Study of the Capacity of Ricean MIMO Channels

Study of the Capacity of Ricean MIMO Channels M.A. Khalighi, K. Raoof Laboratoire des Images et des Signaux (LIS), Grenoble, France Abstract It is well known that the use of antenna arrays at both sides of the communication link can result in high channel capacities provided that the propagation medium is rich scattering.in most previous works presented on MIMO wireless structures, Rayleigh fading conditions were considered.here we study the capacity of MIMO systems under Ricean fading conditions.it is shown that MIMO Rayleigh channels exhibit a larger capacity than the equivalent Ricean channels for high signal-to-noise ratios (SNR) and large number of antennas.for small number of antennas, however, the inverse works, especially for low SNRs. On the other hand, it is shown that if the line-of-sight component is not very significant, the increase in the capacity of Ricean MIMO channels by increasing the number of antenna elements is still considerable.flat (frequency non-selective) fading conditions are considered throughout the paper. Keywords : Antenna arrays, MIMO systems, Ricean fading, channel capacity, channel fading 1 Introduction The ever-growing demand for high data rate communication has favored the use of antenna arrays in wireless communication systems, in order to overcome the problem of limited bandwidth. Of particular interest is the use of antenna arrays at both sides of the radio link, which can result in very large channel capacities. Multipleinputs multiple-outputs (MIMO) systems, thus, have been an important subject of research during the past few years. When there is enough multipath, i.e., in a rich scattering propagation medium, the capacity of a MIMO channel can efficiently be multiplied, by adding antenna elements at both transmitter and receiver [1, 2, 3, 4]. MIMO systems are hence a promising solution for high bit rate applications that will provide a breakthrough in the future wireless communication. Analyses of MIMO channel capacity have mostly considered Rayleigh fading conditions. In fact, the large potential capacity of MIMO systems is for the case where the propagation medium is rich scattering. If there is not enough multipath, a MIMO system loses its advantage over other multi-antenna structures. At the worst case, when there is no multipath, the improvement in the capacity of a MIMO system over a single antenna structure (SISO for single-input single-output) reduces to a constant [2, 3]. When there is a dominant non-fading signal component present, such as a line-of-sight (LOS) propagation path between transmitter and receiver, Ricean fading conditions hold [5, 6]. It is also the case when fixed 1

scatteres/signal reflectors exist in addition to random main scatterers. In this case, the fading envelope is described by a Ricean probability density function. A priori, smaller capacities are expected for Ricean MIMO channels than for Rayleigh ones. Notice that at the limit of purely LOS propagation, we are in fact concerned with a no-multipath channel, where the use of the MIMO structure may be of no interest. The important question is, how the existence of a LOS component in signal propagation affects the MIMO capacity. Are MIMO structures still promising under Ricean fading conditions? The purpose of this paper is to study how the MIMO capacity is affected by the significance of the LOS component, and to see if the increase in the number of antenna elements can still result in a considerable increase in the system capacity. Meanwhile, we will also study single-input multiple-outputs (SIMO) structures and compare the capacity in two cases of Ricean and Rayleigh fading. The paper is organized as follows. Channel model and basic assumptions are given in Section 2. A model for Ricean propagation in relation to a MIMO structure is proposed in Section 3. Next, in Section 4, MIMO capacity expressions are provided when channel is known or not at transmitter. Section 5 considers the case of unknown channel at transmitter, where simulation results are provided comparing the information rate bounds of Ricean and Rayleigh channels. The case of known channel at transmitter is considered in Section 6, where the interest of an optimal power allotment on transmit antennas is studied for the case of Ricean MIMO channels. Discussion on the presented results and conclusions are given in Section 7. 2 Assumptions and channel model The global scheme of a MIMO communication structure is shown in Fig.1. The channel capacity is considered under constraints on signal bandwidth and the total transmit power. No beam forming is considered for the antenna arrays, and the antenna elements patterns are considered as omni-directional with unity gain. Also, the average signal attenuation corresponding to all transmit-receive antenna pairs is considered the same and equal to 1. A discrete-time baseband equivalent channel model is considered, and flat (non-dispersive) fading conditions are assumed. With M R antenna elements at receiver and M T elements at transmitter, the channel matrix H of dimension (M R M T ) will characterize the channel. Entries of H, h ij, which are normalized 1 circularlysymmetric complex random processes, represent the baseband equivalent channel impulse response between jth transmit and ith receive antennas. The statistics of h ij depend, in fact, on the fading conditions. Channel is assumed to be perfectly known at receiver. We will assume quasi-static (also called quasi-stationary) conditions, that is, H is assumed to be constant during one or more communication bursts. Bursts are assumed to be long enough, so that the definition of capacity for a given H is meaningful. In this way, the continuous channel fading process is approximated as piece-wise constant [7, 8]. It is assumed that the antenna elements at both sides of the link are sufficiently spaced apart, so that the mul- 1 In the sense that E{h ij h ij } =1. 2

