Generation of temporal fading envelope sequences for the FSOC channel based on atmospheric turbulence optical parameters

Haifeng Yao; Haifeng Yao; Haifeng Yao; Qun Hao; Qun Hao; Chunyi Chen; Chunyi Chen; Ling Li; Yidi Chang; Silun Du; Xianzhu Liu; Shoufeng Tong; Zhi Liu; Shuqiang Jia; Huilin Jiang; Huilin Jiang

doi:10.1364/OE.465847

1. Introduction

Free space optical communication (FSOC) offers the advantages of good directionality, enormous bandwidth, high speed, and no electromagnetic interference (EMI). It may be employed as a key technology in the field of 5G and 6G communication [1,2]. The effects of atmospheric turbulence disturbances, which cause light wavefront distortion, light intensity random flicker, and communication signal intersymbol interference (ISI), leading in a deterioration in signal-to-noise ratio (SNR) and bit error rate (BER), are the main hurdle for FSOC. In order to analyze and mitigate the effects of turbulence on different FSOC systems, it is necessary to understand the statistical properties of light intensity fluctuations. Reference [3] shows that the atmospheric turbulence channel fading probability density function (PDF) can more accurately describe the statistical characteristics of light intensity fluctuations. These distribution functions, such as lognormal, exponential, K, Gamma-Gamma (G-G), etc., are frequently employed to depict the distribution characteristics of turbulence channel fading. The communication BER performances are also investigated by combining different communication modulation methods, such as on-off keying (OOK), m-ary pulse amplitude modulation (MPAM), m-ary phase shift keying (MPSK) and multiple quadrature amplitude modulation (MQAM). These PDFs are convincingly verified by numerical simulations and actual field experiments, which portray a process of Gaussian laser beam propagation in statistically uniform and isotropic turbulence fading channel [4–6].

The continuous time variation properties of the atmospheric turbulence fading channel are well known. Under the "Tyler Freeze" assumption, there are reasonable theoretical approximation studies. However, the temporal characteristics are neglected. It prevents the generation of turbulence time-domain continuous signals since the present beam propagation model of light intensity random fluctuation cannot be adequately and accurately defined. Therefore, it is challenging to employ Monte Carlo numerical simulation method to obtain time-continuous signal passing through atmospheric turbulence to process FSOC signal. Usually, when verifying the proposed digital processing algorithm, many FSOC experiments are carried out to sample experimental data in this scenario which need a real field experimental environment or a built equivalent simulation environment in the laboratory [7,8]. However, the experimental equipment used in this method, especially for high-speed FSOC systems, is exceedingly harsh and expensive. Furthermore, it is incomplete because all the turbulence states from strong to weak cannot be traversed.

A continuous Markov process can accurately define this distribution property, according to the PDF analysis of radio frequency (RF) communication fading channel [9,10]. The numerical simulation approach allows for any fading parameter values and non-isotropic fading scenarios. Autoregressive (AR) stochastic models can be employed to compute colored noise and non-Gaussian processes, and autocovariance functions (ACFs) can be employed to generate sequences that fit the temporal characteristics of RF fading channels [11–18]. Stochastic differential equations (SDEs) relying on accurate ACF solutions can also be utilized to describe and estimate its power spectral density (PSD) [10]. The above two methods focus on the PDF analysis of the RF fading channel. Their analytical solutions for PSD and ACF are generally straightforward. However, these functions derivation in turbulence fading channels are a process of solving complex analytical equations because the effects of turbulence disturbances under different scales need to simultaneously be considered. Under certain conditions, atmospheric turbulence ACF can be approximated and simplified to a simple form. Based on Markov models of non-Gaussian exponentially correlated processes, D. Bykhovsky approximated the ACF of turbulence fading channel as a simple exponential form $\exp ( - \tau )$ (see [19], Eq. (4); [20], Eqs. (3)–(5)). More than that, he generated a time-domain correlation sequence of lognormal, K, Gamma, and Gamma-Gamma (G-G) by employing stochastic differential equations (SDEs). Assumed that turbulence exists ${l_0} \ll \rho \ll \sqrt {\lambda L}$ (${l_0}$, $\rho$, $\lambda$ and $L$ denote the inner scale of turbulence, the spatial coherence radius of the optical wave at the receiving point, wavelength, and propagation distance, respectively. A. Jurado et al. deduced ACF into the square form of exponent and then used the Fourier transform to obtain PSD which is the filter function (see [21], Eqs. (5)–(6)). Meanwhile, the AR multi-channel generalization model (see [22], Eq. (8)) was used to solve the temporal sequence under various turbulence. According to the assumptions of Ref. [19], our study team also used the SDEs algorithm to obtain a numerical solution for the Jonson $SB_s$ PDF of the Fiber-FSOC system and employed the real-time sequence to investigate the relationships between system time delay and reciprocity (see [23], Eqs. (15)–(19)).

However, the generation accuracy of the aforementioned temporal sequences is entirely dependent on the ACF approximation, which ignores the high-frequency information of the theoretical values. Despite the existence of a definition for coherence time, it fails to adequately integrate the real FSOC system characteristics (such as propagation distance, light wavelength, scintillation index, atmospheric structure constant, etc.). As a result, the temporal performance of the FSOC fading channel cannot be accurately characterized by calculating these simulation sequences under the engineering measurable parameters for a unified experimental procedure. Therefore, we need to investigate a more accurate method for generating the temporal sequence model of the FSOC fading channel, which can provide an important numerical simulation approach for the analysis of the modulated signal’s time-domain characteristics and the BER evaluation. we can provide a signal experimental sample generation technique for advanced FSOC signal processing algorithms under various turbulence fading channel characteristics, and a theoretical basis for algorithm performance evaluation.

