Parametric spherical wave multiple-input and multiple-output model for ray-based simulations

Transcription

1 RADIO SCIENCE, VOL. 42,, doi: /2005rs003303, 2007 Parametric spherical wave multiple-input and multiple-output model for ray-based simulations Jose-Maria Molina-Garcia-Pardo, 1 Jose-Victor Rodriguez, 1 and Leandro Juan-Llacer 1 Received 21 June 2005; revised 18 September 2006; accepted 10 October 2006; published 7 March [1] In this paper, a three-dimensional parametric spherical wave model is presented that constructs the multiple-input and multiple-output (MIMO) channel from a single single-input and single-output realization of a ray-based simulator. This model assumes that rays do not change in the vicinity of the arrays. It uses spherical propagation and image theory to construct the MIMO channel for any array configuration. The main novelty of this model is that it takes into account multiple reflections, diffraction, variations in the reflection coefficient for single-bounce rays, and variations in the diffraction coefficient. Reflection is shown to be sensitive for hard polarization at the Brewster angle, though these rays have very low energy. Diffraction must be considered at the shadow boundaries of the diffraction wedge and assuming a constant diffraction coefficient, leads to inaccurate results at these regions. It is shown that in outdoor environments and for long distances, this model and the planar wave model provide similar results. However, at short distances, the parametric spherical wave model yields more accurate MIMO channels. Finally, the model is compared to some measurements in an outdoor environment. Citation: Molina-Garcia-Pardo, J.-M., J.-V. Rodriguez, and L. Juan-Llacer (2007), Parametric spherical wave multiple-input and multiple-output model for ray-based simulations, Radio Sci., 42,, doi: /2005rs Introduction [2] Digital data transmission will require very high capacities in future generation wireless systems [Correia, 2001]. A lot of attention has been focused in recent years on systems with multiple elements, as they can reach very high spectral efficiencies when multiple antennas are used in both the receiver and the transmitter [Foschini and Gans, 1996]. These techniques are based on the utilization of space diversity at both the transmitter and the receiver, and they are called multiple-input and multipleoutput (MIMO) systems [Telatar, 1999]. The data stream from the source is demultiplexed and transmitted by all the antennas, and provided that a rich scattering exists in the channel, the information can be multiplexed and recovered at the receiver [Foschini, 1996]. [3] Ray-based techniques such as tracing/launching techniques have proved very useful for simulating specific sites. Capacity has been analyzed in a street canyon [Driessen 1 Departamento de Tecnologías de la Información y Comunicaciones, Technical University of Cartagena, Cartagena, Spain. Copyright 2007 by the American Geophysical Union /07/2005RS and Foschini, 1999], in a corridor [Burr, 2002], and in a complex indoor environment [Chuan et al., 2000]. These techniques consider the propagation paths of electromagnetic waves from each transmitting antenna to each receiving antenna. Therefore the computational cost increases with the environment complexity and with the number of transmitting and receiving antennas. [4] Some approaches construct the MIMO channel in the vicinity of the transmitting and the receiving antennas from a single single-input and single-output (SISO) realization. They assume that the rays do not change in number or in precedence. Molisch et al. [2002] and Jiang and Ingram [2002] estimated double-directional parameters in order to construct the MIMO channel matrix for any array configuration using the planar wave assumption. For line of sight (LOS), spherical propagation is shown to be necessary in short-range distances [Jiang and Ingram, 2003]. Hampicke et al. [2002] use spherical propagation within reflections to construct the MIMO channel from measurements. Conrat and Pajusco [2003] and Del Galdo [2003] present two MIMO simulators, assuming planar wave incidence on the electromagnetic fields construction. [5] This paper presents a parametric spherical wave model (PM) that can be used in ray-based techniques to 1of12

