# Power angle spectrum of satellite communication downlink in rain environment at millimeter-wave bands.

1. INTRODUCTIONWith the growing demands for wireless communication, the future communication system needs more advanced technologies to make better use of the limited power, increase the spectrum efficiency and improve the transmission rate and system capacity. In comparison with traditional Single-Input Single-Output (SISO) communication technology, Multiple-Input Multiple-Output (MIMO) technology shows great advantages, such as higher transmission rate, spectrum utilization ratio and power efficiency. Therefore, MIMO communication technology represents a breakthrough in the modern communication field [1-3].

Many reports about MIMO communication have focused on the frequencies below 10 GHz [1]. In contrast, the research on MMW MIMO communication is still in its infancy, although millimeter wave (MMW) communication technology has many advantages [1]. Available publications just explored some particular aspects for MMW MIMO communication. An overview of MMW MIMO communication system was given in [4]. In [5], it was proposed that MMW MIMO should be used for next generation mobile broadband. The investigation and discussion of MMW MIMO communication in indoor environment was carried out in [6-14]. The short-distance transmission performance of the MMW MIMO system in outdoor clear environment was evaluated and discussed in [15-17]. Some reports explored the antenna of MMW MIMO system [18-20]. The publications of [20-34] studied the channel characteristics, the performance of capacity and BER and the Space-Time Coding of MMW MIMO system, but the influences of the troposphere were not considered. In [35-39], it was particularly pointed out that these effects must be taken into account in order to develop MMW MIMO communication technology.

In addition, attention has been paid to the application of MMW MIMO communication technology in earth-satellite communication system in recent years. For example, MIMO over satellite is reviewed in [2], in which MMW MIMO technology is referred to, preliminary explorations of MMW earth-satellite MIMO communication can be found in [40-43]. Similar to MMW MIMO communication, MMW earth-satellite MIMO communication is still in its infancy. Therefore, futher study on MMW earth-satellite MIMO communication is highly desirable. The evaluation of channel characteristics is important in developing MMW earth-satellite MIMO communication system. Among them, PAS is particularly remarkable because it quantifies the received power of receiving antenna as a function of angle. For example, PAS is needed to evaluate angle spread (AS) characteristic, and it is essential to calculate the channel correlation.

Some models of PAS are given in available publications, such as Cosine distribution [44], Uniform distribution [45], Gaussian distribution [46] and Laplacian distribution [47]. Although these models describe well the power angle distribution caused by terrain and the objects around receiving antenna, such as flat surface ground, hills, mountains, buildings, automobiles, trees, vegetation and so on, they are not suitable for MMW systems. The antenna of MMW communication system has higher gain and narrower beam, and it is usually in line-of-sight state between the ground station and the satellite antenna for MMW earth-satellite communication system (especially the earth-satellite fixed communication system). Therefore, in contrast to the propagation and scattering effects induced by the troposphere environments, the effects induced by terrain and the objects around receiving antenna seldom impact on MMW channel characteristics. It can be concluded that the PAS models mentioned above are invalid for the earth-satellite MIMO communication system at MMW bands. The scattering and propagation impacts caused by rain represent one of the most challenging problems in MMW communication. To the best of our knowledge, there has been no report on evaluation models of the PAS caused by rain.

This paper focuses on the PAS of the satellite communication downlink in rain environment at millimeter-wave bands. The two-dimension angle distribution expression of the incoherent intensity of the ground receiving antenna in rain is deduced in detail, and the coherent intensity is discussed according to the first-order multiple scattering approximation theory. The calculation model of PAS is given based on the coherent intensity and the incoherent intensity angle distribution. The PAS model and the simulation results given in this paper will be useful for the quantitative evaluation of the impacts of rain environment on MMW MIMO channel characteristics.

2. DERIVATION OF POWER ANGLE SPECTRUM

As shown in Figure 1, the downlink signal passing through a rain area can be regarded as a plane wave incident on a semi-infinite medium containing a random distribution of many raindrops, because the altitude of the satellite orbit is far greater than the rain height, and the size of the ground receiving antenna is far smaller than that of the rain area.

