A TIME-VARYING MIMO CHANNEL MODEL: THEORY AND MEASUREMENTS

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1 A TIME-VARYING MIMO CHANNEL MODEL: THEORY AND MEASUREMENTS Shuangquan Wang, Ali Abdi New Jersey Institute of Technology Dept. of Electr. & Comput. Eng. University Heights, Newark, NJ 7 Jari Salo, Hassan El-Sallabi, Pertti Vainikainen Helsinki University of Technology Radio Laboratory P. O. Bo 3, FI-5-TKK, Finland ABSTRACT A new narrowband MIMO channel model for microcells is introduced, and different aspects of the model, such as spatial, spatio-temporal, level crossing rate, and average fade duration are compared with the measured data, to verify the proposed model. The results ehibit the utility of the new parametric model. The proposed model serves as a useful tool for design and simulation of MIMO systems in microcellular environment.. INTRODUCTION Over the past few years, many papers have been published on multiple-input multiple-output (MIMO system and channels [][]. Measurement and characterization of MIMO channels have been an active area of research as well [3] [4] [5]. However, still there are many open questions regarding MIMO channel modeling in different propagation environments. For proper design and development of MIMO systems, certainly easy-to-use channel models are required, and their accuracy should be verified via measured data. In this paper, we focus on microcells, where both the base station (BS and mobile station (MS eperience local scattering. Using a simple geometrical model, a closedform epression for the MIMO spatio-temporal cross correlation is derived in the sequel, which transparently includes physical characteristics of the channels such as mean angle of arrivals/departures, angle spreads, and Doppler spread. The space-time correlation structure of a MIMO channel provides useful guidelines for system design such as proper interleaving depth in time and space, required frequency of channel estimation, and so on. The accuracy of the proposed model is verified using measured data, collected at Helsinki University of Technology (HUT, to show the ability of the model in capturing the essential space-time characteristics of the channel. The rest of the paper is organized as follows. The geometry of the model is presented in Section, and a general yet compact result for the space-time correlation between any two subchannels is given, as well as some special cases of interest. In Subsection 3. the distribution of the collected data are discussed, whereas the eperimental/theoretical spatial, spatio-temporal correlations are compared in Subsections 3. and 3.3, respectively. Moreover, the level crossing rates and average fade durations are studied in Subsection 3.4. Finally, the paper concludes with Section 4.. THE TIME-VARYING MIMO CHANNEL MODEL AND ITS CORRELATION STRUCTURE To simplify the notation and without loss of generality, we consider a MIMO system, shown in Fig., where the scatterers local to BS and MS are modeled to be distributed on two separate rings. Let h lp (t, l,p denote the comple low-pass equivalent channel gain of the subchannel between the p th transmit antenna and the l th receive antenna at time t. Mathematical representation of the superposition of plane waves at the MS, after singlebounce scattering, results in double-summation epression for h lp (t, given by ( [6]. By substituting the channel representation into the space-time correlation between h lp (t and h mq (t, defined by ρ lp,mq (τ = E[h lp(th mq (t+τ], such Ωlp Ω mq that Ω lp = E[ h lp (t ], it is shown that [6]: ρ lp,mq (τ = ( η ep { j[b lm cos β acos γ] } I (κ I ( {κ a sin γ b lm sin β c pq b lm c pq sin α sin β + a sin γ[c pq sinα + b lm sinβ] jκ [a sin µ sin γ b lm sin β sin µ c pq cos(α µ ] } + η ep(jc pq cos α I (κ I ( {κ a b lm c pq sin α + c pq sin α [asin γ b lm sinβ]+ab lm cos(β γ jκ[acos(µ γ b lm cos(β µ c pq sin α sin µ] }, ( where and are the maimum angle spreads shown in Fig., d and δ are the antenna spacings at the MS and BS

