MIMO Wireless Channels: Capacity and Performance Prediction

D. Gesbert H. B¨olcskei, D. Gore, A. Paulraj

Gigabit Wireless Inc., Information Systems Laboratory 3099 North First Street, Department of Electrical Engineering San Jose, CA 95134 Stanford University, Stanford, CA 94305 [email protected] {bolcskei, dagore, paulraj}@rascals.stanford.edu

Abstract – We present a new model for multiple- tic assumption in which the entries of the channel ma- input multiple-output (MIMO) outdoor wireless fad- trix are independent complex Gaussian random vari- ing channels which is more general and realistic than ables [2, 6, 8]. The inﬂuence of spatial fading corre- the usual i.i.d. model. We investigate the channel lation on either the TX or the RX side of a wireless capacity as a function of parameters such as the lo- MIMO radio link has been addressed in [3, 15]. In prac- cal scattering radius at the transmitter and the re- tice, however, the realization of high MIMO capacity is ceiver, the distance between the transmit (TX) and sensitive not only to the fading correlation between in- receive (RX) arrays, and the antenna beamwidths dividual antennas but also to the rank behavior of the and spacing. We point out the existence of “pin- channel. In the existing literature, high rank behav- hole” channels which exhibit low fading correlation ior has been loosely linked to the existence of a dense between antennas but still have poor rank proper- scattering environment. Recent successful demonstra- ties and hence low capacity. Finally we show that tions of MIMO technologies in indoor-to-indoor chan- even at long ranges high channel rank can easily be nels, where rich scattering is almost always guaranteed, obtained under mild scattering conditions. conﬁrm this [9]. Despite this progress, several important questions re- garding outdoor MIMO channels remain open and are 1. INTRODUCTION addressed in this paper:

