Rain-induced effects on the envelope probability density functions in multipath channels

Authors


Abstract

[1] The fluctuant peculiarities of a signal passing through rain area are studied with the theory of waves propagating through random media. The characteristics of the probability density distribution of a received envelope in rain region are calculated and discussed. The fluctuant rules of a signal in rain region are found. The effects induced by raindrops scattering on the envelope probability density functions (PDFs) in multipath channels are studied using the results in the work by Durgin et al. (2004, 2002). The detailed formulae are derived for one-path channel and two-path channel. The envelope PDFs are calculated for a narrowband signal with a given carrier frequency at the condition of a certain rain rate and a certain propagating distance, which are contrasted with those calculated at unfluctuant condition. The general results of the effects induced by raindrops scattering on the envelope PDFs in multipath channels are found. The methods evaluating the effects on Rayleigh fading distribution model and Rician fading distribution model induced by raindrops scattering are discussed. The general methods for studying the impacts of meteorological scattering environment (snow, dust storm for example) on the envelope PDFs in multipath channels are searched.

1. Introduction

[2] It is well known that the strength of a received signal in fading wireless channels is the function of time, space and frequency. Furthermore, it is fluctuant between a maximum and a minimum value. Sometimes, a receiver must work under the condition that the strength of received signal is under the threshold of SNR (signal-to-noise). In this case, the quantitative investigation of the power or the voltage envelope of the received signal is imperative to explain the influence of stochastic fading channel on the performance of communication systems [see also Durgin et al., 2004]. The envelope PDFs of many typical fading channels (shadowing, Doppler, multipath propagating and so on) are discussed in many literatures, and they are based on either theoretic or experiment [e.g., Reudink, 1974; Loo, 1996; He et al., 2005]. Multipath propagation is one of the important reasons of received signal fading. In available literatures, the fading effects caused by multipath propagation do not take weather influences (such as rain, snow and dust storm) into account [e.g., Tong and Akaiwa, 1997; Barger, 1996; Bruder and Saffold, 1990]. The impacts of weather environments on wireless communication system in the literatures focused on attenuation, depolarization, and the contribution of rain-induced attenuation to noise temperature [e.g., Alouini et al., 1997; Kassianides and Otung, 2003; Fashuyi and Afullo, 2007]. The fluctuant peculiarities of a received signal in different weather are scanty. And, the authors of this paper could not find literature about the impacts of weather on the envelope PDFs in multipath channels. In fact, when radio wave passes through random media, both the attenuation and the scattering occur, and the scattering field arrived at the receiver is stochastic and fluctuant. So the received signal in random media (such as rain, snow, dust storm, and so on) is stochastic and fluctuant. When rain (or snow, dust storm and so on) happens, the multipath channels become fluctuant. And, the envelope PDFs in multipath channels educed under unfluctuant condition cannot describe the fading characteristics in weather environments. Accurate calculation of envelope PDFs is important to predict the statistical characteristics (for instance, Level Crossing Rate, Average Fade Duration) of fading channel. The general results of the envelope PDF of two-path channel with diffusing power are discussed in the work by Durgin et al. [2004, 2002]. In this paper, the fluctuant peculiarities are studied with the theory of wave propagating through random media. The characteristics of the probability density distribution of received envelope in rain region are calculated and discussed. The fluctuant rules of the signal in rain region are found. The effects induced by raindrops scattering on the envelope PDFs in multipath channels are studied using the results in the work by Durgin et al. [2004, 2002]. The detailed formulae are derived for one-path channel and two-path channel. The envelope PDFs are calculated for a narrowband signal with a given carrier frequency under the condition of a given rain rate and a given propagating distance, which are compared to those calculated under unfluctuant condition. The general results of the effects induced by raindrops scattering on envelope PDFs in multipath channels are found. The methods evaluating the effects of Rayleigh fading distribution model and Rician fading distribution model induced by raindrops scattering are discussed. The general methods for studying the impacts of meteorological scattering environment (for example, snow or dust storm) on the envelope PDFs in multipath channels are searched.