2 construct the MIMO channel matrix from a single SISO realization. The main contribution of this work is the study of the reflection and diffraction coefficient in the construction process. A full methodology is presented to estimate the MIMO channel including LOS, multiple reflections, transmission and diffraction. A ray launching tool is used to test PM, and to compare it with the wellknown planar wave model (PW), and with full MIMO simulations. It is shown that for long distances, PM has similar a accuracy to PW. However, at short-range distances, it is necessary to study the different mechanisms of propagation in detail. [6] The paper is structured as follows: Sections 2 and 3 review briefly the MIMO channel concept and the planar wave model, respectively. The developed PM is explained in great detail in section 4. Sections 5 and 6 present theoretical results in simple and complex environments respectively, and section 7 does the same with measurements. Finally, section 8 summarizes with the conclusions. 2. MIMO Channel Review [7] A narrowband MIMO channel with M transmitting and N receiving antennas can be described as y ¼ Gx þ n ð1þ where x is the transmitted signal M 1 vector, y is the N 1 received signal vector, n is the additive white Gaussian noise (AWGN) N 1 noise vector at the receiver and G is the channel transfer matrix. The channel matrix is usually normalized so as to remove the path loss component, and it shows only the relative variations of the responses among all elements. This means that the average signal-to-noise ratio is constant whatever the position of the receiver is. This normalization can be accomplished by performing the Frobenius norm, so G is normalized to H so as EhjjHjj F 2 i = NM. In our case, as simulations are deterministic, the statistical mean operator is removed. The deterministic MIMO capacity with an averaged signal-to-noise ratio (SNR) of r averaged over all receiving antennas can be obtained as (using equal power allocation) [Paulraj et al., 2004]: C MIMO ¼ log 2 det I n þ r M HHH bit=s=hz ð2þ [8] Single value decomposition (SVD) can be used to resolve the channel into a set of independent subchannels, and then, it is possible to apply SISO Shannon capacity theorem to each one of them. The powers of these subchannels are obtained with the eigenvalues of the covariance matrix of H H H (l i )ifn > M, orhh H (l i ) if N < M. The nonnegative square roots of these eigenvalues are equal to the singular values of H. The maximum number of parallel channels is given by the minimum of N, M and the number of multipath components. Then the MIMO capacity can be calculated as (equal power allocation) C MIMO ¼ Xð min M;N Þ i¼1 log 2 3. Planar Wave Model 1 þ l ir M bit=s=hz ð3þ [9] The objective is to construct the N by M MIMO PW matrix of each ray G MIMO,k. For the kth ray, the doubledirectional channel consists of a channel impulse response matrix (two by two) describing soft and hard polarizations g SISO,k, a direction of departure W tx,k, a direction of arrival W rx,k and a propagation delay t k [Steinbauer, 2001]. The channel response g SISO,k is obtained from g SISO,k and the antenna patterns at the receiver and the transmitter g rx (W rx,k ) and g tx (W tx,k ), where the vector g describes the antenna pattern for soft and hard polarization (two entries) [Molisch, 2004]: g SISO;k ¼ g rx W rx;k gsiso;k g tx W tx;k ð4þ [10] The simplest model that constructs the electric field in the vicinity of the TX and the RX is the planar wave model (PW): it assumes that the distance between the RX and the TX is large enough, the number of rays does not change within antennas, and the incident waves at the receiver are locally plane waves. The field is constructed by estimating the phase difference of the waves at each antenna. [11] The MIMO channel matrix corresponding to the PW PW kth ray is G MIMO,k, whose elements g MIMO k,n,m, before taking the antenna patterns into account and thus obtaining g MIMO k,n,m, are the impulse responses matrices from PW the mth transmitting antenna to the nth receiving antenna [Molisch, 2004]: ¼ g MIMOk;n;m SISO;k ejhk ð W rx;k g PW Þðrx n rx c ð Þi e jhk W tx;k Þðtx m tx c Þi ð5þ where tx m and rx n are the vectors of the mth transmitting antenna and nth receiving antenna respectively, tx c and rx c are the transmitter and receiver fixed array phase centers and k is the wave vector so that hk ri ¼ 2p x cos q cos f þ y cos q sin f þ z sin q l ð Þ ð6þ where q and f refer to the elevation and the azimuth angles. Thomä et al.[2001] summarize other antennas 2of12

