A VECTOR CHANNEL MODEL WITH STOCHASTIC FADING SIMULATION

Transcription

1 A VECTOR CHANNEL MODEL WITH STOCHASTIC FADING SIMULATION Jens Jelitto, Matthias Stege, Michael Löhning, Marcus Bronzel, Gerhard Fettweis Mobile Communications Systems Chair, Dresden University of Technology D-62 Dresden, Germany Abstract The simulation of wireless communication systems with multiple antennas requires a spatial channel model which reasonably characterizes the space and time variant effects of the mobile radio channel. This paper describes a vector channel model with stochastic fading simulation, which combines several advantages from purely stochastic and geometrical channel models. The resulting model is consistent with real world wireless channel characteristics and enables numerically efficient bit-level simulations. The channel model enables the investigation of several aspects of space-time-processing techniques. This includes the performance evaluation of space-time algorithms in dependence on time varying spatial channel characteristics and fading correlation effects between the antennas as well as the user tracking and separation abilities of those algorithms. I. Introduction To be able to satisfy the increasing demand for user capacity as well as the need for high data rate services in mobile communications one important issue is the introduction of advanced air interfaces. This includes research topics such as adaptive antennas, space-time processing and space-time coding techniques. In order to analyze the performance of new concepts an adequate channel model is essential. A common channel modeling strategy is the statistical description of time variant fading effects of physical channels due to moving terminals, moving obstacles and the transmission environment []. However, those scalar stochastic channel models do not provide any directional information. Therefore, they are not directly applicable for systems with multiple antennas. One possible extension of these models is the Directional Gaussian Scattering (DGS) model [2], which assigns directional information to uncorrelated scatterers. This directional information has to be obtained from measurements. Mobility aspects are not considered. Another approach takes into account a certain spatial scatterer distribution and derives the channel characteristics from this distribution, e.g. [3], [4], [5], [6]. For simplicity, these models assume that each multipath component is created by a specular reflection at a remote object, which is often referred to as the geometrically based single bounce (GBSB) assumption. The main advantage of these models is the available spatial information. Once the coordinates of the scatterers are drawn from a random process, all necessary channel characteristics, including angles of arrival (AOA), can be derived. However, a large number of multipaths is required for realistic fading simulation, which limits the applicability of those models for bit-level simulations. The consideration of continuously moving mobiles or scatterers which is important in beamforming performance analysis is even more complex and can hardly be handled with purely geometrical models. Furthermore, slow fading effects can not be handled directly in such models. Remote Reflector Fig.. Remote Reflector p θ p d Base Station Local Scattering around Mobile Typical local scattering and multipath scenario Therefore, a new combined vector channel model (remote reflectors) with stochastic fading simulation (local scattering) is introduced, based on the assumption, that the multipath propagation is characterized by local scatterers around the mobile station and a few dominant spatially well separated reflectors in the far-field (Figure ). This reduces the computational complexity compared with purely geometrical models through the stochastic characterization of local scattering effects. Furthermore, slow fading ef- v

2 fects as well as mobile movement including the appearance and disappearance of remote reflectors are also taken into account. This channel model allows the investigation of beamforming aspects as well as space-diversity concepts without requiring an a-priori assumption about spatial correlation properties between antennas, which is a critical issues for scenarios with small angular spread [7]. II. Signal Model Following a narrow-band space-time channel with one transmit antenna and M receiving antennas is considered. There are multipaths from dominant reflectors. The received signal at the m-th antenna element is given by r m (t) = p= a m (θ p ) (τ p )α p (t)s(t τ p )+z m (t) () for m =...M, where a m (θ p ) corresponds to the phase shift at each antenna element due to array propagation, (τ p ) describes the path attenuation, α p (t) provides the fading characteristic of the space-time channel, s(t) represents the transmitted signal, and z m (t) accounts for interfering waveforms and noise. The characteristics of the time-variant channel which further depend on the AOA θ p and propagation delay τ p of path p are described in more detail in the following. Array propagation factor a m (θ p ): The propagation of a plane wave impinging on the antenna array causes a time delay m at different antenna elements, which results in a phase-shift