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In radio system deployment, the main focus is on assuring sufficient coverage, which can be estimated with path loss models for specific scenarios. When more detailed performance metrics such as peak throughput are studied, the environment has to be modeled accurately in order to estimate multipath behavior. By means of laser scanning we can acquire very accurate data of indoor environments, but the format of the scanning data, a point cloud, cannot be used directly in available deterministic propagation prediction tools. Therefore, we propose to use a single-lobe directive model, which calculates the electromagnetic field scattering from a small surface and is applicable to the point cloud, and describe the overall field as fully diffuse backscattering from the point cloud. The focus of this paper is to validate the point cloud-based full diffuse propagation prediction method at 60 GHz. The performance is evaluated by comparing characteristics of measured and predicted power delay profiles in a small office room and an ultrasonic inspection room in a hospital. Also directional characteristics are investigated. It is shown that by considering single-bounce scattering only, the mean delay can be estimated with an average error of 2.6% and the RMS delay spread with an average error of 8.2%. The errors when calculating the azimuth and elevation spreads are 2.6° and 0.6°, respectively. Furthermore, the results demonstrate the applicability of a single parameter set to characterize the propagation channel in all transmit and receive antenna locations in the tested scenarios.
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Characterizing the propagation channel in environments where radio systems are to be deployed is crucial in order to deal with multipath interference [Lostanlen and Kürner, 2012; Chen and Jeng, 1997]. Channel measurements are widely used for this purpose, but they are often expensive and time-consuming [Rasekh et al., 2009; Kazemi et al., 2012]. Deterministic field prediction methods such as ray tracing is a strong candidate for estimating essential radio channel characteristics in terms of the received power, the power delay profile (PDP), or the delay spread, and can also give statistical properties of the radio channel [Fritzsche et al., 2010; Peter et al., 2007; Bertoni, 2000].
The 60 GHz band has been of interest for years due to the large available bandwidth, which enables multi-Gigabit wireless communication [Tharek and McGeehan, 1988; Gibbins, 1991; Kato et al., 1997; Smulders, 2002]. In addition, the interest has been supported by advances in millimeter-wave hardware [Maltsev et al., 2009] and standardization activities, e.g., Institute of Electrical and Electronics Engineers (IEEE) 802.11ad [Park et al., 2008; Institute of Electrical and Electronics Engineers, 2012].
Due to the short wavelength in the 60 GHz band, the environmental modeling is challenging. In order to achieve the utmost prediction accuracy in deterministic field prediction, the details of the environment has to be modeled with accuracy comparable to the wavelength [Burkholder and Lee, 2005]. In addition, material properties are required in order to solve the scattered fields. Calculating the fields for entire rooms is therefore computationally prohibitive. So far, most of the research regarding 60 GHz propagation prediction has considered only very simple propagation environments, neglecting fixtures and furniture that may affect radio propagation [Collonge et al., 2003; Lim et al., 2007; Peter et al., 2007; Lostanlen and Gougeon, 2007, and references therein]. Our interest is to further improve deterministic propagation prediction by virtue of more accurate structural description of the propagation environment. To this end, we use laser scanning to achieve very accurate modeling of the environment. However, the model cannot directly be used by ray tracing tools since the laser scanning produces a set of points (hereinafter referred to as “point cloud”) and not a surface model. Without the description of the surfaces, the specular component cannot be computed using traditional ray tracing.
We therefore use a single-lobe directive scattering model [Degli-Esposti et al., 2007] for the accurate deterministic field prediction. The model calculates the backscattering from each point in the point cloud [Järvelainen et al., 2012] and the contribution coming from different points is properly combined to give the total field. The use of the single-lobe directive scattering model is motivated by the spatial sampling rate of the laser scanning, which usually is less than the half-the-wavelength at 60 GHz. In contrast to physical optics [Burkholder and Lee, 2005] which assumes isotropic scattering from electrically small surfaces, having coarser spatial sampling leads to electrically larger surfaces that may produce directive scattering patterns.
Furthermore, since it is nearly impossible to know the material properties for every point in the point cloud, we have chosen an experimental approach to define the scattering pattern so that it reflects material properties. In concrete, we use a single-scattering pattern for all points in one room. The value of the scattering coefficient is found by comparing with measured channels.