3 architectures such as uniform rectangular array, circular uniform beam array and stacked circular uniform beam array. Diffraction, reflection and transmission are included in the g SISO,k. The total PW MIMO channel transfer function is obtained as G PW MIMO ¼ Number Xof rays k¼1 G PW MIMO;k 4. Parametric Spherical Wave Model ð7þ [12] The philosophy of the model is to study the different propagation mechanisms that occur in the radio channel for each ray. As in PW, this parametric spherical wave model (PM) assumes that the number of rays does not vary within the antenna array. Now, spherical propagation is considered in the model. Line of sight, multiple reflections, transmission and diffraction are taken into account in the model. The same doubledirectional parameters used in section 3 are going to be used to construct the MIMO matrix for each ray. Once all matrices are constructed for all rays, the total PM MIMO channel transfer function is obtained as G PM MIMO ¼ Number Xof rays k¼1 G PM MIMO;k ð8þ [13] The empirical criterion of Jiang and Ingram [2003] can be used to decide whether or not PM is worthy compared to the classical PW. They calculate the distance where the capacity underestimation error is 50%, and when spherical propagation is used compared with PW for LOS. In our case, for other contributions, one must consider the unfolded distance between the phase centers of the arrays Line of Sight [14] The LOS MIMO matrix is constructed by calculating the Euclidean distances between each pair of antennas. Let us define r c = jrx c tx c j as the distance between the phase centers of both arrays, and r n,m = jrx n tx m j the distance from tx m to rx n (position vectors of the n receiving and m transmitting antenna LOS elements). The g MIMO n,m element of the LOS matrix can be fully constructed using the SISO LOS channel impulse response matrix g LOS SISO as g LOS MIMO n;m ¼ glos SISO e j2p l ðr n;m r c Þ ð9þ LOS [15] As in section 3, the channel response g MIMO n,m has to be computed from (9) using the antenna patters. PW is based on far-field parallel rays approximation whereas PM follows spherical propagation, as in the work by Jiang and Ingram [2002] Multiple Reflections [16] The problem of multiple reflections is easily solved by using the image theory. This consists of building the image of one of the arrays each time the ray bounces against a wall. Assuming that the receiver position is fixed, the problem is reduced to calculating the new positions of all transmitting antennas (image theory). [17] The phase center of the transmitting array tx c is mirrored at tx 0 c ¼ rx c þ tvðcos q rx cos f rx^x þ cos q rx sin f rx^y þ sin q rx^zþ ð10þ where rx c is the phase center of the receiver, t is the propagation delay between the phase centers of the transmitter and the receiver, v the speed of the light, q rx and f rx the elevation and azimuth angle of arrival, and ^x, ^y and ^z are the unitary orthogonal vectors. The images of every transmitting antenna element tx 0 m are found from tx 0 c. The problem is then reduced to a LOS problem in Multiple the new reference system. The g reflected MIMO k,n,m element of the multiple-reflected MIMO matrix is defined using the SISO multiple reflected channel impulse response matrix Multiple g reflected SISO,k : Multiple reflected g MIMO k;n;m ¼ g where r 0 c = jrx c tx 0 0 c j and r n,m response g MIMO k,n,m Multiple reflected e j2p l r0 n;m r0 c SISO;k ð Þ ð11þ = jrx n tx 0 m j. The channel Multiple reflected has to be computed from (11) using the antenna patterns. LOS (9) is a special case of (11) where r 0 n,m = r n,m. [18] A problem arises from the fact that the reflection coefficient depends on the angle of incidence. Two polarizations are distinguished for the reflection coefficient: hard and soft polarization [Balanis, 1989]. In soft polarization, the electric field is perpendicular to the incidence plane, which contains the incidence and reflected rays, and in hard polarization, the electric field is parallel to this plane. For soft polarization, small increments/decrements in the angle of incidence do not have much influence on the reflection coefficient, so the MIMO matrix can be constructed by using (11). However, for hard polarization a phase shift of p is found at the Brewster angle. [19] The incidence angles of rays departing from different antennas in a multiantenna system are different. In the case of single-bounce reflection, the problem of One reflection correcting the phase shift can be solved. g MIMO,k 3of12

4 contains the phase shift caused by the reflection and the phase shift caused by propagation, so the dyadic reflection coefficient for the phase center ray R(a c ) can be recovered. Let us call the impinging angle of two arbitrary rays i and j into a wall a i and a j. The difference between a i and a j can be geometrically obtained, so the constructed MIMO matrix with the reflection coefficient can be calculated as One reflection g MIMO k;n;m Corrected One reflection ¼ g MIMO k;n;m R a n;m ð12þ where R(a n,m ) is the normalized (element by element normalization using R(a c )) dyadic reflection coefficient for the ray that departs from the m transmitting antenna and that arrives at the n receiving antenna Transmission [20] Transmission is