# Polarized Uniform Linear Array System: Beam Radiation Pattern, Beamforming Diversity Order, and Channel Capacity

Authors: Xin Su; KyungHi Chang
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

There have been many studies regarding antenna polarization; however, there have been few publications on the analysis of the channel capacity for polarized antenna systems using the beamforming technique. According to Chung et al., the channel capacity is determined by the density of scatterers and the transmission power, which is obtained based on the assumption that scatterers are uniformly distributed on a 3D spherical scattering model. However, it contradicts the practical scenario, where scatterers may not be uniformly distributed under outdoor environment, and lacks the consideration of fading channel gain. In this study, we derive the channel capacity of polarized uniform linear array (PULA) systems using the beamforming technique in a practical scattering environment. The results show that, for PULA systems, the channel capacity, which is boosted by beamforming diversity, can be determined using the channel gain, beam radiation pattern, and beamforming diversity order (BDO), where the BDO is dependent on the antenna characteristics and array configurations.


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## Body

## 1. Introduction

Recently, there has been a gradual increase in the demand for polarized antenna systems, especially for 5G networks. This is mainly because antenna polarization is an important parameter employed for the design of space-limited wireless devices [1, 2]. Several techniques such as space-time diversity, multiplexing, and array processing can be exploited in order to boost the system throughput for polarized antennas. Thus, many published research articles as well as ongoing projects have focused on antenna polarization [3–5]. However, there have been few reports on the analysis of the channel capacity for polarized antenna systems using the beamforming technique. In [6], the channel capacity is known to be determined by the density of scatterers and the transmission power instead of other system parameters. This is based on the compact uniform linear array (ULA) and assumes that scatterers are uniformly distributed on a 3D spherical scattering model. However, in practical scenarios, the scatterers may not be uniformly distributed, especially in outdoor environments. In addition, [6] does not consider the fading channel, which is important when obtaining the channel capacity in [7, 8]. In this study, we obtain the channel capacity of polarized ULA (PULA) systems using the beamforming technique in a practical scattering environment. Our results indicate that, for PULA systems, the channel capacity, which is boosted by beamforming diversity, is determined mainly by the channel gain, beam radiation pattern, and beamforming diversity order (BDO).In this paper, the beam radiation pattern (including the beamwidth, the number of lobes, and the radiation gain of the beam) of conventional ULA systems is described based on the antenna characteristics and array configurations (i.e., dipole antennas with different lengths, the number of array elements, and the array element spacing). And we characterize the beam radiation pattern for PULA systems in both 2D and 3D forms, where the dipole antenna-based PULA is shown to have a higher beam radiation gain compared with isotropic antenna-based systems. The BDO, which is defined with respect to the number of beam lobes generated by the transceiver, is obtained under different transmission scenarios by considering the line-of-sight (LOS) component and scattering environments. In addition, it is dependent on the antenna characteristics and array configurations and has been shown to be linearly proportional to the number of array elements and array element spacing.The remainder of this paper is organized as follows. In Section2, we analyze the conventional ULA, while the beam radiation patterns of the PULA in both 2D and 3D forms are presented in Section 3. In Section 4, we introduce the BDO concept, and in Section 5 we discuss the channel capacity of PULA systems. Finally, we conclude the paper in Section 6.


## 2. Feature Analysis of Conventional ULA

### 2.1. Array Factor

Figure1 illustrates a conventional ULA, where θ0 denotes the signal incidence angle shifted from the boresight direction (the antenna boresight is the axis vertical to the orientation of the array alignment [9]), d represents the array element spacing, and N is the number of array elements. The array factor F(θ), which represents the far-field radiation of an isotropic ULA system [10], can be expressed in terms of the inner product of the weighting vector and array propagation vector as follows:(1)Fθ=w·v.The weighting vector is given as(2a)w=w1w2⋯wN,and the array propagation vector is given as(2b)v=1,ej2πd/λsinθ,…,ejN-12πd/λsinθT,where λ denotes the signal wavelength. If we use the weighting vector to adjust a beam with a specific angle (shifted from the boresight direction) of θ0 and use the array spacing factor ε to describe the array configuration, that is, d=λ/ε, the array factor can be further expressed as(3)Fθ=∑n=1Nejn-12π/εsinθ-sinθ0.Based on the derivation in Appendix A, the normalized F(θ) is obtained as(4)Fθ=sinπN/εsinθ-sinθ0N·sinπ/εsinθ-sinθ0,θ≠θ01,θ=θ0.Because -2≤sinθ-sinθ0≤2 and ε≥2 are often used to avoid the array aliasing problem [11], F(θ) then approximates to the sinc function as(5)Fθ≈sincπNεsinθ-sinθ0,θ≠θ01,θ=θ0.Figure 1
Configuration of the conventional ULA.