a m and possibly in a magnitude change [8] of the incoming wave. The values of a m for all M antennas form the array propagation vector a(θ p ). For a uniform linear array with antenna spacing d and carrier wavelength λ this vector a(θ p ) can be expressed as a(θ p )=[ e j 2π λ d sin θp...e j(m ) 2π λ d sin θp ] (2) and describes the spatial response of the array to a waveform impinging from direction θ p. Average ath Loss (τ p ): The mean power of each multipath component depends on the propagation delay τ p andisdefinedby an average path loss (τ p )[9]: (τ p )= ref +n log(τ p /τ ref ). (3) The path loss exponent 2 <n<6 depends on the propagation scenario to be simulated. Fading coefficients α p (t): The time variant fluctuations of the path strength are modeled using fading coefficients α p (t) =β p (t) γ p (t). (4) Fast Fading coefficients β p (t): They can be modeled as Rayleigh-distributed random processes. For each dominant reflector one resolvable path is assumed. This path consists of a large number of incoming waves. These waves result from the structure of local scatterers which are assumed to be uniformly distributed around the mobile. The superposition of the waves results in a Rayleigh-faded path [] which is reflected at dominant reflectors (hills, houses,...). Since the dominant reflectors are significantly separated, a different combination of the incoming rays is reflected at each reflector. Therefore, independent fast fading is assumed for each resolvable path p with a specific time delay τ p and AOA θ p. The Rayleigh-fading coefficients are generated from a complex Gaussian random process which is filtered using an IIR-Filter with the typical Jakes- Spectrum []. The verification of the assumptions is described in [8]. Slow fading coefficients γ p : As the mobile moves, (τ p )doesnotstayconstant over time. Sometimes a path is obstructed by an obstacle and the received power from this path is therefore smaller. This effect is usually referred to as shadowing (slow fading). Measurements [2] have shown that the shadowing coefficients γ p are log-normal Gaussian distributed with a variance 3 <σ γ < db. The time correlation of γ p is not known in general. However, measured data in [3] suggest that it can be modeled as simple decreasing correlation function. The time correlation of the shadowing depends on the velocity v of the mobile. To generate the time varying slow fading coefficients γ p a Gaussian random process is filtered using a certain IIR-filter [3], [8]. Transmitted signal s(t): The delayed transmitted signal s(t τ p )isaconvolution of the data symbol sequence {a k } with a pulse shaping filter g(t), s(t τ p )= a k g(t τ p kt) =s p (t), (5) k where T defines the symbol rate. The pulse shaping filter g(t) is often implemented as a Nyquist filter such as the Root Raised Cosine filter with roll off factor α: g(t) = Eg ( 4αt T T )cos(π( + α) t T ) + sin(π( α) t T ) (πt/t)( (4αt/T ) 2. ) (6) ath delay τ p and AOA θ p : The path delay τ p depends on the location of the dominant reflector p and therefore also on the AOA θ p of the path. The geometrically based statistical channel models of Liberti [3] and Lohse [5] generate θ p s based on statistical assumptions which are dependent on τ p. These channel models will be described in more detail in section III.

3 Figure 2 shows how geometrical and statistical assumptions are merged to the proposed channel model. While the model keeps the statistical characteristics of well known taped delay line models (e.g. COST), it is expanded by a geometrical approach to model the spatial dimension. BS θp reflector Rp ψp MS D Stochastic Fading Model Fig. 3. Lohse s geometrical model with exponential scatterer distribution Source Signal Transmit filter Fig. 2. τ τ β (t) γ (t) β (t) γ (t) θ - θ Geometrically Based Single Bounce Model Basic structure of the channel model III. Vector Channel models To draw certain remote reflector positions from predefined statistical spatial scatterer distributions, two different GBSB models are used. Applying the geometrical information all necessary multipath characteristics such as AOA θ p, path delay τ p, and path attenuation (τ p ) can be estimated. Since different scenarios can be characterized by particular scatterer distributions, a common model for all cases can hardly be found. Hence, two distinct models were chosen for modeling different scenarios: (i) the GBSB model from Liberti [3] (Liberti model) for micro cellular environments and (ii) an exponential scatterer model from Lohse [5] (Lohse model) for macro cellular environments. Liberti s model is based on the assumption of uniformly distributed scatterers bounded by ellipses with transmitter and receiver in the focal points. Scatterers on one ellipse are defined by a certain path delay τ p with respect to the single bounce assumption. The adaptation of the model to different scenarios is done through selection of the maximum normalized path delay r max = τ max /τ. Lohse s exponential model assumes a circular distribution of scatterers around the mobile station (Figure 3). The distances R p between mobile station and scatterers follow an exponential distribution considering the physical dimension of the scatterers, whereas the angles of