Atmospheric turbulence-induced fading channel model for space-to-ground laser communications links

Morio Toyoshima; Hideki Takenaka; Yoshihisa Takayama

doi:10.1364/OE.19.015965

1. Introduction

The first theoretical study of scintillation for a ground-to-space laser beam transmission was performed by Fried [1]. Scintillation was measured on a vertical path from a ground-based laser transmitter to the GEOS-III satellite orbiting at 1800 to 2000 km above the Earth. The power spectral density functions were also analyzed, and a larger aperture was found to reduce the higher frequency downlink spectra, as reported by Bufton [2]. The temporal-frequency spectrum associated with atmospheric turbulence has been examined in prior studies [3–5]. Greenwood proposed a bandwidth specification for adaptive optics systems [6], referred to as the Greenwood frequency [7]. The peak frequency of the spectrum, spectra with different apertures, and the dependence of spectra on wavelength were examined [8–11]. The bandwidth considerations of tracking through turbulence were discussed by Tyler [12]. The aperture averaging effect was shown to be effective for mitigating the scintillation in the downlink scenario [13,14]. Churnside evaluated the aperture averaging effects under conditions of weak and strong turbulence [15]. Andrews et al. predicted the frequency shift due to aperture averaging as a function of the characteristic frequency, which they referred to as the Fresnel frequency, defined by the Fresnel scale and mean wind speed [16]. Thus far, however, the frequency characteristics that result from aperture averaging have not been analyzed using actual measurement data of low Earth orbit (LEO)-to-ground laser links.

The frequency spectra of irradiance fluctuations have higher frequency components, particularly in the case of LEO-to-ground laser links. This is because LEO satellites move at high speed (several km/s) and the frequency components of the irradiance fluctuations can reach several kHz as the optical link traverses the air mass while tracking these satellites. The first experiments on laser communications between an optical ground station and an LEO satellite were recently performed [17]. In this scenario, the frequency response due to the presence of a high transverse wind speed were discussed based on the Hufnagel Valley (HV) model, which was dominated by the aperture averaging effect [18,19].

In this paper, we derive the temporal characteristics that result from turbulence spectra and aperture averaging for downlink optical channels. In Section 2, the temporal frequency characteristics based on the spatial frequency spectra for the downlink are derived. The scintillation theory is given in Section 3 with respect to the scintillation index and the rms wind speed. The numerical results for the temporal frequency spectra and the mean frequency are presented and the simulated time-varying signals and the histograms are provided with the estimated scintillation indices in Section 4.

2. Temporal frequency characteristics of laser beam propagation through atmospheric turbulence

2.1. Derivation of the spatial spectrum

Consider an optical receiver that is located at an optical ground station and detects the downlink laser beam transmitted from a satellite. Figure 1 shows the configuration of the optical receiver for measuring the fluctuating laser beam. The receiver, with an aperture pattern of a(r), measures the amplitude of the speckle pattern, s(r), resulting from the atmospheric turbulence at point r. The received electrical signal is thus expressed as

Fig. 1 Configuration of the optical receiver at the optical ground station.

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e (t) = \vec{e} (V t) = τ_{r} \int a (r) s (r + V t) d r .

Here, the receiver area for a(r) is πD ²/4, where the antenna diameter is represented by D, τ_r is the optics loss of the receiver, and V represents the transverse wind velocity between the air mass and the optical link. The function $\vec{e} (V t)$ is the spatial signal form of the electrical signal, e(t), and is given by the following convolution:

\vec{e} (r) = τ_{r} a (r) * s (- r),

Hence, the spatial power spectrum W_s(κ) of s(r) is related to the spatial power spectrum W_a(κ) of a(r) by