Multiple-input multiple-output (MIMO) communica- • What is the capacity of a typical outdoor MIMO tion techniques make use of multi-element antenna ar- channel? rays at both the TX and the RX side of a radio link and have been shown theoretically to drastically improve • What are the key propagation parameters govern- the capacity over more traditional single-input multiple- ing capacity? output (SIMO) systems [2, 3, 5, 7]. SIMO channels in • wireless networks can provide diversity gain, array gain, Under what conditions do we get a full rank MIMO and interference canceling gain among other benefits. In channel (and hence high capacity)? addition to these same advantages, MIMO links can of- • What is a simple analytical model describing the fer a multiplexing gain by opening Nmin parallel spatial capacity behavior of outdoor MIMO wireless chan- channels, where Nmin is the minimum of the number nels ? of TX and RX antennas. Under certain propagation conditions capacity gains proportional to Nmin can be Here we suggest a simple classification of MIMO chan- achieved [8]. Space-time coding [14] and spatial multi- nels and devise a MIMO channel model whose generality plexing [1, 2, 7, 16] (a.k.a. “BLAST”) are popular signal encompasses some important practical cases. Unlike the processing techniques making use of MIMO channels to channel model used in [3, 15], our model suggests that improve the performance of wireless networks. the impact of spatial fading correlation and channel rank Previous work and open problems. The litera- are decoupled although not fully independent, which al- ture on realistic MIMO channel models is still scarce. lows for example to describe MIMO channels with un- For the line-of-sight (LOS) case, previous work includes correlated spatial fading at the transmitter and the re- [13]. In the fading case, previous studies have mostly ceiver but reduced channel rank (and hence low capac- been confined to i.i.d. Gaussian matrices, an idealis- ity). This situation typically occurs when the distance between transmitter and receiver is large. Furthermore, 2.2. Model classification our model allows description of MIMO channels with Let us next introduce the following MIMO theoretical scattering at both the transmitter and the receiver. channel models: We use the new model to describe the capacity behavior as a function of the wavelength, the scattering • Uncorrelated high rank (UHR, a.k.a. i.i.d.) model: radii at the transmitter and the receiver, the distance The elements of H are i.i.d. CN(0, 1). This is the between TX and RX arrays, antenna beamwidths, and idealistic model considered in most studies. antenna spacing. Our model suggests that full MIMO • Uncorrelated low rank (ULR) (or “pin-hole”) capacity gain can be achieved for very realistic values of scattering radii, antenna spacing and range. It shows, in model: H = grx gtx,wheregrx and gtx contrast to usual intuition, that large antenna spacing are independent RX and TX fading vectors ∼ has only limited impact on capacity under fairly general with i.i.d. complex-valued components grx CN ∼CN conditions. Another case described by the model is the (0,IM),gtx (0,IN). Every realization of “pin-hole” channel where spatial fading is uncorrelated H has rank 1 and therefore although diversity is and yet the channel has low rank and hence low capac- present capacity will be much less than in the ULR ity. We show that this situation typically occurs for very model since there is no multiplexing gain. large distances between transmitter and receiver. In the • Correlated low rank (CLR) model: H = × 1 1 case (i.e. one TX and one RX antenna), the pin- grxgtxurxutx where grx ∼CN(0, 1) and gtx ∼ hole channel yields capacities worse than the traditional CN(0, 1) are independent random variables and urx Rayleigh fading channel. Our results are validated by and utx are fixed deterministic vectors of size M × 1 comparing with a ray tracing-based channel simulation. and N × 1, respectively, and with unit modulus en- We find a good match between the two models over a tries. This model yields RX array gain only. wide range of situations. • 1 × 1 HR, defined by the UHR model with M = 2. CAPACITY OF MIMO CHANNELS AND MODEL N = 1, also known as Rayleigh fading channel. CLASSIFICATION • 1 × 1 LR, defined by the ULR or CLR model with We briefly review the capacity formula for MIMO chan- M = N = 1 (double Rayleigh channel). nels and present a classification of MIMO channels. Note that the low rank models (ULR, CLR, 1×1LR) We restrict our discussion to the frequency-flat fading above do not use the traditional normal distribution for case and we assume that the transmitter has no chan- the entries of H but instead the product of two Gaussian nel knowledge whereas the receiver has perfect channel variables. This type of distribution is shown later to knowledge. occur in important practical situations. In the 1×1case, The LR model has worsened fading statistics. This is 2.1. Capacity of MIMO channels due to the intuitive fact that a double Rayleigh channel will fade “twice as often” as a standard Rayleigh channel We assume M RX and N TX antennas. The capacity [4]. in bits/sec/Hz of a MIMO channel under an average transmitter power constraint is given by1 [2] 3. DISTRIBUTED SCATTERING MIMO MODEL h i ρ C =log2 det IM + HH , (1) N We consider non-line-of-sight channels, where fading is induced by the presence of scatterers at both ends of the where H is the M × N channel matrix, IM denotes the radio link. The purpose is to develop a general stochas- identity matrix of size M, and ρ is the average signal- tic channel model that captures separately the diversity to-noise ratio (SNR) at each receiver branch. The el- and rank properties and that can be used to predict ements of H are complex Gaussian with zero mean practically the high rank region of the MIMO chan- and unit variance, i.e., [H]m,n ∼CN(0, 1) for m = nel. The particular case of LOS channels is addressed 1, 2, ..., M, n =1,2, ..., N. Note that since H is ran- in [4], where the authors derive a simple rule predicting dom C will be random as well. Assuming a piece-wise the high rank region. In the following, for the sake of constant fading model and coding over many indepen- simplicity, we consider the effect of near-field scatterers 2 dent fading intervals , EH {C} can be interpreted as the only. We ignore remote scatterers assuming that the Shannon capacity of the random MIMO channel [5]. path loss will tend to limit their contribution to the to- 1The superscript stands for Hermitian transpose. tal channel energy. Finally, we consider a frequency-flat 2 EH stands for the expectation over all channel realizations. fading channel. 3.1. SIMO Fading Correlation Model 3.2. MIMO Correlated Fading Model

We consider a linear array of M omni-directional RX We consider the NLOS propagation scenario depicted in antennas with spacing dr. A number of distributed scat- Fig. 2. terers act as perfect omnidirectional scatterers of a signal which eventually impinges on the RX array. The plane-wave directions of arrival (DOAs) of these signals span an angular spread of θr radians (see Fig. 1). D r dt d r

Or Os O t

M RXs N TXs Dt dr O r

M RXs R

Figure 2: Propagation scenario for fading MIMO channel.