Figure 1: Block diagram of a MIMO communication structure tipath components of the received signals can be considered to fade independently over the receiver antennas. For a randomly time-varying channel, the mutual information (and the capacity) can be regarded as a random quantity, giving rise to capacity-versus-outage considerations. In fact, if the instant channel capacity is less than the pre-assumed value, a channel outage is said to be occurred. In this paper we always consider capacity-versus-outage with P out =0.01 (outage probability). So, the presented capacity values correspond to 99% percentage point of CCDF (Complementary Cumulative Distribution Function) of capacity. 3 Ricean channels Modeling For Ricean channels, the received signal can be considered to be composed of two components, one from LOS and the other from multipath reflections. The former component is deterministic and constant, since it is not subjected to fading, whereas the latter is a randomly time-varying component. So, we can consider the channel matrix as, H Rice = H const + H random (1) Notice that for (1) to hold (concerning H const ), the transmitter and receiver should be almost fixed, which is usually the case in current MIMO structures implementations. We can write the above equation as follows [9]. H Rice = ae jϕ0 H LOS + b H Rayleigh (2) Elements of H Rayleigh are independent normalized (unit-variance) complex circularly symmetric Gaussian random variables. To respect the normalization on the entries of H Rice, we should impose a 2 + b 2 = 1. In this way, the ratio of the averaged received power from LOS and multipath reflections is equal to a 2 /b 2,whichis usually regarded as Ricean K-factor. Instead of K-factor, we will use in our analyses the Ricean Factor defined as RF=a 2 /(a 2 + b 2 )=a 2. The interest of RF for us is that it indicates directly the significance (or in other words, the contribution) of the LOS component in signal transmission. Consider linear arrays broadside to each other and with equal antenna spacings at each side, as shown on Fig.2. In fact, ϕ 0 in (2) takes into account the absolute phase shift between antenna elements #1 at the transmitter and receiver. Consider the first column of H LOS as follows, [ ] T H LOS,1 = 1 e jθ. e j(m R 1)θ = H LOS(:, 1) (3) where. T indicates the transpose operation. Under the condition of R D, that is, large distance between antenna arrays as compared to the antenna spacings, the LOS component of signal propagation can be considered 3

Figure 2: Array geometry considering linear arrays a plane wave in the scenario of Fig.2. So, as the arrays are considered to be broadside to each other, the phase shift θ between signals received on neighbor antennas will be negligible [10] (for a detailed discussion, see the appendix). Neglecting θ, wehave, H LOS,1 (i) 1, i =1,..., M R (4) The same argument is valid for other columns of H LOS under the condition of R D. So, we can write [9] H Rice = ae jϕ0 1 + b H Rayleigh (5) 1 is the unity matrix (with all entries equal to 1). Notice that with the assumptions made above, we have rank(h LOS )=1. 2 To impose equally the effect of LOS (constant) component on real and imaginary parts of the multipath (Rayleigh) component, we will take ϕ 0 = ±π/4, so by e jϕ0 = 1 2 + j 1 2, will have equal average power of LOS component in real and imaginary parts of H Rice entries. In general, however, the capacity of channel depends on the value of ϕ 0, but the general results of this paper are valid for any ϕ 0. Note that the special geometry of Fig.2 and the assumption of R D were considered just to simplify the analysis and to make it possible to get rid of the array-dependent parameters of the LOS propagation component. We will later discuss these assumptions in Section 7, and will explain that the general results to be presented, are valid in the general case. 4 Capacity expressions Let x be the vector of transmitted symbols on M T antennas at one sample time. The vector of corresponding received symbols on the receiver array, z, will be z = Hx + n = y + n (6) where n is the equivalent baseband noise whose elements are considered as zero-mean circularly-symmetric complex additive white Gaussian noise (AWGN) samples with the variance σ 2. We consider the condition that the total transmit power at each sample time is constrained to P T. 2 Here we considered the case of the presence of a LOS in signal propagation. If the Ricean propagation model is due to the existence of fixed dominant reflectors, the same expressions are valid assuming far field conditions to hold. 4