an essential propagation phenomenon at indoor scenarios. Fortunately, a ray transmitted through a wall does not change its direction [Rappaport, 2002] and partition losses are included in the channel Multiple impulse response matrix g transmissions SISO,k. Therefore we can use (9) to construct transmitted rays if we consider that the transmission coefficient is constant. In the case of One one transmission, g transmission SISO,k contains both the phase shifts caused by transmission and that caused by propagation. The transmission coefficient for the phase center ray T (a c ) can be calculated following the same approach One transmission as in single reflection. The g MIMO k,n,m element of the Corrected constructed MIMO matrix with one transmission can be calculated as One transmission ¼ g T a MIMO k;n;m n;m ð13þ One transmission g MIMO k;n;m Corrected where T(a n,m ) is the normalized (element by element normalization using T(a c )) dyadic transmission coefficient for the ray that departs from the m transmitting antenna and that arrives at the n receiving antenna. Simulations show that this correction is only needed under the same circumstances as it was with reflection, i.e., when there is hard polarization around the Brewster angle Diffraction [21] Ling et al. [2001] treat diffraction in a wedge as a keyhole. That statement is true as long as the diffraction coefficient is constant within: (1) the angles formed by the wedge and the incident waves, (2) those formed by the wedge and the diffracted waves, (3) the distances between the TXs (s ) and the wedge, and (4) the distance from the wedge to the RXs (s). Provided that a diffraction coefficient varies sufficiently along different antennas, the assumption that the wedge is a keyhole does no longer apply. [22] In order to solve this propagation phenomenon, the ray-based engine needs to provide the diffraction parameters [see Balanis, 1989]. The algorithm works as follows: the user defines the receiving and transmitting antennas, and from the geometry of the problem the diffraction parameters are obtained for all antennas. Then the diffraction coefficient is recalculated for every pair of antennas. Let us define the diffracted channel impulse response matrix for the phase centre antennas as [Balanis, 1989] g Diffracted SISO;k ¼ D rxc ;tx c A rxc ;tx c e j2p l s rx c ;tx c ð14þ where D rxc,tx c is the diffraction coefficient (usually a dyadic), A rxc,tx c is the spatial attenuation (spreading, divergence factor) and e j2p l s rx c,tx c is a phase factor. The g Diffracted MIMO k,n,m of the MIMO matrix corresponding to a diffraction ray can be expressed using the diffracted channel impulse response matrix g Diffracted SISO,k : 0 1 A rxn g Diffracted ;tx ¼ m e j2p l s B MIMO k;n;m gdiffracted rx D n ;tx m SISO;k rxn ;tx C A A rxc ;tx c e j2p l s rx c ;tx c ð15þ where all terms of (15) have been particularized for every pair of receiving/transmitting antennas, and D rxn,tx m is the dyadic diffraction coefficient normalized element by element using D rxc,tx c. The channel response Diffracted g MIMO k,n,m has to be computed from (15) using the antenna patterns. [23] Multiple reflection and diffraction rays can be treated by placing the mirrored corners: corner tx 0 ¼ tx c þ s 0 ð^x cos q tx cos f tx þ ^y cos q tx sin f tx þ ^z sin q tx Þ corner rx 0 ¼ rx c þ sð^x cos q rx cos f rx þ ^y cos q rx sin f rx þ ^z sin q rx Þ 5. Simulation of Simple Environments ð16þ [24] The layout of Figure 1 has been used to test the PM, and to compare it to the PW and to the full MIMO simulated channel. The layout consists of two regular street corners separated by 13.9 m. The central frequency has been set to 2.1 GHz, and the permittivity and the conductivity to e r = 5 and to s =10 2 S/m, respectively. The transmitter consists of 5 vertically polarized l/2 dipole elements aligned with the y axis. The receiver is 4of12

5 Figure 1. Top view of a simple microcellular environment consisting of two regular corners. also an array of 5 vertically polarized l/2 dipole elements, with the same orientation as the transmitter. [25] In order to test the models, full 5 5 MIMO channel matrices are simulated at each point using a raylaunching engine. The angular step between the launched rays is set to 0.1 degree. A reception sphere is then defined at the receiver; its radius is proportional to the angular step and the total raypath length [Schaubach and Davis, 1994]. Then at each point (using the phase centers of the arrays) a SISO simulation is performed in order to extract the double-directional channel parameters (using the same ray-launching engine). From this information, the virtual MIMO channel matrix is determined using both PW (construction