Figure 1: Propagation scenario for SIMO fading corre- The propagation path between the two arrays is ob- lation. Each scatterer transmits a plane-wave signal to structed on both sides of the link by a set of significant a linear array near-field scatterers (such as buildings and large objects) refered to as TX or RX scatterers. Scatterers are mod- Several distributions can be considered for the DOAs, eled as omni-directional ideal reflectors. The extent of including uniform, Gaussian, Laplacian etc. [10, 11]. the scatterers from the horizontal axis is denoted as Dt The addition of different plane-waves causes space- and Dr, respectively. When omni-directional antennas selective fading at the RX antennas. It is well known are used Dt and Dr correspond to the TX and RX scat- that the resulting fading correlation is governed by the tering radius, respectively. On the RX side, the signal angle spread, the antenna spacing and the wavelength. reflected by the scatterers onto the antennas impinge on The RX array response vector h can now be modeled as the array with an angular spread denoted by θr,where θr is function of the position of the array with respect ∼CN h (0,Rθr,dr ) or equivalently to the scatterers. Similarly on the TX side we define (2) 1/2 an angular spread θt. The scatterers are assumed to be h = Rθ ,d g with g ∼CN(0,IM), r r located sufficiently far from the antennas for the plane- where Rθr,dr is the M × M correlation matrix. For wave assumption to hold. We furthermore assume that uniformly distributed DOAs, we find [10, 12] Dt,Dr R (local scattering condition).

i= S− X 1 π −2πj(k−m)dr cos( +θr,i) [Rθr,dr ]m,k = e (3) 3.2.1. Signal at the Receive Scatterers S S− i=− We assume S scatterers on both sides, where S is an where S (odd) is the number of scatterers with corre- arbitrary, large enough number for random fading to sponding DOAs θr,i. For “large” values of the angle occur (typically S>10 is sufficient). The exact dis- spread and/or antenna spacing, Rθr,dr will converge tribution of the scatterers is irrelevant here. Every TX to the identity matrix, which gives uncorrelated fad- scatterer captures the radio signal and re-radiates it in ing. For “small” values of θr,dr, the correlation ma- the form of a plane wave towards the RX scatterers. trix becomes rank deficient (eventually rank one) caus- The RX scatterers are viewed as an array of S virtual ing (fully) correlated fading. For the sake of simplicity, antennas with average spacing 2Dr/S,andassuchex- we furthermore assume the mean DOA to be orthogo- perience an angle spread defined by tan(θS /2) = Dt/R. nal to the array (bore-sight). Note that the model pro- We denote the vector signal originating from the n-th vided in (2) can readily be applied to an array of TX TX antenna and captured by the S RX scatterers as T antennas with corresponding antenna spacing and signal yn =[y1,n, y2,n, ..., yS,n] . Approximating the RX departure angle spread. scatterers as a uniform array of sensors and using the correlation model of (3.1), we find After proper power normalization3 and replacing Y by (6), we obtain the following new MIMO model ∼CN yn (0,RθS,2Dr/S) or equivalently 1/2 √1 1/2 1/2 1/2 yn = R gn with gn ∼CN(0,IS). H = Rθ ,d GrRθ ,2D /SGtRθ ,d . (9) θS ,2Dr/S S r r S r t t (4)