4.1 Channel known at transmitter If channel state information (CSI) is available at transmitter, the available power can be distributed optimally over the transmit antenna, the solution known as water filling (WF). The WF capacity is given by [2, 4, 11, 12], C WF = M i=1 log 2 ( 1+ λ X,i σ 2 λ 2 H,i ) bps/hz (7) ( ) + λ X,i = ψ σ2 λ 2 H,i (8) where, (s) + = s for s>0and zero otherwise. λ H,i are singular values of the matrix H, andm = min(m T,M R ). Also, λ X,i are the eigenvalues of the transmit-symbols autocorrelation matrix, R X. ψ is determined so as to satisfy the constraint on the total transmit power, M T λ X,i = P T (9) i=1 For details on implementation aspects, as well as the transmitter and receiver structures see [10, 13]. 4.2 Channel unknown at transmitter If the CSI is not available at transmitter, P T is distributed uniformly over the transmit antennas. In this case, the MIMO channel capacity is given by [1, 4] C no-wf = M i=1 ( log 2 1+ P ) T M T σ 2 λ2 H,i bps/hz (10) In this paper, unless otherwise mentioned, channel is assumed unknown at transmitter, and the channel capacity is considered according to (10). Under the same condition, for a single-input multiple-outputs (SIMO) channel, the capacity expression is as follows [1, 4]. ( ) C SIMO = log 2 1+ P M R T σ 2 H i 2 bps/hz (11) i=1 Considering the assumptions of the previous section and the model considered for the arrays, if the channel is completely LOS and there is no multipath, we have rank(h) = 1. In this case, H=H LOS = 1 and assuming M R = M T = M, wehaveλ H,1 = M and λ H,i =0; i =2,..., M. Therefore, the capacity will be, ( C LOS = log 2 1+M P ) T σ 2 bps/hz (12) Here the capacity is a deterministic value, and the use of antenna arrays has only the effect of a gain in SNR at receiver. In other words, we gain nothing in channel capacity by employing multiple antennas at transmitter. It can be easily seen that the same expression holds for a SIMO structure (with M R = M) for a purely LOS channel under the conditions explained in Section 3. 5

Figure 3: Comparing capacity of Rayleigh and Ricean MIMO channels; M=M T =M R, P out=0.01 for RF 1, SNR=10dB 5 Unknown-CSI at transmitter 5.1 Comparing Ricean and Rayleigh MIMO capacities Using the capacity expressions given in Subsection 4.2, simulations are made to study the capacity of Ricean channels and to compare it with the Rayleigh channel capacity. The results to be presented are obtained using at least 10 5 channel realizations. We expect that with an increase in RF, smaller capacity be achieved for MIMO channels. Let us first consider a moderate signal-to-noise ratio, SNR= PT σ =10dB. Fig.3 shows curves of capacity of Ricean channels (at 2 P out =0.01) versus M T =M R =M (the number of antennas at both sides), for different cases of RF =0, 30, 70, 90, and 100%. RF=0% represents the pure Rayleigh fading, while RF=100% represents the purely LOS channel. Remember that for a purely LOS channel, the capacity is a deterministic value. It is seen that with an increase in RF, smaller capacities are obtained, but is is not the case for small M (here M = 2). On the other hand, the increase in the MIMO capacity with increase in M is still considerable even for RFs about less than 70%. To see better the effect of RF, curves of MIMO channel capacity versus RF are shown on Fig.4 for SNR=10dB and M=M T =M R =1,2,4,6 (M = 1 represents the SISO channel case). It is seen that for M=4,6, the capacity decreases with increase in RF, however, it is not the case for M=2. CCDF curves of the capacity are given on Fig.5 for SNR=10dB, several values of RF, and two cases of M=M T =M R =2,6. Similar results as in Fig3 are shown in Fig.6 for the case of SNR=0dB, a relatively low-snr. It is seen that for M<5, the capacity increases with increase in RF. 6

