Copyright: 2003 by WPMC Steering Board

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1 Copyright: 2003 by WPMC Steering Board Copyright and reproduction perission: All rights are reserved and no part of this publication ay be reproduced or transitted in any for or by any eans, electronic or echanical, including photocopy, recording, or any inforation storage and retrieval syste, without perission in writing fro the publisher. Notwithstanding, instructors are peritted to photocopy isolated articles for noncoercial classroo use without fee.

2 Deficiencies of apopular Stochastic MIMO Radio Channel Model Ernst Bonek,HüseyinÖzcelik,MarkusHerdin,Werner Weichselberger and JonWallace 2 ernst.bonek, hueseyin.oezcelik, arkus.herdin, wall@ieee.org Institut für Nachrichtentechnik und Hochfrequenztechnik, Technische Universität Wien Gußhausstrasse 25/389, A-040 Wien, Austria 2 Departent of Electrical and Coputer Engineering, Brigha Young University Provo, UT , USA Abstract Making a detailed review of the Kronecker MIMO radio channel odel we present iportant iplications and deficiencies of this channel odel. We show that the Kronecker odel does not render the ultipath structure correctly, but it introduces artifact paths that are not present in the underlying easureent data. Moreover, the Kronecker odel underestiates the utual inforation of the MIMO channel systeatically which we deonstrate by analyzing easureent data of an 8 8 MIMO syste in indoor non-line-of-sight scenarios. Introduction Multiple-input ultiple-output (MIMO) systes proise large inforation-theoretic capacities [], enabling high data rate transission, especially in rich ultipath indoor scenarios. However, to develop such systes, it is essential to have an accurate description of the underlying channel. An iportant paraeter of MIMO channels is the correlation between the channel atrix eleents because correlation is known to significantly influence the capacity/utual inforation. A well-known stochastic MIMO radio channel odel that creates channel realizations based on correlations is the so-called Kronecker odel [2] [3], also known as IST-METRA odel. This odel was already investigated in several publications, e.g. McNaara et al. in [4] or [5], where the perforance of the odel was found to be satisfactory. However, we show that this odel does not render the ultipath structure correctly, but it introduces artifact paths that are not present in the underlying easureent data and that therefore the utual inforation estiated fro this odel is systeatically below the utual inforation of the easured channel. The paper is organized as follows: Section 2 presents a review of the Kronecker odel, stating the assuptions, the iplications and the deficiencies of the odel. Section 3 briefly explains the easureent environent and the easureents we used to investigate the Kronecker odel and Section 4 shows the results of the easureent evaluation, finally followed by Section 5 where we present soe conclusions. 2 Review of the Kronecker Model 2. Signal Model and Notation We consider in the following frequency-flat fading MIMO channels with transit and n receive antennas where each single realization of the channel can be described by the n channel atrix H. The received signal vector is given by y Hx n () where x denotes the transit signal vector and n the noise at the receiver. The entries of the channel atrix H are denoted by h ik. Here, we will neglect the noise for the sake of siplicity. In the following, vec denotes the vector operator stacking all eleents of a atrix colun-wise into a single vector, T eans transpose, coplex conjugate, H Heritian transpose and E denotes the expectation operator. For an eleent-wise atrix notation we use A a i j which eans that A is build up fro the eleents a i j. Furtherore, we use a atrix square root 2 which fulfills 2 2 H and the Kronecker product denoted by. 2.2 Derivation of the Kronecker Model In [2] and [3], a popular stochastic MIMO radio channel odel was presented that allows the generation of MIMO channel realizations based on the receive and transit correlation atrix. The basic idea, in slightly different