For uncorrelated TX antennas, the S×N channel matrix 3.3. Interpretation & The Pin-Hole Channel describing the propagation between the N TX antennas In (9), the spatial fading correlation between the TX and the S scatterers Y =[y1,y2, ..., yN ] simply writes antennas, and therefore the TX diversity gain, is gov- 1/2 1/2 erned by the deterministic matrix Rθ ,d and hence im- Y = R Gt, (5) t t θS ,2Dr /S plicitly by the local TX angle spread, the TX antenna beamwidth and spacing. On the RX side, the fading where Gt =[g1,g2, ..., gN ]isanS×Ni.i.d. Rayleigh correlation is similarly controled by the RX angle spread fading matrix. However, there is generally correlation 1/2 and antenna spacing through R . between the TX antennas because of finite angle spread θr,dr The rank of the MIMO channel is primarily controled and insufficient antenna spacing. Therefore, a more ap- 1/2 through R . The model in (9) shows that it is propriate model becomes θS,2Dr /S well possible to have uncorrelated fading at both sides, 1/2 1/2 t and yet have a rank deficient MIMO channel with re- Y = RθS,2Dr/S G Rθt,dt , (6) duced capacity. Such a channel is dubbed a “pin-hole” 1/2 × because scattering (fading) energy travels through a where Rθt,dt is the N N matrix controlling the TX antenna correlation as suggested in the TX form of model very thin air pipe, preventing the rank to build up. In (2). practice, this occurs when the product DtDr is small compared to the range R, making θS small, and causing 1/2 the rank of R to drop. Note that Dt, Dr play a 3.2.2. The MIMO Model θS,2Dr/S role analogous to dt, dr in the green field case, as shown Like the TX scatterers, the Rx scatterers are assumed in [4]. here to ideally reradiate the captured energy. As shown Eq. (9) suggests that in the scattering case the rank in Fig.2, a set of plane waves, with total angle spread behavior of the MIMO channel is mainly governed by θr, impinge on the RX array. Denoting the distance the scattering radii and by the range. Scatterers can between the s-th scatterer and the m-th RX antenna as be viewed as virtual antenna arrays with very large ds,m, the vector of received signals from the n-th TX spacing and aperture. Unlike the usual intuition, the antenna can be written as physical antenna spacing has limited impact on the ca-   pacity provided antennas remain uncorrelated, which e−2πjd,/λ ...... e−2πjdS,/λ   occurs at λ/2 spacing for reasonably high local angle zn = ::yn. (7) spread/antenna beamwidth. Note that if scattering is e−2πjd,M /λ ...... e−2πjdS,M /λ | {z } absent at one end of the link, the relevant parameter Φ on that particular end driving the MIMO rank becomes the antenna spacing. Collecting all RX and TX antennas according to Z = When either the TX or the RX antennas are fully [z1, z2, ..., zN ], we obtain correlated due to small local angle spread, the rank of Z =ΦY, (8) the MIMO channel also drops. In this situation, both the diversity and multiplexing gains vanish, preserving where Φ is the M × S matrix in (7). The problem with only the RX array gain. Note that there is no TX array the expression in (8) is the explicit use of deterministic gain since we assumed that the channel is unknown in phase shifts in the matrix Φ which makes the model in- the transmitter. convenient. The simple equivalence result below allows From the remarks above it follows that antenna cor- us to get rid of this inconvenience and obtain a new and relation causes rank loss but the converse is not true. entirely stochastic MIMO model. The new model contains not one but the product of Lemma. For S →∞,Z=ΦYhas the same p.d.f. two random Rayleigh distributed matrices. This is in 1/2 r r contrast with the traditional Rayleigh MIMO model of as Rθr,dr G Y where G is an i.i.d. Rayleigh fading × 3 matrix of size M S. We use a normalization to fix the channel energy regardless Proof. See the appendix. of how many scatterers are considered. 1/2 Comparison of CDF Curves [2, 8]. Depending on the rank of R ,there- 1 θS ,2Dr/S −.− MIMO Channel Model __ Ray Tracing Channel sulting MIMO fading statistics ranges “smoothly” from 0.9 .... UHR Uncorrelated High Rank ULR Uncorrelated Low Rank Gaussian to product of two independent Gaussians. In 0.8 1/2 0.7 the high rank region, RθS ,2Dr/S becomes the identity matrix. Using the central limit theorem, the product 0.6 intermediate GrGt approaches a single Rayleigh distributed matrix, 0.5

0.4 which justiﬁes the traditional model in that particular ULR Prob. Capacity < abscissa UHR 1/2 0.3 case. In the low rank (i.e. rank one) region, RθS ,2Dr /S is the all one matrix. The MIMO channel becomes 0.2 1/2 1/2 0.1 rx Rθr,dr g gtxRθt,dt , an outer-product with independent 0 0 2 4 6 8 10 12 14 TX and RX Rayleigh fading vectors. In this case we Capacity in Bit/Sec/Hz have no multiplexing gain, but there is still diversity gain with the exact amount depending on the TX and Figure 3: Capacity c.d.f. obtained with MIMO model RX fading correlation. for three sets of parameters. From left to right. Set 1: In practice depending on local angle spread and an- Dt = Dr =30m, R = 1000km. Set 2: Dt = Dr =50m, tenna spacing, the model will range smoothly from the R =50km. Set 3: Dt = Dr = 100m, R =5km. CLR to UHR models.

