Physical Modelling of a Dispersed Multi Channel System

Transcription

1 Physical Modelling of a Dispersed Multi Channel System Rudolf Sprik Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65-67, 118 XE Amsterdam, The Netherlands - sprik@science.uva.nl ABSTRACT Scattering of waves in the medium between transmitters and receivers in a multiple-in-multiple-out (MIMO) system can in principle lead to a large enhancement of the channel capacity. To obtain the best results, the scattering of the waves in the medium needs to be strong enough to decouple the available connections. The physics of the wave propagation in a scattering medium is in general complex and when correlation between channels exists, hard to model accurately. The effects of limited scattering and channel correlation are studied here using an exact t-matrix based model to represent the medium in a two dimensional space. As an example, the effect of deliberately enhancing the scattering by placing the transmitters and receivers in a small strongly scattering surrounding enclosure is studied. The influence of the enclosure on the available channel capacity is modelled as function of enclosure geometry. the scattering properties of the medium. The transmitter and receiver arrays (circle symbols) have a characteristic spacing d and extent D. The wavelength λ, the distance L between transmitters and receivers, and the array extent D determine the maximum obtainable focusing in a non-scattering medium. With scatterers (+ signs) the waves travel un-scattered for a distance of the mean-free-path l. In a rich scattering environment λ < l < L and d λ. The optimal condition for MIMO data transfer is obtained when all channels are sufficiently decoupled as can be the case in a rich scattering environment []. I. INTRODUCTION Recent developments in wireless multi-in-multiple-out (MIMO) communication systems proved that significant data transfer enhancements can be achieved in strongly scattering environments [1][][3][]. Theoretically the optimum situation is obtained when the scattering decouples all the possible transmitter-receiver connections. According to the generalized Shannon analysis [1][5], the channel capacity strongly depends on the channel matrix H specifying the connectivity between n tr transmitters and n rc receivers. The detailed physical properties of the medium that influence the wave propagation from the transmitters to the receivers is hidden in H. For example [], if the spacings in the antenna arrays are small and also the spatial extent of the array is limited, decoupling the channels in a non-scattering environment is hampered. Whatever wave distribution is chosen to feed the transmitter array, the spatial resolution at the receiving and will not be good enough to discriminate between different channels. Scattering in the medium will effectively enlarge the spatial extent of the arrays and on average decouple the channels. The relevant physical parameters associated with the multiple scattering of waves between the transmitter and the receiver arrays is given schematically in Fig. 1. The mean-free-path l specifies the characteristic path length for a wave before being scattered and represents Figure 1. The outline of the relevant physical parameters to describe the multi channel connectivity. The phenomena associated with strong scattering of waves have been intensely studied in condensed matter physics and other areas such as imaging and seismology [6]. In particular the problem of strong localization of waves in region where λ l attracted attention and also the strong scattering in photonic band gap crystals [7]. In many of these studies the essential physical properties of the scattering medium can be modelled by a collection of point-like elastic scatterers. The use of an elastic t-matrix [8] to represent the scatterer is an efficient way to model strong multiple scattering and is frequently exploited in studies of multiple scattering of light [9][1]. Here this approach is used as an alternative to propagation models developed for wireless communication [11]. Under conditions where the medium is not scattering enough, transfer enhancement can still be achieved by enclosing the transmitter and/or receiver antennas (further abbreviated as TR) in an artificial scattering shell. The effect of such a shell of strongly material around