2 y ν BS δ O α O MS β γ BS MS d Fig.. Geometrical configuration of a channel with local scatterers around the MS and BS. arrays, respectively, γ is the direction of MS movement, α and β show the direction of BS and MS arrays, respectively. µ,µ [ π,π account for the mean direction of AoD and AoA respectively, and κ,κ are the parameters of the von Mises distributions for angle-of-departure (AoD and angle-of-arrival (AoA, respectively[6]. η shows the contribution of MS local scatterers and f D is the maimum Doppler frequency. a=πf D τ, b lm =π(l md/λ, c pq =π(p qδ/λ, I n (z = π π ez cos θ cos(nθdθ is the n th -order modified Bessel function. In what follows, we show most eisting correlation models can be considered as special cases of our generic parametric spatio-temporal MIMO model, given by (. If there is no scatterer around the BS, then ( reduces to ρ lp,mq (τ η= = ep(jc pq cos α I (κ I ( {κ a b lm c pq sin α + c pq sin α [asin γ b lm sin β]+ab lm cos(β γ jκ[acos(µ γ b lm cos(β µ c pq sin α sin µ] }, ( which is consistent with ( in [7]. Special cases of ( can be found in [7]. With l = m and p = q, the temporal correlation of the single subchannel h lp (t, can be derived from ( ρ lp,mq (τ l=m,p=q = ( ηep(jacos γ I κ a sin γ jκ a sinµ sin γ I (κ ( κ ηi a jκacos(µ γ +. (3 I (κ If η = and κ = (isotropic scattering around the MS, (3 simplifies to the well-known Clarke s correlation, i.e., J (πf D τ [8], where J ( is the Bessel function of the first kind of order zero. When the MS does not move, one gets f D =, which reduces ( to the following spatial correlation between h lp and h mq ρ lp,mq = ( η e jb lm cos β ( {κ I (κ I b lm sin β c pq b lm c pq sin α sin β + jκ [b lm sinβ sin µ + c pq cos(α µ ] } cos α ( ejcpq {κ + η I b lm I (κ c pq sin α b lm c pq sin α sin β + jκ[b lm cos(β µ+c pq sin α sin µ] }, (4 with isotropic scattering around both the BS and MS, i.e., κ = κ =, and parallel arrays with α = β = π, the spatial correlation in (4 further simplifies to ρ lp,mq = ( ηj (b lm + c pq + ηj (b lm + c pq. (5 3. COMPARISON OF THE PROPOSED MODEL WITH COLLECTED DATA The layout of the location where the data is collected is shown in Fig. 3. The measurement setup and subchannels are shown in Fig. (a, whereas the configuration of the receive array is presented in Fig. (b. The transmit element spacing is one wavelength λ and the receive element spacing is.785λ. The mobile speed is.4m/s, which with the.54ghz carrier frequency, result in f D = v/λ =.87Hz. The spatial sample spacing is λ 4, which is equivalent to 87.5 ms sampling spacing in time. Further details of the data and measurement setup can be found in [4][9].

3 T h h T R h h (a The system R Fig.. The measurement setup. y R 7 N Travel Direction R (b R configuration In order to compare the correlations, LCR and AFD of the model with the data in the following subsections, the directions α, β, γ should be determined first. As shown in Fig. and 3, it is not realistic to get the eact values for α, β and γ over the entire measurement route. In this paper, only part of the data around the intersection of streets Rauhankatu and Unioninkatu are used, and one obtains their approimate values α π, β 4π 5 and γ π from Fig Statistical Distribution of the Data When deriving the compact space-time correlation epression in ( [6], it has been assumed that the number of scatterers around both BS and MS are large enough. This translates into comple Gaussian distribution for each subchannel h lp (t. To verify this, the data is normalized [6] such that each subchannel has zero mean with unit variance. As shown in Fig. 4, the real and imaginary parts are very close to Gaussian. Furthermore, the empirical CDF of the amplitude and phase are nearly Rayleigh and uniform, respectively. Cummulative Distribution Function (CDF Phase Real and Imaginary Parts Amplitude CDF of CDF of CDF of Amplitude CDF of Phase Theoretical Gaussian CDF Theoretical Rayleigh CDF Theoretical Uniform CDF Normalized Level Fig. 4. and theoretical distribution of the data. 3.. Spatial Correlations For a system, we consider four different types of spatial correlations, i.e., parallel, crossing, transmit and receive, defined and given by and ρ parallel E[h (th (t] = ρ,, (6 ρ crossing E[h (th l(t] = ρ,, (7 ρ T E[h (th (t + h (th (t] = (ρ, + ρ,, (8 y about 5 meters N ρ R E[h (th (t + h (th (t] = (ρ, + ρ,, (9 TX 9 deg 47 meters Rauhankatu where ρ lp,mq is given in (4. Using the estimated parameters discussed in the net subsection, the theoretical spatial correlation obtained from (6-(9 are (ρ parallel, ρ crossing, ρ T, ρ R =(.87,.76,.496,., whereas the spatial correlations estimated directly from the normalized data are (.388,.7,.6364,.85. Clearly the theoretical and empirical correlations are close Spatio-Temporal Correlations Fig. 3. The environment where the data is collected. To estimate the unknown parameters of the model, i.e. Λ= (,,κ,κ,µ,µ,η, the space-time correlation in ( is fitted to the data, via a numerical least-square search. The estimated parameters are given by ˆΛ = ( π 6, π 4,7,, 9π 8, 9π 8,.7. As shown in Fig. 5-8, theoretical space-time correlations of the model ehibits good fit to empirical correlations.

