Implementation of a Space-Time-Channel-Filter

Transcription

1 Implementation of a Space-Time-Channel-Filter Nadja Lohse, Clemens Michalke, Marcus Bronzel and Gerhard Fettweis Dresden University of Technology Mannesmann Mobilfunk Chair for Mobile Communications Systems D-0062 Germany lohse@ifn.et.tu-dresden.de Abstract: Investigations on novel space-time transceivers have driven a redesign of space-time channel models, which compute spatial and temporal correlated radio signals jointly. A low complex space-timechannel-filter is presented in this paper, which enables a direct modelling of this joint correlated channel process. It is realized by the Karhunen-Lóeve-Transform (KLT), which is based on separate knowledge of temporal fading processes and the spatial correlation. The spatial correlation and its corresponding measure, the coherence distance, were analysed for real scenarios. A narrowband realisation of the represented channel modelling approach was implemented and tested for a unified linear array with 8 antenna elements. Introduction The growing demand of spectrally efficient wireless communications systems has driven the development of transceivers, which utilize the spatial domain by applying multiple antennas. The performance evaluation of such space-time transceivers requires channel models, which accurately models the spatial characteristics of the mobile radio channel. A varity of spatial channel models exist. The spatial characteristics are modelled either deterministically, geometrically or stochastically []. However, the design and required complexity of space-time trans ceivers depends on the correlation between the spatially separated antenna elements. The need of low complex channel modelling has stimulated the development of spatial channel models, which are based on spatial correlation processing [2,3]. The modelling approach that is presented in the following, uses a simple multistage spacetime-channel-filter method to generate a joint spatial and temporal correlated process of a wireless channel. This approach enables a direct modelling of spatial correlation functions, which affect the performance of space-time transceivers. The space-time channel filter has been implemented on the COSSAP simulation platform.

2 2 Multistage Channel Filter Signal domain Correlation domain The mobile radio channel can be described completely by transfer functions that relate the space, time and frequency radio signal parameters in signal and correlation domain, as outlined in the following. The plane wave propagation in direction of the position vector r and in time t with corresponding angular frequency? (a measure of the signal waves per time) and spatial frequency k (a measure of the signal waves per space) can be characterized by [4]: ω dt = kr d. () The spatial frequency corresponds to a directional vector?, which also points in the direction of propagation. In polar coordinates it is a function of azimuth ϕ and elevation θ of the incident angle: sinθ cosϕ 2π k = sinθ sinϕ = k?. (2) λ cosθ Parameter k defines the wave number. In a timevariant multi-path propagation environment signals arrive with different time delays t l (defined by r l /c, where c is the speed of light), directional vectors? l and velocity-dependent Doppler shifts f Dl : k S(k,τ,f D ) t t t f f D r T(f,x,t) 3D cross corr. k S(k,τ,f D ) t t t f Dr f D R( f, x, t) Figure : Duality of space, time and frequency signal parameters in signal domain and correlation domain. τ ξ fd 2π f = k dx= 2π t. (3) τl ξl f DL The dependences between these so-called Fourier pairs time t and Doppler frequency f D, space r and spatial frequency k and time delay t and signal frequency f can be derived from (3). In the correlation domain these radio signal parameters are extended by a corresponding shifted value. They can be reduced if the spatially extended WSSUS assumption holds and therefore the correlation functions only depend on differences?t,?f and?x and on the dual multi path parameters f D, t and k [5]. All signal parameters are shown in Figure. Each corner of both hand cubes represents a possible three-dimensional transfer function in signal or correlation domain. For these the spatially extended Bello-functions [5] can be used. Figure shows the scatterer function S(k, t, f D ), which is identical in signal and correlation domain for uncorrelated scatter- The parameters of one Fourier pair are related by the Fourier Transform.