The focus of this paper is to validate the propagation prediction method by comparing measured and predicted channel characteristics in the 60 GHz band. The validation is performed in two different scenarios, a small office room and an ultrasonic inspection room inside a hospital. The scattering model parameters are tuned in order to minimize the error between the measured and predicted PDP, and the mean delay and the root-mean-square (RMS) delay spread are calculated along with the azimuth and elevation spreads.
The paper is organized as follows: Section 2 presents the point cloud-based full diffuse propagation prediction method, applicable to the single-lobe directive model. Also, the used channel metrics for validation are shown. In section 3, we briefly cover the channel measurements performed in order to allow the validation of the prediction method, and section 4 concentrates on optimizing the model parameters based on measurement data. The results are presented in section 5. We derive delay and angular characteristics and prove that good agreement between measured and predicted channels can be achieved. Finally, we conclude the main findings of the point cloud-based full diffuse propagation prediction method in section 6.
2 Propagation Prediction
2.1 Prediction Model
Structural models of the small office room and the ultrasonic inspection room were acquired with laser scanning in the form of a point cloud [Järvelainen et al., 2012]. The point cloud for the small office room consisted of 19,947 points, as shown in Figure 1, and for the ultrasonic inspection room, the size of the point cloud was 15,736 points. The radio channel for the environment of interest is estimated as the sum of all signal paths between the transmit (Tx) and receive (Rx) sides. The signal path refers to the propagation path along which the radio wave travels from the Tx to the Rx antennas, and in this work we assume that these paths consist of the line of sight (LOS) and of single-bounce scattering from each point in the point cloud. The single-bounce assumption is made due to several reasons. First, it limits the computation time and the memory required leading to time-efficient prediction of channels. Second, signals being scattered several times are generally weaker than single-bounce components due to larger scattering losses and the longer traveling distance leading to larger path loss, and therefore, their effective contributions to the conveyed energy from the transmitter to the receiver are marginal.
The scattering is predicted using the single-lobe directive scattering model [Degli-Esposti et al., 2007] shown in Figure 2. The amplitude of the scattered wave is computed with
where is the incident field, is the scattering coefficient in the range [0,1], ri and rs are the distances between the scatterer and Tx and Rx antennas, respectively, λ is the wavelength, dS is the area of the scattering surface element, θi is the incident angle, ψR is the angle between the direction of the specular reflection and the scattering, αR determines the width of the scattering lobe, and is a scaling coefficient introduced to maintain the total energy preserved in a scattering process [Degli-Esposti et al., 2007]. In order to determine the angles θi and ψR, the normal vectors for each surface element need to be determined [Järvelainen et al., 2012]. The area of the surface element dS, whose sum equals the area of the room surface, is calculated as , where dmean is the mean distance to the four neighboring points. The power of each signal path is affected by scattering only, but no transmission loss is taken into account.
Each path can now be described as
where τ is the propagation time (delay) determined by the distance of each path, is the polarimetric complex amplitude computed with (1), Ω=[φϑ] and Ω′=[φ′ϑ′] are vectors composed of azimuth and elevation angles on the Tx and Rx sides, respectively. The notation al stands for the parameter value a for the lth path, 1 ≤ l ≤ L. The parameters S and αR are defined separately for each polarization combination. Due to restrictions in available measurement results, the main focus in this work is on vertical polarized fields on the Tx and Rx sides.
2.2 Channel Metrics
In order to compare predicted and measured channels, we need metrics that can be derived from both the prediction method and channel measurements. These metrics include the PDP and the power angular spectrum (PAS).
2.2.1 Power Delay Profile
Using the Fourier transform, we can calculate the radio channel transfer function (CTF) as a sum of the paths with
where gl(Ωl) and gl′(Ωl′) are the complex amplitudes of the antenna radiation patterns on the Tx and Rx sides, respectively, fi is the radio frequency, which ranges from 61 to 65 GHz with 2001 frequency steps, 1≤i≤I, and ξ is a phase uniformly distributed over [02π). When the radiation patterns of the Tx and Rx antennas are known, the complex amplitude for a single path can be determined with the knowledge of Ω and Ω′. By converting the CTF back to delay domain, the channel impulse response (CIR) is obtained as
where I is the number of frequency steps and τn is the delay ranging from 0 to 500 ns with a resolution of 0.25 ns (1≤n≤2001). The PDP is obtained from
where K is the number of CIRs representing small-scale realizations. The small-scale channel realizations are obtained by altering the phase of the scattered paths ξ randomly. The value K has to be consistent between prediction and measurements in order for a fair comparison of the PDP.