departure ψ p are uniformly distributed within the interval [;2π]. The resulting joint scat- Base Station Antenna Array terer density function can be written as f R,ψ (R p,ψ p )= 2π }{{} f ψ (ψ p) e Rp R } R {{} f R(R p) (7) with R as the mean scatterer distance. Since the desired joint pdf of relative path delay and AOA f τ,θ (τ p,θ p ) is a rather complex expression [5] and the calculation of an analytic expression of its probability function is difficult, a different approach is used for the channel model. Using the separability of the marginal densities f ψ (ψ p )andf R (R p )in(7)thecumulative probability functions can be calculated as { F R (R p )= e Rp R for R p (8) otherwise F ψ (ψ p )= ψ p for ψ p < 2π. (9) 2π The generation of samples for the given random variables is done by a common random number generator which draws samples x p and y p from uniformly distributed random variables x and y in the interval [;]. These samples are then transformed to create R p (x p )andψ p (y p ) using and R p = R ln( x p ) () ψ p =2πy p. () Finally, the values for path delay τ p and AOA θ p can be calculated for a predefined distance D between base and mobile station where c is the speed of light. τ p = ( ) R p + R p c 2 + D 2 2R p D cos ψ p (2) arctan θ p = arctan ( ) Rp sin ψ p D R p cos ψ p for R p cos ψ p D ( ) Rp sin ψ p D R p cos ψ p + π for R p cos ψ p >D (3) As stated earlier, the Liberti model is assumed to be more valid for micro cellular environments, whereas

4 the Lohse model assumes scatterers mainly around the mobile station which is more suitable in a macro cellular environment. Both models were analyzed and compared by Monte Carlo simulations. Figure 4 compares the mean angular spread θ vs. mean delay spread τ e for both models. angular spread θ in degrees Liberti model Lohse model 2 D = 3 m D = 5 m D = 5 m mean delay spread τe in ns Fig. 4. Comparison of the angular spread for the Liberti vs. Lohse model As can be seen, Lohse s exponential model generates much smaller angular spreads even for high delay spread values. Liberti s model, on the other hand, generates much higher angular spread values even for small delay spreads. For both models the angular spread reduces with increasing distance D between base and mobile station. IV. Mobility model A crucial subject for the simulation of space-time channels are the moving mobiles. In case of a conventional beamformer the smart antenna array must steer its beams towards the moving mobiles. Therefore, the performance of a beamformer can not be evaluated without simulating mobility. Although mobility is already taken into account for the calculation of fading coefficients (see II), no specific movement is considered so far for changes in τ p and θ p. However, the distances to the dominant reflectors change when the mobile moves. New reflecting objects appear while others become less important. The emerging new paths (with new τ p and θ p ) due to mobility influence the multipath structure of the spatial and temporal channel impulse response. This shall be illustrated by an example: The positions of dominant reflectors drawn from GBSB models are more likely to be distributed in the vicinity of the mobile. If the mobile moves away from its original position, this assumption is no longer valid. In fact, if a mobile moves into a given direction, no dominant reflector will be close to the mobile after some time, which violates the assumptions used to derive the GBSB models. Therefore, it is reasonable to model vanishing old paths and emerging new ones in order to account for mobility effects more accurately. Here, we follow a rather pragmatic approach, which takes care of the mobility implications on τ p and θ p in the statistical model while keeping the simulation model as simple as possible: A path is discarded, if the corresponding slow fading coefficient γ p falls below a given threshold Γ min. A new path is generated from the underlying GBSB model for each discarded old multipath. The channel model will continually replace old paths by new ones. The parameter Γ min affects the replacement of old paths and must be chosen carefully depending on the scenario. In discrete time simulations, the continuous movements of the mobile are approximated using small steps. The step size has to be small enough to ensure that most of the dominant reflectors are still present over a number of iterations. For implementation efficiency, some simplifications are used to model the effects of mobility which only slightly affect the characteristics of the space-time cannel: For a given multipath the values of θ p and τ p are kept constant although they change slightly due to changes of the position of the mobile. Only the AOA for the line of sight component is adjusted when the mobile moves. Changes of the mobile position are approximated as movement on a given trajectory with small time increments. The number of multipath components is currently kept constant, but can easily be replaced by some arbitrary (e.g. random) number. V. Simulation results In this section, some basic simulation results are shown to evaluate the channel model performance. The vector channel model with stochastic fading simulation is designed to handle a great variety of possible mobile