W_{\vec{e}} (κ) = τ_{r} W_{a} (κ) W_{s}^{*} (κ),

where κ is the spatial frequency vector and W_s*(κ) is the conjugate of W_s(κ). The inverse Fourier transform of Eq. (3) becomes

\begin{matrix} \vec{e} (r) = ℑ^{- 1} [W_{\vec{e}} (κ)], \\ = \frac{1}{2 π} \int W_{\vec{e}} (κ) \exp (j 2 π k r) d κ, \\ = \frac{τ_{r}}{2 π} \int W_{a} (κ) W_{s}^{*} (κ) \exp (j 2 π k r) d κ . \end{matrix}

The temporal frequency spectrum is given by

\begin{matrix} W_{e} (f) = \int e (t) \exp (- j ω t) d t, \\ = \int \vec{e} (V t) \exp (- j ω t) d t . \end{matrix}

Here, substituting Eq. (4) into Eq. (5), we have

\begin{matrix} W_{e} (f) = \int [\frac{τ_{r}}{2 π} \int W_{a} (κ) W_{s}^{*} (κ) \exp (j 2 π k V t) d κ] \times \exp (- j ω t) d t, \\ = \frac{τ_{r}}{2 π} \int W_{a} (κ) W_{s}^{*} (κ) d κ \times \int \exp [j (2 π k V - ω) t] d t, \\ = \frac{τ_{r}}{2 π} \int W_{a} (κ) W_{s}^{*} (κ) d κ δ (2 π k V - ω) . \end{matrix}

Let κ and ϕ be the polar components of the two-dimensional vector κ = (κcosθ, κsinθ). We then obtain

W_{a} (κ) = W_{a} (κ, ϕ) .

If the direction of the transverse wind velocity is represented by one axis, Eq. (6) can be written as

\begin{matrix} W_{e} (f) = \frac{τ_{r}}{2 π} \int_{0}^{\infty} W_{a} (κ, ϕ) W_{s}^{*} (κ, ϕ) κ d κ \times \int_{0}^{2 π} 2 π δ (V κ \cos ϕ - f) d ϕ, \\ = τ_{r} \int_{0}^{\infty} W_{a} (κ, ϕ) W_{s}^{*} (κ, ϕ) κ d κ \times \int_{0}^{2 π} δ (V κ \cos ϕ - f) d ϕ, \end{matrix}

where V is the transverse wind speed. Here, the integration of the impulse function is

\int_{- \infty}^{\infty} f (t) δ (t - t_{0}) = f (t_{0}) .

Therefore,

\begin{matrix} W_{e} (f) = τ_{r} \int_{f / V}^{\infty} W_{a} (κ) W_{s}^{*} (κ) κ d κ \frac{1}{V \sqrt{κ^{2} - {(f / V)}^{2}}}, \\ = \frac{τ_{r}}{V} \int_{0}^{\infty} W_{a} (\sqrt{κ^{2} + \frac{f^{2}}{V^{2}}}) W_{s}^{*} (\sqrt{κ^{2} + \frac{f^{2}}{V^{2}}}) d κ . \end{matrix}

2.2. Derivation of the temporal frequency spectrum

The power spectral component can be viewed as the probability of each frequency; we therefore consider the expected value of the frequency. When the spectrum W_s²(κ) can be described by the von Kármán spectrum,

W_{s}^{2} (κ) = \frac{0.033 C_{n}^{2} \exp (- κ^{2} / κ_{m}^{2})}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}},

where

κ_{m} = 5.92 / l_{0}

and

κ_{0} = 2 π / L_{0}

. l ₀ is an inner scale and L ₀ is an outer scale. The window function a(r) is circular with a diameter D, as below:

{\begin{matrix} \begin{matrix} a (r) = 1 & | r | \leq D / 2 \end{matrix} \\ \begin{matrix} a (r) = 0 & | r | > D / 2 \end{matrix} \end{matrix} .