### 2.2. Off-Boresight Angle

As discussed in [9], the beamwidth is increased significantly when the beam steers to an angle far off the boresight direction, such as the case when θ0 reaches 90°. Note that the beamwidth generally refers to the width of the main-lobe of a beam. To illustrate this, we obtain the half power beamwidth (HPBW) by setting F(θ)=1/2. In addition, from sinc⁡1.391≈1/2, the HPBW can be given as(6)HPBW≈2arcsin⁡1.391επN+sinθ0.This illustrates that the beamwidth is inversely proportional to the number of array elements N and array element spacing d but is directly proportional to the off-boresight angle θ0. Figure 2 shows HPBW examples of isotropic ULA systems that were obtained by varying θ0 for three different angles, that is, θ0= 0°, 45°, and 90°. The figure shows that the HPBW extends about four times when θ0 changes from 0° to 90°. Note that, in this illustration, we set N=8 and ε=2 and the boresight direction along the horizontal axis. To focus the beamwidth when there is a large off-boresight angle effect, [9] proposes asensor delay line-based array system, which can effectively prevent the larger off-boresight angle effect by increasing the dimensions of the array system.Figure 2
Illustration of the off-boresight angle effect.


### 2.3. Number of Lobes for a Beam

We previously discussed the beamwidth, which is affected by the number of array elements, the array element spacing, and the off-boresight angle. In this section, we analyze the relationship between these factors and the number of lobes present in a beam. In the case where relays from different directions are used to contribute space diversity in an effort to boost the capacity of a ULA system [12], all of the lobes of a beam need to be considered as the BDO, which is eventually used to derive the channel capacity.According to (4), there are zero values at sinθ=±ε·n/N+sinθ0, where n=0,…,N-1 and θ∈0,π. Therefore, the lobe-widths of the ULA beam are given by(7)Wn=θ:±εnN+sin⁡θ0≤sin⁡θ<±εn+1N+sin⁡θ0.Because each lobe has two zero points and there are total 2Nε-1 zero points at θ∈0,π, we can find 2Nε-1-1 lobes. Due to the symmetrical beam pattern of ULAs, the overall number of lobes is given as 4Nε-1-2, which is unrelated to the off-boresight angle θ0. Figure 3 depicts the number of lobes when N is incremented with ε=2, where the number of lobes can be easily observed as 6, 14, 30, and 62 when N equals 4, 8, 16, and 32, respectively. Based on the above analysis, we can make the following proposition.Figure 3
The number of beam lobes varies with differentN values.Proposition 1.
The beamwidth and number of lobes of a beam are relevant to the array configuration. The beamwidth is inversely proportional to the number of array elements and the array element spacing but is directly proportional to the off-boresight angle. Meanwhile, the number of lobes of a beam is proportional to the number of array elements and the array element spacing but has no relationship with the off-boresight angle.


## 2.1. Array Factor

Figure1 illustrates a conventional ULA, where θ0 denotes the signal incidence angle shifted from the boresight direction (the antenna boresight is the axis vertical to the orientation of the array alignment [9]), d represents the array element spacing, and N is the number of array elements. The array factor F(θ), which represents the far-field radiation of an isotropic ULA system [10], can be expressed in terms of the inner product of the weighting vector and array propagation vector as follows:(1)Fθ=w·v.The weighting vector is given as(2a)w=w1w2⋯wN,and the array propagation vector is given as(2b)v=1,ej2πd/λsinθ,…,ejN-12πd/λsinθT,where λ denotes the signal wavelength. If we use the weighting vector to adjust a beam with a specific angle (shifted from the boresight direction) of θ0 and use the array spacing factor ε to describe the array configuration, that is, d=λ/ε, the array factor can be further expressed as(3)Fθ=∑n=1Nejn-12π/εsinθ-sinθ0.Based on the derivation in Appendix A, the normalized F(θ) is obtained as(4)Fθ=sinπN/εsinθ-sinθ0N·sinπ/εsinθ-sinθ0,θ≠θ01,θ=θ0.Because -2≤sinθ-sinθ0≤2 and ε≥2 are often used to avoid the array aliasing problem [11], F(θ) then approximates to the sinc function as(5)Fθ≈sincπNεsinθ-sinθ0,θ≠θ01,θ=θ0.Figure 1
Configuration of the conventional ULA.