Based on the assumption mentioned above, we can consider an arbitrary polarization plane wave incident upon a semi-infinite medium containing a random distribution of many raindrops. In the coordinate axes xyz (see Figure 1 and Figure 2), the unit vector of incidence direction is [[??].sub.0], the field [??]([??]) = [[[u.sub.H]([??]) [u.sub.V]([??])].sup.T] at [??] = (0,0, L) consists of the average or coherent field <[??] ([??])> = [[<[u.sub.H]([??])> <[u.sub.V]([??])>].sup.T] and the fluctuation field or incoherent field [[??].sub.f]([??]) = [[[u.sub.f_H]([??]) [u.sub.f_V]([??])].sup.T], where the subscripts H and V respectively denote the horizontal polarization component and the vertical polarization component. [??] ([??]) can be written as (1) [48]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

And, the total intensity [I.sub.t]([??]) of the receiving antenna at [??] = (0,0, L) is presented as:

[I.sub.t]([??]) = [I.sub.c]([??]) + [I.sub.f]([??]) (2)

where, [I.sub.c]([??]) is the coherent intensity, and [I.sub.f]([??]) is the incoherent intensity.

It is reasonable to evaluate the rain-induced propagation and scattering effects with the first-order multiple scattering approximation theory [48]. [[??].sub.f]([??]) is a sum of the scattered fields from all the raindrops in the range 0 < z < L, neglecting the backscattering contribution from the region z > L, and [I.sub.f] ([??]) is a sum of the scattered intensity from all the raindrops in the range 0 < z < L [48].

According to the first-order multiple scattering approximation theory, the average or coherent field and the coherent intensity can be obtained by [48]

<[u.sub.H](V)([??])> = [u.sub.0_H(V)] exp(ikL/cos[[theta].sub.i] - [[tau].sub.c_H(V)/2) (3)

[I.sub.c] = [u.sup.2.sub.0_h] exp(-[tau].sub.c_H]) + [u.sup.2.sub.0_V] exp(- [[tau].sub.c_V]) (4)

where, [[theta].sub.i] is the angle between the unit vector [[??].sub.0] and the axis z, [u.sub.0_H(V)] the amplitude of the horizontal polarization component (the vertical polarization component) of the incident wave on the rain area, and k = 2[pi]/[lambda] the wave number. In Equations (3) and (4), [[tau].sub.c_H(V)] is the optical distance of the horizontal polarization wave (the vertical polarization wave), which is written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where, N(D) denotes raindrop size distribution function, and [[sigma].sub.t_H(V)](D) denotes the extinction cross section of the raindrop with the diameter D illuminated by the horizontal polarization wave (the vertical polarization wave), [[theta].sub.i] is the angle between the unit vector [[??].sub.0] and the axis z, [rho] and <[[sigma].sub.t_H(V)]> are respectively denoted as [48,49]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The coherent intensity is a constant value and can be readily calculated by (4) for an incidence direction. Therefore, the coherent intensity has little contribution to the angle distribution of the received intensity [I.sub.t]([??]). And the angle distribution of the coherent intensity [I.sub.c]([??]) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where, [??] is the concerned arriving direction in the coordinate axes xyz. In what follows, we will deduce the angle distribution of the incoherent intensity.

As mentioned before, [[??].sub.f]([??]) is a sum of the scattered fields from all the raindrops in the range 0 < z < L, and [I.sub.f]([??]) is a sum of the scattered intensity from all the raindrops in the range 0 < z < L. Let us first consider the scattering field [??].sub.f] (D, [??]) = [[[u.sub.f_H](D, [??]) [u.sub.f_V](D, [??])].sup.T] at [??] = (0,0, L) from a single raindrop with the diameter D in dV located at (x, y, z) in the coordinate axes xyz. According to the scattering theory and the first order multiple scattering approximation theory, [[??].sub.f](D, [??]) = [[[u.sub.f_H](D, [??]) [u.sub.f_V] (D, [??])].sup.T] can be written as [48,49]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