The spatial spectrum of W_a(κ) is given by

W_{a} (κ) = \frac{π D^{2}}{4} \frac{2 J_{1} (π D κ)}{(π D κ)},

where κ is the scalar spatial frequency. Using Eqs. (11) and (13), the temporal power spectrum is given by

\begin{matrix} W_{e}^{2} (f) = {| \frac{τ_{r}}{V} \int_{0}^{\infty} W_{a} (\sqrt{κ^{2} + \frac{f^{2}}{V^{2}}}) W_{s}^{*} (\sqrt{κ^{2} + \frac{f^{2}}{V^{2}}}) d κ |}^{2}, \\ = \frac{τ_{r}^{2}}{V^{2}} \int_{0}^{\infty} {(\frac{π D^{2}}{4})}^{2} \frac{2^{2} J_{1}^{2} (π D \sqrt{κ^{2} + f^{2} / V^{2}})}{{(π D \sqrt{κ^{2} + f^{2} / V^{2}})}^{2}} \times \frac{0.033 C_{n}^{2} \exp [- (κ^{2} + f^{2} / V^{2}) / κ_{m}^{2}]}{{(κ^{2} + f^{2} / V^{2} + κ_{0}^{2})}^{11 / 6}} d κ, \\ = \frac{0.033 C_{n}^{2} τ_{r}^{2} D^{2}}{4 V^{2}} \int_{0}^{\infty} \frac{J_{1}^{2} (π D \sqrt{κ^{2} + f^{2} / V^{2}})}{(κ^{2} + f^{2} / V^{2})} \times \frac{\exp [- (κ^{2} + f^{2} / V^{2}) / κ_{m}^{2}]}{{(κ^{2} + f^{2} / V^{2} + κ_{0}^{2})}^{11 / 6}} d κ . \end{matrix}

The normalized frequency components can be viewed as the probability of occurrence at each frequency. The normalized power spectrum corresponding to the probability function is formulated as follows

P_{W} (f) = f W_{e}^{2} (f) / \int_{0}^{\infty} W_{e}^{2} (x) d x .

The expected value of the frequency corresponding to the mean frequency can be obtained by multiplying the frequency by the weighting spectra:

{\bar{f}}_{W} = \int_{0}^{\infty} P_{W} (f) d f .

If we modulate the amplitude and phase randomly in each complex frequency component of W_e(f), the power spectral density (PSD) for generating the random time-varying signals can be calculated from

W_{e}^{'} (f) = W_{e} (f) \times \exp [j θ_{r a n d} (f)],

where θ_rand(f) is the random phase perturbation from 0 to 2π. Therefore, the time-varying electrical signals can be given by the inverse Fourier transform by

e (t) = | ℑ^{- 1} [W_{e}^{'} (f)] | .

3. Scintillation theory

3.1. Gamma-gamma distribution

The probability density function (PDF) of the received signal for the moderate-to-strong-turbulence regime can be well explained by the gamma-gamma distribution, which was also in good agreement with the simulation data [20–24]. The gamma-gamma distribution is given by [24]

p_{G} (I) = \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} I^{(α + β) / 2 - 1} \times K_{α - β} (2 \sqrt{α β I}), I > 0,

where I is the intensity, Γ() is the gamma function and K_a(b) is the modified Bessel function of the second kind. The parameters of the gamma-gamma distribution with the large-scale and small-scale scintillation are given by

α = \frac{1}{σ_{x}^{2}} and β = \frac{1}{σ_{y}^{2}},

where σ_x ² and σ_y ² are the normalized variances for the large-scale and small-scale scintillation, respectively. The scintillation index is related to these parameters by

σ^{2} = \frac{1}{α} + \frac{1}{β} + \frac{1}{α β} .

3.2. Rms wind speed in the HV model

The structural parameter of the HV model is given by [24] and the C_n ² model modified for the optical ground station at National Institute of Information and Communications Technology (NICT) is given by [25]

\begin{matrix} C_{n}^{2} (h) = M \times 0.00594 {(v / 27)}^{2} {(10^{- 5} h)}^{10} \exp (- h / 1000) + 2.7 \times 10^{- 16} \exp (- h / 1500) \\ + A \exp (- h / 100), \end{matrix}

where M is the new parameter added for the NICT optical ground station and h is the altitude of the optical ground station. When C_n ² becomes larger than 10⁻¹⁴, the scintillation can be considered to be in the moderate-to-strong-turbulence regime. The rms wind speed in meters per second, v, is determined from

v = {[\frac{1}{15 \times 10^{3}} \int_{5 \times 10^{3}}^{20 \times 10^{3}} V_{B}^{2} (h) d h]}^{1 / 2},

where V_B(h) is described by the Bufton wind model, which is obtained as

V_{B} (h) = ω_{g} h + v_{g} + 30 \exp (- \frac{h - 9400}{4800}) .