In this paper, our purpose is to properly employ the ACF of the atmospheric turbulence channel, which significantly restores turbulence disturbance information at multiple frequency bands, to generate a temporal sequence related to the FSOC parameters without simplifying them. Section $1$ is the introduction. a time-domain correlation sequence generation model for turbulence fading channel is given in Section $2$. Section $3$ indicates the experiments and analysis for temporal sequence generation. Finally, the conclusion is elaborated in Section $4$.

2. Time-domain correlation sequence generation model for turbulence fading channel

We begin to consider ${{B}_{\ln X}}\left ( \rho \right )$ large-scale and ${{B}_{\ln Y}}\left ( \rho \right )$ small-scale log-irradiance covariance functions, the ACF model for optical turbulence is given by [24]

(1)$${B_I}\left( \rho \right){\rm{ = exp}}\left[ {{B_{\ln X}}\left( \rho \right) + {B_{\ln Y}}\left( \rho \right)} \right] - 1,$$

In Ref. [25], complete expressions for ${{B}_{\ln Y}}\left ( \rho \right )$ and ${{B}_{\ln X}}\left ( \rho \right )$ are, respectively,

(2)$$\begin{aligned} {B_{\ln X}}(\rho ) &= 1.06\sigma _R^2\int_0^1 {\int_0^\infty {{\eta ^{ - 11/6}}} } \exp \left( { - \eta /{\eta _X}} \right){J_0}(\rho \sqrt {k\eta /L} )(1 - \cos \eta \xi )d\eta d\xi\\ &\cong 0.16\sigma _R^2\eta {_X^{7/6}}{_1{F_1}}\left( {\frac{7}{6};1; - \frac{{k{\rho ^2}{\eta _X}}}{{4L}}} \right), \end{aligned}$$

(3)$$\begin{aligned} {B_{\ln Y}}(\rho ) &= 1.06\sigma _R^2\int_0^1 {\int_0^\infty {{{\left( {\eta + {\eta _Y}} \right)}^{ - 11/6}}} } {J_0}(\rho \sqrt {k\eta /L} )(1 - \cos \eta \xi )d\eta d\xi\\ &\cong 1.27\sigma _R^2{\left( {\frac{{k{\rho ^2}}}{{L{\eta _Y}}}} \right)^{5/12}}{K_{5/6}}\left( {\sqrt {\frac{{k{\rho ^2}{\eta _Y}}}{L}} } \right). \end{aligned}$$

where, $\sigma _{R}^{2}=1.23C_{n}^{2}{{k}^{7/6}}{{L}^{11/6}}$ is Rytov variance, $C_n^2$, $L$, $k={2\pi }/{\lambda }$ and $\lambda$ are atmospheric structure constant, propagation distance, number of waves and wavelength, respectively. ${_1{F}_{1}}\left ( \cdot \right )$ and ${{K}_{{5}/{6}\;}}\left ( \cdot \right )$ denote a confluent hypergeometric function and a modified Bessel function of the second kind. ${{\eta }_{X}}$ and ${{\eta }_{Y}}$ are given by

(4)$${\eta _X} = \frac{{2.61}}{{1 + 1.11\sigma _R^{12/5}}},$$

(5)$${\eta _Y} = 3\left( {1 + 0.69\sigma _R^{12/5}} \right).$$

Substitute Eqs. (2)–(5) into Eq. (1), and let $\rho ={{V}_{\bot }}\tau$, we can deduce the ACF of time

(6)$$\begin{aligned} {{B_I}\left( \tau \right) = }&{\exp \left[ {\frac{{0.49\sigma _R^2}}{{{{\left( {1 + 1.11\sigma _R^{12/5}} \right)}^{7/6}}}}{}_1^{}{F_1}\left( {\frac{7}{6};1; - \frac{{kV_ \bot ^2{\tau ^2}{\eta _X}}}{{4L}}} \right)} \right.}\\ &{\left. { + \frac{{0.50\sigma _R^2}}{{{{\left( {1 + 0.69\sigma _R^{12/5}} \right)}^{5/6}}}}{{\left( {\frac{{kV_ \bot ^2{\tau ^2}{\eta _Y}}}{L}} \right)}^{5/12}}{K_{5/6}}\left( {\sqrt {\frac{{kV_ \bot ^2{\tau ^2}{\eta _Y}}}{L}} } \right)} \right] - 1.} \end{aligned}$$

where ${{V}_{\bot }}$ and $\tau$ represent transverse wind speed and time index, respectively. We also define normalized ACF by the expression

(7)$${\psi _I}\left( \tau \right) = \frac{{{B_I}\left( \tau \right)}}{{{B_I}\left( 0 \right)}}.$$

According to the Ref. [26], Eq. (7) is a stationary stochastic process and we can derive its power spectrum form

(8)$${S_I}(\Omega ) = 4\int_0^\infty {{\psi _I}\left( \tau \right)} \cos \left( {\Omega \tau } \right)d\tau.$$

where, $\Omega = 2\pi \upsilon$, and $\upsilon = {1 \mathord {\left / {\vphantom {1 \lambda }} \right. } \lambda }$ denotes light frequency. ${{f}_{s}}$ is sampled to obtain the normalized digital frequency as $\omega ={2\pi }/{{{f}_{s}}}$ in order to further discretize the data. When additive white noise (AWGN) with a power of $\sigma ^{2}$ through a transfer function composed of rational filter $H\left ( \omega \right )$, the temporal signal can be produced [27]. Hence, Eq. (8) can be written as

