The Impact of Antenna Height on 3D Channel: A Ray Launching Based Analysis

Abstract

Three-dimensional (3D) multi-input-multi-output (MIMO) is one of the enabling technologies for next-generation mobile communication. As the elevation angle in the 3D MIMO channel model might vary against the height of the base station (BS) antenna, it should be considered within channel modeling. In this paper, the impact of antenna height on the channel characteristics of the 3D MIMO channel is investigated by using the intelligent ray launching algorithm (IRLA). Three typical street scenarios, i.e., the straight street, the forked road, and the crossroad, are selected as benchmarks. The joint and marginal probability density functions (PDFs) of both the elevation angle of departure (EAoD) and the elevation angle of arrival (EAoA) are obtained through simulations. Moreover, the elevation angle spread (AS) and the elevation delay spread (DS) under various antenna heights are jointly discussed. Simulation results show that the characteristics of the PDFs of EAoD will vary under different street scenarios. It is observed that in order to obtain the maximum or minimum value of the AS and the DS, the BS antenna should be deployed at half of the building height.

1. Introduction

One common feature of the research works on future wireless communication technologies is the pursuit of high spectral efficiency whilst multiple mobile stations (MSs) access the network. Multi-user (MU), multi-input-multi-output (MIMO) technology is expected to play a key role to achieve high spectral efficiency [1]. To thoroughly investigate MIMO, it is essential to have a good understanding of radio-propagation characteristics of transmission path between the base station (BS) and the MS [2]. Several models are proposed to describe the radio channel [3,4,5]. In [3,4], the authors analyzed the channel characteristics by assuming the scatters are located uniformly within a circle or an ellipse. In [5], the authors argued that stochastic geometry and random graph theory are indispensable tools for the analysis of wireless networks. For the conventional study on channel model, linear antenna arrays were employed in the horizontal direction [6,7,8]. This assumption is valid since the distance between the MS and the BS is much larger than the distance between the height of the BS antenna and the scatterers around the MS [9].

However, considering 3D distributed MSs, a better coverage and capacity can be achieved by using a 3D antenna array [10]. Under this condition, the elevation angle of departure (EAoD) and elevation angle of arrival (EAoA), which significantly affect the performance of MU-MIMO, should be considered for channel modeling. The original EAoD and EAoA model was proposed in [11]. In 1990, a measurement in Tokyo based on 900 MHz observed that the distribution of elevation angle is extremely close to be Gaussian [12]. In 2002, the PDF of the EAoD for multipath at the BS was derived by assuming that the scattering objects were located uniformly within a circle around the MS [13]. Then, more research works started to appear over 10 years later. In [14], the spatial channel model (SCM) was extended to 3D by considering the elevation angle at the MS. In [15], a 3D scattering model was generalized for the macrocell environment with an MS located at the center of a 3D scattering semispheroid and a BS employing a directional antenna, which is located outside of the semispheroid. Recently, the scattering behavior was studied in [16]. Some other works focusing on the elevation angle in different propagation environments can be found in [17,18]. In [17], a new expression, which directly relates the PDF of the elevation angle, was proposed to the power spectral density of the received signal. In [18], the authors applied a novel technique for measuring the angular distribution in different environments. Although previous works have analyzed the distribution of the EAoD and the EAoA, the impact of antenna height on the EAoD and the EAoA has not been considered.

Ultra-dense small cells, with various antenna heights, are widely investigated in 5G networks. Due to small coverage areas, the impact of antenna height on parameters of wireless propagation, especially the EAoD and the EAoA, becomes significant. Previous work indicated that the heights of the antennas at both the BS and the MS impact \ the distribution of EAoD and EAoA significantly [19]. However, they did not consider how the antenna height would affect the distribution of elevation angle.

The main contributions of this paper are as follows. Firstly, the EAoD and the EAoA are characterized under typical scenarios against heights of antennas. Secondly, the PDFs of EAoD and EAoA are derived in closed-forms under each scenario. From the derived PDFs, we observe that the PDFs of EAoD and EAoA are almost linearly correlated with the antenna height under the straight street scenario. Meanwhile, the characteristics of PDFs change dramatically when the BS antenna height is half of the building’s height under the forked road and crossroad scenarios. Finally, the effects of antenna height on the AS and the DS under the three typical scenarios are also discussed. We conclude that the AS and the DS reach their maximum or minimum value when the antenna height is half of the building’s height, except the AS under the crossroad scenario.