Here, v_g is the ground wind speed, and ω_g is the slew rate associated with satellite motion relative to an observer on the ground. As shown in Fig. 2 , the rms wind speed can exceed 200 m/s when the antenna slew rate is greater than 0.8°/s. The transverse wind velocity V in Section 2 is regarded as the rms wind speed v in this paper. Equation (23) is the conventional equation; therefore, it is useful to create linkages between the rms wind speed and the antenna slew rate for LEO satellites. As the outer scale is very sensitive for the peak frequency components in the power spectra, the outer scale can be properly fitted with the measured power spectra at the desired rms wind speed.

Fig. 2 Rms wind speed as a function of the antenna slew rate at the optical ground station based on the Bufton wind model.

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4. Numerical results

4.1. Temporal frequency spectrum

Figures 3(a) and 3(b) show the temporal power spectra fW_e ²(f) calculated for aperture diameters of 5 cm and 32 cm, and an rms wind speed of 76 m/s. The peak frequency for the 5-cm diameter is higher than that for the 32-cm diameter. The temporal power spectra are calculated based on Eq. (15). The power spectra at different rms wind speeds of 76 m/s and 117 m/s are plotted in Fig. 3(b). The peak frequency component for v = 117 m/s becomes higher than that for v = 76 m/s.

Fig. 3 Normalized power spectra of fW_e ²(f) as functions of the rms wind speed and the receiver aperture diameter. (a) Rms wind speed of v = 76 m/s and the receiver aperture diameters of 5 cm and 32 cm. (b) Rms wind speeds of 76 m/s and 117 m/s and the receiver aperture diameter of 32 cm.

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4.2. Mean frequency

The mean frequencies can be estimated using Eq. (26). The power spectra fW_e ²(f) are integrated and the expected values of the frequencies are plotted in Fig. 4 . The two curves were calculated with aperture diameters of 5 cm and 32 cm, indicated as “We(f) model” in Fig. 4. The measurement results obtained in space-to-ground laser communications experiments [19] are also plotted in the figure and these values were measured by a pin-photo diode (PD), an avalanche photo diode (APD), and a quadrant detector (QD) with aperture diameters of 5 cm, 31.8 cm and 31.8 cm, respectively.

Fig. 4 Mean frequency of the power spectra fW_e ²(f) for receiver aperture diameters of 5 cm and 32 cm as a function of the antenna slew rate.

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4.3. Scintillation index

The normalized variance, which is called the scintillation index, can be calculated using the C_n ² model in Ref [25]. The parameters used for the simulation are shown in Table 1 . From Eq. (22), C_n ² at an altitute of 122 m for the NICT optical ground station in Koganei can be estimated to be 2.68 x 10⁻¹⁴ m^-2/3. The atmospheric coherence lengths, known as Fried’s parameter, are estimated to be 4.2 cm and 4.7 cm at elevation angles of 23.5° and 35.2°, which correspond to seeing angles of 5.3 arcsec and 4.7 arcsec, respectively. The scintillation indices can be estimated to be 0.093 and 0.083 for the aperutre diameter of 32 cm at elevation angles of 23.5° and 35.2° based on the parameters in Table 1. The scintillation indices are 0.61 and 0.71 for the aperture diameter of 5 cm.