using (5)) and PM. Simulations are performed using uniform power allocation [Foschini and Gans, 1996]. The three models are compared in capacity terms using (3), and by using a signal-to-noise ratio of 20 db at all receive branches. By nominal capacity, PW capacity and PM capacity we understand those obtained: by the full MIMO simulation, by the PW model, and by the PM model, respectively LOS [26] To test the LOS construction, a simulation considering only the direct ray has been run. We have not considered other contributions such as ground reflection, reflection or diffraction. The transmitter is placed in Figure 2. Capacity considering only LOS for the environment of Figure 1 when the receiver is moved along the line y =0. 5of12

6 Figure 3. Capacity considering LOS and one reflection for the environment of Figure 1 when the receiver is moved along the line y =0. (0, 0, 4 m) and the receiver performs a run from position (1 m, 0, 2 m) to position (100 m, 0, 2 m) (see Figure 1). Elements of both arrays are spaced by l/2. Figure 2 shows the capacity in bit/s/hz at each position of the receiver. [27] Capacity of PM (construction using (9)) and nominal capacity of the MIMO channel are exactly the same as expected. Using PW, capacity is always 8.97 bit/s/hz for 20 db, as the rank of the matrix is one. This was observed by Jiang and Ingram [2003], where an empirically based threshold distance is derived. It is seen that the capacity converges to a limit, where the approximation of PW holds. Note that PM has only constructed phase differences, but not amplitude differences (they are very small). [28] Regarding the computational cost, in our simulations PW and PM construct the 5 5 MIMO channel in 65% and in 102% of the time of a SISO realization, respectively, while a full MIMO simulation lasts 25 times this duration Multiple Reflections [29] The transmitter is now placed at ( 10 m, 5 m, 4 m), and the receiver is moved from ( 70 m, 0, 2 m) to (0, 0, 2 m) (Figure 1). In this simulation, the direct ray and a single-bounce ray have been considered (we have not considered any ground reflections). The elements at the transmitter and receiver are spaced by l. Capacity of the PM (construction of reflection using (11)) follows quite precisely the nominal capacity (Figure 3). Again, PW is accurate when the distance from the transmitter to the receiver is large enough. The decrease of capacity at x = 10 is due to the array orientations [Almers et al., 2003], which corresponds to the worst case. The extra computational cost of the PM depends on the number of antennas, and it consists of calculating the distance from each antenna element of one array to each antenna element of the imaged array. It is important to note that the number of impinging waves needs to remain constant within the antennas of the same array. In this case, PW and PM construct the 5 5 MIMO channel in a 72% and in a 137% of the time of a SISO realization respectively. [30] The impact of the reflection coefficient is here studied. As discussed in the previous section, the phase shift depends on the angle of incidence. The worst case is that of hard polarization at the Brewster angle. The horizontal polarized transmitter is placed at ( 10 m, 5 m, 0), facing the wall at the Brewster angle of the wall (the Brewster angle calculated from these electric parameters of the wall is 24.1 [Balanis, 1989]). The receiver faces this angle of reflection, and performs a run along the Brewster line of the wall reflection of Figure 1. Figure 4 shows MIMO capacity for 20 positions when a single-bounce ray at Brewster angle occurs (we have only considered the reflection on the wall). The distance between adjacent positions is 1 m. [31] We have used (11) for PM with R constant and (12) for PM (R corrected). Correction refers to recalculating R(a n,m ) for each ray. It is shown that using the correction PM provides a more accurate result. At 6of12

7 Figure 4. Capacity considering LOS and one reflection for the environment of Figure 1 when the receiver is moved along the Brewster angle line. the Brewster angle there is no reflection, and around the Brewster angle the amplitude of the reflection is very low, so it does not contribute a great deal to capacity. As for the PW and PM without correction, their results are not very good. It is then concluded that the reflection coefficient around the Brewster angle affects capacity. When there are high-order reflections, correction is more complicated because more information is needed to recalculate the accumulated Fresnel reflection coefficient Diffraction [32] Finally, the diffraction coefficient is studied. It is interesting to separate the diffraction phenomenon from other