2.2.2 Power Angular Spectrum
Deriving the PAS requires a description of a radio channel with well-defined antenna arrays at the Tx and Rx sides. Well-defined antenna arrays in this paper mean that the relative locations of the antennas in the array are known and that they share the same polarimetric radiation pattern. We denote a radio channel matrix at radio frequency fi with , where M and M′ are the number of antenna elements in the Tx and Rx antenna arrays, respectively. Assuming an ideal antenna array with identical radiation patterns of antenna elements and no mutual coupling effects between them, an entry of the channel matrix is denoted as
where Wm′mi is a phase rotation term due to antenna location displacement relative to a coordinate origin denoted as
where u(Ω)=[cosϑ cosφ cosϑ sinφ sinϑ]T is a unit vector at the Tx and Rx antenna arrays for the azimuth and elevation angles of interests, dm=[dxdydz]T is a position vector of the mth antenna element at the Tx and Rx antenna arrays relative to the coordinate origin, 〈x,y〉 is an inner product of two vectors x and y, ·T is a transpose operation, and c denotes the velocity of light. The antenna orientation effect is implicitly included in the antenna radiation patterns g(Ω) and g′(Ω′) on the Tx and Rx sides, respectively.
Having defined the channel matrix and the phase rotation term due to antenna displacement, the PAS is defined by
where ·H denotes an operation of Hermitian transpose. The PAS is derived by an incoherent average of those for the frequency range from 62.9 to 63.1 GHz with I=101 frequency samples. In order for fair comparisons of the PAS between prediction and measurements, the antenna array configurations, e.g., the number of antenna elements, the antenna element type, and antenna array geometry, have to be the same.
In comparing the measured and predicted PAS, we reduced the dimension of the PAS (8) for simpler visualization and analysis. Two double-directional PASs defined for (1) the Tx and Rx azimuth angles and (2) the Tx and Rx elevation angles are used for the comparison in particular. The azimuth PAS, PAS(φ,φ′),is derived by setting the Tx and Rx elevation angles to 0 in (8), and one of the horizontal linear arrays is chosen from the Rx planar array. The elevation PAS, PAS(ϑ,ϑ′), is derived by a marginal integral of the PAS in (8) over the Tx azimuth angles φ, while only one of the vertical linear arrays is chosen from the Rx planar array.
3 Channel Measurements for Validation
3.1 Measurement Equipment and Setup
The radio channels for the small office room and the ultrasonic inspection room have previously been measured with a channel sounder consisting of a local oscillator (LO), vector network analyzer (VNA), and frequency converters. The VNA frequency range was set from 5 to 9 GHz with 2 MHz intervals, and the baseband signal was mixed with the 14 GHz LO signal to produce a radio frequency signal from 61 to 65 GHz. A back-to-back calibration was performed before the measurements in order to compensate for the effect of cables, frequency converters, and waveguides [Haneda et al., 2011]. Phase stability was maintained by minimizing cable bending during the measurement.
3.1.1 Small Office Room
In the small office room, one Rx location and 64 Tx locations were measured as illustrated by the floor plan in Figure 3. The dimensions of the room were 4.5×4.3×2.9 m3, and the Tx and Rx antenna heights were 1.1 m and 1.4 m, respectively. On the Tx side, a M=7×7 virtual planar array was scanned on the horizontal plane using a two-dimensional electromechanical positioner. On the Rx side, a M′=7×7 virtual planar array was scanned similarly in the vertical plane. This measurement setup was chosen based on the IEEE 802.11ad usage model “Wireless display,” where a user portable device such as a DVD player or a game console transmits an image to a TV [Perahia et al., 2010; Myles and de Vegt, 2008]. The user device may contain an antenna array implemented on the horizontal plane and the display may be able to accommodate a vertical antenna array. The antenna elements were azimuth-angle-omnidirectional vertical biconical antennas on both Tx and Rx sides, and the interelement spacing of the virtual antenna arrays was 2 mm. The elevation pattern of the biconical antenna is presented in Figure 4. The CTFs for K=M′M=2401 antenna element combinations were then measured for each Tx location.