This paper is organized as follows. In Section 2, we briefly introduce the IRLA and the experiment scenarios. In Section 3, we derive the PDFs of EAoD and EAoA under experiment scenarios through simulations. Meanwhile, the AS and DS are also analyzed. Finally, some conclusions are drawn in Section 4.

2. Methods and Numerical Results

2.1. Intelligent Ray Launching Algorithm (IRLA)

Radio wave propagation prediction modeling is critical in wireless network planning and optimization [20]. Types of propagation models have been presented in decades which can be classified as empirical models and deterministic models [21,22,23]. On the one hand, the channel characteristics in empirical models are determined by empirical factors, such as the carrier frequency and the distance between the BS and the MS. These models are time-efficient but with lower accuracy. On the other hand, deterministic models mainly consider the environmental information, such as the position and the reflection factor of the buildings. Thus, they are time consuming, but with higher accuracy [24]. The IRLA in our research can be categorized as a deterministic approach. The IRLA is used to compute all the possible reflections and diffractions of the emitting rays from the BS. Once the rays undergo reflections or diffractions, the additional loss will be added to the corresponding signal strength [25].

In [26], the authors originally presented the IRLA model, which relays on the cubic data obtained from discretization of the environment. The building walls, ground, and trees can be regarded as a cube element. Each cube is associated with a set of information. Then the authors proposed and implemented the parallel IRLA based on parallel object-oriented programming in C++ [27]. The IRLA contains three main components: light-of-sight (LOS), vertical diffraction (VD) and horizontal diffraction and reflection (HDR). The LOS component is used to calculate the path which is visible to emitter. The VD component is responsible for finding the roof-top diffractions rays. The HDR component is the most complicated part which launches a complete set of combination of diffraction and reflection rays. In [28], the authors implemented the angular dispersion of ray launching to improve its accuracy. Moreover, the model was extended to indoor scenarios in [29] and the combination with other methods was given in [30,31]. The correction of the IRLA model is justified in [29] and [32,33,34]. A measurement test in [29] showed the indoor IRLA model is promising with accurate results and time saving. Meanwhile, the accuracy of the model under an indoor-to-outdoor scenario and outdoor-to-indoor scenario were validated in measurements [32,33]. Recently, analysis and comparison between the simulation results and measurements showed that the ray-launching methods are efficient and accurate for massive MIMO channel modelling [34].

2.2. Numerical Results

The simulation scenario is part of the center of Paris, as shown in Figure 1. The size of the area is 500 m × 300 m, and the average building height is approximately 21 m [35]. We configure the transmit power of the BS as 0 dBm and the carrier frequency as 2.4 GHz (these two values can be set as any reasonable values at low frequency). For sake of convenience, we regard there are 4 × 1 omnidirectional antennas arranged in horizontal plane in the BS. The maximum reflection and diffraction numbers are 3 and 7, respectively. The parameters are listed in

Table 1. Parameter settings of the experimental scenarios.

Size of Area	Average Buildings Height	BS Power	Carrier Frequency	BS Antenna Type	Max Reflection Number	Max Diffraction Number
500 m × 300 m	21 m	0 dBm	2.4 GHz	Omnidirectional 4 × 1	3	7

.

Figure 1 illustrates that there are three typical street scenarios: a straight street (red rectangular area), a forked road (black rectangular area), and a crossroad (dark red rectangular area) [36]. These three typical scenarios are displayed in Figure 2 and the parameters for each scenario are listed in

Table 2. Parameters of the three scenarios.

	Average Buildings Height	Length of Wider Street	Width of Wider Street	Length of Lower Street	Width of Lower Street
Straight street	20 m	80 m	20 m	0 m	0 m
Forked road	25 m	80 m	20 m	40 m	12 m
Crossroad	22 m	40 m	20 m	40 m	12 m

. During simulations, h is the height of the BS, and n is the index of MSs from 1 to 100,000, which means 100,000 receivers are uniform distributed around the BS in total. We deployed the BS at (0, 0, h) and the receivers at (x_n, y_n, z_n) in the global coordinate system. We regard the angles above the antenna height as having a negative degree (<0°), and the angles below the antenna height as a positive degree (>0°). Without loss of accuracy, we have not considered the direct component.