propagation mechanisms, as diffraction normally has very low energy. The transmitter is placed at ( 10 m, 5 m, 4 m), and the receiver performs a run from (0, 20 m, 2 m) to (0, 20 m, 2 m) considering the diffraction of the two regular corners of Figure 1 (we have not considered ground reflection). The antenna elements are separated by l/2. In Figure 5, the following is shown: full MIMO simulations, PW, PM without considering variations in the diffraction coefficient (using (15) with D constant), and, finally, PM correcting the diffraction coefficient (15). [33] It can be seen that it is very important to correct the diffraction coefficient at the shadow boundaries of each corner, as both PW and PM with D constant fail to construct the MIMO channel. The first two peaks of Figure 5 correspond to the reflection shadow boundary (RSB) of the first (left) corner and the second (right) corner of Figure 1, while the third one is due to the incident shadow boundary of the first corner. [34] The computational cost of the algorithm consists of recalculating the diffraction coefficient from each transmitting antenna to all receiving antennas, using the diffraction parameters obtained from the raybased engine. In our example PW and PM construct the 5 5 MIMO channel in a 3.6% and in a 210% of the time of a SISO realization respectively. PW is much more efficient than PM, however, it fails to predict the electric field in the transition zones. 6. Simulation of an Outdoor Environment [35] The layout of Figure 6 is used to test the PM and PW in a more complex environment. The layout corresponds to a zone of the city of Murcia (Spain). It corresponds to a microcellular environment, completely flat, surrounded by buildings of five or six stories. The buildings are generally made of brick, and other obstacles included lampposts, signs and trees. The frequency is set to 2.1 GHz, and permittivity and conductivity are set to e r = 5 and to s =10 2 S/m respectively. The transmitter consists of a linear array with 5 vertical polarized l/2 dipole elements separated by 3l. The receiver is a linear array of 3 vertical polarized l/2 dipole elements, separated by l. In the simulation, up to 10 reflections are considered, 2 reflections after diffraction and ground reflection for all rays (three- 7of12

8 Figure 5. Capacity considering the diffraction of the two wedges for the environment of Figure 1 when the receiver is moved along the line x =0. Figure 6. Top view of an outdoor microcellular environment. 8of12

9 Figure 7. Capacity in the environment of Figure 6 when the transmitter is placed in TX 1 and the receiver performs an 80 m run. dimensional simulation). The transmitter height is 4 m and the receiver height 1.7 m. [36] The transmitter is placed in two positions, TX 1 and TX 2. From the first one, the minimum distance between the transmitter and the receiver is about 40 m. For the second position, the minimum distance is 7.5 m. The receiver is moved along 80 m (1 m spaced). The transmitter broadside direction is perpendicular to the receiver run, and the receiver broadside direction is aligned to its run. [37] For the first simulation (Figure 7), the transmitter is placed at TX 1. The receiver starts its run in non line of site (NLOS), then at 17 m it enters in LOS and finally at 47 m it enters again in NLOS (in NLOS capacity is higher than in LOS for the same SNR). As it can be observed, both PM and PW predict very accurately the MIMO capacity. The mean relative error, compared to the nominal capacity, is 3.24% for PW and it is 2.94% for PM. It has been observed that in the full MIMO simulation the number of rays changed from one antenna element to another of the array. Because of this, it is not possible to predict the nominal capacity by constructing the MIMO channel using a single SISO realization. However, results show that the prediction is very accurate (less than 3% in our case). [38] The second simulation places the transmitter at TX 2. The receiver starts its run in LOS, and then, at 49 m, it enters in NLOS (Figure 8). The distance between the transmitter and the receiver has been significantly reduced. Because of this, PW does not predict capacity very well from position 30 to position 50, where the distance is minimum (Figure 6). The mean relative error for PW is 6.8% and 4.6% for PM. [39] Finally, because of the assumption of a constant SNR whatever the position of the receiver is, it is observed that capacity is much higher in the deep shadow region (NLOS) than in the vicinity of the transmitter (LOS). 