3.1.2 Ultrasonic Inspection Room
The measurement floor plan for the ultrasonic inspection room is presented in Figure 5, which shows the location of the 2 Tx and 16 Rx locations. The room dimensions were 7.2×6.1×2.6 m3, the Tx antenna height was 0.8 m, and the Rx antenna height was 1.6 m. A biconical and an open waveguide antenna were used as Tx and Rx antenna elements, respectively. No virtual array was used in this scenario, but the transfer function for each Tx-Rx link was measured K=100 times while the Rx location was varied by 0.3 m in the horizontal plane.
3.2 Channel Metrics
The same channel metrics, i.e., PDP and PAS, are derived from the measurements to compare them with prediction. The PDP was derived in both the small office room and in the ultrasonic inspection room. The number of frequency steps in the measurement of the CTF was 2001. By means of inverse discrete Fourier transform, the CTF is converted to the delay domain, and the PDP is obtained from (5) with K=100 in the ultrasonic inspection room and K=2401 in the small office room. This value of K was used to derive both measured and predicted PDPs in order to assure consistency when evaluating the prediction accuracy. Since the well-defined antenna arrays are available in the measurements of the small office room only, the PAS was derived only for this scenario using (8). The width of the angular bin was 3°.
4 Optimization of Scattering Model Parameters
The scattering model described by (1) has two parameters that control the nature of the scattering; S affects the magnitude of the scattering, and αR defines the width of the scattering lobe. In this work, S and αR are optimized by comparing the measured and predicted PDP and minimizing the average error in the −30 dB range compared to the maximum level of the measured PDP. The parameters were found by a brute force search to minimize the following cost function,
where PDPmeas and PDPpred are the measured and predicted PDPs, respectively, and 1 ≤ j ≤ J denotes the indices for which the following inequality is satisfied:
4.1 Small Office Room
The optimization for the small office room is performed by varying αR between 1 and 20 and S between 0.1 and 1 with a step size of 0.1. The optimization interval for S is based on the definition given in section 2.1 which ensures that the scattered power cannot be larger than the incident power. The interval for αR has been determined heuristically based on physically meaningful values. The average error is found to depend almost solely on the parameter S in the studied parameter range. It is noticed that many different αR values in the studied interval produce approximately the same ε, and among these, we choose the smallest value, αR=11. The S parameter is then optimized separately for all Tx locations while keeping αR fixed at 11. Figure 6 shows the optimum value Sopt for all Tx locations as a function of the Tx-Rx distance. Figure 7 illustrates the PDP comparison for Tx4, for which Sopt=0.6. The average Sopt over all the Tx locations is calculated as 0.5. With the local Sopt choice, the minimum and maximum ε values are 2.0 dB at Tx 4 and 6.2 dB at Tx 78, respectively, with the average ε being 4.1 dB. In general, it is noticed that ε slightly increases as the Tx-Rx distance increases.
4.2 Ultrasonic Inspection Room
In the ultrasonic inspection room, we only consider the −20 dB range compared to the maximum PDP value due to poor agreement between measured and predicted curves at longer delay range. An optimization is carried out in the same manner as for the small office room, by varying αR between 1 and 20 and S between 0.1 and 1 with a step size of 0.1. The results highlight similarly to the small office room that S is the more dominant parameter compared to αR when minimizing ε. Although αR does not have a large impact on ε, it can be observed that in general, a smaller αR gives slightly better agreement. Therefore, we choose αR=1 and optimize S for all Rx locations. Figure 8 shows Sopt for all Rx locations as a function of the Tx-Rx distance. The PDP for Tx1-Rx72, Sopt=0.8, is presented in Figure 9. The average ε is found to be 4.3 dB, and the minimum and maximum ε values are 3.2 dB and 6.0 dB, respectively. The average Sopt is 0.9.
The two scenarios were found to be slightly different in terms of optimized model parameter values. Although it was noticed that the optimum choice of αR is not evident, visual inspection showed that the shape of the measured PDPs in the small office room were notably more spiky compared to the ultrasonic inspection room where diffuse propagation prevails. The scattering parameter S was clearly a more dominant parameter than αR. The difference between the optimized values in the two tested scenarios was obvious. This result suggests that the surfaces in the small office room does not scatter fields as those in the ultrasonic inspection room in terms of power, which can be explained by the fact that there are two large windows in the small office room. However, it must be kept in mind that the uncertainty in the surface element dS calculation affects the value of S. If the total dS area is underestimated, S appears to be higher. The point cloud in the ultrasonic inspection room has a few regions along the ceiling in which the point density is very sparse compared to the rest of the point cloud and the total dS may be slightly underestimated.
