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Error rate analysis of few-mode fiber based free-space optical communication

Xingjie Fan, Jianyu Niu, Jing Ma, Yiming Bian, and Julian Cheng

Open Access Open Access

Abstract

Few-mode fiber (FMF) based receiver emerges as a promising solution for free-space optical (FSO) communication due to its excellent performance in the presence of turbulence. We propose a theoretical model that uses the coupling efficiency of FMF to evaluate the performance of FMF-based FSO system in the presence of turbulence. The series solutions and asymptotic solutions to bit-error rate (BER) of such system are derived for maximal-ratio combining (MRC) scheme and equal gain combining (EGC) scheme over the Gamma-Gamma turbulence channels. Simulation results show that for the FMF-based FSO system, the asymptotic BER of MRC and EGC are highly accurate in the large transmitted optical power regimes.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Free-space optical (FSO) communication is susceptible to atmospheric turbulence. In the presence of turbulence, the sensitivity for the traditional photodetector based receiver is thermal noise limited, leading to a performance bottleneck [1]. By contrast, few-mode fiber (FMF) based receiver can provide better performance in the presence of turbulence [2,3] and it has attracted much attention from the academia [4,5]. On the one hand, FMF-based receiver has higher coupling efficiency [5] and sensitivity [1], resulting in better performance. On the other hand, the development of mode demultiplexers enables FMF-based receiver to be compatible with current single-mode fiber (SMF) based optical communication systems [2].

The promising trends of large-scale applications of FMF-based FSO systems require systematic analyses to lay a solid theoretical foundation for further performance improvements. Recently, a theoretical model of the FMF-based FSO system based on non-mode-selective mode demultiplexers (NMSM) was proposed [5]. The NMSM is a kind of mode demultiplexers, in which each mode of FMF is distributed to random SMF, thus reducing system complexity. However, the coupling effects of different modes of the FMF after passing through the turbulent channel have distinct characteristics [6]. Thus, the NMSM cannot distinguish different modes of FMF, thus limiting the performance of FMF-based FSO systems. Another kind of mode demultiplexers is mode-selective mode demultiplexers (MSM), in which each mode of FMF corresponds to a specific SMF. The theoretical analysis based on MSM can provide a more accurate performance evaluation of the FMF-based FSO system than the NMSM. Experimental results also show that the MSM, such as mode-selective photonic lantern [7] and multi-plane light converter [8], can shed light on the application of FMF-based FSO system.

In this work, we aim to evaluate the performance of the FMF-based FSO system with MSM in the presence of turbulence. In particular, the impact of turbulence on mode coupling, which may significantly deteriorate the system performance, is measured by mode coupling efficiency of each mode of FMF to characterize the performance of the FSO system. We need to address two key challenges. First, we cannot obtain the instantaneous coupling efficiency of each mode of the FMF due to the uncertainty of atmospheric turbulence. Although we can use average coupling efficiency to analyze the error rate of the system for simplicity, this method is still inaccurate. Second, we have to consider the influence of mode demultiplexer on mode coupling even that the existing mode demultiplexer can achieve low-loss reception [9].

To address these two challenges, we use the average coupling efficiencies of each mode of FMF as mathematical expectations of a truncated multivariate Gaussian distribution to describe the power distribution at the output of the demultiplexer. Specifically, as the truncated multivariate Gaussian distribution is used to describe small power fluctuation [5,10], the instantaneous coupling efficiency is assumed to obey the truncated multivariate Gaussian distribution with the average coupling efficiency of the specific mode of FMF as the mathematical expectation. Moreover, the variances of the truncated multivariate Gaussian distribution are used to characterize the power fluctuation caused by the instantaneous coupling efficiency and the mode demultiplexer. It should be noted that, as the coupling efficiency can be considered as the degree of mode field matching between the incident light field and the fiber mode, the coupling efficiency is usually assumed to be unaffected by the optical irradiance. To the best of the authors’ knowledge, the probability density function (PDF) of instantaneous coupling efficiency does not have a widely accepted model despite that the PDF of the received optical irradiance can be well described, such as Gamma-Gamma distribution model [11].

We consider an intensity-modulated direct detection (IM/DD) FSO system with on-off keying (OOK) modulation. For simplicity, we let the FMF input represent the input of the mode demultiplexer and let the SMFs output represent the output of the mode demultiplexer. We utilize the inherent physical characteristics of mode demultiplexer to obtain a reliable estimation of weighting coefficients of maximal-ratio combining (MRC) scheme. The traditional method used to estimate the weighting coefficients of the MRC scheme depends on previous signals [12], which may lead to a significant delay and high complexity of the system hardware in the presence of turbulence. The weighting coefficient of a specific SMF output can be estimated through the average coupling efficiency of the corresponding mode of FMF input. We also evaluate the impact of the estimation error on MRC scheme. Due to formidable mathematical complexity, it is challenging to obtain accurate closed-form error rate expressions. In particular, we propose two approximate methods to obtain the analytical expressions of BER and a lower bound of BER. We also analyze the BER performance of FMF-based system with MRC in the presence of turbulence and compare the results with that of the equal gain combining (EGC) scheme.

This paper is organized as follows. Section 2 presents the system description. In Section 3 and Section 4, we derive the signal-to-noise ratio (SNR) and the BER theoretical formulations. The integral solution, series solution and asymptotic solution of the BER are presented. Numerical results and analysis are shown in Section 5. Section 6 concludes the paper.