(9)$${S_I}(\omega ) = \sum_{n = 1}^\infty {{\psi _I}\left( n \right)} {e^{ - j\omega n}} = \left| {\frac{{B\left( \omega \right)}}{{A\left( \omega \right)}}} \right|\sigma^2 = H\left( \omega \right)\sigma^2.$$

where $A\left ( \omega \right ) = 1 + {a_1}{e^{ - j\omega }} + \cdots {a_n}{e^{ - jn\omega }}$, $B\left ( \omega \right ) = 1 + {b_1}{e^{ - j\omega }} + \cdots {b_m}{e^{ - jm\omega }}$. Observing the covariance structure of the autoregressive moving average (ARMA) process [28,29], it can be deduced

(10)$$\begin{aligned} \left\{ \begin{array}{l} {\psi _I}\left( 0 \right) + \sum\limits_{n = 1}^N {{a_i}{\psi _I}\left( { - n} \right)} = {\sigma ^2}\\ {\psi _I}(m) + \sum\limits_{n = 1}^N {{a_n}{\psi _I}\left( {m - n} \right)} = 0 \end{array} \right., \end{aligned}$$

(11)$$\begin{aligned} {{\boldsymbol{\Psi} _{I,{(n + 1)} \times {(n + 1)}}}}\left[ {\begin{array}{c} 1\\ {{\boldsymbol{\theta} _{n \times 1}}} \end{array}} \right] = \left[ {\begin{array}{c} {{\boldsymbol{\sigma} _{n\times 1} ^2}}\\ 0 \end{array}} \right] = \left[ {\begin{array}{cccc} {{\psi _I}\left( 0 \right)} & {{\psi _I}\left( { - 1} \right)} & \cdots & {{\psi _I}\left( { - n} \right)}\\ {{\psi _I}\left( 1 \right)} & {{\psi _I}\left( 0 \right)} & {} & \vdots \\ \vdots & {} & \ddots & {{\psi _I}\left( { - 1} \right)}\\ {{\psi _I}\left( n \right)} & \cdots & {} & {{\psi _I}\left( 0 \right)} \end{array}} \right]\left[ {\begin{array}{c} 1\\ {{a_1}}\\ \vdots \\ {{a_n}} \end{array}} \right] = \left[ {\begin{array}{c} {{\sigma ^2}}\\ 0\\ \vdots \\ 0 \end{array}} \right]. \end{aligned}$$

where ${a_i}$ represents the coefficient of the filter, which can be regarded as the transfer function of the turbulence fading channel. Let ${\alpha _n} = {\psi _I}\left ( {n + 1} \right ) + \boldsymbol {\widetilde {\Psi _{I,1 \times n}^ * }} {\boldsymbol {\theta _{n \times 1}}}$, Eq. (11) can be written as

(12)$$\begin{aligned}\left[ {\begin{array}{cc}{{\boldsymbol{\Psi} _{I,{(n + 1)} \times {(n+1)}}}} &{\begin{array}{c}{{\psi _I}\left( { - n - 1} \right)}\\{{\boldsymbol{\Psi} _{I,n \times 1}}}\end{array}}\\{\begin{array}{ccc}{{\psi _I}\left( {n + 1} \right)} &{\widetilde {\boldsymbol{\Psi} _{I,1 \times n}^ * }}\end{array}} &{{\psi _I}\left( 0 \right)}\end{array}} \right]\left[ {\begin{array}{c}1\\{{\boldsymbol{\theta} _{n \times 1} }}\\0\end{array}} \right] = \left[ {\begin{array}{c}{ \boldsymbol{\sigma} _{n \times 1} ^2}\\0\\{{\alpha _n}}\end{array}} \right].\end{aligned}$$

Here, ${\boldsymbol {\Psi } _{I,n \times 1}} = {\left [ {\begin {array}{ccc}{{\psi _I}\left ( 1 \right )} & \cdots & {{\psi _I}\left ( n \right )} \end {array}} \right ]^T}$, ${\widetilde {{\boldsymbol {\Psi } _{I,n \times 1}}}} = {\left [ {\begin {array}{ccc} {{\psi _I}^ * \left ( 1 \right )} & \cdots & {{\psi _I}^ * \left ( n \right )} \end {array}} \right ]^T}$. According to Ref. [30], we employ the Yule-Walker equation to solve Eq. (12) by using the order recursive solution method, we can get

(13)$$\begin{aligned} \begin{array}{ccc} {{\boldsymbol{\theta} _{(n + 1) \times 1}} = \left[ {\begin{array}{c} {{\boldsymbol{\theta} _n}}\\ 0 \end{array}} \right] + {\xi _{n + 1}}\left[ {\begin{array}{ccc} {{\boldsymbol{\theta} _n}}\\ 1 \end{array}} \right],} &{\sigma^2(n+1) = \sigma^2(n)\left( {1 - {{\left| {{\xi( n + 1)}} \right|}^2}} \right),} &{{\sigma ^2}(n){\theta(n)} = {Y(n)}}, \end{array} \end{aligned}$$

(14)$$\sigma^2(n + 1) = \sigma^2(n)\left( {1 - {{\left| {{\xi(n + 1)}} \right|}^2}} \right),$$

(15)$${\sigma ^2}(n)H\left( \omega \right) = {Y(n)}.$$

where $\xi \left ( {n + 1} \right ) = {{ - {\alpha _n}} \mathord {\left / {\vphantom {{ - {\alpha _n}} {{\sigma ^2}(n)}}} \right. } {{\sigma ^2}(n)}}$. In this way, we can deduce the coefficients ${\theta (n)}$ of the transfer function of the turbulence fading channel in turn, and can calculate the time-domain correlation sequence $Y$ employing Eqs. (13)–(15). The above is a time-dependent calculation based on AWGN as random values. Still, the PDF is not taken into account, resulting in the generated sequence not obeying the turbulence disturbance theory. For this reason, assuming this system is a free-space receiver and the aperture smoothing effect is not considered, this paper uses the G-G function as the PDF of the turbulence fading channel, its parameters $\alpha$ and $\beta$ can be related to the turbulence structure constant $C_n^2$ , which is given by [31]