Figure 4: Capacity of Ricean MIMO channels versus Ricean Factor; P out=0.01 for RF 1, SNR=10dB 5.2 Discussion In fact, the gain in the capacity of a MIMO structure compared to a SISO one, can be considered to be composed of two components [14]; the array gain at receiver which corresponds to the gain in the average power of the signal combination on M R antennas, and the diversity gain which corresponds to the gain from increasing the system dimensionality (rank of H) and depends highly on the correlation between h ij. The array gain is obviously the same for any RF, but the diversity gain decreases with an increase in RF, because the correlation between h ij increases. It is similar to the concept of correlated fading discussed in [15, 16, 17]. On the other hand, a greater RF corresponds to less signal fading at receiver. We know that the fading is more destructive at low SNR. The results of Fig.3 and Fig.6 show that for low SNR and relatively small number of antennas, a greater RF (and so, less fading) is better that a small RF (and hence a greater diversity gain). As a matter of fact, concerning the Ricean MIMO channel capacity, there is a compromise between the diversity gain and reduced fading. For large number of antennas, the dominant factor is the diversity gain, and the capacity increases with decrease in RF. Results of Fig.3,6 are not indeed contradictory with the previous statements on MIMO systems. MIMO systems are known to be promising in a rich-scattering propagation medium, and the capacity increases almost linearly with the number of antennas at both sides, in high-snr [2, 3]. From the results presented here, it is expected that the limit of high-snr condition depends on the number of antennas. Fig.7 confirms this, showing the curves of capacity versus SNR for M=M T =M R =2,3,4 for Rayleigh and purely LOS MIMO channels. Notice that the corresponding curves for Ricean channels lay between the curves of Rayleigh and LOS. It is seen that for M =2, for example, the MIMO capacity does not rely on multipath propagation under the limit of SNR<26dB. This limit is much lower for greater M, such as M=4 (about 1.7dB). Indeed, our choice of SNR=0,10dB was just to reveal this dependency on SNR of the difference between Rayleigh and Ricean capacities. 7

Figure 5: Comparing CCDF curves of capacity for Rayleigh channels and Ricean channels with different Ricean Factors; SNR=10 db; (a) M=M T =M R=2, (b) M=M T =M R=6 8

Figure 6: Comparing capacities of Rayleigh and Ricean MIMO channels; M=M T =M R, P out=0.01 for RF 1, SNR=0dB 30 25 C (bps/hz) 20 15 10 Rayleigh M=4 Rayleigh M=3 LOS, M=4 M=3 M=2 5 Rayleigh M=2 0 5 0 5 10 15 20 25 30 SNR (db) Figure 7: Comparing capacities of Rayleigh and LOS MIMO channels as a function of SNR; M=M T =M R, P out=0.01 for Rayleigh case 9

Figure 8: Capacity of Ricean SIMO channels versus Ricean Factor; P out=0.01 for RF 1, SNR=10dB 5.3 SIMO Ricean capacity To see how the capacity of SIMO channels is affected by the presence of a LOS, simulation results are performed for this case too. Fig.8 shows the capacity curves versus RF for M R =1,2,4,6 and SNR=10dB. As expected, the capacity increases with an increase in RF. In fact, for SIMO channels, the purpose of using multiple antennas at receiver is to combat signal fading, and to gain in average received SNR. Evidently, for greater RF values, fading is less significant, and so, greater capacities are resulted. 6 Known-CSI at transmitter 6.1 Comparing Ricean and Rayleigh MIMO capacities In the previous sections, we considered the unknown-csi channel capacity. It is also interesting to study the WF capacity of Ricean channels and to compare it with the case of Rayleigh fading. For the case of Rayleigh fading and for equal number of antennas at receiver and transmitter (M R =M T ), it is known that the optimal WF solution is of interest for low SNR and relatively small number of antennas [13, 18]. The case is somehow different for Ricean channels, depending on RF. Fig.9 shows curves of MIMO channel capacity, with and without WF as a function of RF. Also, Fig.10 shows the increase in capacity (called WF-gain) as a function of RF. Two cases of M=M T =M R =2 and M=4 are considered. Remember from the results of Fig.3,4 that for M=4, it is the diversity gain which has the dominant effect, whereas for M=2, the fading reduction has the major role. We have also chosen SNR=10dB, so as to be able to make a comparison with the case of Rayleigh fading. It is seen that the WF-gain is very important, regardless of M. Even, for great RF, the obtained gain is more important for a greater M. 10

9 8 7 WF, M=4 no WF, M=4 C(bps/Hz) 6 5 4 WF, M=2 no WF, M=2 3 2 0 10 20 30 40 50 60 70 80 90 100 RF(%) Figure 9: WF solution for a Ricean MIMO channel with M T =M R=M, SNR=10dB, P out=0.01 2 1.8 1.6 WF gain (bps/hz) 1.4 1.2 1 M=4 0.8 0.6 M=2 0.4 0 10 20 30 40 50 60 70 80 90 100 RF(%) Figure 10: WF-gain in capacity, same conditions as in Fig.9 11