Comparison of CDF Curves for a 1x1 system @10dB SNR In the 1 × 1 case, meaningful high rank and low rank 1 1x1 double Rayleigh (theoretical) (dash−dotted) models can still be deﬁned, according to the rank taken 0.9 1/2 0.8 by RθS ,2Dr /S. The high rank model is the traditional Rayleigh channel. The low rank model has “double 0.7 1x1 Rayleigh (theoretical) (dash−dotted) Rayleigh” distribution. Note that the model does not 0.6 suggest the existence of a “correlated high rank” MIMO 0.5 channel, which corresponds also to intuition. 0.4

Prob. Capacity < abscissa Channel Model (continuous) 0.3 4. MONTE CARLO SIMULATIONS 0.2 0.1

0 0 1 2 3 4 5 6 7 8 The capacity distribution predicted by the proposed Capacity in Bit/Sec/Hz stochastic MIMO model for various values of the key parameters is compared to that achieved by an actual Figure 4: Capacity c.d.f. obtained for the 1 × 1model. ray tracing channel with the same parameters. We use two sets of parameters: from left to right. Set The ray tracing model follows the scenario depicted 1: Dt = Dr =30m, R = 1000km.Set2:Dt=Dr= in Fig. 2. In all examples we used S =20TXand 100m, R =5km. RX 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 bps/Hz in all cases and becomes almost exact as we ap- distribution of the scatterers as long as Dt,Dr and the proach UHR and ULR regions. angular spreads remain fixed. We used M = N =3and Finally, we look at the capacity (rank) build-up as placed the scatterers at a distance Rt from the TX array function of the scatering radius. and Rr from the RX array. We use Rr = Rr = Dt = Fig. 5 is a plot of average capacity for varying Dt = Dr in all simulations in order to maintain a high local Dr with R fixed at 10 km. The high capacity region is angle spread and hence low antenna correlation. The quickly attained, even for a very large range. Existing frequency was set to 2GHz and the SNR was 10 dB. measurements suggest practical scattering radiuses of To introduce random fading we use small random per- around 100 meters [11]. turbations of the TX and RX antenna array positions. We plot the capacity distribution (model and ray trac- 5. CONCLUSION ing) for three separate sets of control parameters, cover- ing the region between the UHR and the ULR models. We introduced a model for describing the capacity be- The curves obtained are shown in Fig. 3. havior of outdoor MIMO channels. The model describes 1/2 Fig. 4 illustrates the impact of the rank of RθS ,2Dr /S the effect of certain propagation geometry parameters in on the capacity in the 1 × 1 case. The proposed chan- scattering situations such as the scattering radius and nel model predicts the capacity distribution up to one the range. Our model predicts excellent performance Validation of Formula Predicting Knee in Capacity Curve (D = D ) t r 8.5 [2] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi- 8 element antennas,” Bell Labs Tech. J., pp. 41–59, Autumn 7.5 Capacity Curve 1996. R = 10 km λ = 0.15 m 7 M = 3 N = 3 [3] H. Bölcskei, D. Gesbert, A. Paulraj, “On the capacity of SNR = 10 d = 3λ 6.5 t wireless systems employing OFDM-based spatial multiplex- d = 3λ r ing”, IEEE Transaction on Communications, revised Sept. 6 Capacity (bits) 2000. 5.5 [4] D. Gesbert, H. Bölcskei, D. Gore, A. Paulraj, “Outdoor 5 MIMO wireless channels: Models and performance predic-