Table 1. Parameters for Calculating the Scintillation Index

View Table

4.4. Generation of random time-varying signals

Figures 5(a) and 5(b) show the random time-varying signals generated from the proposed model of Eq. (18) with aperture diameters of 5 cm and 32 cm, and rms wind speed of v = 76 m/s. At the smaller apertures, the time-varying signals have higher frequency fluctuations. At higher rms wind speeds, the random time-varying signals have higher frequency fluctuations, as shown in Figs. 5(c) and 5(d). The normalized intensities are rescaled by the scintillation indices estimated in Subsection 4.3.

Fig. 5 Time-varying signals with different circular apertures and different rms wind speeds with the random phase pertubation in the PSDs. (a) Rms wind speed of 76 m/s and the receiver aperture diameters of 5 cm. (b) Rms wind speed of 76 m/s and the receiver aperture diameters of 32 cm. (c) Rms wind speed of 76 m/s and the receiver aperture diameter of 32 cm. (d) Rms wind speed of 117 m/s and the receiver aperture diameter of 32 cm.

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In this simulation, two Fast Fourier Transforms (FFTs) and some integral calculations are required for generating the time-varying signals. The calculation is not especially complex, and so it is possible to generate the random time-varying signals in real time. Actual implementation will be the next step for this study. The proposed simulation model will assist in examining the error correcting codes in the laboratory level test and can be used to test equipment for free-space laser propagation through atmospheric turbulence.

4.5. Histogram of the time-varying signals

Figure 6 shows histograms of the random time-varying signals with different wind speeds, which are the same data as in Figs. 5(c) and 5(d). The dynamic range of the signal fluctuation is about 20 dB at a probability of 10⁻². Comparing Figs. 6(a) and 6(b), if the wind speed changes, the PDF has almost the same profile. The gamma-gamma distrubtion is also plotted in Fig. 6. The simulated data has good agreement with the gamma-gamma distribution; however, there is some difference in the vicinity at lower intensities. From Figs. 6(a) and (b), the simulated results have longer tails. The longer tails will cause deeper fading in the signals, and will contribute to a worst-case senario for evaluation tests using the proposed model. We already examined the comparison between the calculated and measured data in Figs. 9–14 in the reference [25]. The probability density functions for the uplink and downlink measured in the ground-to-satellite laser communication experiments had good agreement with the gamma-gamma distribution.

Fig. 6 Histograms of the time-varying signals with the receiver aperture diameter of 5 cm and simulated gamma-gamma distribution. (a) Elevation angle of 23°. (b) Elevation angle of 35°.

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5. Conclusion

The temporal characteristics resulting from aperture averaging were discussed for downlink optical channels and the simulation model for the fading channel was developed. The temporal power spectral densities were derived and the random time-varying signals were generated as a function of the receiver aperture size and rms wind speed. At smaller apertures and higher rms wind speeds, the random time-varying signals have higher frequency fluctuations. The results proposed here will assist in examining the optimal error correcting codes and the appropriate interleaver given the communication purpose, and also in evaluating equipment using the proposed model for free-space laser propagation through atmospheric turbulence.

Acknowledgments

The authors would like to express their sincere gratitude to the Japan Aerospace Exploration Agency (JAXA) and NEC Toshiba Space Systems, Ltd., for conducting the NICT optical ground station to the Optical Inter-orbit Communications Engineering Test Satellite (OICETS) experiments.

References and links

1. D. L. Fried, “Scintillation of a ground-to-space laser illuminator,” J. Opt. Soc. Am. 57(8), 980–983 (1967). [CrossRef]

2. J. L. Bufton, “Scintillation statistics measured in an earth-space-earth retroreflector link,” Appl. Opt. 16(10), 2654–2660 (1977). [CrossRef] [PubMed]

3. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58(10), 1523–1545 (1970). [CrossRef]

4. A. Ishimaru, “Temporal frequency spectra of multifrequency waves in turbulent atmosphere,” IEEE Trans. Antenn. Propag. 20(1), 10–19 (1972). [CrossRef]

5. S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. 61(10), 1285–1292 (1971). [CrossRef]

6. D. P. Greenwood and D. L. Fried, “Power spectra requirements for wave-front-compensative systems,” J. Opt. Soc. Am. 66(3), 193–206 (1976). [CrossRef]