6.2 Discussion To explain these results, we remember that the MIMO channel can be regarded as to be composed of a set of equivalent parallel independent subchannels, whose gain is given by the singular values of the channel matrix H, and their number is equal to the rank of H. The capacity of (10) is indeed the sum of the capacities of these subchannels. WF solution, in fact, consists of distributing the power optimally on these subchannels, as given by (8) [13, 18]. By WF, more power is assigned to better subchannels, those with greater gain (corresponding to greater λ H,i ), and less or probably no power to worse subchannels, the noisiers (corresponding to smaller λ H,i ). For a Ricean channel, for increased RF, one of the singular values of H becomes dominant, and the others approach to zero. By uniformly distributing the available power P T on the equivalent parallel subchannels (i.e., λ X,i = P T /M ), we lose power by assigning it to bad ones, i.e. subchannels corresponding to small singular values of H. Consequently, the WF gain is more important for a greater RF. Notice that for a LOS channel, H has only one non-zero singular value, and the optimal solution is to distribute the total power on the corresponding subchannel. 7 Discussion and conclusions We studied in this paper the capacity of Ricean MIMO channels. The presented results were particularly based on the assumption of using linear arrays which are far from and broadside to each other. Our results revealed that the effect of LOS component contribution on the MIMO channel capacity depends on SNR and the number of antennas. For high SNR values (usually the case in indoor applications), the dominant factor which affects the channel capacity is the diversity gain. Here, an increase in RF increases the correlation between the channel coefficients, and as a consequence, the capacity decreases as a result of decreased diversity gain. Our results correspond well with those of [19]. We also showed that the increase in capacity by increase in the number of antennas is still considerable for RF< 70%. For low SNR values (usually the case in outdoor applications) and relatively small number of antennas, it is not the diversity gain which has the dominant effect. For low SNR, fading is more destructive, and so, an increase in RF may result in an increase in the MIMO channel capacity. In other words, fading reduction affects more the channel capacity than the diversity gain. Meanwhile, for large number of antennas, there is potentially a large diversity gain which dominates the effect of fading reduction on capacity. In this case, the capacity again increases with decrease in RF. In fact, here we can speak of a compromise between fading reduction at receiver and the diversity gain given by the MIMO structure. For a SISO or a SIMO channel, the diversity gain equals 1, and the only factor affecting the capacity is fading reduction. So, the capacity increases with an increase in RF. In particular, the considerable increase in the capacity of SIMO channels for RF> 70% was shown in Fig.8. 12

Concerning MIMO structures, with the presented model, where no multipath exists and the channel is completely LOS, employing multiple antennas at transmitter is of no use and the capacity is equal to that of a SIMO structure, as given in (12). If the LOS component is very significant, special antenna arrangements may be employed (if there is such a freedom for the system designer) to obtain large capacities. The idea is to arrange the antenna elements in order to produce special phase shifts between the signals received on different antennas, in a way that we obtain M = rank(h LOS ) orthogonal SISO subchannels. Some examples of such antenna arrangements are presented in [19] where, in general, large antenna spacings are required for most of them. With such structures, a considerable increase in capacity can be achieved by increase in M. On the other hand, if there is a limitation on adopting such an antenna geometry, channel capacity can be increased by adding some reflectors in the propagation medium, so as to weaken the LOS contribution. Although the results presented in these papers were for a special and simple channel model, they are valid in general case. Notice that the decrease in diversity gain is because of the increased correlation between the channel coefficients. This correlation comes form the LOS component, and is independent of the parameters such as the angle-of-departure, angle-of-arrival, or the distance between the arrays. So, we expect that the same general results are valid for a different scenario than that in Fig.2 (such a particular situation can be the case in indoor applications where notably, the distance between the arrays may not be very great). For example, we can speak of the results presented in [20] which are for a particular angle-of-arrival at the receiver array. Notice also that we assumed that the LOS component is constant and deterministic. Hence, our case differs a little from the concept of correlated fading, studied in [15, 16, 17] for example. In the latter case, multipath signals arrive at the receiver from a given direction. In other words, it may be regarded as a randomly varying LOS contribution. References [1] G.J. Foschini and M.J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications, vol.6, 1998, pp.311-335 [2] G.G. Raleigh and J.M. Cioffi, Spatio-temporal coding for wireless communication, IEEE Transactions on Communications, vol. COM-46, No.3, Mar. 1998, pp.357-366 [3] G.G. Raleigh and V.K. Jones, Multivariate modulation and coding for wireless communication, IEEE Journal on Selected Areas in Communications, vol. SAC-17, No.5, May 1999, pp.851-866 [4] M.A. Khalighi, K. Raoof, and G. Jourdain, Capacity of wireless communication systems employing antenna arrays, a tutorial study, Journal of Wireless Personal Communications, accepted for publication, June 2002 [5] B. Sklar, Rayleigh fading channels in mobile digital communication systems; Part I: Characterization; Part II: Mitigation, IEEE Communication Magazine, vol.35, No.7, 1997, pp. 90-109 13