(16)$${p_I}\left( I \right) = \frac{{2{{\left( {\alpha \beta } \right)}^{\left( {\alpha + \beta } \right)/2}}}}{{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)I}}{\left( {\frac{I}{{\left\langle I \right\rangle }}} \right)^{\left( {\alpha + \beta } \right)/2}}{K_{\alpha - \beta }}\left( {2\sqrt {\frac{{\alpha \beta I}}{{\left\langle I \right\rangle }}} } \right), I > 0,$$

(17)$$\begin{array}{cc} {\alpha = \frac{1}{{\sigma _X^2}} = \frac{1}{{\exp \left( {\sigma _{\ln X}^2} \right) - 1}},} &{\beta = \frac{1}{{\sigma _Y^2}} = \frac{1}{{\exp \left( {\sigma _{\ln Y}^2} \right) - 1}}}, \end{array}$$

(18)$$\begin{array}{cc} {\sigma _{\ln X}^2 = \frac{{0.49\sigma _R^2}}{{{{\left( {1 + 1.11\sigma _R^{12/5}} \right)}^{7/6}}}},} & {\sigma _{\ln Y}^2 = \frac{{0.51\sigma _R^2}}{{{{\left( {1 + 0.69\sigma _R^{12/5}} \right)}^{5/6}}}}.} \end{array}$$

where $\left \langle {{I^n}} \right \rangle$ represents the $n^{th}$ moment of the light intensity $I$, ${\rm {\Gamma }}\left ( \cdot \right )$ denotes Gamma function. Note that here $I$ is the theoretical light intensity sequence. When the normalized random light intensity sequence ${\left \{ {{{{{\hat I}_n}} \mathord {\left / {\vphantom {{{{\hat I}_n}} {\left \langle I \right \rangle }}} \right. } {\left \langle I \right \rangle }}} \right \}_{1 \times N}}$ obtained from the simulation obeys the Gamma-Gamma PDF, it can be expressed as

(19)$$\begin{array}{ll} {{{\left[ {{{{{\hat I}_1}} \mathord{\left/ {\vphantom {{{{\hat I}_1}} {\left\langle {\hat I} \right\rangle }}} \right. } {\left\langle {\hat I} \right\rangle }},{{{{\hat I}_2}} \mathord{\left/ {\vphantom {{{{\hat I}_2}} {\left\langle {\hat I} \right\rangle }}} \right. } {\left\langle {\hat I} \right\rangle }}, \ldots {{{{\hat I}_n}} \mathord{\left/ {\vphantom {{{{\hat I}_n}} {\left\langle {\hat I} \right\rangle }}} \right. } {\left\langle {\hat I} \right\rangle }}} \right]}^T} = \hat I \in PD{F_{Gamma - Gamma}},} & {i = 1,2, \ldots ,n} \end{array}.$$

According to Ref. [11], $\vec I$ is obtained after $\hat I$ rearranging in order of $Y$ rank, the correlation coefficient (CC) of the sequence ${\rho _{\vec I ,Y}}$ can be calculated by

(20)$${\rho _{\vec I ,Y}} = \frac{{\left\langle {\left( {\vec I \left( i \right) - \left\langle {\vec I } \right\rangle } \right)\left( {Y\left( i \right) - \left\langle Y \right\rangle } \right)} \right\rangle }}{{\sqrt {\left\langle {{{\left( {\vec I \left( i \right) - \left\langle {\vec I } \right\rangle } \right)}^2}} \right\rangle \left\langle {{{\left( {Y\left( i \right) - \left\langle Y \right\rangle } \right)}^2}} \right\rangle } }} = 1.$$

Therefore, we perfectly map the temporal correlation information of $Y$ to $\vec I$ and satisfy

(21)$$\left\{ {\begin{array}{c} {\vec I \in PD{F_{Gamma - Gamma}}}\\ {\underbrace {\left\langle {\left( {\vec I \left( i \right) - \left\langle {\vec I } \right\rangle } \right)\left( {\vec I \left( {i - m} \right) - \left\langle {\vec I } \right\rangle } \right)} \right\rangle }_{\vec \psi \left( m \right)} = {\psi _{\vec I }}\left( m \right)}. \end{array}} \right.$$

Equation (21) shows that the $\vec I$ generated ACF ${\psi _{\vec I }}\left ( m \right )$ is equal to the numerical simulation result of the theoretical solution Eq. (7) ${\psi _I}\left ( m \right )$, and the PDF information is preserved. The modeling process of the above Eqs. (1)–(21) can be depicted in Fig. 1. That is, the ACF created is the same as the numerical solution of the theoretical solution (6), and the PDF information is preserved. Figure 1 depicts the modeling method as mentioned above Eqs. (1)–(21).

Fig. 1. Implement route for modeling turbulent time-domain correlation signal generation. Where all modeling procedures correspond to Eqs. (1)–(21). The ACF and PSD are determined first, and then the turbulence fading channel function structure is designed using spectral estimation theory. The filter coefficients are calculated utilizing the Yule-Walker algorithm. Moreover, a sequence of temporal Gaussian correlation is generated by combining AWGN. Finally, according to the rank mapping principle of the time-domain Gaussian correlation, the random light intensity sequence obeying the G-G PDF is reordered, which highly restores the turbulence disturbance information in different frequency bands.