7. Comparison With Measurements [40] The channel sounder is based on a multiport network analyzer configured as one transmitting port and four receiving ports, where the four receiving antenna elements are directly connected. The transmitting port is fed to a 30 db low-noise amplifier, then to a 20 m long cable and finally to a fast switch, where the four transmitting antenna elements are connected. The control of the measurements is automatically carried out by a laptop, which controls the fast switch and the ENA via GPIB and LAN, respectively. [41] Eight Cisco Aironet omnidirectional mast mount antennas (AIR-ANT2506) have been used, with a 5 dbi gain each. The masts have a height of 1.8 m, and the antenna element spacing is fixed to 2l at 2.45 GHz. [42] Measurements have been carried out at the campus of the Universidad Politécnica de Cartagena. The measurements were taken all along a straight line for 10 positions of the receiver (see Figure 9). Both arrays were oriented in parallel for the first receiving position. 9of12

10 Figure 8. Capacity in the environment of Figure 6 when the transmitter is placed in TX 2 and the receiver performs an 80 m run. [43] During the measurement there was no movement in the environment and that is why the channel is assumed to be quasi-stationary. Furthermore, ten measurements were taken for each position in order to check the stationarity as well as to average the measurements to reduce the noise floor. The central frequency was 2.45 GHz with an intermediate frequency of 3 KHz. The transmitted power was 0 dbm at each antenna element and the dynamic range was over 100 db. [44] We have plotted in Figure 10 nominal, PW and PM MIMO capacities using (3) of the simulated environment of Figure 9, considering the above mentioned measurement configuration. In the simulation, LOS, reflection, diffraction and ground reflection of all con- Figure 9. Top view of the measured environment. 10 of 12

11 Figure 10. Capacity in the environment of Figure 9. tributions have been considered. Besides, identical electric parameters as those considered in the previous sections have been selected. On the one hand, PM obtains the nominal MIMO capacity, except for x =1, where not all the antenna elements receive the same number of contributions. On the other hand, one can see that PW fails to predict the nominal capacity. Then we have also plotted the mean capacity obtained from the 10 measurements (Frobenius normalized) with triangles. There is a difference of less than 1 bit/s/hz between the simulations and the measurements. This difference is due first to the simplicity of the environment considered in Figure 9 and second to possible far reflections (for example, there is a small hill at 4 m behind the receiver). 8. Conclusions [45] In this paper a parametric spherical wave model that constructs the MIMO channel from a single SISO realization of a ray-based engine has been presented. The model assumes that the number of rays does not change within the antennas of the transmitter and the receiver; neither does the origin of those rays. The method assumes spherical propagation, and it constructs the MIMO channel using image theory. It studies variations in the reflection coefficient and variations in the diffraction coefficient. [46] For LOS, the parametric spherical wave model provides the same results as the full MIMO simulation. Planar wave approximation is only valid for long distances between the transmitter and the receiver. [47] When multiple reflections occur, the model also follows the nominal capacity, provided that the number of rays remains constant within all antennas of the array. Also, it was shown that it is not necessary to consider variation in the reflection coefficient due to different incident angles in a wall, except for hard polarization and around the Brewster angle. However, this ray has very low energy, so it can be neglected when there is another ray with a higher power. PM presents an order of complexity that doubles that of PW; however, they both construct the MIMO channel in the time of a computation of a SISO realization. [48] The diffraction coefficient is shown to be very sensitive along incident shadow and reflected shadow boundaries. This means that apart from taking into account all paths from all antennas to the diffraction wedge, variations in the diffraction coefficient must be considered to construct the MIMO channel at the vicinity of the diffraction wedge. In this case, PM is about 50 times more complex than PW, but much less than a full MIMO simulation. [49] Then, the parametric spherical wave model is tested in an outdoor environment, for two positions of the transmitter. In the simulated example, when the transmitter is far from the receiver, PW and PM predict similar capacity with a relative error at around 3%. It is at close distances where PM is more advantageous and yields more accurate MIMO channels. [50] Finally, measurements performed at 2.45 GHz were compared to simulations. While PW fails to predict the nominal MIMO capacity, PM achieved a good 11 of 12