Since only single-bounce scattering are accounted for in the channel prediction, the deviation between measurements and prediction will increase with longer delay range. This can be noticed in the ultrasonic inspection room, where the deviation between measured and predicted PDPs starts to increase after approximately 12 ns for the Tx1-Rx72 link as observed in Figure 9. When comparing the measured and predicted PDPs in the small office room as shown in Figure 7, the deviation starts later, which also supports the assumption that the walls in the ultrasonic inspection room scatter signals better, and thus, the difference between measurements and prediction is larger.
The field prediction method does not currently consider transmission, but only scattering. The results imply that this is sufficient in the 60 GHz band due to the high penetration attenuation at this frequency. However, this might prevent the method from being used at lower frequencies. It must also be noted that scattering objects have been present only in the far-field regions of the antennas during the channel measurements. Modeling non-LOS scenarios is beyond the scope of this paper.
5 Results and Discussion
5.1 Delay Spread
As metrics to characterize the PDP when comparing measured and predicted results, we use the mean delay τm and the RMS delay spread τRMS. The mathematical formulation and use of these can be found in [Molisch, 2011]. The above mentioned metrics are calculated and compared between measurements and by considering the part of the PDP which satisfies (10).
5.1.1 Small Office Room
The measured and predicted mean delay and RMS delay spread as a function of Tx-Rx distance is presented in Figure 10. The result highlights that the RMS delay spread is generally underestimated while the mean delay shows better agreement. Table 1 presents the average mean delay and RMS delay spread considering all Tx locations along with the average difference between measured and predicted metrics. When the mean delay is derived from both measured and predicted PDPs, the average difference is only 0.23 ns, or 4.4%. Comparing the RMS delay spread results, we see that the error is larger, 0.63 ns (20.8%).
Table 1. Average Measured and Predicted Mean Delay and RMS Delay Spreads for All Tx Locations in the Small Office Room
Average of All Tx Locations
5.1.2 Ultrasonic Inspection Room
The measured and predicted mean delay and RMS delay spread as a function of Tx-Rx distance is presented in Figure 11. Table 2 presents the mean delay and the RMS delay spread averaged over all Rx locations, and the average difference between these. The average difference between measured and predicted mean delay is 0.48 ns, or 7.2%, and is clearly overestimated as illustrated by Figure 11. On the contrary, the RMS delay spread is underestimated by 0.21 ns (11.2%).
Table 2. Average Measured and Predicted Mean Delay and RMS Delay Spreads for All Rx Locations in the Ultrasonic Inspection Room
Average of All Rx Locations
5.2 Angular Spread
The directional characteristics are studied in the small office room by comparing azimuth and elevation spreads for both measured and predicted channels.
Exemplary measured and predicted PASs with Tx and Rx azimuth angles for Tx31 are presented in Figures 12 and 13. The PASs with Tx and Rx elevation angles are presented in Figure 14. The azimuth spread at the Tx and Rx sides, Sφ and Sφ′, are then calculated with [Molisch, 2011]
where φ is the azimuth angle and
The elevation spread at the Tx and Rx sides, Sϑ and Sϑ′, are calculated similarly from the elevation PAS with angles ϑ and ϑ′ on the Tx and Rx sides, respectively. Figure 15 portrays the measured and predicted Sφ and Sφ′, and Table 3 presents measured and predicted azimuth and elevation spreads along with the average difference between measured and predicted values. The results demonstrate good agreement between predicted and measured channels also in angular domain. It is also noted that the azimuth spread increases slightly as the Tx-Rx distance increases, but the difference between measured and predicted azimuth spread does not have a clear distance dependency. The elevation spread does not vary over the distance since all Tx locations are on the same horizontal plane and the main part of the energy is propagating at elevation angles close to 0° regardless of the Tx position.