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3. Experiments and analysis for temporal sequence generation

As shown in Fig. 2, we first investigate the normalized ACF performance under various turbulence intensities, following the implement route of Fig. 1. The blue, black and red lines represent $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, respectively. The light wavelength, propagation distance and transverse wind speed at this time are given by $\lambda = 1550 \ {\rm {nm}}$, $L=5000 \ \rm {m}$, and ${{v}_{\bot }}=1 \ \rm {m/s}$, respectively. Therefore, the Rytov variances $\sigma _R^2$ of turbulence fading channel obtained by our numerical simulation are 19.03, 1.903 and 0.19, respectively, corresponding to the three cases of strong, medium and weak turbulence. The normalized ACF decreases as turbulence intensity increases, and the entire line trend is slanted towards the $Y$-axis. According to Refs. [23,32], the coherence time ${\tau _d}$ is defined as

(22)$$\psi_I \left( {{\tau _d}} \right) = {e^{ - 1}}.$$

Fig. 2. Normalized ACF and PDF performances for different turbulence fading channels. (a) normalized ACF performances for different turbulence intensities, the blue, black and red lines represent $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, respectively. The coherence times ${\tau _d}$ are $6.4 \ \rm {ms}$, $26.0 \ \rm {ms}$ and $35.2 \ \rm {ms}$, respectively. When ACF tends to $0$, i.e., ${\psi _I}\left ( \tau \right ) \to 0$, the blue, black and red lines denote ${\tau _d} = 0.3616 \ \rm {s}$, ${\tau _d} = 0.1181 \ \rm {s}$ and ${\tau _d} = 0.0821 \ \rm {s}$, respectively. The parameters of corresponding normalized light intensity G-G PDF are (b) $\alpha =7.2306$, $\beta =1.0453$,(c) $\alpha =3.9935$, $\beta =1.7438$ and (d) $\alpha =12.1495$, $\beta =10.6132$, respectively. The light wavelength, propagation distance and transverse wind speed at this time are given by $\lambda = 1550 \ {\rm {nm}}$, $L=5000 \ \rm {m}$, and ${{v}_{\bot }}=1 \ \rm {m/s}$, respectively.

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After statistical calculation, we can see that the coherence times ${\tau _d}$ corresponding to strong, medium and weak turbulence are $6.4 \ \rm {ms}$, $26 \ \rm {ms}$ and $35.2 \ \rm {ms}$, respectively. The coherence time ${\tau _d}$ becomes smaller as turbulence intensity increases, which is demonstrated in Fig. 2(a). Equations (16)–(18) of the aforesaid atmospheric parameters are numerically analyzed for further studying the distribution performance of normalized light intensity at this period. Its form is skewed from exponential to "bell-shaped," and the normalized light intensity distribution is more concentrated, indicating that the scintillation variance is lesser at this time, as illustrated in Figs. 2(b)–2(d), The light intensity scintillation effect mainly consists of large-scale and small-scale scintillation when the turbulence intensity is moderate to weak. The large-scale logarithmic scintillation index tends to zero as turbulence intensity increases, i.e., $\sigma _{\ln x}^2 \to 0$, while the small-scale turbulence logarithmic scintillation index tends to be saturated, i.e., $\sigma _{\ln y}^2 \to 0.69$. The small-scale is mostly affecting the turbulence disturbance, see Eq. (18), which is one of the reasons for using the G-G PDF as turbulence fading channel function in this paper.

According to Eq. (6), the transverse wind speed ${v_ \bot }$ is another important element impacting the coherence time ${\tau _d}$, as it can invoke atmospheric movement and trigger random turbulence medium fluctuations on the light wave propagation path. Therefore, the coherence time ${\tau _d}$ performances under various wind speeds ${v_ \bot }$, and propagation distances $L$ and atmospheric turbulence structure constants $C_n^2$ are depicted in Fig. 3(a). For the convenience of plotting, the turbulence state at this time is set to medium $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$. When the wind speed ${v_ \bot }$ remains constant, the coherence time ${\tau _d}$ increases as the propagation distance $L$ grows. This is because the coherence time ${\tau _d}$ is a function of $1/L$, indicating that the random medium can be comparable to a sizeable random function in the airspace as the propagation distance $L$ rises. The higher correlation between their media and the longer distance, which reflects the more substantial blocking effect in the time-domain and the larger system robustness, that is, with the increase of distance, the optical time variation characteristics of the entire system weaken, as plotted in Fig. 3(b). However, when the propagation distance $L$ is constant, the coherence time ${\tau _d}$ decreases as the wind speed ${v_ \bot }$ increases. The wind speed accelerates the unpredictability of the turbulence random phase and diminishes spatial coherence under the "Taylor freeze" assumption, resulting in a drop in coherence time, as illustrated in Fig. 3(c).

Fig. 3. Contributions of transverse wind speed ${v_ \bot }$ and light wave propagation distance $L$ for the coherence time ${\tau _d}$ of atmospheric turbulence fading channel. (a) Coherence time ${\tau _d}$ distribution performances of atmospheric turbulence fading channel under different transverse wind speeds ${v_ \bot }$ and light wave propagation distances $L$, the turbulence state at this time is set to medium $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$; (b) the relationships between light wave propagation distance $L$ and coherence time ${\tau _d}$ when the transverse wind speed ${v_ \bot } = 1 \rm {m/s}$; (c) the relationships between transverse wind speed ${v_ \bot }$ and coherence time ${\tau _d}$ when the propagation distance $L=5000 \ \rm {m}$; (d) the relationships between Fresnel frequency ${{\omega }_{\tau }} = {{{v}_{\bot }}} / {\sqrt {{L}/{k}}}$ and coherence time ${\tau _d}$. The blue, black and red lines represent $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, respectively. The light wavelength is set to $\lambda = 1550 \ {\rm {nm}}$.