[6] G.J. Proakis, Digital Communications, McGraw Hill, second edition, 1989 [7] G.J. Foschini, Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas, Bell Labs Technical Journal, vol.1, No.2, Autumn 1996, pp.41-59 [8] T.L. Marzetta and B.M. Hochwald, Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading, IEEE Transactions on Information Theory, vol. IT-45, No.1, Jan. 1999, pp. 139-157 [9] M.A. Khalighi, J.M. Brossier, G. Jourdain, and K. Raoof, On capacity of Ricean MIMO channels, in Proceedings of PIMRC 2001, 30 Sept. - 3 Oct. 2001, San Diego, CA, vol.a, pp. 150-154 [10] M.A. Khalighi, Study of Multiple Antenna Communication Systems ; Channel Capacity and Iterative Detection, Ph.D. Thesis, Oct. 2002, INPG University, Grenoble, France [11] E. Telater, Capacity of multi-antenna Gaussian channel, AT&T Bell Labs, Tech. Memo., June 1995 [12] E. Telatar, Capacity of multi-antenna Gaussian channels, invited paper, European Transactions on Telecommunications, vol. ETT-10, No.6, Nov.-Dec. 1999, pp. 585-595 [13] M.A. Khalighi, K. Raoof, and G. Jourdain, Increase in the capacity of transmit diversity systems by optimal power allotment at transmitter, EURASIP Journal on Signal Processing, submitted [14] J.B. Anderson, Array gain and capacity for known random channels with multiple element arrays at both ends, IEEE Journal on Selected Areas in Communications, vol. SAC-18, No.11, Nov. 2000, pp. 2172-2178 [15] W.C. Jakes, Microwave Mobile Communications, IEEE Press, 1998. [16] J. Salz and J.H. Winters, Effect of fading correlation on adaptive arrays in digital mobile radio, IEEE Transactions on Vehicular Technology, vol. VT-43, No.4, Nov.1994, pp.1049-1057 [17] D. Shiu, G.J. Foschini, M.J. Gans, and J.M. Kahn, Fading correlation and its effect on the capacity of multi-element antenna systems, IEEE Transactions on Communications, vol. COM-48, No.3, Mar. 2000, pp.502-513 [18] M.A. Khalighi, J.M. Brossier, G. Jourdain, and K. Raoof, Water Filling Capacity of Rayleigh MIMO channels, in Proceedings of PIMRC 2001, 30 Sept. - 3 Oct. 2001, San Diego, CA, vol.a, pp. 155-158 [19] D.F. Driessen and G.J. Foschini, On the capacity formula for multiple input-multiple output wireless channels: a geometric interpretation, IEEE Transactions on Communications, vol. COM-47, No.2, Feb. 1999, pp.173-176 [20] F.R. Farrokhi, G.J. Foschini, A. Lozano, and R.A. Valenzuela, Link-optimal space-time processing with multiple transmit and receive antennas, IEEE Communications Letters, vol.5, No.3, Mar. 2001, pp. 85-87 14

Appendix: Hypothesis of plane wave at receiver array of LOS component In this appendix we explain the rationality of neglecting θ in (3) when considering Fig.2, we assumed that the arrays are linear and broadside to each other. Since equal antenna spacings are assumed, the phase shifts between signals received on neighbor antenna elements are equal. We also assumed that R D, with D the distance between the antenna elements at the receiver, usually greater than λ/2. That is, it is assumed that the transmitter and the receiver are positioned far from each other. This assumption is well satisfied in many applications. we have, θ = R R 2π (13) λ λ is the wavelength. If antenna arrays are positioned broadside to each other, θ can be negligible. For example, for the special configuration considered on Fig.2, we can write, θ = 4πD sin2 δ/2 λ sin δ (14) δ is the angle between two lines from the transmit antenna #1 to the receive antennas #1, 2. For R D, δ is a very small angle and we can write (sin δ = D R δ), θ πdδ λ πd2 λr 1 (15) 15