Table 3. Average Measured and Predicted Tx and Rx Azimuth and Elevation Spreads for All Tx Locations in the Small Office Room
Average of All Tx Locations
5.3 Optimization With a Single S Parameter
As an outcome of the optimization process in the small office room shown in Figure 6, we see that Sopt shows similar values for all locations in the small office room. It is therefore tempting to consider if the channel at all locations inside one scenario can be estimated with one single Sopt. Among the used Sopt parameter values during the optimization process described in section 4.1, Sopt=0.5 provides the best agreement considering all Tx locations. By using one single parameter, Sopt=0.5 for all locations in the small office room, the average ε in (9) increases by 0.2 dB. The average delay and angular characteristics are shown in Tables 4 and 5. It is discovered that the average performance improves, the mean delay error is only 0.14 ns (2.6%) and the RMS delay spread error is 0.46 ns. In the angular domain, the prediction error does not improve significantly.
Table 4. Average Measured and Predicted Mean Delay and RMS Delay Spreads for All Tx Locations in the Small Office Room With S=0.5
Average of All Tx Locations
Table 5. Average Measured and Predicted Tx and Rx Azimuth and Elevation Spreads for All Tx Locations in the Small Office Room With Sopt=0.5
Average of All Tx Locations
When considering the ultrasonic inspection room, Sopt=0.9 provides the minimum average ε when optimizing the prediction method with one single Sopt. This parameter choice increases the average ε by 0.2 dB. The average mean delay error increases to 0.55 ns (8.1%), but the average RMS delay spread error decreases to 0.15 ns (8.2%) as shown in Table 6.
Table 6. Average Measured and Predicted Mean Delay and RMS Delay Spreads for All Rx Locations in the Ultrasonic Inspection Room With Sopt=0.9
Average of All Rx Locations
The results clearly demonstrate that we can use the same S and αR parameter values in all locations in one scenario in order to describe the channel characteristics. This conclusion can be physically explained by the fact that the major scatterers such as surrounding walls and other scatterers contributing to multipaths are the same for all locations inside one room.
In this paper, a 60 GHz point cloud-based radio wave propagation prediction method, where backscattering is calculated with a single-lobe directive model, has been proposed. Two different indoor scenarios, a small office and an ultrasonic inspection room, were studied, and the model parameters were optimized by comparing measured and predicted PDPs and minimizing a cost function that describes the average error between PDPs. In the small office room, αR=11 was chosen as the optimum value, although changing αR did not have a strong impact on the average error. The optimized S showed similar values for different Tx locations, and the average Sopt was 0.5. Similar behavior was discovered also in the ultrasonic inspection room, where αR=1 was chosen and the average Sopt considering all Rx locations was 0.9. It was demonstrated that the same Sopt parameter can be used to predict the propagation in all locations inside one scenario, with Sopt=0.5 and Sopt=0.9 providing the minimum average ε in the small office room and the ultrasonic inspection room, respectively. Results show that the power deviation between measured and predicted PDPs is at its best under 2 dB and approximately 4 dB on average. Delay characteristics were also evaluated in both scenarios. In the small office room, the mean delay and the RMS delay spread could be estimated with an error of 0.14 ns (2.6%) and 0.46 ns (15.1%), respectively, by using Sopt=0.5 for all Tx locations. The same values calculated in the ultrasonic inspection room were 0.48 ns (7.2%) and 0.15 ns (8.2%). Angular characteristics were considered by comparing the measured and predicted azimuth spread at both Tx and Rx sides in the small office room. The average difference between measured and predicted azimuth spreads considering all Tx locations were 4.0° and 2.6° at the Tx and Rx sides, respectively. The difference between measured and predicted elevation spreads were 0.6° and 2.0° at the Tx and Rx sides. The results prove that the proposed propagation prediction method in question is a suitable candidate for estimating channel features in indoor scenarios in the 60 GHz band both in terms of delay and azimuth and elevation angles.
Future work involves parametrization of the prediction method for multipolarized channels, feasible modeling of shadowing and studying the influence of double-bounce components on the prediction accuracy for longer delay range. Also, validating the method in system simulation will be performed.
The authors would like to thank M. Kyrö and V.-M. Kolmonen in Aalto University School of Electrical Engineering, Finland, for assisting the radio propagation measurements, C. Gustafson in Lund University for providing measured antenna radiation patterns, and V. Degli-Esposti and E. M. Vitucci in University of Bologna, Italy, along with J. Takada in Tokyo Institute of Technology, Japan, for their useful comments in the subject of this paper.