In Equation (9), [u'.sub.0_h] and [u'.sub.0_V] are the amplitude of the horizontal polarization component and the vertical polarization component of the incident wave on dV, respectively; [f.sub.12], [f.sub.21], and [f.sub.22] are the elements of the scattering amplitude function matrix for the raindrop with the diameter D, which are the functions of scattering angle [THETA] (cos = [THETA] = [[??].sub.0] X [??]), and are related to the scattering functions [S.sub.1], [S.sub.2], [S.sub.3] and [S.sub.4] used by Van de Hulst and in Mie solution for a sphere [48, 50]; [??] is the unit vector of a scattering direction in the coordinate axes xyz. [r.sub.1] is the distance from dV to (0,0, L). [u'.sub.0_H] and [u'.sub.0_V] are given as (see Figure 2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where, [[tau].sub.0_H(V)] is given as

[[tau].sub.0_H(V)] = p<[[sigma].sub.t_H(V)]>(L - [r.sub.1] cos [theta])/cos [[theta].sub.i] (11)

In Equations (10) and (11), 9 is the angle between the unit vector [??] and the axis z.

[[tau].sub.1_H(V),] in Equation (9), is written as

[[tau].sub.1_H(V)] = [rho]<[[sigma[.sub.t_H(V)])>[r.sub.1] (12)

[f.sub.11], [f.sub.12], [f.sub.21] and [f.sub.22] are presented as [48, 51]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[[sigma].sub.t_H(V)](D) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Note that if the particle is spherically symmetric, [f.sub.12] = [f.sub.21] = 0, which is approximatively suitable for raindrop [48]. Therefore, (9) is simplified as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where, cos [THETA] = [[??].sub.0] x [??]. In the coordinate axes xyz, [[??].sub.0] and [??] can be written as

[[??].sub.0] = sin[[theta].sub.i] cos[[phi].sub.i][??]] + sin[[theta].sub.i] sin[phi[].sub.i] [??]] + cos [theta[].sub.i] [??]] (16)

[??] = sin[theta] cos[phi][??] + sin[theta] sin[phi][??] + cos[theta][??] (17)

In Equations (16) and (17), [[theta].sub.i] is the angle between the unit vector [[??].sub.0] and the axis z, [[psi].sub.i] is the angle between the projection of [[??].sub.0] in the plane xy and the axis x, and [theta] is the angle between the unit vector [??] and the axis z, [phi] is the angle between the projection of [??] in the plane xy and the axis x. Therefore, cos[THETA] = [[??].sub.0] x [??] is derived as

cos [THETA] = sin [theta] cos[psi] sin[[theta].sub.i] cos[psi].sub.i] + sin[theta] sin[psi].sub.i] + cos[theta]cos [[theta].sub.i] (18)

Consequently, the received incoherent intensity d[I.sub.f]([??]) from the raindrops in dV can be presented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Define <[absolute value of [f.sub.11(22)](cos[THETA])].sup.2]> as [48,52]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Equation (19) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

For convenience, analyse Equation (21) in the new coordinate axes [x.sub.1] [y.sub.1] [z.sub.1], whose origin is the point (0, 0, L) in the coordinate axes xyz (see Figure 2). In the coordinate axes [x.sub.1] [y.sub.1] [z.sub.1], dV is located at ([x.sub.1], [y.sub.1], [z.sub.1]), which is also ([r.sub.1],[theta],[psi]),] so dV can be written as

dV = [r.sup.2.sub.1] sin[theta] d[theta] d[phi] d[r.sub.1] = [r.sup.2.sub.1]d[OMEGA] d[r.sub.1] (22)

Define [I.sub.f_[??]]([theta], [psi]) as

[I.sub.f_[??]]([theta], [psi]) = [[integral].sup.R.sub.O] [d[I.sub.f]([??])/d[OMEGA]] [r.sup.2.sub.1] d[r.sub.1] (23)

Equation (23) means the received incoherent intensity at [??] = (0, 0, L) coming from the direction of ([theta], [psi]) in the coordinate [x.sub.1] [y.sub.1] [z.sub.1] also in the coordinate xyz, i.e., [I.sub.f_[??]([theta], [psi]) is the angle distribution function of the incoherent intensity. [I.sub.f_[??]([theta], [psi]) can be derived from Equations (21) (23) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

By using Equations (11), (12) and (18), Equation (24) can be simplified as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

In Equation (25)

[mu] = cos[theta] [[mu].sub.i] = cos[[theta].sub.i] (26)