3. Results

3.1. Scenario A: Straight Street in the Center of Paris

Under the straight street in Figure 2a, the PDFs of EAoD and EAoA are investigated firstly. Due to reflection and diffraction during the signal transmission, each receiver can obtain multiple rays with different path losses, EAoDs and EAoAs. The PDFs of EAoD and EAoA can then be calculated after adding the weighting value (path loss).

Two simulation results, antenna heights at 8 m and 12 m, are indicated above. Figure 3 indicates that the PDF values of both the EAoD and EAoA around 0 radians are obviously greater than the values away from 0 degrees, and the sub-curves at the negative and positive are asymmetrical, which have been verified in [18]. The author observed that a double-sided exponential function is more suitable for the PDF of elevation angles which can be represented as:

$$f_{EAOD}\left(x\right)=\{\begin{array}{c}{\lambda }_{1EAOD}{e}^{-{\lambda }_{1EAOD}x}, x<0\\ {\lambda }_{2EAOD}{e}^{-{\lambda }_{2EAOD}x}, x>0\end{array}$$

(1)

$$f_{EAOA}\left(x\right)=\{\begin{array}{c}{\lambda }_{1EAOA}{e}^{-{\lambda }_{1EAOA}x}, x<0\\ {\lambda }_{2EAOA}{e}^{-{\lambda }_{2EAOA}x}, x>0\end{array}$$

(2)

Here, the unit of x is degrees. λ_1EAoD, λ_2EAoD, λ_1EAoA, and λ_2EAoA are the rate parameters of these double-sided exponential functions. By logarithm fitting the PDFs of EAoD and EAoA (for example: λ^′_1EAoD = $-10\times {\log }_{10}e\times {\mathsf{\lambda }}_{1EAoD}$), we can finally obtain the values of λ^′_1EAoD, λ^′_2EAoD, λ^′_1EAoA, and λ^′_2EAoA. In Figure 3: λ^′_1EAoD = 26.83, λ^′_2EAoD = −36.48, λ^′_1EAoA = 16.43, λ^′_2EAoA = −15.69. Obviously, the λ^′ will be different with various antenna heights.

In Figure 4, the λ^′ with various antenna heights (4–18 m) are listed. This illustrates that the logarithm exponent value are almost linearly correlated with the antenna height. Then, we can derive that: λ^′_1EAoD = 1.64 × m + 14.35, λ^′_2EAoD = 1.60 × m − 48.83, λ^′_1EAoA = $-$0.63 × m + 22.84, λ^′_2EAoA = $-$0.76 × m − 9.55, where m is the antenna height which unit is meter. As we can see from Figure 4, the logarithm exponent values λ^′_1EAoD and λ^′_2EAoD (as shown in Equation (1)) linearly increase with the antenna height growing, and the exponent values λ^′_1EAoA and λ^′_2EAoA (as shown in Equation (2)) linearly decrease with the antenna height growing.

Finally, the PDFs of EAoD and EAoA related to antenna height can be represented as:

$$f_{EAOD}\left(x\right)=\{\begin{array}{c}-\left(0.38m+3.30\right)\exp \left(0.38m+3.30\right), x<0\\ -\left(0.37m-11.24\right)\exp \left(0.37m-11.24\right), x>0\end{array}$$

(3)

$$f_{EAOA}\left(x\right)=\{\begin{array}{c}-\left(-0.15m+5.26\right)\exp \left(-0.15m+5.26\right), x<0\\ -\left(-0.18m-2.20\right)\exp \left(-0.18m-2.20\right), x>0\end{array}$$

(4)

Using the PDFs, the AS and the DS, which represent the elevation angle dispersion and the time dispersion of multipath components respectively, can be evaluated [13].

$${\mathsf{\sigma }}_{AS}=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^{L-1}{P}_i{\theta }_i^2}{{{\displaystyle \sum }}_{i=1}^{L-1}{P}_i}-{\left(\frac{{{\displaystyle \sum }}_{i=1}^{L-1}{P}_i{\theta }_i}{{{\displaystyle \sum }}_{i=1}^{L-1}{P}_i}\right)}^2}$$