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We define Fresnel frequency to better assess the effect of factors on temporal correlation [33].

(23)$${\omega _\tau } = \frac{{{v_ \bot }}}{{\sqrt {{L \mathord{\left/ {\vphantom {L k}} \right. } k}} }}.$$

Equation (23) takes into account the effects of wind speed, propagation distance and light wavelength, as shown in Fig. 3(d). The coherence time reduces from $25.1331 \ \rm {ms }$ to $18.2480 \ \rm {ms }$ when ${\omega _ \bot } = 40{\ \rm { rad/s}}$ and the turbulence intensity increases from $C_n^2 = 5 \times {10^{ - 16}}{{ \ \rm {m}}^{ - 2/3}}$ to $C_n^2 = 5 \times {10^{ - 15}}{{ \ \rm {m}}^{ - 2/3}}$. The coherence time at this moment drops to $4.4143 \ \rm {ms}$ if the turbulence level is further enhanced to $C_n^2 = 5 \times {10^{ - 14}}{{ \ \rm {m}}^{ - 2/3}}$. Because large-scale turbulence clusters lose energy and become small-scale turbulence clusters when the turbulence intensity rises. The diffraction effect of light increases at this time, increasing the phase randomness of the medium of the light wave propagation route, destroying the light wavefront, and shortening the coherence time. As the Fresnel frequency increases further, the coherence time approaches zero.

The atmospheric characteristics of Eq. (6) can be precisely represented, allowing us to construct turbulence fading channel random signals that directly mirror the real turbulence state, as observed in Fig. 3. Therefore, combining Eqs. (1)–(21), the light wavelength $\lambda$, transverse wind speed ${v_ \bot }$ and transverse wind speeds ${v_ \bot }$ are set to $\lambda = 1550 \ {\rm {nm}}$, ${v_ \bot } = 1{\ \rm {m/s}}$ and $L = 5000 \ \rm {m}$, respectively. Atmospheric turbulence state are also given by $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, respectively. These parameters are easily measured directly in the actual FSOC system. We can simulate and create time-domain correlated continuous sequence signals in the different turbulent fading channels using the Monte Carlo approach, as illustrated in Figs. 4(a)–4(c). According to Ref. [24] (see Chapt. 8, Eq. (9)), the scintillation index can be expressed as

(24)$$\sigma _{\vec I }^2 = {{\left( {\left\langle {{{\vec I }^2}} \right\rangle - {{\left\langle {\vec I } \right\rangle }^2}} \right)} \mathord{\left/ {\vphantom {{\left( {\left\langle {{{\vec I }^2}} \right\rangle - {{\left\langle {\vec I } \right\rangle }^2}} \right)} {{{\left\langle {\vec I } \right\rangle }^2}}}} \right. } {{{\left\langle {\vec I } \right\rangle }^2}}}.$$

Despite the insufficient sampling interval and generated data capacity, the scintillation index of Figs. 4(a)–4(c) can be approximated by $1.2210$, $0.9667$, and $0.1839$, respectively. It reveals that with the increase of the turbulence degree, the temporal signal jitter degree increases, and the random fluctuation trend increases. These signals are all time continuity, and the signal envelopes are highly similar to the previous experiments done by our research team (see Ref. [23]).

Fig. 4. Generation of time-domain correlated continuous sequence signals in different turbulence fading channel. (a) blue, (b) black and (c) red lines represent $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, respectively. The light wavelength $\lambda$, transverse wind speed ${v_ \bot }$ and propagation distance ${L}$ are set to $\lambda = 1550 \ {\rm {nm}}$, ${v_ \bot } = 1{\ \rm {m/s}}$ and $L = 5000 \ \rm {m}$.

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Moreover, we know that Eq. (24) is the scintillation index obtained by engineering calculation, which is mainly composed of the refraction of large-scale turbulence and the diffraction of small-scale turbulence. It can be written as

(25)$$\sigma _I^2 = \exp \left[ {\frac{{0.49\sigma _R^2}}{{{{\left( {1 + 1.11\sigma _R^{12/5}} \right)}^{7/6}}}} + \frac{{0.51\sigma _R^2}}{{{{\left( {1 + 0.69\sigma _R^{12/5}} \right)}^{5/6}}}}} \right] - 1,\quad 0 \le \sigma _R^2 < \infty.$$

The theoretical scintillation indices of Figs. 4(a)–4(c) can be obtained using theoretical numerical computation, and they are $1.2273$, $0.9675$, and $0.1843$, respectively. Meanwhile, we found that the data deviations are $0.005$, $0.0008$, $0.006$, respectively, all of which are lower than $0.006$, manifesting that for different turbulence fading channels, the generation mode of time-domain correlated continuous sequence signal conforms to the theoretical calculation Eq. (25).

We plot Fig. 5 in order to verify the matching degree between the time-domain signals in Fig. 4 and the theoretical normalized ACFs, Under the three turbulence states of strong, medium and weak (corresponding to Figs. 5(a)–5(c), respectively), the normalized ACFs ${\psi _{\vec I }}\left ( \tau \right )$ after rank matching are well matched with the ACFs ${\psi _I}\left ( \tau \right )$ calculated theoretically.

Fig. 5. ACF performance of time-domain correlated continuous sequence signals in Different turbulence fading channel. (a), (b) and (c) represent $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, respectively. The light wavelength $\lambda$, and transverse wind speeds ${v_ \bot }$ are set to $\lambda = 1550 \ {\rm {nm}}$, ${v_ \bot } = 1{\ \rm {m/s}}$ and $L = 5000 \ \rm {m}$, respectively. The blue dotted, black solid and red dashed line denote theoretical calculation $I$, after rank mapping $\vec{I}$ and before rank mapping $\hat {I}$, respectively.