2 the TR is studied using the t-matrix approach confined in a two dimensional geometry. II. MODEL a. Wave propagation The propagation of an electromagnetic wave is modelled by using a Green s function approach [8][9]. The Green s function G(ω, r 1, r ) describes the propagation from a wave with frequency ω at point r 1 to position r. The problem can in general be worked out for vector waves and in a three dimensional space. However, to enhance the efficiency of the modelling the presented calculations are restricted to a two dimensional space. I.e. the third direction is ignored and the scatterers are effectively parallel cylinders. In such a geometry the two polarization directions can be treated separately as two uncoupled scalar wave problems. The free space scalar Green s function G for the two dimensional Helmholtz equation describing the scalar wave transport is give by [8]: G (ω, r 1, r ) = i H (k r r 1 ), (1) with the H the Hankel function defined in terms of the zero-order Bessel functions J and Y : H (u) = J (u) + iy (u), and k = π/λ = ω/c the wave vector. The unit system with c 1 will be used here and length or time units will not be mentioned explicitly. To efficiently model the wave propagation in a strongly scattering environment the scattering elements are represented by a line-like scatterer with a frequency dependent cross-section as was developed in studies of multiple scattering of light [9][1]. The scatterer is represented by an isotropic elastic t-matrix. The energy of the wave is scattered in all directions, but the total wave energy is conserved. This enables to evaluate the intrinsic effects of multiple scattering without energy loss. The method gives exact solutions without the need to solve the wave equation with more calculation intensive methods such as e.g. numerically integrating a finite element representation. The elastic t-matrix for a line-like scatterer in the two dimensional space is given by [1]: t(ω, µ, Λ) = µω 1 µω π ln( Λ ω ) i µω, () where µ and Λ are two parameters characterizing the t-matrix. Parameter µ is a measure of the strength of the scatterer and parameter Λ is a cut-off parameter to avoid divergence at high frequencies. The t-matrix describes elastic scattering and obeys the optical theorem in the two dimensional space: Im(t(ω)) = t(ω) = ωσ, (3) with σ the scattering cross-section in units of length 1 in the D problem. An example of the t-matrix for a particular choice is given in Fig.. The transition region near ω.5 behaves as an internal resonance in the scattering and is located where: µω ln( Λ ω ) = π. The associated cross-section is given by: ωσ =. Hence, the scattering strength can not be chosen arbitrarily large and still fulfill the physical properties of an elastic scatterer. From the cross-section of a single scatterer the meanfree-path l for waves in a medium with a low surface density of scatterers η is: l = 1 ησ. () For the parameters chosen in Fig. with η =.1 per unit surface area at ω =.5 (i.e. λ = π/ω = 1.) is: l = and fulfills the condition: λ < l L in the examples that are given later. For higher densities when η σ estimate () fails. The scatterers can not be treated independently anymore. The approach outlined below includes these dependent scattering exactly. Re[t(ω)], Im[t(ω)] ω Figure. The two dimensional elastic t-matrix () for the choice µ =.1, Λ = 1. Full line Im[t(ω)]; dashed line Re[t(ω)]. The wave propagation in the strongly scattering environment can be formulated as a Lippman-Schwinger equation for the total Green s function operator Ĝ(ω) in terms of the t-matrix operator ˆT (ω) for the whole system and the free wave propagation Green s function operator Ĝ(ω) as is often used in treating quantum mechanical problems [8]: Ĝ(ω) = Ĝ(ω) + Ĝ (ω) ˆT (ω)ĝ(ω)). The ˆT (ω) is the t-matrix of all the scatterers together and includes all multiple scattering in the medium. In general ˆT (ω) is very complicated. However, in a system with point-like scatterers the solution is considerably simplified. The propagation of the waves from a transmitter u tr at position r tr to a position r in a space with n s scatterers p 1,..,ns at positions p 1,.., p ns can be calculated using the t-matrix for the individual scatterers () and the free

3 space Green s function (1). The solution of the multiple scattering problem reduces to inversion of the matrix: { 1 i = j; D i,j = (5) G (ω, p i, p j )t(ω) i j. and gives the full t-matrix of the system: T = t(ω)d 1. (6) b. Multi channel transfer Without any scatterers the Line-of-Sight channel matrix is: H LOS i,j = G (ω, r tr,i, r rc,j ), (7) with r tr,i and r rc,j the positions of transmitter i and receiver j. The unit response from transmitter i to receiver j with the scatterers present is calculated by starting with the incoming field u in at the scatterer n with position r sc,n : u in n = G (ω, r tr,i, r sc,n ). (8) The multiple scattering between the scatterers is then evaluated by using the T matrix in (6): u out = T u in. (9) Finally the field at the receiver is the sum of the direct field and the fields produced by the scatterers: H SCA i,j = G (ω, r tr,i, r rc,j ) + n G (ω, r sc,n, r rc,j )u out n. c. Estimate of the channel capacity (1) Based