(5)

$${\mathsf{\sigma }}_{DS}=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^{L-1}{P}_i{\tau }_i^2}{{{\displaystyle \sum }}_{i=1}^{L-1}{P}_i}-{\left(\frac{{{\displaystyle \sum }}_{i=1}^{L-1}{P}_i{\tau }_i}{{{\displaystyle \sum }}_{i=1}^{L-1}{P}_i}\right)}^2}$$

(6)

where $P_i$ is the ray power, ${\theta }_i$ and ${\tau }_i$ are the angle and delay of $i$th ray. The statistic results of the AS and the DS are represented in Figure 5 and Figure 6, respectively.

Figure 5 indicates that the value of the AS of EAoD is smaller than that of EAoA, and the variation range of the AS of EAoD is also smaller. The AS reaches its maximum value when the antenna height (10 m) is half of the building’s height. Figure 6 exhibits that the value of the DS is nearly unchangable when the antenna height is around half of the buildings height and increases conspicuously when the antenna is near the bottom or the top of the buildings.

By using the linear least square regression, we can obtain the correlation of the AS and DS with antenna height:

$${\mathsf{\sigma }}_{AS,EAOD}=\{\begin{array}{c}0.1m+9.4, m\le 10\\ -0.11m+11.5, 10< m\le 20 \end{array}$$

(7)

$${\mathsf{\sigma }}_{AS,EAOA}=\{\begin{array}{c}0.383m+23.07, m\le 10\\ -0.4625m+31.525, 10< m\le 20 \end{array}$$

(8)

$${\mathsf{\sigma }}_{DS}=\{\begin{array}{c}-2.905m+53.74, m<8\\ 30.5, 8\le m\le 12 \\ 2.3667m+2.1, 12< m\le 20\end{array}$$

(9)

where m is the antenna height. ${\mathsf{\sigma }}_{AS,EAOD}$, ${\mathsf{\sigma }}_{AS,EAOA}$, and ${\mathsf{\sigma }}_{DS}$ are the AS of EAoD, EAoA, and the DS, respectively.

3.2. Scenario B: Forked Road in the Center of Paris

Then, by using the same research steps in Section 3.1, we investigate the PDFs of EAoD and EAoA under the forked road scenario, as shown in Figure 2b. The simulation results when the antenna is located at 8 m and 12 m are indicated in Figure 7.

After fitting the curves under different antenna heights (4 m, 6 m, 8 m, 10 m, 12 m, 14 m, 16 m and 18 m), the PDFs of EAoD and EAoA can be obtained as indicated in

Table 3. The statistic results of logarithm exponent values under fork road scenario with various antenna heights (4 m, 6 m, 8 m, 10 m, 12 m, 14 m, 16 m, and 18 m). Here, EAoD down, EAoD up, EAoA down, and EAoA up represent λ^′_1EAoD, λ^′_2EAoD, λ^′_1EAoA, and λ^′_2EAoA, respectively.

	4 m	6 m	8 m	10 m	12 m	14 m	16 m	18 m
EAoD down	40.19	32.86	32.31	33.16	33.16	33.25	34.58	36.2
EAoD up	−32.49	−35.89	−34.23	−33.67	−32.63	−32.99	−32.93	−32.94
EAoA down	17.4	20.92	19.11	18	17.69	17.94	17.95	18.03
EAoA up	−18.06	−16.2	−15.77	−16	−17.06	−16.64	−16.45	−18.68

.

From Table 3, two conclusions can be reached:

• For the PDF of EAoD: the logarithm exponent value λ^′_1EAoD is nearly unchangeable when the antenna height is lower than half of the building’s height, but has a dramatic increase when the antenna height grows. However, the logarithm exponent value λ^′_2EAoD enhances when the antenna height is lower than half of the building’s height and then remains stable.
• For the PDF of EAoA: the logarithm exponent value λ^′_1EAoA decreases rapidly when the antenna height is lower than half of the building’s height and then varies slightly. However, the logarithm exponent value λ^′_2EAoA is almost stable when the antenna height is lower than half of the building’s height, but has a significant drop when the antenna height increases.