3 ers, the signal transfer function T(f, x, t), which is also known as the fading process, and the signal correlation function R(?f,?x,?t). The received signal can be obtained by multiplying and convoluting the transmitted signal. This correlation was analysed for a real scenario and will be mapped onto the temporal fading process using the Karhunen-Lóeve-Transform, which results in the joint space-time fading process. It is possible to model the transfer functions as a three-dimensional filter using spatial and temporal sampling of a band-pass signal. Since the deterministic signal is generally unknown, only a stochastic description of the channel transfer functions exists, that can also be modelled with the three-dimensional filters. To avoid the difficult implementation of a multidimensional filter it will be implemented using a multistage approach: A one-dimensional filter is extended to a two-dimensional filter stage to stage. (This could be continued according to the necessary channel filter dimension.) The Karhunen-Lóeve- Transform can be used as a possible realization of this approach: It converts a set of onedimensional uncorrelated processes into a twodimensional correlated process. Here, the mobile radio channel will be described by the narrow-band signal transfer function T(x,t). We characterize the coefficients of that joint space-time fading process with β(x,t). Their modelling by a two-stage space-time channel filter based on the separate modelling of a wellknown temporal fading process, the Rayleigh fading, and of the spatial correlation function. 3 Spatial Correlation Analysis For notational convenience and to simplify the analysis, we will constrain the following considerations to the one-dimensional case in the spatial domain. Let the antenna elements of an uniform linear array be aligned on the x-axis. Then, the spatial frequency k is reduced to a scalar k. In order to obtain environment-dependent spatial correlation properties, we will start with the probability density of the received or transmitted signal energy in the spatial frequency domain k. Applying the spatial Fourier Transform yields a probability density function of the signal energy in the one-dimensional spatial domain x: + jkx Px ( ) = Pke ( ) dk, (4) Since the spatial frequency is a function of the incident angle f in (2), the discrete spatial Fourier transform (4) can be formulated as a function of the distribution of the incident signal energy. Here, we use P(f ) instead of P(k) for finite antenna arrays, as suggested in [6]:

4 L jklxi P( x ) = P( ϕ ) e, (5) i i l l = Correlation coefficients? ij determine the spatial correlation of signal energy between two spatially separated points, which can be described using the probability of the signal energy at these points: ρ = P ( x, x ) = P ( x x ). (6) ij ij i j ij i j If the spatial variable x is substituted in (5)by the difference x i -x j, the correlation coefficient can be determined by: ρ ij L l= jkl( xj) jklxi = P( ϕ ) e e, (7) l where i, j [.. M], and M indicate the number of spatially separated points, which correspond to the number of the antenna elements. The spatial correlation matrix R xx contains the spatial correlation coefficients? ij : R xx ρ ρm =. (8) ρm ρmm From (7) and (8), R xx can also be obtained using the antenna propagation vector a l for each multi path components: jklx T jklxm = al e e, (9) L H xx = P( ϕl) l l l= R aa. (0) The coherence distance x c is characteristic for the spatial autocorrelation function (ACF), which provides a more general measure for the spatial signal change as the spatial correlation matrix. The coherence distance determines, for which spatial distance x the radio signals are considered as coherent or fully correlated. The spatial transfer function is considered spatially flat within the coherence distance. In other words: If antenna elements are arranged spatially more dense than the coherence distance x << x c, () the channel is not space-selective. The coherence distance corresponds to a spatial distance after which the ACF drops under a given threshold. Typical threshold values for frequency or temporal coherence are 50% or 90%. In order to investigate the coherence distance the spatial ACF needs to be determined: N n xcor( n) = Px ( i xi+ n) (2) N n 0 ACF( n) = xcor( n N)

5 for n=,...,2n-, where N indicates the number of spatial samples. The ACF can be derived form the spatial correlation coefficients of the matrix R xx as given in (7): N n xcor( n) = R xx(, ii+ n). (3) N n 0 4 Computation of Spatial Correlation Figure 3: Correlation coefficient for different cell sizes. Antenna height is 5m. The spatial correlation can be obtained numerically using Monte Carlo simulations: A large number L of incidence signal angles was drawn according to a sum of two normal distributions. This choice of this particular distribution is based on a statistical evaluation of incidence angles carried out with simulation tool Radio Propagation Simulator (RPS) for a specific scenario campus of the University of Technology in Dresden [7]. Figure 4: Correlation coefficient for different antenna heights. Cell size is 50m. pdf pdf of angular spread fitting curve φ MS Figure 2: pdf of incidence angle at MS for f=ghz, antenna height=0m and cell size=50m. Figure 2 shows the pdf of incidence angles at the mobile, which is moved within this scenario. Now, the spatial correlation matrix can be calculated using (0). These numeric computations were carried out for two different environment parameters (antenna height, cell size) and for different carrier frequencies. Figures 3 and 4 represent results for the correlation coefficients (the first column of the correlation matrix), which indicate the spatial correlation with re-