[g.sub.H(V)] = [[[u.sub.0_H(V)].sup.2][exp(-[[gamma].sub._H(V)] L/[mu].sub.i]) - exp(-[[gamma].sub._H(V)] L/[mu])]/ - [[gamma].sub._H(V)] (27)

[g.sub.1_H(V)] = [[[u.sub.0_H(V)].sup.2] exp[-[[gamma].sub._H(V)] L/[mu].sub.i] (28)

where,[gamma]H(V) = [rho]([[sigma].sub.t_H(V)]>. By the way, if a raindrop is regarded as an isotropic, homogeneous sphere, [gamma]H = [gamma]V, which is not well suitable for rain environment (especially in the case of a large rain rate). [gamma]H and [gamma]V can be calculated by the model given in ITU-R P.838 [53]. Rewrite (8) using [mu] and [psi] as

[I.sub.c_[??]]([mu], [psi]) = [I.sub.c][delta]([mu] - [[mu].sub.i])[delta]([psi] - [[psi].sub.i)] (29)

Now, the angle distribution [I.sub.t_[??]] ([mu], [psi]) can be given as

[I.sub.t_[??]] ([mu], [psi]) = [I.sub.c_[??]] ([mu], [psi]) + [I.sub.f_[??]] ([mu], [psi]) (30)

PAS [P.sub.[??]] ([mu], [psi]) of the antenna located at [??] = (0, 0, L) can be given as

[P.sub.[??]]([theta], [psi]) = [I.sub.t_[??]] ([mu], [psi]) A ([theta], [psi])[m.sub.P] (31)

where, [m.sub.P] is the polarization mismatch factor and a real number between zero and one, which depends on the degree of match of the polarization state of the wave and the antenna [54], A([theta], [psi]) is the effective aperture in the direction ([theta], [psi]), which can be calculated by

A([theta], [psi]) = [[lambda].sup.2]/4[pi]]G([theta], [psi]) (32)

where, [lambda] is the wave length of the incident wave and G([theta], [phi]) the gain of the receiving antenna.

3. NUMERICAL RESULTS AND DISCUSSION

The model given in Section 2 is suitable for a practical downlink with an arbitrary polarization incident wave in an arbitrary incident direction in the coordinate axes similar to those in Figure 1 and Figure 2. The incident direction for a real downlink is determined by the link geometry structure. In this section, [gamma]H(V) = [rho]<[[sigma].sub.t_H(V)]>, <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> (which is for horizontal polarization), <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> (which is for vertical polarization), and the PAS are calculated and discussed by Mie method based on the Marshall-Palmer raindrop size distribution [55]. Note that PAS is analyzed under the conditions of A([theta],[psi]) = 1, [m.sub.p] = 1 and the intensity of incident wave is equal to 1.

Figure 3 is the results of [gamma] = [gamma]H = [gamma]V varying with the corresponding rain rate and frequency. Figure 4 gives the results of <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> and <[[absolute value of [f.sub.22](cos[THETA])].sup.2]> varying with the scattering angle for different frequencies in different rain environments. Figure 5 gives a two-dimension PAS example of the vertical polarization incident wave at 90 GHz. Figure 6-Figure 8 show the one-dimension incoherent power varying with the rain height L, rain rate, and the frequency, the polarization and the incident angle of the incident wave.

Figure 3 shows that [gamma] approximately increases linearly with the rain rate for the same frequency on log paper; [gamma] increases with the frequency for the same rain rate, but [gamma] is close to a constant value with frequency increasing, and the larger the rain rate is, the greater the constant value is.

It can be concluded from Figure 4 that <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> and <[[absolute value of [f.sub.22](cos[THETA])].sup.2]> differently change with the scattering angle for different frequencies and rain rates. If [THETA][not equal to] 0, <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> is greater than <[[absolute value of [f.sub.22](cos[THETA])].sup.2]> for the same frequency and rain rate; for the same frequency, <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> distinctly exceeds <[[absolute value of [f.sub.22](cos[THETA])].sup.2]> for the larger rain rate; for the same rain rate, <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> distinctly exceeds <[[absolute value of [f.sub.22](cos[THETA])].sup.2]> for the lower frequency. If [THETA] = 0, <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> is equal to <[[absolute value of [f.sub.22](cos[THETA])].sup.2]> for the same frequency and rain rate; for the same frequency, the larger the rain rate is, the greater <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> and <[[absolute value of [f.sub.22](cos[THETA])].sup.2]>]; for the same rain rate, the higher the frequency is, the greater <[[absolute value of [f.sub.11](cos[THETA])].sup.2]> and <[[absolute value of [f.sub.22](cos[THETA])].sup.2]> are.