Finally, the PDFs of EAoD and EAoA related to antenna height can be calculated:

$$f_{EAOD}\left(x\right)=\{\begin{array}{c}-\left(7.64\right)\exp \left(7.64\right), x<0, m\le 12.5 \\ -\left(0.14m+5.89\right)\exp \left(0.14m+5.89\right), x<0, 12.5<m<25\\ -\left(0.07m-8.43\right)\exp \left(0.07m-8.43\right), x>0, m\le 12.5 \\ -\left(-7.58\right)\exp \left(-7.58\right), x>0, 12.5<m<25\end{array}$$

(10)

$$f_{EAOA}\left(x\right)=\{\begin{array}{c}-\left(-0.06m+4.87\right)\exp \left(-0.06m+4.87\right), x<0, m\le 12.5 \\ -\left(4.13\right)\exp \left(4.13\right), x<0, 12.5<m<25\\ -\left(-3.80\right)\exp \left(-3.80\right), x>0, m\le 12.5 \\ -\left(-0.06m-3.01\right)\exp \left(-0.06m-3.01\right), x>0, 12.5<m<25\end{array}$$

(11)

With the results obtained above, the AS and the DS of EAoD and EAoA can be gained as displayed in Figure 8 and Figure 9.

It is obvious from Figure 8 that the ASs of EAoD and EAoA arrive at the peak value when the antenna height (12 m) is half of the building’s height. In Figure 9, we can conclude that the minimum value of the DS can be achieved when the antenna height is half of the building’s height and the value arises when the antenna is near the ground or the top of the buildings.

Finally, the AS and the DS related to antenna height can be represented as:

$${\mathsf{\sigma }}_{AS,EAOD}=\{\begin{array}{c}0.0824m+9.2706, m\le 12.5\\ -0.1273m+11.8909, 12.5< m\le 25\end{array}$$

(12)

$${\mathsf{\sigma }}_{AS,EAOA}=\{\begin{array}{c}0.2412m+22.6853, m\le 12.5\\ -0.2m+28.2, 12.5< m\le 25 \end{array}$$

(13)

$${\mathsf{\sigma }}_{DS}=\{\begin{array}{c}-0.5647m+32.5588, m\le 12.5\\ 0.4364m+20.0455, 12.5< m\le 25\end{array}$$

(14)

3.3. Scenario C: Crossroad in the Center of Paris

Finally, we research the PDFs of EAoD and EAoA under cross road scenario as shown in Figure 2c. The simulation results are obtained when the antenna located at 8 m and 12 m as shown in Figure 10.

The logarithm exponent values of EAoD and EAoA can be obtained by fitting the curves through different antenna heights (4 m, 6 m, 8 m, 10 m, 12 m, 14 m, 16 m, and 18 m) which are exhibited in

Table 4. The statistic results of logarithm exponent values under fork road scenario with various antenna heights (4 m, 6 m, 8 m, 10 m, 12 m, 14 m, 16 m, and 18 m). Here, EAoD down, EAoD up, EAoA down, and EAoA up represent λ^′_1EAoD, λ^′_2EAoD, λ^′_1EAoA, and λ^′_2EAoA, respectively.

	4 m	6 m	8 m	10 m	12 m	14 m	16 m	18 m
EAoD down	199	149.8	118.3	110.5	88.88	79.08	85.5	79.05
EAoD up	−79.57	−87.87	−76.54	−85.21	−103	−113.1	−131.2	−172.7
EAoA down	19.52	17.72	14.08	12.08	12.87	10.91	9.77	11.1
EAoA up	−9.556	−12.86	−10.7	−11.25	−13.12	−13.46	−15.32	−19.56

.

From Table 4, we can conclude that:

• For the PDF of EAoD: the logarithm exponent value λ^′_1EAoD reduces rapidly when the antenna height is lower than half of the building’s height, and gains the minimum value at 14 m, and then the value remains the same. In contrast, the logarithm exponent value λ^′_2EAoD is nearly unchangeable when the antenna height is lower than half of the building’s height, but has a dramatic decrease when the antenna height grows.
• For the PDF of EAoA: the exponent value decreases λ^′_1EAoA as the antenna height rises. In contrast, the exponent value λ^′_2EAoA is nearly unchanged when the antenna height is lower than half of the building’s height, but has a significantly drop when the antenna height increases.