on the generalized Shannon analysis [1][5] of the information transfer in a system with n tr transmitters and n rc receivers, the channel capacity C can be expressed in terms of the channel matrix H as: C = log det(i nrc + ρ n tr HQH ), (11) where I nrc the n rc dimensional unit matrix, ρ the available signal-to-noise, and H the transpose conjugate of H, and Q a measure for the correlation in the transmitted signal. The estimate for the channel capacity is obtained by examining the singular value decomposition of the H matrix with dimension n tr n rc. The decomposition results in: H = USV with U and V two unitary matrices and S a diagonal matrix with the positive singular values S 1,...,ns on the diagonal. The n s nonzero singular values are identical to the square root of the eigenvalues λ i = Si, i = 1,..., n s of the matrix HH. The number n s of non-zero values λ is at most min(n tr, n rc ). The largest singular value is associated with the strongest coupling connection. The distribution of amplitudes over the transmitter and the receiver associated with this singular value is given by the columns in U and V. The optimum distribution of transmitter powers to give the maximum channel capacity as specified by Q in (11) can be found by the water filling algorithm as [1]: C(µ) = under the constraint: P (µ) = i=1,..,n s max(log (µλ i ), ), (1) for the total available power. i=1,...,n s max(µ 1 λ i, ) (13) III. RESULTS a. Implementation of the model The scattering model and channel capacity estimate is implemented in Matlab [13] using several hundred scatterers, but can easily be recoded to run on a parallel computer to represent configurations with thousands of scatterers. The main limitation is the matrix inversion and multiplication. An example of the influence of scatterers is given in Fig. 3 where the calculated field as function of receiver position for 3 transmitter positions at L = 1 with D = is shown. The response without scatterers (top curves) shows a slowly oscillating field as function of position for each of the transmitters and reflects spatial dependence of the phase of the direct signals. The scatterers add speckle like oscillations on a shorter length scale (bottom curves, using 18 scatterers in a 3 3 square with µ =.1 and Γ = 1 at ω = π). These rapid oscillations indicate decorrelation between transmitted channels even when the receivers are placed at short distances from each other. Line of sight Field Scattered Field Receiver y position Figure 3. The received field pattern from a transmitter placed at three different positions. Top: the response without scatterers. Bottom: the response with scatterers.

4 The channel matrix for a given TR and scatterer configuration is calculated for the situation without scatterers to obtain the Line-of-Sight channel matrix H LOS and with scatterers to obtain the response H SCA. From the two channel matrices the channel capacity without scatterers C LOS and with scatterers C SCA can be estimated and averaged over many realization of the random placing of the scatterers in the configuration. b. Examples The model enables to calculate various TR geometries, scatterer properties, and densities. Here the results for only a few such examples will be discussed. In Fig. shows the result for the singular value decomposition of the channel matrix for a 9 9 TR configuration with D = 5, d =.5, and L = 1 at ω = π embedded in a scattering environment. The square of the singular values for LOS and scattering conditions are calculated for randomly generated configurations and averaged. The top and middle group of lines are with 56 and scatterers homogeneously distributed in a square of 3 3 centered around the TR with µ =.1 and Γ = 1. The lowest dotted line is the LOS reference. S Singular Value number Figure. The square of the singular values for LOS and scattering conditions calculated randomly generated scatterer configurations. The largest singular value for the LOS reference and the scattering situation are comparable with slightly higher values for the SCA case for an increasing number of scatterers. Here l remains larger than L for all shown densities. The other singular values in the scattering case increase for an increasing number of scatterers and show less spreading for the different configurations. The estimated channel capacity as function of total power P using the water filling algorithm (1) is given in Fig. 5. The conditions are as in Fig.. From low to high maximum capacity,, 8, 16, 3, 6, 18, 56, and 51 scatterers were used respectively. For the smaller powers only the highest singular values contribute and C SCA C LOS. At higher total powers, some power is also distributed over connections associated with the smaller singular values and the relative capacity increases further. Upon