier. The model does not suggest the existence of a correlated high rank MIMO channel, which corresponds also to usual intuition. 5 Numerical Evaluation In order to verify our approach, we use a comparison with an explicit ray-tracing model. In every simulation 500 independent Monte-Carlo realizations of the ray-tracing channel are generated. The capacity distribution predicted by the proposed stochastic MIMO model for various values of the key model parameters is compared to that achieved by the actual ray tracing channel with the same parameters. The ray tracing model follows the scenario depicted in Fig. 5. In all examples we used S =20 transmit and receive scatterers which are randomly distributed uniformly around a line perpendicular to the x-axis. We found that the final capacity results are insensitive to the particular distribution of the scatterers as long as D t,d r and the angular spreads remain fixed. We used M = N =3and 12

13 placed the scatterers at a distance R t from the transmit array and R r from the receive array. For simplification we use R r = R r = D t = D r in all simulations in order to maintain a high local angle spread and hence low antenna correlation. The frequency was set to 2GHz and the SNR was 10dB. To introduce random fading we use small random perturbations of the transmit and receive antenna array positions in each of the Monte Carlo experiments. The capacity distribution predicted by our channel model for the corresponding set of parameters is plotted alongside for comparison. This process was carried out for three separate sets of control parameters, covering the region between the UHR and the ULR models. The curves obtained are shown in Fig. 6. Fig. 7 illustrates the impact of the rank of R 1/2 θ S,2D r/s on the capacity in the 1 1 case. The proposed channel model predicts the capacity distribution up to one bps/hz in all cases and becomes almost exact as we approach UHR and ULR regions. Finally, another validation looks at predicting the high rank behavior of the channel using an extension of (4). Fig. 8 is a plot of average capacity for varying D t = D r with R fixed at 10 km. A possible generalized high rank prediction formula is where 2Dt N 1 2D t 2D r N 1 M 1 Rλ M, is the maximum spacing achieved by virtual antennas distributed over the distance of 2D t. This formula predicts the high rank region to start around 23m of scattering radius, a little before but close to the knee in the figure. The result shows how capacity builds up with scattering. It also suggests that the effective aperture of the virtual antenna made of scatterers is slightly less than what is predicted by D t. It confirms though how fast the high capacity region is attained, even for a very large range. 6 Conclusion We introduced a model for describing the capacity behavior of outdoor MIMO channels. The model describes the effect of certain propagation geometry parameters in LOS and fading (NLOS) situations. Our model allows to study the behavior of channel rank as a function of antenna spacing and range in LOS situations, or more practically, as a function of scattering radius and the range in fading situations. The model predicts excellent performance outdoors for very reasonable values of scattering radius, almost regardless of how large the antenna spacing is. We pointed out the existence of pinhole channels for which antennas are perfecly uncorrelated at the transmitter and the receiver and 13

14 yet the rank properties are poor and hence capacity will decrease. This typically occurs for very large values of the range R. Ongoing work includes real-world data analysis and model fitting to extract estimates of the key propagation parameters and extensions to frequency-selective channels. Acknowledgment The authors would like to thank Prof. J. Bach Andersen for his helpful comments on an earlier version of this manuscript. Appendix (Proof of the Lemma) Let R 1/2 θ S,2D = r/s UΣU be the eigendecomposition of R 1/2 θ S,2D r/s. According to (9) Z = ΦY = ΦUΣU G t R 1/2 θ t,d t. (13) When S is large enough, the central limit theorem applies to the product F = ΦU which tends to be normally distributed. Hence, [F] m,s CN(0, 1). The correlation between the rows of ΦU is governed by the receive angle spread θ r and the antenna spacing through R θr,d r. Because the columns of U are orthogonal, we easily show that in addition the columns of F are independent. It can furthermore be shown that F R 1/2 θ r,d r G r, (14) where G r is an M S i.i.d. Rayleigh distributed matrix. Hence, for large S, we have Z R 1/2 θ r,d r G r ΣU G t R 1/2 θ t,d t. Finally, the distribution of G r is unchanged if we right-multiply G r by the unitary matrix U and hence Z R 1/2 θ r,d r G r UΣU G t R 1/2 θ t,d t R 1/2 θ r,d r G r R 1/2 θ S,2D G r/s tr 1/2 θ t,d t. 2 References [1] Proceedings of Annual Workshop for Smart Antennas in Wireless Communications, Stanford University, CA, July [2] Wireless takes to the high wire, Business Week, pp , Feb. 14th [3] A. J. Paulraj and T. Kailath, Increasing capacity in wireless broadcast systems using distributed transmission/directional reception, U. S. Patent, no. 5,345,599,

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17 1 Capacity Distribution of Theoretical MIMO Channels at 10 db SNR Prob [Capacity bps/hz < abscissa] x3 CLR 3x3 ULR 3x3 UHR Capacity in bps/hz Fig. 1. The uncorrelated low rank (ULR) model shows the impact of rank loss on capacity. The correlated low rank (CLR) loses both multiplexing and diversity gains. 1 Capacity Distribution of Theoretical SISO Channels at 10 db SNR Prob [Capacity bps/hz < abscissa] x1 LR 1x1 HR Capacity in bps/hz Fig. 2. Capacity curves for the 1 1 high rank (Rayleigh) and 1 1 low rank (double Rayleigh) channels. The double Rayleigh model has worsened fading statistics. 17

18 M dr dt N l 2 1 RX Array R O l TX Array 1 Fig. 3. N input M output MIMO green-field model considered. dr O r M RXs Fig. 4. Propagation scenario for SIMO fading correlation. Each scatterer transmits a plane-wave signal to a linear array. 18

19 d r D r dt O r Os O t M RXs Dt N TXs R Fig. 5. Propagation scenario for fading MIMO channel. We assume plane-wave propagation. Scatterers are ideal reflectors MIMO Channel Model Ray Tracing Channel... UHR Uncorrelated High Rank ULR Uncorrelated Low Rank Comparison of CDF Curves 0.7 Prob. Capacity < abscissa ULR UHR intermediate Capacity in Bit/Sec/Hz Fig. 6. Capacity c.d.f. obtained with MIMO model for three sets of parameters. From left to right. Set 1: D t = D r =30m, R = 1000km. Set 2: D t = D r =50m, R =50km. Set 3: D t = D r = 100m, R =5km. 19

20 x1 double Rayleigh (theoretical) (dash dotted) Comparison of CDF Curves for a 1x1 SNR 0.8 Prob. Capacity < abscissa x1 Rayleigh (theoretical) (dash dotted) Channel Model (continuous) Capacity in Bit/Sec/Hz Fig. 7. Capacity c.d.f. obtained for the 1 1 model. We use two sets of parameters: from left to right. Set 1: D t = D r =30m, R = 1000km. Set2:D t = D r = 100m, R =5km. 8.5 Validation of Formula Predicting Knee in Capacity Curve (D t = D r ) 8 Capacity (bits) Capacity Curve R = 10 km λ = 0.15 m M = 3 N = 3 SNR = 10 d t = 3λ d r = 3λ D t (m) Fig. 8. Mean capacity as a function of D t = D r. The range R is fixed to 10km. The capacity builds up quickly as the scattering radius increases. 20