3 +! notation, is to generate a correlated Rayleigh fading channel atrix H C n according to where vec H R 2 H vec G (2) R H E vec H vec H H (3) is the n n channel correlation atrix containing the coplex correlation between all n eleents of the n channel atrix H. G is a rando fading atrix with unity power independent entries deterining the single eleent fading statistics of H. In case of Rayleigh fading, it has zero-ean i.i.d. coplex Gaussian entries with unity variance. Note that for a Rayleigh fading channel, the second order statistics R H contains the full channel knowledge. There are two assuptions necessary and sufficient for a Kronecker structure of the full channel correlation atrix: (I) The receive antenna correlation between any pair of receive antennas i j given by ρ Rx ij k E h ik hjk (4) has to be independent of the transit antenna k. In atrix notation we can now define the receive correlation atrix by R Rx E HH H ρ Rx ij "! (5) Analogously, the transit antenna correlation between each transit antenna pair k l given by ρ Tx kl i E h ik hil (6) has to be independent of the receive antenna i. In atrix notation we can now define the transit correlation atrix by R Tx E H T H n ρ Tx kl (7) E # h ik # 2 of all chan- This leads to equal power P h nel atrix eleents. (II) The correlation coefficient between any pair of channel atrix eleents, as given by the full channel correlation atrix R H, has to be the product of the corresponding receive and transit antenna correlation coefficient noralized by the power of one channel atrix eleent: ρ H ik jl E h ik hjl P h ρ Rx i j ρ Tx kl (8) This can be equivalently forulated in atrix notation with tr R Rx tr R Tx n P h : R H tr R Rx With these assuptions and the identities R Tx R Rx! (9) $ B T A% vec D& vec ADB (0) A B ' C D ( AC )* BD ' () equation (2) siplifies to the so-called Kronecker odel H tr R Rx R 2 2 Rx G R Tx T! (2) Beside siplified analytical treatent or siulation of MIMO systes, (2) allows for independent array optiization at Tx and Rx. Therefore, and because of the siplicity of this approach, the Kronecker odel has becoe popular. 2.3 Iplications of the Kronecker Assuption To consider the effect of the Kronecker assuption on the directions of arrival (DOAs) and directions of departure (DODs), we have to look at the structure of the receive and transit correlation atrix for different transit and receive weights, respectively. Using (2), the receive correlation atrix R Rx w for beaforing at Tx side, i.e. using specific transit weights w is given by R Rx w E Hww H H H tr R Rx --,. E --/ R 2 Rx G 0 R 2 T Tx wwh 0 R 2 Tx X G H 0 R 2 Rx 8-- H (3) where we use X as an abbreviation for the inner part. It is straight-forward to show that E GXG H is always an identity atrix scaled by a coplex scalar factor K, if G is an unity power, independently distributed coplex Gaussian rando atrix (see Appendix). Therefore the receive correlation atrix R Rx w for the transit weights w is equal to

4 R Rx w tr R Rx R 2 2 Rx KI 0 R Rx H K: R Rx (4) This eans that the structure of the receive correlation atrix does not change if we use different transit weights, the receive correlation atrix is only scaled by a coplex constant. Using for exaple a beaforing vector to transit to a specific direction leads always to a scaled version of the sae receive correlation atrix and, therefore, to a scaled version of the sae DOA profile. This holds also true for the transit correlation atrix and the corresponding DODs. Transitter Receiver Figure : The Kronecker odel enforces that all directions of departure are linked to all directions of arrival, the joint DOA-DOD spectru of a Kronecker synthesized channel is the product of the average DOA and the average DOD spectru Considering the joint DOD-DOA spectru we can therefore say that the Kronecker odel enforces it to be the product of the DOD and the DOA spectru. In other words (as illustrated in Figure ), this eans that all directions of departure are linked to all directions of arrival, only the total power of the DOAs depends on the chosen DOD, analogously only the total power of the DODs depends on the chosen DOA. 2.4 Physical Interpretation of the Kronecker Assuption A physical interpretation of the Kronecker assuption is this: Irrespective of which transit weight vector is chosen, the scatterers surrounding the receiver are illuinated by one and the sae power distribution. Therefore, the sae DOA spectru is generated. The sae DOA spectru eans identical receive correlation. The total receive power ay, of course, vary. Any arbitrary transit weight vector does include the case of a single antenna transitting, which is assuption (I) in our paper. Other than stated in [3] and other papers, this condition alone is not sufficient. The sae holds true for the reverse link. 2.5 Deficiencies of the Kronecker Model The ain deficiency of the Kronecker odel is that the ultipath structure of the synthesized channel is