2. Fluctuant Peculiarities of a Received Signal in Rain

[3] When radio wave passes through random media, both attenuation and scattering occurs, and the scattering field arrived at receiver is stochastic and fluctuant as the result of random distributed raindrops with different diameters. As shown in Figure 1, the received field u(equation image) at equation image (x, 0, L) consists of average field 〈u(equation image)〉 and fluctuant field uf(equation image), the received voltage V(equation image) consists of average voltage 〈V(equation image)〉 and fluctuant voltage Vf(equation image); for further details, see Ishimaru [1978].

equation image
equation image

Assume that uf(equation image) is the sum of scattering field from the raindrops in the region of 0 ≤ zL and the contributions of the back scattering field from the raindrops in the region of z > L are neglected; for further details, see Ishimaru [1978].

equation image
equation image

where

equation image
equation image

In the same way, the total intensity It is the sum of average intensity Ic and fluctuant intensity Ii, It, Ic and Ii can be expressed as [see Ishimaru, 1978]

equation image
equation image
equation image

In equations (2)–(5), Ei is the amplitude of incident wave, k = nek0 is the wave number in rain, ne is the equivalent complex refractive index of rain [Li and Huang, 2002], k0 is wave number of free space. f(equation image, equation image) is the scattering function, D is the diameter of raindrop, σt(D) is the extinction cross of the raindrop with diameter of D, f(θ, D) is the scattering function of the raindrop with diameter of D, and θ is the angle between equation image and equation image, N(D) is sizes distribution spectrum of raindrops. γ and g(γ, θ) are shown in (6) and (7), respectively [see Ishimaru, 1978]

equation image
equation image

where μ = cos θ in formula (7).

Figure 1.

Radio wave passes through rain region.

[4] Fluctuant field uf(equation image) can be written as

equation image

where, ϕ is uniform distribution from 0 to 2π, A follows Rayleigh distribution [see also Ishimaru, 1978]

equation image
equation image

σ2, in equation (9), denotes variance of stochastic amplitude, pdiff = 2σ2 denotes average power of fluctuant field, and pdiff can be computed by pdiff = IiAe, where Ae denotes effective aperture of the receiving antenna. It is concluded from (1)–(8) that P(A) is a function of frequency. For a narrowband signal, P(A) can be computed using carrier frequency. It is necessary to perform Fourier transform for evaluating P(A) for a broad-band signal. Figures 27are some calculated results under the condition that N(D) is assumed as Weibull spectrum, f(θ, D) is calculated by Mie theory [see Hao et al., 1999], and the directivity of receiving antenna is isotropy. Figures 2–7 also showed that the fluctuant intensity Ii cannot be neglected and the effects of rain-induced fluctuant field must be considered, which is discussed in detail in section 4.

Figure 2.

The relation of Ic/Ii changing with optics deepness.

Figure 3.

The relation of Ic/Ii changing with rain rate.

Figure 4.

The relation of Ic/Ii changing with frequency of radio wave.

Figure 5.

The PDF of Vf/〈V〉 (the amplitude of incident wave is “1,” and the directivity of receiving antenna is isotropy).

Figure 6.

The PDF of Vf (the amplitude of incident wave is “1,” and the directivity of receiving antenna is isotropy).

Figure 7.

The PDF of Vf/σ (the amplitude of incident wave is “1,” and the directivity of receiving antenna is isotropy).

3. Effects Induced by Rain on the Envelope PDFs in Multipath Channels

[5] Accurate forecast of envelope PDFs is important for predicting all statistical characteristics of fading channel (for instance Level Crossing Rate, Average Fade Duration and so on). In the work by Durgin et al. [2004, 2002], the closed-form solutions of the envelope PDF for five types of multipath channels are educed, and the general results of envelope probability density functions of two-path channel with diffusing power are discussed. The results used in this paper are listed in Table 1.

Table 1. Summary of Canonical Wave Fading Envelope PDFsa
PDFExpression for Envelope PDF ρ ≥ 0Characteristic FunctionE{R} (volts)E{R2} (volts2)
One-Waveδ(ρ − V1)J0(V1v)V1V12
Two-Waveequation image ∣V1 − V2∣ ≤ ρ ≤ V1 + V2J0(V1v) J0(V2v)equation imageV12 + V22
Rayleighequation imageequation imageequation imagePdiff
Ricianequation imageJ0(V1ν) × equation imageNo closed-form solutionPdiff + V12
TWPDequation imageΓ(ρ, Pdiff, K, Δ)J0(V1ν) J0(V2ν) · equation imageNo closed-form solutionV12 + V22 + Pdiff

[6] K, Δ and Γ(equation image, K, Δ) in TWPD are expressed as [Durgin et al., 2004, 2002]

equation image
equation image
equation image

aj in formula (12) is given in the work by Durgin et al. [2004, 2002], M = [KΔ/2] is the order of sum, and Ψ(x, K, α) presents as

equation image

[7] The results listed in Table 1 cannot describe the case in bad weather environment, while the process of studying TWPD in the work by Durgin et al. [2004, 2002] gives a clew for discussing the effects induced by raindrops scattering on the envelope PDFs in multipath channels. The envelope PDFs in rain region for One-Wave, Two-Wave, Rician and Rayleigh are educed and discussed based on the fluctuant peculiarities discussed in section 1.