6 spect to the first antenna element. From these Figures, it is apparent that the correlation increases with cell size and with antenna height of the base transceiver station. Analysing the signal correlation over the incident angle, the following results were observed: The correlation decreases with cell size and with antenna heights. This inverse tendency is based on the Fourier transform relation between the signal correlation function in the angular domain and in the spatial domain. Table : Coherence distance and angular spread in the downlink for the campus scenario in Dresden. Cell size 50m 50m >500m Angular spread σ ϕ Coh. distance x c 0,24λ 0,28λ 0,84λ The coherence distance for a 50 percent correlation is shown in Table for different cell sizes and compared with the angular spread σ ϕ at the mobile in the downlink for the campus scenario in Dresden. The angular spread represents the root mean square value of the probability density function of the signal energy in the angular domain carried out from the RPS simulations [7]. The decrease of angular spread for increasing cell sizes can be explained by the simultaneous increase in distance between transmitter and receiver. The coherence distance tends to reciprocal results. With respect to the above mentioned Fourier relation between spatial and angular correlation, the channel is considered to be spatially not selective, if: x <<. (4) σ ϕ 5 Karhunen-Lóeve-Transform The space-time channel filter is realized by means of the Karhunen-Lóeve-Transform, which is essentially an orthogonal transformation of coordinates based on a principal component analysis of correlated stochastic signals [8]. A number of M uncorrelated processes c i (k) is transformed into a M-dimensional process ß(k) by projection onto an orthogonal M-dimensional space with coordinates u, u 2,.., u m : M ß( k) = ci( k) u i. (5) Vector ß(k) has the following correlation matrix H R = ß( k) ß ( k). (6) Let us determine the transformation which leads to a given correlation matrix R: R = U H? U, (7) where L denotes the diagonal matrix of the singular values? i and U represents the matrix which columns contain the corresponding eigen-

7 vectors u i. Then, equation (7) can also be formulated as: M H R = λiuu i i. (8) r ( t) = ß( t) s( t). (2) 6 Implementation of Space-Time Channel Filter The coefficients c i (k) are characterised by: { c k } Ε ( ) = 0,,...,M, i * λ i Ε { ci( kc ) j( k) } = 0 i j. (9) j In order to use the KLT as a space-time channel filter, the processes c i (k) are replaced by temporal fading processes q D i (k). This results in the analytical space-time channel filter description: M D ß( k) = λiqi ( k) u i. (20) Figure 5: Realisation of space-time-filter. The KLT as a signal dependent transform provides a suitable tool to convert M spatial uncorrelated temporal correlated processes q D i (t) into M spatial and temporal correlated processes ß(t) if the M-dimensional orthogonal coordinate system is determined by eigenvectors u of a given spatial correlation matrix R xx. Two preliminary tasks have to be accomplished before we can implement the space-time channel filter by means of the KLT as shown in Figure 5.. Generate M temporal fading processes q D i (k) 2. Calculate the singular values? i and eigenvectors u i by singular value decomposition of given spatial correlation matrix R xx Hence, the received signal vector r(t) can be written for flat fading channels as the product of the transmitted signal s(t) and the M-dimensional vector of space-time fading coefficients ß(t): The computation of the temporal fading processes q D i (t) is adapted from the modelling of GSM radio channels and implemented as a complex valued Rayleigh distributed distortion. The