4.5 tion” IEEE Transaction on Communications, submitted July 2000. 4 0 10 20 30 40 50 60 70 80 90 100 D (m) t [5] I. E. Telatar, “Capacity of multi-antenna gaussian channels,” Tech. Rep. #BL0112170-950615-07TM, AT & T Bell Figure 5: Mean capacity as a function of Dt = Dr. The Laboratories, 1995. range R is fixed to 10km. The capacity builds up quickly [6] J. Bach Andersen, “Array gain and capacity for know ran- as the scattering radius increases. dom channels with multiple element arrays at both ends,” to appear in the IEEE Journal on Selected Areas in Com- munications, 2000. outdoors for very reasonable values of scattering radius, [7] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding almost regardless of how large the antenna spacing is. for wireless communication,” IEEE Trans. Comm., vol. 46, We pointed out the existence of pin-hole channels which no. 3, pp. 357–366, 1998. can occur for very large values of the range R. [8] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple anten- ACKNOWLEDGMENT nas,” Wireless Personal Communications, vol. 6, pp. 311– 335, 1998. The authors would like to thank Prof. J. Bach An- [9] “Experimental results for MIMO technology in an indoor-to- dersen for his helpful comments. indoor environment”, Internal Tech. Report, Gigabit Wire- less, March 1999. APPENDIX (Proof of the Lemma) [10] D. Asztély, “On antenna arrays in mobile communication 1/2 systems: Fast fading and GSM base station receiver algo- Let RθS ,2Dr/S = UΣU be the eigendecomposition of 1/2 rithms,” Tech. Rep. IR-S3-SB-9611, Royal Institute of Tech- nology, Stockholm, Sweden, March 1996. RθS ,2Dr /S. According to (6) [11] J. Fuhl, A. F. Molisch, and E. Bonek, “Unified channel 1/2 t model for mobile radio systems with smart antennas,” IEE Z =ΦY=ΦUΣU GRθt,dt . (10) Proc.-Radar, Sonar Navig., vol. 145, pp. 32–41, Feb. 1998. When S is large enough, the central limit theorem ap- [12]R.B.Ertel,P.Cardieri,K.W.Sowerby,T.S.Rappaport, plies to the product F =ΦUwhich tends to be nor- and J. H. Reed, “Overview of spatial channel models for an- mally distributed. Hence, [F]m,s ∼CN(0, 1). The cor- tenna array communication systems,” IEEE Personal Com- relation between the rows of ΦU is governed by the munications, pp. 10–22, Feb. 1998. RX angle spread θr and the antenna spacing through [13] P. Driessen, G. J. Foschini, “On the capacity formula for multiple input multiple output wireless channels: A geomet- Rθr,dr . Because the columns of U are orthogonal, we ric interpretation,” IEEE Transactions on Communications, easily show that in addition the columns of F are in- pp. 173–176, Feb. 1999. dependent. It can furthermore be shown that F ∼ 1/2 [14] V. Tarokh, N. Seshadri, A. R. Calderbank, “Space-time R Gr Gr × θr,dr ,where is an M S i.i.d. Rayleigh dis- codes for high data rate wireless communication: Perfor- tributed matrix. Hence, for large S,wehaveZ∼ mance criterion and code construction,” IEEE Transactions 1/2 1/2 on Information Theory, March 1998, vol. 44, no. 2, pp. 744- Rθ,d GrΣU GtRθ ,d . Finally, the distribution of Gr r r t t 765. is unchanged if we right-multiply Gr by the unitary 1/2 1/2 ∼ r t ∼ [15] D. Shiu and G. J. Foschini and M. J. Gans and J. M. matrix U and hence Z Rθr,dr G UΣU G Rθt,dt 1/2 1/2 1/2 Kahn, “Fading correlation and its effect on the capacity r t of multi-element antenna systems,” IEEE Trans. Comm., Rθr,dr G RθS,2Dr /SG Rθt,dt . March 2000, vol. 48, no. 3, pp. 502-513. REFERENCES [16] G. D. Golden, G. J. Foschini, R. A. Valenzuela, P. W. Wolni- ansky, “Detection Algorithm and Initial Laboratory Results [1] A. J. Paulraj and T. Kailath, “Increasing capacity in using the V-BLAST Space-Time Communication Architec- wireless broadcast systems using distributed transmis- ture,” Electronics Letters, Vol. 35, No. 1, Jan. 1999, pp. 14- sion/directional reception,” U. S. Patent, no. 5,345,599, 15. 1994.