4 3.4. Level Crossing Rate and Average Fade Duration LCR and AFD of the signal envelope are two important temporal statistical features, which bear useful information on dynamic behavior of time-varying fading channels. To calculate the LCR and AFD of a subchannel, one needs its temporal correlation, which is given in (3. With γ = π, direction of the MS movement in the eperiment, it simplifies to ( ηi κ a jκ a sinµ ρ(τ = I (κ ( ηi κ a jκasin µ +, I (κ where a = πf D τ. The n th spectrum moment b n is defined by [] b n = dn ρ(τ j n dτ n τ=, ( is also required to calculate LCR and AFD. From ( and ( we get b =, [ ( ηi (κ sin µ b = πf D I (κ b = 4π f D + ηi ] (κsin µ, I (κ { η κ[i (κ + I (κ]sin µ + I (κcos µ κi (κ + ( η κ [I (κ +I (κ ]sin µ +I (κ cos µ } κ I (κ. ( The theoretical LCR and AFD at the amplitude threshold r for Rayleigh fading are given, respectively, by [] b b N(r = re r, (3 π and π(e r t(r = r. (4 b b Using the estimated parameters given in Subsection 3.3, (3 and (4 are compared in Fig. 9 and with the empirical LCR and AFD from data. The close match verifies the accuracy of the proposed model. 4. CONCLUSION In this contribution, a parametric model for narrowband MIMO channels is proposed. In addition, a variety of the characteristics of the model such as statistical amplitude/phase distribution, spatial correlation, spatio-temporal cross correlation, level crossing rate, and average fade duration are compared with measured data. The results show the accuracy of the model in describing the data. The model provides useful information for efficient design of multi-antenna systems in mircocellular environments. 5. REFERENCES [] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge, UK: Cambridge University Press, 3. [] D. Gesbert, M. Shafi, D. shan Shiu, P. J. Smith, and A. Naguib, From theory to practice: an overview of MIMO space-time coded wireless systems, IEEE J. Select. Areas Commun., vol., pp. 8 3, 3. [3] K. Yu and B. Ottersten, s for MIMO propagation channels, a review, Wirel. Commun. Mob. Comput., vol., pp ,. [4] K. Sulonen, P. Suvikunnas, J. Kivinen, L. Vuokko, and P. Vainikainen, Study of different mechanisms providing gain in MIMO systems, in Proc. IEEE Veh. Technol. Conf., Orlando, FL, 3, pp [5] J. Ø. Nielsen, J. B. Andersen, P. C. F. Eggers, G. F. Pedersen, K. Olesen, E. H. Sørensen, and H. Suda, Measurements of indoor 6 3 wideband MIMO channels at 5.8 GHz, in Proc. IEEE Int. Symp. on Spread Spec. Tech. and App., Sydney, Australia, 4, pp [6] S. Wang, K. Raghukumar, A. Abdi, J. Wallace, and M. Jensen, Indoor MIMO channels: A parametric correlation model and eperimental results, in Proc. IEEE Sarnoff Symp., Princeton, NJ, 4, pp. 5. [7] A. Abdi and M. Kaveh, A space-time correlation model for multielement antenna systems in mobile fading channels, IEEE J. Select. Areas Commun., vol., pp ,. [8] G. L. Stüber, Principles of Mobile Communications (nd ed.. Boston: Kluwer,. [9] K. Kalliola, H. Laitinen, L. I. Vaskelainen, and P. Vainikainen, Real-time 3-D spatial-temporal dualpolarized measurement of wideband radio channel at mobile station, IEEE Trans. Instrum. Meas., vol. 49, pp ,. [] A. Abdi, K. Wills, H. A. Barger, M. S. Alouini, and M. Kaveh, Comparison of the level crossing rate and average fade duration of Rayleigh, Rice, and Nakagami fading models with mobile channel data, in Proc. IEEE Veh. Technol. Conf., Boston, MA,, pp

5 .5.5 Temporal Autocorr Temporal Cross Corr Fig. 5. The autocorrelation of h (t. Fig. 8. The space-time cross correlation between h (t and h (t..5 The LCR of h (t The AFD of h (t Temporal Autocorr LCR/f D.4.3 AFD*f D Fig. 6. The autocorrelation of h (t Fig. 9. LCR and AFD of h (t..5.5 The LCR of the Envelop(Average over All Channels The AFD of the Envelop(Average over All Channels Temporal Cross Corr LCR/f D.4.3 AFD*f D Fig. 7. The space-time cross correlation between h (t and h (t Fig.. LCR and AFD, averaged over all subchannels.