12 agreement as long as the number of contribution was the same within all antenna elements. Measurements differed from the prediction by less than 1 bit/s/hz, mainly due to not considering far contributions. [51] Acknowledgment. The authors would like to thank the Ministerio de Educación y Ciencia of Spain for funding this work (TEC C04-04/TCM). References Almers, P., F. Tufvesson, P. Karlsson, and A. F. Molisch (2003), The effect of horizontal array orientation on MIMO channel capacity, in Proceedings of IEEE Vehicular Technology Conference, vol. 1, pp , IEEE Press, Piscataway, N. J. Balanis, C. A. (1989), Advanced Engineering Electromagnetics, John Wiley, Hoboken, N. J. Burr, A. G. (2002), Evaluation of the capacity of the MIMO channel in a corridor using ray tracing techniques, in 2000 International Zurich Seminar on Broadband Communications: Accessing, Transmission, Networking, pp , IEEE Press, Piscataway, N. J. Chuan, C.-N., G. J. Foschini, R. A. Valenzuela, D. Chizhik, J. Ling, and J. M. Kahn (2000), Capacity growth of multielement arrays in indoor outdoor wireless channels, in IEEE Wireless Communications and Networking Conference, vol. 3, pp , IEEE Press, Piscataway, N. J. Conrat, J.-M., and P. Pajusco (2003), A versatile propagation channel simulator for MIMO link level simulation, COST 273 Proj. Rep. TD (03)120, Elektrobit, Oulunalo, Finland. Correia, L. M. (Ed.) (2001), COST 259: European Co-operation in Mobile Radio Research, Wireless Flexible Personalised Communications, John Wiley, Hoboken, N. J. Del Galdo, G. (2003), IlmProp: A flexible geometry-based simulation environment for multiuser MIMO communications, COST 273 Proj. Rep. TD (03)188, Elektrobit, Oulunalo, Finland. Driessen, P.-F., and G. J. Foschini (1999), On the capacity formula for multiple input-multiple output wireless channels: A geometric interpretation, IEEE Trans. Commun., 47, Foschini, G. J., and M. J. Gans (1996), On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Commun., 6, Foschini, J. (1996), Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas, Bell Lab. Tech. J., 1, Hampicke, D., M. Landmann, C. Scheider, G. Sommerkorn, R. Thomä, T. Fügen, J. Maurer, and W. Wiesbeck (2002), MIMO capacities for different antenna array structures based on double directional wideband channel measurements, paper presented at IEEE Vehicular Technology Conference Fall 2002, Inst. of Electr. and Electron. Eng., Vancouver, B. C., Canada. Jiang, J.-S., and M. Ingram (2002), Path model and MIMO capacity for measured indoor channels at 5.8 GHz, paper presented at IEEE International Symposium on Antenna Technology and Applied Electromagnetics, Inst. of Electr. and Electron. Eng., Montreal, Que., Canada. Jiang, J.-S., and M. A. Ingram (2003), Distributed source model for short-range MIMO, in VTC2003-Fall Orlando: 2003 IEEE 58th Vehicular Technology Conference, vol. 1, pp , IEEE Press, Piscataway, N. J. Ling, J., D. Chizhik, and R. A. Valenzuela (2001), Predicting multi-element receive and transmit array capacity outdoors with ray tracing, paper presented at IEEE Vehicular Technology Conference, Inst. of Electr. and Electron. Eng., Rhodes, Greece. Molisch, A. (2004), A generic model for MIMO wireless propagation channels in macro- and micro cells, IEEE Trans. Signal Process., 52, Molisch, A. F., M. Steinbauer, M. Toeltsch, E. Bonek, and R. S. Thomä (2002), Capacity of MIMO systems based on measured wireless channels, IEEE J. Select. Areas Commun., 20, Paulraj, A. J., D. A. Gore, R. U. Nabar, and H. Bölcskei (2004), An overview of MIMO communications A key to gigabit wireless, Proc. IEEE, 92, Rappaport, T. S. (2002), Wireless Communications: Principles and Practice, Prentice-Hall, Upper Saddle River, N. J. Schaubach, K. R., and N. J. Davis (1994), Microcellular radio channel propagation prediction, IEEE Trans. Antennas Propag., 36, Steinbauer, M. (2001), The radio propagation channel A nondirectional, directional and double-directional point-of-view, Ph.D. thesis, Tech. Univ., Vienna. Telatar, I. E. (1999), Capacity of multi-antenna Gaussian channel, Eur. Trans. Telecommun., 10, Thomä, R. S., D. Hampicke, A. Richter, and G. Sommerkorn (2001), MIMO vector channel sounder measurements for smart antenna system evaluation, Eur. Trans. Telecommun., 12, L. Juan-Llacer, J.-M. Molina-Garcia-Pardo, and J.-V. Rodriguez, Departamento de Tecnologías de la Información y Comunicaciones, Technical University of Cartagena, Campus Muralla del Mar s/n, E Cartagena, Spain. (josemaria. molina@upct.es) 12 of 12