[8] For one-wave model which is the approximate case of line-of-sight propagating in free space, the PDF should be Rayleigh fading or Rician fading if rain happens, and the scattering effect caused by raindrops is taken into account.

equation image

Pdiffr and V1r in equation (14) are written as

equation image
equation image

where Ae denotes effective aperture of receiving antenna, γ1 is optics deepness, V1 is the received voltage at clear sky condition, and V1r is the received voltage at raining condition using the same antenna.

[9] Two-wave fading model depict intervening fading at clear sky condition. However, the scattering effect induced by rain makes the envelope PDF of two-wave model showing Rayleigh fading, or Rician fading, or TWPD fading at different propagating instances. If the fluctuant intensities of both paths roughly equal the average intensities or be greater than those, the PDF should be Rayleigh distribution; If fluctuant intensity of only one path approximately equal the average intensity or be greater than that, the PDF should be Rician distribution; If fluctuant intensities of both paths are obviously less than the average intensities, the PDF should be TWPD distribution.

equation image

The parameters in (16) are given in (17)

equation image
equation image
equation image
equation image

[10] For Rician distribution model, if the average intensity of main wave is greater than fluctuant intensities, the fading distribution model is still Rician, but attenuation to diffusing power existed in unfluctuant situation and the scattering power caused by raindrops should be taken into account. And, if fluctuant intensity of main wave approximately equal the average intensities or be greater than that, the envelope PDF should be Rayleigh distribution.

equation image

In equation (18), P′diff denotes diffusion power faded by rain, Pdiffr, the power arrived at the receiver which is scattered by raindrops when main wave pass through rain region, can be presented as

equation image

[11] The envelope PDF of Rayleigh distribution is shown in (20)

equation image

In equation (20), P′diff denotes diffusion power faded by rain. P′diff in equations (18) and (20) should be calculated by the statistical peculiarities of propagating environment in given wireless communication system.

[12] Figures 810 are some results computed under assumed parameters that the amplitude of incident wave is “1,” and the directivity of receiving antenna is isotropy and other parameters are labeled in the figures, to visualize the impacts induced by raindrops scattering on the envelope PDFs in multipath channels. We can draw a conclusion from Figures 8–10 that the effects caused by rain environment can markedly influence the envelope PDFs in multipath channels, and the impacts of rain environment must be considered in antifading measures study, which is discussed in detail in section 4.

Figure 8.

The PDF of one-wave model in rain environment (the amplitude of incident wave is “1,” and the directivity of receiving antenna is isotropy).

Figure 9.

An example of the PDF for two-wave model only attenuation caused by rain having been taken into account.

Figure 10.

An example of the PDF for two-wave model in rain environment.

4. Conclusions

[13] It is concluded from Figures 2–4 that the ratio of average intensity to fluctuant intensity at receiving antenna is the function of rain rate, the distance between transmitting antenna and receiving antenna, the frequency of radio wave. And, Figures 2–4 show that the ratio decrease with optics deepness increase and with frequency of radio wave increase, which implies that fluctuant character is almost concomitant with rain. Figures 5–7 show that the PDF of fluctuant field is the function of not only rain rate, the distance between transmitting antenna and receiving antenna, the frequency of radio wave, but also the amplitude of incident wave and the effective aperture of a receiving antenna. And, the envelope PDF of a received signal is decided by the variance of fluctuant field, and the variance can be computed by fluctuant intensity Ii. As some examples, Figures 5–7 are simulated under the condition that the amplitude of incident wave is 1, and the directivity of receiving antenna is assumed as isotropy. Figures 8–10 illuminate the effects induced by raindrops scattering on the envelope PDFs in multipath channels. The impacts are obvious which concludes that the impacts of raindrops scattering on the envelope PDFs in multipath channels cannot be neglected. Figure 10 also shows the envelope PDF in two-path channel changing to TWPD distribution, Rician distribution and Rayleigh distribution with rain rate increasing. And the change is faster at higher frequency. The conclusion is that the impacts of weather environments on the envelope PDFs in multipath channels is significant and cannot be neglected. And, it is necessary to adopt appropriate antifading measures according to specific weather condition and specific communication system.

Acknowledgments

[14] This work is supported by Graduate Innovation Fund, Xidian University, State Key Laboratory of Integrated Service Network (ISNZJ1-01) and State Natural Sciences Foundation (60741003).

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