significantly different to the ultipath structure of the underlying easured channel. Whereas it is very likely that single DODs are linked to single DOAs, the Kronecker odel enforces, as shown before, the synthesized channel to be such that every DOD couples into every DOA by enforcing the joint DOD-DOA spectru to be the product of the DOD and the DOA spectru. This odification of the ultipath structure leads to lower utual inforation estiates when using the synthesized channels than when using the easureent data directly. Instead, the diversity of the channel is increased. We will show the utual inforation isatch by using easureent data in the results section. An explanation for this behavior can be found in [6]. 3 Measureent 3. Measureent Setup For the easureents, we used the wideband vector channel sounder RUSK ATM [7] with a easureent bandwidth of 20 MHz at a center frequency of 5.2 GHz. At the transit (Tx) side, a onopole antenna was ounted on a 2D positioning table foring a λ; 2 spaced virtual array of 20x0 positions, where we used only a 0 5 subatrix for the evaluation to keep clear of large-scale effects. The receiver (Rx) was a directional 8-eleent unifor linear array (ULA) with 0! 4λ intereleent spacing and two additional duy eleents. The antenna eleents were printed dipoles with a backplane having 20< 3dB beawidth. For each Tx position the channel sounder easured 28 teporal snapshots of the frequency dependent transfer function between the Tx onopole and all Rx antennas. Within the easureent bandwidth of 20MHz, 93 equidistant frequency saples of the channel coefficients were taken giving a 4-diensional coplex channel transfer atrix (snapshot, frequency, Rx and Tx position). Since the easureent of the whole 4-diensional channel transfer atrix took about 0 inutes, we easured at night to ensure stationarity. 3.2 Environent The easureents were carried out in the offices of the Institut für Nachrichtentechnik und Hochfrequenztechnik, Technische Universität Wien. In total, 24 Rx positions were easured: one in a hallway - with line-of-sight (LOS) to Tx, 23 of the in several office roos connected to this hallway - with non-line-of-sight (NLOS) to Tx. The (virtual) Tx array was always positioned in the sae place in the hallway. Soe roos were aply, others sparsely furnished with wooden and etal furniture

5 and plants. At each position, we rotated the Rx antenna to three different broadside directions D, D2 and D3 that were angularly spaced by 20<. Thereby, we get 72 different scenarios, i.e. cobinations of Rx positions and directions. 3.3 Evaluation For each Rx position and direction we created 5 spatial realizations of the 8 8 MIMO channel atrix by oving a virtual 8-eleent unifor linear array (ULA) over the 0 5 Tx grid. Taking the 93 frequency bins, the total set of channel saples is fored by different spatial and frequency realizations. Considering a channel unknown at Tx, the utual inforation of the MIMO channel for each realization was calculated by [] C log 2 det I ρ n HHH (5) where I denotes the unity atrix, ρ the ean receive SNR, n the nuber of Tx antennas and H the noralized n MIMO channel atrix. Here, n 8. The noralization was done such that the average receive SNR for each scenario was 20dB [8]. Additionally, we synthesized different realizations of H according to the Kronecker odel (2), noralized the and copared the resulting utual inforation for both easured and Kronecker odel channel atrices. 4 Results In Figure 2, the average utual inforation when using the Kronecker odel vs the average utual inforation estiated directly fro the easureent data for each scenario is shown. As can be seen, the Kronecker odel always underestiates the true utual inforation, i.e. the points lie below the identity (dashed) line. The dashdotted and dashed lines correspond to 0% and 20% relative error, respectively. Moreover, the isatch in utual inforation gets worse with decreasing easured utual inforation. Since decreasing easured utual inforation is related to increasing correlation at either or both receive and transit sides [9] [8], it shows that the Kronecker odel fails in correlated environents. Why is Kronecker utual inforation always lower than that one estiated directly fro the easureent? Figure 3 provides a copelling answer by coparing the joint DOD-DOA Fourier spectru of a typical scenario for easured channels (top) and channels synthesized with the Kronecker odel (botto). In the easured channel, specific DODs are clearly linked to specific DOAs, such that the joint spectra is not