## 2.2. Off-Boresight Angle

As discussed in [9], the beamwidth is increased significantly when the beam steers to an angle far off the boresight direction, such as the case when θ0 reaches 90°. Note that the beamwidth generally refers to the width of the main-lobe of a beam. To illustrate this, we obtain the half power beamwidth (HPBW) by setting F(θ)=1/2. In addition, from sinc⁡1.391≈1/2, the HPBW can be given as(6)HPBW≈2arcsin⁡1.391επN+sinθ0.This illustrates that the beamwidth is inversely proportional to the number of array elements N and array element spacing d but is directly proportional to the off-boresight angle θ0. Figure 2 shows HPBW examples of isotropic ULA systems that were obtained by varying θ0 for three different angles, that is, θ0= 0°, 45°, and 90°. The figure shows that the HPBW extends about four times when θ0 changes from 0° to 90°. Note that, in this illustration, we set N=8 and ε=2 and the boresight direction along the horizontal axis. To focus the beamwidth when there is a large off-boresight angle effect, [9] proposes asensor delay line-based array system, which can effectively prevent the larger off-boresight angle effect by increasing the dimensions of the array system.Figure 2
Illustration of the off-boresight angle effect.


## 2.3. Number of Lobes for a Beam

We previously discussed the beamwidth, which is affected by the number of array elements, the array element spacing, and the off-boresight angle. In this section, we analyze the relationship between these factors and the number of lobes present in a beam. In the case where relays from different directions are used to contribute space diversity in an effort to boost the capacity of a ULA system [12], all of the lobes of a beam need to be considered as the BDO, which is eventually used to derive the channel capacity.According to (4), there are zero values at sinθ=±ε·n/N+sinθ0, where n=0,…,N-1 and θ∈0,π. Therefore, the lobe-widths of the ULA beam are given by(7)Wn=θ:±εnN+sin⁡θ0≤sin⁡θ<±εn+1N+sin⁡θ0.Because each lobe has two zero points and there are total 2Nε-1 zero points at θ∈0,π, we can find 2Nε-1-1 lobes. Due to the symmetrical beam pattern of ULAs, the overall number of lobes is given as 4Nε-1-2, which is unrelated to the off-boresight angle θ0. Figure 3 depicts the number of lobes when N is incremented with ε=2, where the number of lobes can be easily observed as 6, 14, 30, and 62 when N equals 4, 8, 16, and 32, respectively. Based on the above analysis, we can make the following proposition.Figure 3
The number of beam lobes varies with differentN values.Proposition 1.
The beamwidth and number of lobes of a beam are relevant to the array configuration. The beamwidth is inversely proportional to the number of array elements and the array element spacing but is directly proportional to the off-boresight angle. Meanwhile, the number of lobes of a beam is proportional to the number of array elements and the array element spacing but has no relationship with the off-boresight angle.


## 3. Beam Radiation Pattern of PULA

In this section, we discuss the beam pattern associated with the antenna characteristics, such as the use of dipole antennas, where the beamwidth and beam lobe gain are different from the case that uses the isotropic ULA.


### 3.1. Cross-Array Beam Generation by PULA Configuration

The dipole antenna or doublet is the most widely used class of antenna, and the radiation pattern analysis throughout this paper is based mainly on the dipole antenna. Figure4 demonstrates a dipole antenna with a length of L. aθ,ϕ,γ,κ denotes the far-field radiation pattern of the dipole, where γ is the angle of the dipole shifted from the z-axis and κ is the factor used to determine the dipole length; that is, L=κλ. The half-wavelength dipole is defined when κ=1/2. Figure 5 shows our proposed orthogonal triple-polarized dipole ULA, where an array and branch (A&B) multiple antenna configuration is depicted. In other words, the PULA system is composed of several paralleled array elements, and there are three orthogonal colocated branches/dipoles at each array element. Three branch sets of AnB1, AnB2, and AnB3 are implemented along the z-axis, y-axis, and x-axis, respectively.Figure 4
Configuration of the dipole antenna.Figure 5
A
&
B multiple antenna configuration of the triple-polarized dipole ULA.Because three colocated branches at each array element are fixed with zero spacing, as shown in Figure5, which makes the beamwidth scale up to 360° according to Proposition 1, the beams need to be generated by using corresponding cross-array branches rather than the colocated branches at each array element. Therefore, we can obtain three orthogonal beams generated by a PULA as follows:(i)
The beam varied in thex-y plane is generated by the set of branches of AnB1, where n=1,2,…,N.(ii)
The beam varied in thex-z plane is generated by the set of branches of AnB2.(iii)
The beam varied in they-z plane is generated by the set of branches of AnB3.