8 time domain process q D i (k) is modelled M times, which has to correspond with the dimension of the correlation matrix. We use the complex valued Singular Value Decomposition (SVD) for decomposing the correlation matrix according to (7) or (8). In contrast to other routines ( e.g. LU or Gaussian decomposition), the singular values and eigenvectors are calculated in one step. The complex valued SVD is solved by reducing the matrix into real matrices. The Karhunen-Lóeve-Transform itself is the implementation of (20). Figure 6: Comparison of the defaulted and simulated spatial correlation coefficients. In order to validate the implementation of the KLT based space-time channel filter, a spatial correlation matrix was calculated from ray tracing simulations of the campus scenario at TU Dresden, which have previously been confirmed with measurements [7]. The spatial and temporal correlation properties were determined from the resulting output of the KLT. Figure 6 shows the close match between the KLT based spatial correlation and the given target correlation. As apparent from Figure 7, the KLT does not affect the given temporal correlation of the Rayleigh fading process, which was used to generate the spatially correlated signals. It can be outlined, that the implemented narrowband space-time correlation filter was verified successful. Figure 7: Comparison of the temporal correlation coefficients with and without KLT. 7 Computation of joint Spatial and Temporal Fading The resulting joint space-time fading coefficients of the proposed channel filter were calculated for spatial fully correlated signals and for uncorrelated signals as shown in Figure 8 and Figure 9, respectively. If the signals at spatially separated antennas are fully correlated, the real correlation

9 matrix R xx has only one singular value and all fading coefficients b i (k) are of the same size. For completely uncorrelated antenna signals the correlation matrix is real and has full rank with equal singular values. For that case, each antenna element can be considered separate (antenna diversity). Additionally, the space-time fading coefficients for the campus scenario at University of Technology in Dresden are shown in Figure 0. The resulting space-time fading profile matches the given spatial correlation as depicted in Figure 6. Figure 8: Space-time fading coefficients for spatially full correlated signals. 8 Conclusions A method has been presented to determine an environment-dependent spatial correlation matrix from the distribution of the incidence signal energy over spatial frequency. Both, spatial correlation matrix and coherence distance, were numerically estimated for different scenarios based on previously investigated angular distributions of the incidence signal energy. The KLT was used to generate joint space-time fading coefficients for a computed spatial correlation and a given temporal fading. The resulting space-time channel filter was implemented on the COSSAP-simulation platform. It can be used to evaluate the design of mobile radio transceivers, which utilize temporal and spatial processing. Figure 9: Space-time fading coefficients for spatially uncorrelated signals. Figure 0: Space-time fading coefficients for real spatial correlation.

10 However, the channel correlation matrix is not only a function of space and time but also frequency-selective. For MIMO (Multiple Input Multiple Output) channels, the spatial selectivity at the transmitter needs to be considered as well. This will result in a joint space (transmitter/receiver)-time-frequency fading processes, which can be implemented using multistage KLTs. [6] John D. Kraus: Antennas. McGraw-Hill, 2 rd edition, 988 [7] Lohse, N., Huebner, J., Bronzel, M.: Parameter Evaluation for Space-Time Channel Models. Technical Symposium on Wireless Personal Communications, Blacksburg, USA, Jun. 4-6, 2000 [8] Haykin, S.: Adaptive Filter Theory. Information and System Series, Prentice Hall, 3 rd edition, Literature [] Steinbach, M. et al.: Mission Report- Modelling Unification Workshop. COST 259, Vienna, Austria, April 22-23, 999 [2] Hammerschmidt, J. S.: Spatio-Temporal Channel models for the Mobile Station; Concept, Parameters, and Canonical Implementation. VTC 2000-Spring, Tokyo, May [3] M. Stege, J. Jelitto, M. Bronzel and G. P. Fettweis: A Multiple Input - Multiple Output Channel Model for Simulation of Txand Rx-Diversity Wireless Systems. VTC 2000-Fall, Boston, Sep [4] Johnson, D. H.: Array Signal Processing: Concepts and Techniques. Signal Processing Series, Prentice Hall, 993 [5] Kattenbach, Ralf: Statistical Modeling of Short-Term Fading Effects for Directional Radio Channels. COST 259, Leidschendam, Netherlands, Sept , 999