Then, the PDFs of EAoD and EAoA can be represented as:

$$f_{EAOD}\left(x\right)=\{\begin{array}{c}-\left(-2.95m+52.22\right)\exp \left(-2.95m+52.22\right), x<0, m\le 11 \\ -\left(19.80\right)\exp \left(19.80\right), x<0, 11<m<22\\ -\left(-20.72\right)\exp \left(-20.72\right), x>0, m\le 11 \\ -\left(-1.90m+0.15\right)\exp \left(-1.90m+0.15\right), x>0, 11<m<22\end{array}$$

(15)

$$f_{EAOA}\left(x\right)=\{\begin{array}{c}-\left(-0.17m+5.15\right)\exp \left(5.15\right), x<0, 0<m<22\\ -\left(-2.76\right)\exp \left(-2.76\right), x>0, m<11 \\ -\left(-0.15m-1.08\right)\exp \left(-0.15m-1.08\right), x>0, 11<m<22\end{array}.$$

(16)

The AS and the DS of EAoD and EAoA can be gained by the above results.

Figure 11a shows that the AS of EAoD remains almost stable with different antenna heights. Figure 11b illustrates the AS of EAoA gains its maximum value when the antenna height is half of the building’s height (11 m) and drops gradually to the ground or the top of the buildings. The DS shows a similar phenomenon as that described in Section 3.1 and Section 3.2 and as shown in Figure 12.

Finally, the AS and the DS related to antenna height can be represented as:

$${\mathsf{\sigma }}_{AS,EAOD}=4.05$$

(17)

$${\mathsf{\sigma }}_{AS,EAOA}=\{\begin{array}{c}0.9857m+17.7571, m\le 11\\ -0.9m+23, 11< m\le 22 \end{array}$$

(18)

$${\mathsf{\sigma }}_{DS}=\{\begin{array}{c}-0.3571m+18.4286, m\le 12.5\\ 0.3m+11.2, 11< m\le 22\end{array}$$

(19)

3.4. Discussion

Based on the simulation results in Section 3.1, Section 3.2 and Section 3.3, some insights and observations regarding the 3D wireless channels are given as follows. On the one hand, to obtain the maximum and minimum values of the AS and the DS, BS antennas need to be located at half of the building’s height under three typical street scenarios. On the other hand, the PDFs of EAoD and EAoA are approximated as double-sided exponential functions. Meanwhile, the parameters of the double-sided exponential function are linearly correlated with the antenna height under the straight street scenario, but change dramatically when the BS antenna height is half of the buildings height under forked road and crossroad scenarios.

Furthermore, from Figure 3, Figure 7, and Figure 10 we can find that most numbers of EAoD and EAoA concentrate at 0 radians, even though we ignore the direct component. Generally, the path distance of a single reflection angle is smaller than the multi-path which makes the weighting value higher. Thus, we can conclude that the single reflection path plays a leading role for researching the distributions of EAoD and EAoA.

4. Conclusions

In this paper, the impact of antenna height on EAoD and EAoA is investigated under typical outdoor scenarios: straight street, fork road, and crossroad. At the beginning, we briefly introduced the IRLA and the simulation scenario (the center of Paris). Then, the closed-form expressions of the PDF of EAoD and EAoA were derived under various street scenarios, the related AS and DS are also presented. Finally, the discussion of the results was given. According to the results, we conclude that the PDFs of EAoD and EAoA are modelled as exponential functions whose parameters are observed as a linear function of the antenna height under the straight street scenario. Meanwhile, the characteristics of EAoD and EAoA change dramatically when the BS antenna height is half of the building’s height in the forked road and crossroad scenarios. Moreover, the results of the AS and the DS show that in order to obtain the maximum or minimum value of the AS and the DS, the BS antenna should be deployed at half of the building’s height. We also conclude that the single reflection path dominates the distribution of EAoD and EAoA. In our future works, the PDFs of EAoD and EAoA with different antenna patterns and tilt will be considered under typical outdoor scenarios.