### 3.2. PULA Beam Radiation Pattern in 2D Form

First, the PULA radiation pattern is analyzed in 2D form, where only two orthogonal colocated branches (vertical and horizontal) are implemented at each array element. By fixing the vertical branch along with the vertical axis whenγ=0, we have the following: the vertical dipole radiation pattern avθ,κ in 2D form whereϕ=0 is presented as(8)avθ,κ=cos⁡⁡πκsin⁡θ-cos⁡⁡πκcos⁡⁡θ.Figure 6 depicts the dipole radiation pattern of the vertical branch with κ different values. We observe that the dipole does not radiate in the longitudinal direction of the antenna structure and consequently has a higher gain compared with that of an isotropic antenna. In addition, the radiation pattern has a higher gain and narrower width as κ increased, where the case of κ=1 achieves the lowest angle of radiation and the highest horizontal range. Note that when κ>1, the radiation pattern of the dipole does not follow the current shape in Figure 6 [13].Figure 6
Radiation patterns of the vertical dipole in 2D form.The radiation pattern of the colocated dipole can be easily obtained based on (8) as(9)ahθ,κ=cos⁡⁡πκsinθ-γ-cos⁡⁡πκcos⁡⁡θ-γ,where it is shifted by γ degrees and the horizontal dipole pattern is obtained as γ= 90°. The PULA 2D beam radiation pattern is expressed by multiplying the array factor and the radiation pattern of a single polarized antenna. Let Pv(θ,κ,θ0,v) and Ph(θ,κ,θ0,h) denote the beam radiation patterns generated by the sets of vertical dipoles and horizontal dipoles, respectively. We have(10)Pvθ,κ,θ0,v=avθ,κsin⁡πN/εsin⁡θ-sin⁡θ0,vN·sin⁡π/εsin⁡θ-sin⁡θ0,v,Phθ,κ,θ0,h=ahθ,κsin⁡πN/εcos⁡⁡θ-sin⁡θ0,hN·sin⁡π/εcos⁡⁡θ-sin⁡θ0,h,where θ0,v and θ0,h are the off-boresight angles corresponding to the sets of vertical branches and horizontal branches, respectively.Figure7 is drawn when N=4 and ε=2. We observe that the beam radiation patterns generated by the sets of vertical dipoles and horizontal dipoles have a 90° rotation. The beamforming gain can be extended by increasing the dipole length. However, the trade-off regarding the antenna size needs to be considered in the antenna design. The 2D beam radiation pattern for an isotropic ULA is also given in Figure 7 for comparison purposes, where it has a larger sidelobe and smaller main-lobe pattern compared with the dipole-based array system [14]. This is mainly because the dipole does not radiate in the longitudinal direction of the antenna structure and maintains a higher radiation gain compared with that of the isotropic antenna.Figure 7
Beam radiation patterns of the PULA in 2D form.


### 3.3. PULA Beam Radiation Pattern in 3D Form

In [3], the 3D dipole radiation pattern is found to have a donut shape. In addition, the 3D radiation pattern of the dual-polarized half-wavelength dipole was investigated in [4, 15] with different inclination angles of γ, for example, 0°, 45°, and 90°. In this paper, we further obtain the 3D radiation pattern for a triple-polarized system, where three colocated dipoles (B1, B2, and B3) are deployed along the z-axis, y-axis, and x-axis, respectively. The formulae for the radiation pattern derived based on (31) in [4] for these three colocated dipoles are given as(11)aB1θ,ϕ=cos⁡⁡θcos⁡⁡ϕcos⁡⁡π/2sin⁡θcos⁡⁡ϕ1-sin2⁡θcos⁡2⁡ϕ,aB2θ,ϕ=sin⁡θcos⁡⁡ϕcos⁡⁡π/2sin⁡θcos⁡⁡ϕ1-sin2⁡θcos⁡2⁡ϕ,aB3θ,ϕ=sin⁡ϕcos⁡⁡π/2sin⁡θcos⁡⁡ϕ1-sin2⁡θcos⁡2⁡ϕ.Based on (11) the radiation patterns of triple-polarized half-wavelength dipoles are plotted in Figure 8, where the three patterns affect each other and the donut radiation shape of the dipole antenna is reshaped due to the presence of colocated dipoles.Figure 8
Radiation patterns of the triple-polarized half-wavelength dipole in 3D form.Similar to the 2D case, the PULA 3D beam radiation pattern was also obtained by multiplying the array factor and radiation pattern of a single polarized antenna. LetPB1(θ,ϕ,θ0,1), PB2(θ,ϕ,θ0,2), and PB3(θ,ϕ,θ0,3) represent the beam radiation patterns generated by the three dipole sets AnB1, AnB2, and AnB3, respectively. We have(12)PB1θ,ϕ,θ0,1=∑n=1NaB1·ejn-12π/εsin⁡θ-sin⁡θ0,1=aB1sinπN/εsinθ-sinθ0,1N·sin⁡π/εsin⁡θ-sin⁡θ0,1,PB2θ,ϕ,θ0,2=∑n=1NaB2·ejn-12π/εsin⁡ϕ-sin⁡θ0,2=aB2sinπN/εsinϕ-sinθ0,2N·sin⁡π/εsin⁡ϕ-sin⁡θ0,2,PB3θ,ϕ,θ0,3=∑n=1NaB3·ejn-12π/εsin⁡π/2-ϕ-sin⁡θ0,3=aB3sinπN/εcos⁡ϕ-sinθ0,3N·sin⁡π/εcos⁡⁡ϕ-sin⁡θ0,3,where θ0,1, θ0,2, and θ0,3 are the off-boresight angles corresponding to AnB1, AnB2, and AnB3, respectively.By settingN=4 and ε=2, we plot the beam radiation pattern of a PULA in 3D form, as illustrated by Figure 9. We can see that the main-lobe petal of each beam has a shell shape, which is mainly formed by the interaction of the electromagnetic fields of colocated beams.Figure 9
Beam radiation patterns of PULA in 3D form.