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According to Eq. (20), the correlation coefficients ${\rho _{I,\vec I }}$ between the three turbulence states can be calculated as $0.9990$, $0.9993$ and $0.9998$ , respectively. It shows that the time-domain signals after rank matching contain strong time-domain correlation and atmospheric turbulence disturbance characteristic information. However, the statistical value of this correlation coefficient is not 1, indicating an error in the AR random process. We also draw the ACF curve ${\psi _{\hat I}}\left ( \tau \right )$ of the time domain signal $\hat I$ before rank matching for explaining the error source. The theoretical correlation coefficients ${\rho _{I,\hat I}}$ can be calculated as $0.9992$, $0.9998$ and $0.9998$. Obviously, $\hat I$ and $\vec I$ share the same time characteristics. The ACF ${\psi _I}\left ( \tau \right )$ temporal information of signal $I$ is not lost during rank matching, and it precisely inherits the ACF ${\psi _{\hat I}}\left ( \tau \right )$ of $\vec I$, i.e., ${\psi _{\vec I }}\left ( \tau \right ) = {\psi _{\hat I}}\left ( \tau \right ) = {\psi _I}\left ( \tau \right )$, proving that our proposed model Eq. (20) is correct.

It is worth noting that one of the key parameters used to evaluate FSOC system BER is the statistical distribution properties of time-domain light intensity signals. Therefore, the PDF cures of the time-domain continuous light intensity signal of Fig. 4 and Fig. 5 are plotted in Fig. 6. According to Eq. (20), we calculate that the correlation coefficients of Figs. 6(a)–6(c) are $0.9999$, $0.9998$ and $0.9999$, respectively. We can deduce from Fig. 6 that the time-domain continuous light intensity signals $\vec I$ generated under various turbulence fading channels conform to the statistical characteristics of G-G PDFs, proving that our proposed time-domain continuous light intensity signal generation model Eq. (21) is correct.

Fig. 6. PDF performances of time-domain correlated continuous sequence signals and theoretical calculations in different turbulence fading channels. (a), (b) and (c) represent $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, respectively. The light wavelength $\lambda$, transverse wind speed ${v_ \bot }$ and propagation distance ${L}$ are set to $\lambda = 1550 \ {\rm {nm}}$, ${v_ \bot } = 1{\ \rm {m/s}}$ and $L = 5000 \ \rm {m}$, respectively. The blue (a), black (b) and red (c) lines represent the theoretical PDFs, respectively, and the fitting efficiencies are 0.9999, 0.9998, and 0.9999, respectively.

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In addition, PSD is also one of the most essential characteristics of time signals. We obtain ${\vec I _n}$, however, by ACF ${\psi _I}\left ( \tau \right )$ Monte Carlo simulation. It’s challenging to offer an analytical solution to Eq. (8) because of its intricate nature. According to the Ref. [24,33,34], we employ the weak turbulence approximation theory to simplify Eq. (8) into

(26)$$\begin{aligned} {S_I}(\omega ) = \frac{{6.95\sigma _R^2}}{{{\omega _{_\tau }}}}{\mathop{\rm Re}\nolimits}& \left\{ {{{\left( {\frac{\omega }{{{\omega _\tau }}}} \right)}^{ - 8/3}}\left[ {1{ - _1}{F_1}\left( { - \frac{5}{6}; - \frac{1}{3}; - \frac{{i{\omega ^2}}}{{2\omega _\tau ^2}}} \right)} \right]} \right.\\ & \left. { - 0.72{i^{4/3}}_1{F_1}\left( {\frac{1}{2};\frac{7}{3}; - \frac{{i{\omega ^2}}}{{2\omega _\tau ^2}}} \right)} \right\}. \end{aligned}$$

Equation (26) is taken logarithm, the attenuation envelope tends to a straight line with $0$ slope when $\omega <\omega _t$, i.e., $\lg \left ( {{S_I}\left ( \omega \right )} \right ) = \lg \left ( {6.95\sigma _R^2/{\omega _{_\tau }}} \right )$. Moreover, Eq. (26) can be equal to a function with a slope of −8/3 when $\omega >\omega _t$, i.e., greater than the Fresnel frequency, indicating that the power spectrum attenuation here obeys the −8/3 rule. We have plotted theoretical fitting curves in Fig. 7(a) for turbulence intensities $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, propagation distance $L = 5000 \ \rm {m}$, and wind speed ${v_ \bot } = 1 \rm {m/s}$, for example, the red, black and blue lines denote $\lg \left ( {{S_I}\left ( \omega \right )} \right ) = - \left ( {8/3} \right )\lg \left ( {{\omega \mathord {\left / {\vphantom {\omega {{\omega _\tau }}}} \right. } {{\omega _\tau }}}} \right ) + 5.2749$, $\lg \left ( {{S_I}\left ( \omega \right )} \right ) = - \left ( {8/3} \right )\lg \left ( {{\omega \mathord {\left / {\vphantom {\omega {{\omega _\tau }}}} \right. } {{\omega _\tau }}}} \right ) + 5.62749$, $\lg \left ( {{S_I}\left ( \omega \right )} \right ) = - \left ( {8/3} \right )\lg \left ( {{\omega \mathord {\left / {\vphantom {\omega {{\omega _\tau }}}} \right. } {{\omega _\tau }}}} \right ) + 7.2749$, respectively. The above relationships can be written as