channel scenarios. It is able to handle the combined vector and stochastic modeling as well as the stochastic fading or vector modeling cases alone. Therefore, the model could for instance be used as pure scalar model (no AOA information available) or the fading could be introduced without the stochastic process through the usage of a large number of paths. The intention of the model development was a good approximation of various scenarios while keeping the computational complexity moderate. Therefore, we focus our attention on cases where local scattering and the resulting fading are modeled as stochastic processes and channel characteristics such as angular spread, space and frequency selectivity are modeled by a GBSB model using only few remote reflectors. The model parameters were chosen to fit measurement scenarios in order to compare the model with

5 measurement results [4]. The simulated velocity was v = m/s, the number of simulated multipaths was = 2 and maximum excess delay τ emax = τ max τ was limited to 2ns. Figure 5 shows the space-time fading characteristics of the channel for the Liberti model. The resulting angular spread was estimated to θ 78. The corresponding results for the Lohse model are shown in Figure 6. For this model the angular spread was estimated to θ. Comparing the Figures normalized signal power in db Fig space d/ λ (normalized distance) normalized signal power in db Liberti model ( τe max = 2ns) 5 time t [ms] Space-time selective fading for the Liberti model 2 space d/ λ (normalized distance) Fig Lohse model ( τe max = 2ns) 5 time t [ms] Space-time selective fading for the Lohse model 5 and 6 it can be concluded that Lohse s exponential model generates a weaker space selective fading. This effect is caused by the relatively small value of θ. The application of Liberti s model for the multipath generation causes a much stronger space selective fading through due to larger angular spread values. The proposed vector channel model for space-time processing can easily incorporate different geometrical models for the remote reflector generation, which provides a very flexible tool for simulations of various scenarios. VI. Conclusions In this paper a vector channel model with stochastic fading simulation has been introduced. The combination of stochastic and geometrical assumptions results in a mathematically tractable and computationally efficient channel model which handles a great variety of vector channels. Micro and macro cellular type environments can be simulated using different GBSB model implementations. The model enables the simulation of beamforming as well as space diversity concepts and handles both spatially narrowband and wideband signals. Moreover, the actual performance of smart antenna and space-time receiver concepts considering spatial correlation effects can be evaluated. The introduction of a suitable mobility model enables simulations of the smart antenna tracking performance in multi-user scenarios. Finally, for investigations of space-time coding or transmit diversity concepts the model can easily be extended to multiple input and multiple output channels. References [] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers - Synchronization, Channel Estimation and Signal rocessing, Wiley Series in Telecommunications and Signal rocessing. John Wiley & Sons, Inc., 998. [2] U. Martin, Statistical mobile radio channel simulator for multiple antenna reception, in IEICE 996 International Symposium on Antennas and ropagation, Chiba, Sept. 996, pp [3] Joseph C. Liberti and Theodore S. Rappaport, A geometrically based model for line-of-sight multipath radio channels, in VTC 96, 996, pp [4] M.Bronzel,J.Jelitto,N.Lohse,G.Fettweis,R.Thomä, G. Sommerkorn, D. Hampicke, U. Trautwein, and A. Richter, Experimental verification of vector channel models for simulation and design of adaptive antenna array receivers, in ACTS Mobile Communication Summit 98, 998, pp [5] N.Lohse,M.Bronzel,J.Jelitto,D.Hunold,andG.Fettweis, Analyse des Delay/Doppler Spread bei räumlicher Filterung anhand eines Kanalmodells und Messungen, in ITG-Fachtagung Intelligente Antennen, Karlsruhe, June 998. [6] G. Raleigh, N. Diggavi, A. Naguib, and A. aulraj, Characterization of fast fading vector channels for multiantenna communication systems, in 28th Asilomar Conference on Signals, Systems and Computers, acific Grove, Nov. 994, vol. II. [7] A. F. Naguib, Adaptive Antennas for CDMA Wireless Networks, h.d. thesis, Stanford University, Aug [8] M. Stege, J. Jelitto, M. Bronzel, and G. Fettweis, A stochastic vector channel model - implementation and verification, in Vehicular Technology Conference, 999, accepted. [9] T.S. Rappaport, Wireless Communication- rinciples and ractice, rentis Halls Communications Engineering Technologies Series. rentice Hall TR, 996. [] W. C. Jakes, Microwave propagation, John Wiley, New York, 97. [] S. A. Fechtel, A novel approach to modelling and efficient simulation of frequency-selective fading radio channels, Journal on Selected Areas in Communications, vol., no. 3, Apr [2] R. Steele, Mobile Radio Communications, John Wiley & Sons, 992. [3] M. Gudmundson, Correlation model for shadow fading in mobile radio systems, IEE Electronics Letters, vol. 23, pp , nov 99.