increasing the density of scatterers, the enhancement becomes larger. However, the maximum relative increase is still smaller than expected for fully Gaussian de-correlated sources [1]. Most likely due to remaining correlation between connections. C sca / C los P Figure 5. The relative channel capacity as function of total power P determined by the water filling algorithm (1). To test the effect of a shell of scatterers around the transmitters and the receivers the configuration sketched in Fig. 6 is used. The scatterers (+ symbols) are placed in squares around the transmitters and the receivers. The total number of scatterers is fixed, but the size of the shell is varied. The result for the estimated capacity is given in Fig. 7. A 9 9 TR configuration with D = 5, d =.5, and L = at ω = π was used with 6 scatterers (µ =.1 and Γ = 1) and averaged over 1 realizations for the scatterers. The size of the square around the TR is gradually increasing from W = 1 to W =. The results clearly show that the enhancement is higher when the scatterers are distributed over a larger area, but decreases again when the scattered signal reduces for too large spreading. For the smaller W values l W. The results are consistent with the picture that the decoupling of the transfer channels works better if the effective aperture of the transmitting and receiving areas increases. Further analysis and calculations is in progress to establish a more quantitative relation between the effective area, the mean-free-path, and the channel capacity enhancement. IV. CONCLUSION The elastic t-matrix model to calculate the wave propagation in a scattering medium provides a flexible and efficient model to study the effects of channel correlations based on the physics of wave scattering rather than a model based on statistical assumptions. The t-matrix model can easily be extended to include dynamical changes such as temporal fading of connections

5 y x Figure 6. Example of a configuration with scatterers placed in squares around the transmitters and receivers. C sca / C los Log 1 (P) 1 W Figure 7. Relative capacity for the configurations sketched in Fig. 6 as function of W. [11] by implementing motion of the scatterers. Recently the use of time-reversal symmetry of the wave propagation in scattering media to optimize transfer has been demonstrated in acoustics [1]. The model medium consisting of elastic scatterers obeys time-reversal symmetry and reciprocity for the wave transport and can be used to efficiently model the connection between timereversal symmetry, multiple scattering and optimized data transfer. V. ACKNOWLEDGEMENTS The author thanks Martijn Wubs for providing the exact elastic D t-matrix and discussions. This work is part of the research programme of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). REFERENCES [1] E. Telatar, Capacity of multi-antenna gaussian channels, AT&T-Bell Technical Memorandum, (see: and Capacity of multi-antenna Gaussian channels, Eur. Trans. Telecom., vol. 1, no. 6, Nov. 1999, pp [] G.J. Foschini, Layered space-time architecture for wireless communications in a fading environment when using multiple antennas, Bell Tech. J., vol. 1, 1996, pp. 1-59; 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 [3] A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Communication Through a Diffusive Medium: Coherence and Capacity, Science, vol. 87,, pp [] S.H. Simon, A.L. Moustakas, M. Stoytchev, and H. Safar, Communication in a disordered world, Physics Today, september 1, pp [5] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., vol. 7, 198, pp. 379 and 63. [6] P. Sheng, Introduction to wave scattering, localization, and mesoscopic phenomena, Boston: Academic Press, [7] Proceedings of the NATO Advanced Study Institute Photonic Crystals and Light Localization in the 1st century, ed. C.M. Soukoulis, Kluwer:Dordrecht, 1. [8] R.G. Newton, Scattering theory of waves and particles, New York:McGraw Hill, [9] P. de Vries, D.V. van Coevorden, and A. Lagendijk, Point scatterers for classical waves, Rev. Mod. Phys., vol. 7, 1998, pp [1] D. van Coevorden, R. Sprik, A. Tip, and A. Lagendijk, Photonic Bandstructures of Atomic Lattices, Phys. Rev. Lett., vol. 77, 1996, pp [11] T.S. Rappaport, Wireless Communications: Principles and Practice, nd edition, Prentice Hall: Upper Saddle River,. [1] M. Wubs, Quantum optics and multiple scattering in dielectrics, Thesis, University of Amsterdam, june 3; M. Wubs, L. Suttorp, and A. Lagendijk, Spontaneous-emission rates in finite photonic crystals of plane scatterers, Phys. Rev. E, vol. 69,, pp :17. [13] Matlab: [1] A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, Taking Advantage of Multiple Scattering to Communication with Time- Reversal Antennas, Phys. Rev. Lett., vol. 9, 3, pp. 131:1-.