## 3.1. Cross-Array Beam Generation by PULA Configuration

The dipole antenna or doublet is the most widely used class of antenna, and the radiation pattern analysis throughout this paper is based mainly on the dipole antenna. Figure4 demonstrates a dipole antenna with a length of L. aθ,ϕ,γ,κ denotes the far-field radiation pattern of the dipole, where γ is the angle of the dipole shifted from the z-axis and κ is the factor used to determine the dipole length; that is, L=κλ. The half-wavelength dipole is defined when κ=1/2. Figure 5 shows our proposed orthogonal triple-polarized dipole ULA, where an array and branch (A&B) multiple antenna configuration is depicted. In other words, the PULA system is composed of several paralleled array elements, and there are three orthogonal colocated branches/dipoles at each array element. Three branch sets of AnB1, AnB2, and AnB3 are implemented along the z-axis, y-axis, and x-axis, respectively.Figure 4
Configuration of the dipole antenna.Figure 5
A
&
B multiple antenna configuration of the triple-polarized dipole ULA.Because three colocated branches at each array element are fixed with zero spacing, as shown in Figure5, which makes the beamwidth scale up to 360° according to Proposition 1, the beams need to be generated by using corresponding cross-array branches rather than the colocated branches at each array element. Therefore, we can obtain three orthogonal beams generated by a PULA as follows:(i)
The beam varied in thex-y plane is generated by the set of branches of AnB1, where n=1,2,…,N.(ii)
The beam varied in thex-z plane is generated by the set of branches of AnB2.(iii)
The beam varied in they-z plane is generated by the set of branches of AnB3.


## 3.2. PULA Beam Radiation Pattern in 2D Form

First, the PULA radiation pattern is analyzed in 2D form, where only two orthogonal colocated branches (vertical and horizontal) are implemented at each array element. By fixing the vertical branch along with the vertical axis whenγ=0, we have the following: the vertical dipole radiation pattern avθ,κ in 2D form whereϕ=0 is presented as(8)avθ,κ=cos⁡⁡πκsin⁡θ-cos⁡⁡πκcos⁡⁡θ.Figure 6 depicts the dipole radiation pattern of the vertical branch with κ different values. We observe that the dipole does not radiate in the longitudinal direction of the antenna structure and consequently has a higher gain compared with that of an isotropic antenna. In addition, the radiation pattern has a higher gain and narrower width as κ increased, where the case of κ=1 achieves the lowest angle of radiation and the highest horizontal range. Note that when κ>1, the radiation pattern of the dipole does not follow the current shape in Figure 6 [13].Figure 6
Radiation patterns of the vertical dipole in 2D form.The radiation pattern of the colocated dipole can be easily obtained based on (8) as(9)ahθ,κ=cos⁡⁡πκsinθ-γ-cos⁡⁡πκcos⁡⁡θ-γ,where it is shifted by γ degrees and the horizontal dipole pattern is obtained as γ= 90°. The PULA 2D beam radiation pattern is expressed by multiplying the array factor and the radiation pattern of a single polarized antenna. Let Pv(θ,κ,θ0,v) and Ph(θ,κ,θ0,h) denote the beam radiation patterns generated by the sets of vertical dipoles and horizontal dipoles, respectively. We have(10)Pvθ,κ,θ0,v=avθ,κsin⁡πN/εsin⁡θ-sin⁡θ0,vN·sin⁡π/εsin⁡θ-sin⁡θ0,v,Phθ,κ,θ0,h=ahθ,κsin⁡πN/εcos⁡⁡θ-sin⁡θ0,hN·sin⁡π/εcos⁡⁡θ-sin⁡θ0,h,where θ0,v and θ0,h are the off-boresight angles corresponding to the sets of vertical branches and horizontal branches, respectively.Figure7 is drawn when N=4 and ε=2. We observe that the beam radiation patterns generated by the sets of vertical dipoles and horizontal dipoles have a 90° rotation. The beamforming gain can be extended by increasing the dipole length. However, the trade-off regarding the antenna size needs to be considered in the antenna design. The 2D beam radiation pattern for an isotropic ULA is also given in Figure 7 for comparison purposes, where it has a larger sidelobe and smaller main-lobe pattern compared with the dipole-based array system [14]. This is mainly because the dipole does not radiate in the longitudinal direction of the antenna structure and maintains a higher radiation gain compared with that of the isotropic antenna.Figure 7
Beam radiation patterns of the PULA in 2D form.