7. D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. 67(3), 390–393 (1977). [CrossRef]

8. E. Ryznar, “Dependency of optical scintillation frequency on wind speed,” Appl. Opt. 4(11), 1416–1418 (1965). [CrossRef]

9. T. J. Gilmartin and R. R. Horning, “Spectral characteristics of intensity fluctuations on a laser beam propagating in a desert atmosphere,” IEEE J. Quantum Electron. 3(6), 254–256 (1967). [CrossRef]

10. D. H. Höhn, “Effects of atmospheric turbulence on the transmission of a laser beam at 6328 A. II-frequency spectra,” Appl. Opt. 5(9), 1433–1436 (1966). [CrossRef] [PubMed]

11. T. S. Chu, “On the wavelength dependence of the spectrum of laser beams traversing the atmosphere,” Appl. Opt. 6(1), 163–163 (1967). [CrossRef] [PubMed]

12. G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11(1), 358–367 (1994). [CrossRef]

13. D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57(2), 169–175 (1967). [CrossRef]

14. P. J. Titterton, “Scintillation and transmitter-aperture averaging over vertical paths,” J. Opt. Soc. Am. 63(4), 439–444 (1973). [CrossRef]

15. J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30(15), 1982–1994 (1991). [CrossRef] [PubMed]

16. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10(1), 53–70 (2000). [CrossRef]

17. M. Toyoshima, Y. Takayama, H. Kunimori, T. Jono, and K. Arai, “Data analysis results from the KODEN experiments,” Proc. SPIE 6709, 1–8 (2007).

18. M. Toyoshima, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, Y. Takayama, N. Kura, K. Ohinata, K. Arai, and K. Shiratama, “Ground-to-OICETS laser communication experiments,” Proc. SPIE 6304, 63040B (2006).

19. M. Toyoshima, H. Takenaka, Y. Shoji, and Y. Takayama, “Frequency characteristics of atmospheric turbulence in space-to-ground laser links,” Proc. SPIE 7685, 76850G, 76850G-12 (2010). [CrossRef]

20. R. J. Hill and R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14(7), 1530–1540 (1997). [CrossRef]

21. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999). [CrossRef]

22. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39(12), 3272–3280 (2000). [CrossRef]

23. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39(30), 5426–5445 (2000). [CrossRef] [PubMed]

24. L. C. Andrews and R. L. Phillips, eds., Laser Beam Propagation through Random Media, 2^nd ed. (SPIE Press, 2005).

25. M. Toyoshima, Y. Takayama, H. Kunimori, S. Yamakawa, and T. Jono, “Characteristics of laser beam propagation through the turbulent atmosphere in ground-to-low earth orbit satellite laser communications links,” IEICE Trans. Commun. J94-B(3), 409–418 (2011).

Parameters		Values
Parameters for C_n ²	A =	9.0x10⁻¹⁴ m^-2/3
	M =	0.2
Ground station altitude	h =	122 m
Satellite altitude		570 km
Propagation distance		1281 km and 978 km
Elevation angle		23.5° and 35.2°
Wavelength		847 nm
Receiver aperture diameter		5 cm and 32 cm
Rms wind speed		76 m/s and 117 m/s
Inner scale		4 mm
Outer scale		1.6 m

Atmospheric turbulence-induced fading channel model for space-to-ground laser communications links

Abstract

1. Introduction

2. Temporal frequency characteristics of laser beam propagation through atmospheric turbulence

2.1. Derivation of the spatial spectrum

2.2. Derivation of the temporal frequency spectrum

3. Scintillation theory

3.1. Gamma-gamma distribution

3.2. Rms wind speed in the HV model

4. Numerical results

4.1. Temporal frequency spectrum

4.2. Mean frequency

4.3. Scintillation index

4.4. Generation of random time-varying signals

4.5. Histogram of the time-varying signals

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (24)

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