(27)$$\lg \left( {{S_I}\left( \omega \right)} \right) \propto \left\{ {\begin{array}{ll} { - \frac{8}{3}\lg \left( {\frac{\omega }{{{\omega _\tau }}}} \right) + \lg \left( {\frac{{6.95\sigma _R^2}}{{{\omega _{_\tau }}}}} \right),} & {\omega \ge {\omega _\tau }}\\ {\lg \left( {\frac{{6.95\sigma _R^2}}{{{\omega _{_\tau }}}}} \right),} & {\omega < {\omega _\tau }} \end{array}} \right.$$

where $\propto$ represents the envelope trend. We perform a fitting analysis on the PSD envelopes of the time-domain continuous light intensity signals ${\vec I _n}$ generated in Figs. 4(a)–4(c), as shown in Figs. 7(b)–7(d). The first segments of the PSD envelopes are discovered to be a comparatively smooth straight line with a slope of approximately 0 that conform to the law that the value of a straight line reduces dramatically as turbulence weakens. The curves are fitted to the second segments in Figs. 7(b)–7(d), for example blue cure $\lg \left ( {{S_{\vec I }}\left ( f \right )} \right ) = - \left ( {{8 \mathord {\left / {\vphantom {8 3}} \right. } 3}} \right )\lg \left ( f \right )$ - $0.5$, black cure $\lg \left ( {{S_{\vec I }}\left ( f \right )} \right ) = - \left ( {{8 \mathord {\left / {\vphantom {8 3}} \right. } 3}} \right )\lg \left ( f \right ){-}1.5$, red cure $\lg \left ( {{S_{\vec I }}\left ( f \right )} \right ) = - \left ( {{8 \mathord {\left / {\vphantom {8 3}} \right. } 3}} \right )\lg \left ( f \right ) - 2.5$, which are an attenuation with a slope of −8/3 and conform to the theory shown in Eq. (27). In the comparison of Fig. 7(a), Eq. (21) highly restores the attenuation information of the atmospheric turbulence fading channel in different frequency bands. Synthesizing the information presented in Figs. 1–7, it turns out that when analyzing the FSOC performance, our proposed time-domain signal generation model may help researchers estimate the channel state information of the simulated FSOC in real-time based on the actual system and atmospheric parameters. This provides a strong theoretical basis and simulation method for signal regeneration [35], as well as being beneficial for real-time signal advanced algorithms processing [1], temporal characteristics study and adaptive optics control algorithm optimization [7], and real-time simulation of turbulent environments [20].

Fig. 7. Fitted PSD cure performances of time-domain correlated continuous sequence signals and theoretical calculations in different turbulence fading channels. (a) PSD curves of theoretical calculation for (b) - (d), see Eqs. (26)–(27); (b), (c) and (d) represent $C_n^2 = 5 \times {10^{ - 14}}{{\ \rm {m}}^{ - 2/3}}$, $C_n^2 = 5 \times {10^{ - 15}}{{\ \rm {m}}^{ - 2/3}}$ and $C_n^2 = 5 \times {10^{ - 16}}{{\ \rm {m}}^{ - 2/3}}$, respectively. The light wavelength $\lambda$, transverse wind speed ${v_ \bot }$ and propagation distance ${L}$ are set to $\lambda = 1550 \ {\rm {nm}}$, ${v_ \bot } = 1{\ \rm {m/s}}$ and $L = 5000 \ \rm {m}$, respectively. The blue (a), black (b) and red (c) cures represent the fitted PSDs, respectively.

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4. Conclusion

This paper proposes a continuous-time FSOC channel fading sequence generation model obeying G-G PDF, which incorporates the ACF time characteristic information of atmospheric parameters. First, the normalized channel fading function ACF model and PSD analytical formula are established under the influence of parameters such as transmission distance, optical wavelength, scintillation factor, and atmospheric structure constant. The Yule-Walker function is utilized to calculate the filter coefficients in the AR stochastic process, and the numerical solution of the time-domain Gaussian correlation sequence is derived using white Gaussian noise. Moreover, after rank mapping, a light intensity signal generation model that obeys time-domain correlation with PDF information is established, taking into account the association between the G-G parameters and the large and small scale turbulence parameters. The performances of ACF in different atmospheric conditions are then analyzed using the Monte Carlo numerical approach, and the corresponding PDFs are given. The investigations reveal that the defined Fresnel frequency ${\omega _d}$ may comprehensively characterize the physical meaning of ACF, and that the FSOC fading channel coherence time degradation rises as ${\omega _d}$ grows. We generate the sample signals of the temporal sequence under various turbulence sequences that match the waveform time characteristics reported in our previous experiments, and the scintillation indexes are highly consistent with the theoretical solution. The correlation coefficients are greater than $0.9996$, and the PSDs obey the decay trend of −8/3 in case of various turbulence situations. The results show that the temporal sequences we generated effectively restore the fading time-domain and frequency-domain informations for the FSOC turbulence channels, providing an important theoretical basis and numerical analysis methods for the construction of real-time simulations of turbulent environments, as well as in the performances evaluation of communication BER, channel estimation, and the sample data formation of advanced modulation and demodulation algorithms.

Funding

Natural Science Foundation of Chongqing (cstc2021jcyj-msxmX0457); State Key Laboratory Foundation of applied optics (SKLA02022001A11); China Postdoctoral Science Foundation (2021M700415, 2021TQ0035); National Natural Science Foundation of China (61775022, 62105029, U2141231).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Generation of temporal fading envelope sequences for the FSOC channel based on atmospheric turbulence optical parameters

Abstract

1. Introduction

2. Time-domain correlation sequence generation model for turbulence fading channel

3. Experiments and analysis for temporal sequence generation

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (27)

Optics Express