## 3.3. PULA Beam Radiation Pattern in 3D Form

In [3], the 3D dipole radiation pattern is found to have a donut shape. In addition, the 3D radiation pattern of the dual-polarized half-wavelength dipole was investigated in [4, 15] with different inclination angles of γ, for example, 0°, 45°, and 90°. In this paper, we further obtain the 3D radiation pattern for a triple-polarized system, where three colocated dipoles (B1, B2, and B3) are deployed along the z-axis, y-axis, and x-axis, respectively. The formulae for the radiation pattern derived based on (31) in [4] for these three colocated dipoles are given as(11)aB1θ,ϕ=cos⁡⁡θcos⁡⁡ϕcos⁡⁡π/2sin⁡θcos⁡⁡ϕ1-sin2⁡θcos⁡2⁡ϕ,aB2θ,ϕ=sin⁡θcos⁡⁡ϕcos⁡⁡π/2sin⁡θcos⁡⁡ϕ1-sin2⁡θcos⁡2⁡ϕ,aB3θ,ϕ=sin⁡ϕcos⁡⁡π/2sin⁡θcos⁡⁡ϕ1-sin2⁡θcos⁡2⁡ϕ.Based on (11) the radiation patterns of triple-polarized half-wavelength dipoles are plotted in Figure 8, where the three patterns affect each other and the donut radiation shape of the dipole antenna is reshaped due to the presence of colocated dipoles.Figure 8
Radiation patterns of the triple-polarized half-wavelength dipole in 3D form.Similar to the 2D case, the PULA 3D beam radiation pattern was also obtained by multiplying the array factor and radiation pattern of a single polarized antenna. LetPB1(θ,ϕ,θ0,1), PB2(θ,ϕ,θ0,2), and PB3(θ,ϕ,θ0,3) represent the beam radiation patterns generated by the three dipole sets AnB1, AnB2, and AnB3, respectively. We have(12)PB1θ,ϕ,θ0,1=∑n=1NaB1·ejn-12π/εsin⁡θ-sin⁡θ0,1=aB1sinπN/εsinθ-sinθ0,1N·sin⁡π/εsin⁡θ-sin⁡θ0,1,PB2θ,ϕ,θ0,2=∑n=1NaB2·ejn-12π/εsin⁡ϕ-sin⁡θ0,2=aB2sinπN/εsinϕ-sinθ0,2N·sin⁡π/εsin⁡ϕ-sin⁡θ0,2,PB3θ,ϕ,θ0,3=∑n=1NaB3·ejn-12π/εsin⁡π/2-ϕ-sin⁡θ0,3=aB3sinπN/εcos⁡ϕ-sinθ0,3N·sin⁡π/εcos⁡⁡ϕ-sin⁡θ0,3,where θ0,1, θ0,2, and θ0,3 are the off-boresight angles corresponding to AnB1, AnB2, and AnB3, respectively.By settingN=4 and ε=2, we plot the beam radiation pattern of a PULA in 3D form, as illustrated by Figure 9. We can see that the main-lobe petal of each beam has a shell shape, which is mainly formed by the interaction of the electromagnetic fields of colocated beams.Figure 9
Beam radiation patterns of PULA in 3D form.