Figure 5 shows that the PAS is mainly affected by the incoherent intensity or the incoherent power, which is coming from different directions. Therefore, AS (angle spread) is caused by the incoherent power. Although the exhibited results are for the vertical polarization incident wave at 90 GHz, the patterns for horizontal polarization incident wave and other frequencies are similar. Consequently, Figure 6-Figure 8 mainly discuss the one-dimension incoherent power angle distribution changing with rain environment parameters, and the frequency, the polarization and the incident angle of the incident wave.

It can be concluded from Figure 6 that if the rain environment parameters and the incident wave characteristics are given, the incoherent power focuses on the forward scattering direction (namely "forward scattering character" in this paper) more obviously for higher frequency, which also means the incoherent power has a narrow angle distribution. Figure 6 also shows that the "forward scattering character" can be completely destroyed at the lower frequency if the other conditions are the same.

Figure 7 shows that a larger rain rate or rain height can make the "forward scattering character" more obvious for the same characteristics of the incident wave for the lower frequency. Figure 8 shows that the peak of the incoherent power is determined by the rain rate and rain height for the same incident wave characteristics.

As shown in Figure 6-Figure 8, the incoherent power angle distribution has complicated relationship with the rain environment parameters (such as rain rate, rain height) and the incident wave characteristics (such as frequency, polarization, and incident angle). It can be concluded that the raindrop size distribution also has impacts on the result of PAS. So, it is necessary to calculate the PAS under a specific link and the rain environment parameters for investigating MMW MIMO channel characteristics.

4. CONCLUSIONS

We have investigated the power angle spectrum (PAS) of the satellite communication downlink in rain environment at millimeter-wave bands. The coherent intensity is discussed according to the first-order multiple scattering approximation theory, and the two-dimension angle distribution expression of the incoherent intensity of the ground receiving antenna in rain environment is deduced in detail. The calculation model of PAS is given based on the coherent intensity and the incoherent intensity angle distribution model. The model given in this paper is suitable for a practical downlink with an arbitrary polarization incident wave in an arbitrary incident direction in the coordination similar to those in Figure 1 and Figure 2.

In this paper, [[gamma].sub.H(V)] = [rho]([[sigma].sub.t_H(V)], <[[absolute value of [f.sub.11](cos[THETA])].sup.2]>, <[[absolute value of [f.sub.22](cos[THETA])].sup.2]>, and the PAS are calculated and discussed by Mie method based on the Marshall-Palmer raindrop size distribution. Note that PAS is analyzed under the conditions of A([theta], [phi]) = 1, [m.sub.P] = 1 and the intensity of incident wave is equal to 1. It can be concluded based on the simulation results in this paper that PAS has complicated relationship with the rain environment parameters (such as rain rate, rain height, and raindrop size distribution) and the incident wave characteristics (such as frequency, polarization, and incident angle). Therefore, it is necessary to calculate PAS under a specific link and the environment parameters using the model given in this paper for investigating MMW MIMO channel characteristics.

ACKNOWLEDGMENT

The authors would like to thank the National Natural Science Foundation of China under Grant No. 61001065, and the Fundamental Research Funds for the Central Universities for financial support of this work.

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Shu Hong Gong *, Wei Meng, and Jing Yang

School of Science, Xidian University, Box 273, Xi'an 710071, China

* Corresponding author: Shu Hong Gong (ljbrp2003@163.com).

Received 27 July 2013, Accepted 22 August 2013, Scheduled 1 September 2013

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Author: | Gong, Shu Hong; Meng, Wei; Yang, Jing |
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Publication: | Progress In Electromagnetics Research B |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | Sep 1, 2013 |

Words: | 5168 |

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