separable into a product of the DOD and the DOA spectru. The Mutual inforation of the Kroncker odel [bits/s/hz] overestiation underestiation Mutual inforation of the easured channel [bits/s/hz] 0% 20% Figure 2: Average utual inforation of Kronecker odel generated and easured channel for an 8 8 MIMO syste for 72 different scenarios Kronecker factorization, however, forces the joint spectru to be separable, thus introducing artifact paths lying at the intersections of DOD and DOA spectral peaks. As explained in [6], these artifacts lower average utual inforation but increase diversity order. These effects will likely be ore pronounced for large arrays with high angular resolution. 5 Conclusion We reviewed the Kronecker MIMO radio channel odel and considered the assuptions, the iplications and the deficiencies that the factorization into separated receive and transit correlation atrices causes. We showed that the Kronecker assuption forces the joint DOD-DOA spectru of the channel odel to be the product of the DOA and the DOD spectru and therefore enforces that each DOD is linked to each DOA. However, in reality, as we show with easureents, it is likely to happen that single DODs are linked to single DOAs, which the odel is not able to reproduce. Additionally, the utual inforation predicted by the Kronecker odel is always below the utual inforation estiated fro the easured channel directly. By analyzing a large set of broadband indoor easureents, we found a capacity isatch, which increases with increasing correlation, of up to 27% for an assued average receive SNR of 20dB. The reason for this is again incorrect reproduction of the ultipath structure, which introduces artifact paths in the joint DOD-DOA spectru, resulting necessarily in lower capacity but higher diversity. Note that our results are not in contradiction to [5], where the authors claied to have validated Kronecker factorization and odeling for a 2 2 and 3 3 syste, since these deficiencies are not so pronounced because of the reduced spatial resolution of their systes.

6 ! Or equivalently with P g E # g ik # 2 Y FE P g k> x kkg I n (9) References [] I.E. Telatar, Capacity of ultiantenna gaussian channels, AT&T Bell Labs, online available: [2] K.I. Pedersen, J.B. Andersen, J.P. Keroal, and P. Mogensen, A stochastic ultiple-input-ultipleoutput radio channel odel for evaluation of spacetie coding algoriths, IEEE VTC Fall 2000, Sept Figure 3: Joint DOD-DOA Fourier spectru (in db) of the easured channel atrix (top) and of the Kronecker odel (botto) - exaple scenario Acknowledgent We would like to thank Helut Hofstetter (ftw.) for help with the easureents and T-Systes Nova GbH for providing an eight eleent unifor linear array of printed dipoles. Appendix Let G be a n coplex Gaussian rando atrix with unity power and independently distributed eleents g i j. For such a atrix Y E GXG H (6) is always equal to a scaled identity atrix K I n. Writing the atrix ultiplication eleent-wise, the eleents of Y becoe y i j E = ik x kl gjl? k> g l> g ik x kl gjl A k> l> (7) Since the eleents of G are independent and identically distributed, the expectation of the right hand side of (7) is non-zero only for i j and k l, and therefore (7) siplifies to y i j,. / E B # g ik # 2 x kk C i j k> 0 i D j (8) [3] J.P. Keroal, L. Schuacher, K.I. Pedersen, P.E. Mogensen, and F. Frederiksen, A stochastic MIMO radio channel odel with experiental validation, IEEE Journal on Selected Areas in Counications, vol. 20, pp , Aug [4] D.P. McNaara, M.A. Beach, and P.N. Fletcher, Spatial correlation in indoor MIMO channels, Personal Indoor and Mobile Radio Counications, vol., pp , [5] K. Yu, M. Bengtsson, B. Ottersten, D. McNaara, P. Karlsson, and M. Beach, Second order statistics of NLOS indoor MIMO channels based on 5.2 GHz easureents, IEEE Globeco 200, vol., pp [6] A.M. Sayeed, Deconstructing ultiantenna fading channels, IEEE Transactions on Signal Processing, vol. 50, pp , Oct [7] R.S. Thoä, D. Hapicke, A. Richter, G. Soerkorn, A. Schneider, U. Trautwein, and W. Wirnitzer, Identification of tie-variant directional obile radio channels, IEEE Transactions on Instruentation and Measureent, vol. 49, pp , April [8] H. Özcelik, M. Herdin, H. Hofstetter, and E. Bonek, A coparision of easured 8 8 MIMO systes with a popular stochastic channel odel at 5.2 GHz, ICT, vol. 2, pp , [9] D.S. Shui, G.J. Foschini, M.J. Gans, and J.M. Kahn, Fading correlation and its effect on the capacity of ultieleent antenna systes, IEEE Transactions on Counications, vol. 48, no. 3, pp , March 2000.