## 4. Beamforming Diversity Order (BDO)

In the relevant studies, for example, [16, 17], the interference reduction by beamforming is studied, where the sidelobe radiation of a beam is suppressed via the leaky wave antenna and particle swarm optimization in order to decrease the sidelobe interferences caused at neighbor cells. However, in this paper, we focus on the theoretical analysis under a perfect scenario without any interference. The radiation of sidelobes can also contribute to the SNR that enhances the capacity, because the BDO is increased with the additional number of sidelobes. Figure 10 describes a radio transmission scenario when beamforming technology is applied at both the transmitter (Tx) and receiver (Rx). In this case, we define the signal-transmission cutting surface (STCS), which is the surface that holds the signal from the Tx to the Rx. According to Figure 10(a), three lobes are simultaneously held by the STCS at the Tx (STCSTx), and three lobes are simultaneously held by the STCS at the Rx (STCSRx). And we define the BDO as the multiplication of the number of lobes cut by STCSTx and STCSRx. For example, in the case of Figure 10(a), the BDO is 9. On the other hand, the BDO may decrease because of the large off-boresight angle effect, as illustrated in Figure 10(b); for example, the BDO is one where only one lobe is held by the STCSTx and STCSRx.Figure 10
Transmission scenario with beamforming technology.
(a)
BDO without large off-boresight angle
(b)
BDO with large off-boresight anglePractically, the scatterers are often distributed into clusters [18, 19]. A group of local scatterers within a cluster have relatively close coordinates on the sphere representing the incidence signal scattered by the roofs, walls, trees, windows, doors, and so on from a correlated direction. Meanwhile, global scatterers among different clusters have independent coordinates, which denote the incidence signal scattered from uncorrelated directions. Given a cluster of local scatterers, for the ith scatterer with a polar coordinate of Ciri,θi, the signal response after scattering is given as(13)hit=Ait-τTxCi-τRxCi·exp⁡⁡-j2πfcτTxCi+τRxCi,where τTx(Ci) is the signal propagation delay from the Tx to Ci, τRx(Ci) is the signal propagation delay from Ci to the Rx, and fc denotes the signal carrier frequency. Ai represents the channel gain and exp⁡⁡(·) gives the signal phase rotation after scattering. Because the local scatterers in this cluster are very close to each other, we have(14)τTxCi+τRxCi≈τTxC0+τRxC0fori=1,…,I,where C0 represents the coordinates of the center of gravity for the cluster. Then, h0t can be used to represent the scattered signal as follows:(15)h0=hiCi=C0.When the scatterers in different clusters are not close to each other, (τTx(C0,q)+τRx(C0,q)) is independent of (τTx(C0,q+1)+τRx(C0,q+1)) for q=1,…,Q, where C0,q is the coordinate of the center of gravity for the qth cluster. Then, the received signal power can be denoted as(16)Pt=∫0∞∫θmin⁡θmax⁡A0,qt-τTxC0,q-τRxC0,q·exp⁡⁡-j2πfcτTxC0,q+τRxC0,q2dθ0,qdr0,q.The fully scattered environment refers to the scattered signal coming from all directions; that is,θmin⁡=-π and θmax⁡=π, and dense urban and indoor scenarios are usually studied under such an environment. The non-fully–scattered environment is realized when θmin⁡>-π and θmax⁡<π. Most rural and remote scenarios, where wide open spaces and flat terrain fulfill the LOS conditions, are studied in this environment [20].We assume the scenario where the Tx, which steers its main-lobe toward Rx, is located far from Rx. Consequently, the BDO at the Tx side can be assumed to be one, and the overall BDO of the system is determined only by the Rx. In addition, suppose several clusters of local scatterers are located on a sphere that encircles the Rx, as illustrated in Figure11; we can summarize the system BDO as given by Table 1. From Table 1, we note that BDO equals the number of lobes generated by the Rx in a fully scattered scenario. In the case where the LOS component exists in a non-fully–scattered environment, the main-lobe of the Tx covers half of the surface of the sphere around the Rx. Therefore, the BDO is varied from 2Nε-1-1 to 4Nε-1-2. For the case of the non-LOS (NLOS) in a non-fully–scattered scenario, the BDO is varied from one to 2Nε-1-1. Compared with the diversity order found in [3], our findings in Table 1 clearly describe the diversity order associated with the factor of the array configuration in practical scattering environments. Moreover, the fact that each lobe has two zero points was overlooked in [3].Table 1
BDO values depending on scenarios.
Scenario
BDO
LOS + NLOS(fully scattered)
4
N
ε
-
1
-
2
Rx
NLOS(fully scattered)
4
N
ε
-
1
-
2
Rx
LOS + NLOS(non-fully–scattered)
2
N
ε
-
1
-
1
≤
BDO
≤
4
N
ε
-
1
-
2
Rx
NLOS(non-fully–scattered)
1
≤
BDO
≤
2
N
ε
-
1
-
1
RxFigure 11
Scattering environment.


## 5. Channel Capacity by PULA with Beamforming

For a fully scattered environment under the assumption that there is no interference, [6] derives the channel capacity for an array system as follows:(17)C=ρ·ηN0log⁡⁡ebit/s,where ρ=Is/82π/λ2RS2. ρ refers to the density of the scatterers located on the 3D spherical model, Is is the number of scatterers, RS is the radius of the scattering sphere, η is the transmission power, and N0 indicates the additive noise power density. From (17), we observe that the channel capacity is only related to the scatterer density and transmission power as opposed to other system parameters. Equation (7) concludes in two ways. At first, it describes that the capacity is regardless of the system bandwidth, where a high throughput can be achieved even by a negligibly small bandwidth. And second, it shows that the capacity is irrespective of the antenna array configuration. However, the conclusions in [6] were obtained using a ULA compact antenna with ε≫2, and based on Table 1 the diversity order thus decreases significantly. In [6], we also see that (17) is employed when the scatterers are uniformly distributed on the sphere. This is not the case practically, and the scatterers’ distribution may follow the example shown in Figure 11. In addition, the fading channel gain, which is usually used to derive the channel capacity, was not considered by (7). Therefore, we provide a new method to obtain the channel capacity for PULA systems, and this method is based mainly on the fading channel gain, beam radiation pattern, and BDO.In [7, 8], the channel capacity with Rx diversity is given by(18)C=Elog⁡2⁡1+h2SNR2,where h denotes a gain of fading channel. Here, we define the beam lobe signal to noise power ratio (LSNR) based on the beam radiation pattern as(19)LSNRθ^k≜Pθ^k,κ2nk2,where k denotes the lobe index of the beam, θ^k (can be found in Appendix B) holds the condition for P(θ,κ) having peak values, and nk refers to the added noise power at the kth lobe. Based on (19), the effect of the off-boresight angle can be disregarded according to Proposition 2.Proposition 2.
The off-boresight angle does not affect the lobe power but affects the lobe-width (proved by AppendixB).The channel capacity for a PULA system under the Rayleigh fading channel can be given as(20)CRayleigh=Eξ∑k=14Nε-1-2log⁡2⁡1+hkPθ^k,κ2nk2,where ξ=6 is the degree-of-freedom of the triple-polarized antenna in [3]. hk denotes a gain of fading channel corresponding to the kth lobe. Then, the capacity upper bound is found by Jensen’s inequality for concave functions as follows:(21)CRayleigh≤2ξ∑k=12Nε-1-1log⁡2⁡1+EhkPθ^k,κ2nk2.By assuming that LOS components exist and the Rx PULA steers its main-lobes towards the LOS signal, the capacity upper bound under the Rician fading channel can be given as(22)CRician≤ξlog⁡2⁡1+EK/K+1h0+1/K+1h1Pθ^1,κ2n12+ξlog⁡2⁡1+Eh1Pθ^1,κ2n12+ξ∑k=24Nε-1-2log⁡2⁡1+EhkPθ^k,κ2nk2,where the first term contains the capacity contributed by the main-lobe that steers the LOS component (K is the Rician K-factor). The second term represents the capacity obtained by this main-lobe for the NLOS component, while the third term indicates the capacity from other lobes of the NLOS component. Note that (21) and (22) apply to the scenario involving a fully scattered environment. In the case of a non-fully–scattered environment, the PULA capacity is decreased due to the decrement of BDO.Figure12 gives the normalized channel capacity for PULA with beamforming. From Figure 12, we observe that, by extending the dipole length (e.g., L=1λ), we can effectively increase the capacity of the PULA system. This is because the radiation gain increases as κ increases, where the case of κ=1 achieves the highest gain, as discussed in Section 3. The channel capacity is also increased when more array elements are implemented (e.g., N=8), and this is mainly due to the increment of the BDO. Compared with the capacity increment obtained by extending the dipole length, the capacity increment due to the BDO is small because the array factor is normalized by the number of array elements in (4). Finally, Figure 12 depicts the channel capacity under different channel models, where the AWGN has a higher capacity than in the case of fading channels. In addition, the Rician fading channel with K=9 dB can achieve a higher channel capacity compared with the Rayleigh fading channel due to the presence of the LOS component.Figure 12
The normalized channel capacity for PULA with beamforming.


## 6. Conclusions

In this paper, we derive the channel capacity for PULA systems using the beamforming technique. Our results show that the channel capacity is determined mainly by three system indexes, that is, the fading channel gain, beam radiation pattern, and BDO, where the BDO is dependent on the antenna characteristics and array configurations. In contrast to conventional works, we also consider the fading channel in a practical scattering environment, and the numerical results confirm the validity of the channel capacity obtained in this paper.



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*Source: 371236-2015-05-19.xml*