2. System description

2.1 Coupling efficiency of FMF

The fiber coupling efficiency is defined as the ratio of the average power coupled into the fiber to the average available power in the receiver aperture plane. In the presence of atmospheric turbulence, the average coupling efficiency of the $i$-th mode can be expressed as [13]

$$a_i= \frac{4}{\pi d_R^{2}} \int\!\!\!\int_A\!\exp\left(-|\vec{r}_1-\vec{r}_2|^{2}/\rho^{2}\right)E_{i}^{*}(\vec{r}_1)E_{i}(\vec{r}_2)\mathop{}\!\mathrm{d}\vec{r}_1\mathop{}\!\mathrm{d}\vec{r}_2$$
where $A$ is the plane of the receiver lens; $d_R$ is the receiver lens diameter and $E_{i}^{*}(\vec r)$ is the complex conjugate of the $i$-th backpropagated fiber mode field on plane $A$. The notation $\rho$ is the spatial coherence distance of a plane wave distorted by atmospheric turbulence under weak fluctuation, and it is given by [14]
$$\rho=\left(1.46C_n^{2}k^{2}L\right)^{{-}3/5}$$
where $C_n^{2}$ is the refractive-index structure constant; $k$ is the wavenumber of the optical field and $L$ is the communication link distance.

For a weakly guided step-index FMF, the electric field distribution of the guided modes at the end face of fiber can be represented by the solution to the scalar Helmholtz equation, which is also known as linearly polarized (LP) modes. Due to formidable mathematical complexity, it is challenging to obtain an exact solution of the coupling efficiency directly from the LP mode of a step-index fiber. However, the LP mode of a step-index fiber can be well approximated by a scale-adapted set of Laguerre-Gaussian (LG) modes [15]. The solution of LG modes at the plane $A$ can be represented as

$$\mathrm{LG}_{pl}(r,\phi)=\frac{B_{pl}}{\omega}\left(\sqrt{2}\frac{r}{\omega}\right)^{l}L^{l}_p\left(\frac{2r^{2}}{\omega^{2}}\right)\exp\left({-\frac{r^{2}}{\omega^{2}}}\right)e^{{-}jl\phi}$$
where $l$ and $p$ are the indices for the guided azimuthal and radial components and these indices vary with different modes (i.e., the $\mathrm {LP}_{lp}$ mode corresponds the $\mathrm {LG}_{p-1,l}$ mode [15]); $B_{pl}=\left (\frac {2p!}{\pi (l+p)!}\right )^{\frac {1}{2}}$ is a normalization factor; $\omega$ is the radius of backpropagated fiber modes at the receiver lens (To simplify the analysis, we assume the radii of different backpropagated fiber modes are the same); $L^{l}_p$ are the associated Laguerre polynomials; $j$ is the unit imaginary number. Substituting Eq. (3) into Eq. (1), after some algebraic manipulations [6], we can obtain
$$\begin{aligned} a_i=&2\pi B^{2}_{pl}\int_0^{1}\!\!\!\int_0^{1}\!I_{l}\left(\frac{A_R}{A_C}2x_1x_2\right)\exp\left(-\frac{A_R}{A_C}(x_1^{2}+x_2^{2})\right)\\ &\times \,\,L^{l}_p\left(2\tau^{2}x_1^{2}\right)L^{l}_p\left(2\tau^{2}x_2^{2}\right)\left(2\tau^{2}x_1x_2\right)^{l+1}\exp\left(-\tau^{2}(x_1^{2}+x_2^{2})\right)\mathop{}\!\mathrm{d} x_1\mathop{}\!\mathrm{d} x_2. \end{aligned}$$
Here $A_R=\pi d_R^{2}/4$ is the aperture area; $A_C=\pi \rho ^{2}$ is the spatial coherence area of the incident plane wave, also known as speckle size; $\tau =d_R/2\omega$ is the coupling geometry parameter.

In this work, we ignore the correlation terms between modes in coupling efficiency from free space to FMF, and the total average coupling efficiency can be expressed as

$$a_{tot}=\sum_{i=1}^{N}a_i$$
where $N$ is the total number of modes held by the FMF.

2.2 Power distribution at the SMFs output

The incident optical signal is coupled by the FMF after passing through atmospheric turbulence and then transmitted to the mode demultiplexer. For the incident light field that has already coupled into an $N$-mode FMF, the coupling efficiency $a_i\ (i=1,2,\ldots ,N)$ for the $i$-th mode has been determined and satisfies Eqs. (4) and (5). We can obtain the power distributed for each mode at FMF input as

$$P_{F,i}=a_iP_R=\frac{a_i}{a_{tot}}P_F,\qquad 0\leq a_i<1$$
where $P_{R}$ is the power received at the FMF aperture plane; $P_{F}$ is the total power coupled into the FMF input, i.e., $P_F=a_{tot}P_R$.

We assume that each mode of FMF maps to a specific SMF, and the number of modes hold by FMF is equal to the number of SMFs. Let $\zeta \ (0< \zeta \leq 1)$ denote the insertion loss factor from the FMF input into the SMFs output, then the total output power of the SMFs output is $P_S=\zeta P_F$. Similarly, we denote the power distributed at SMFs output by $P_{S,i}$ for each SMF and denote the ratio of $P_{S,i}$ to $P_R$ by $b_i$, i.e., $P_{S,i}=b_iP_R=\frac {b_i}{\zeta a_{tot}}P_S$. In addition to the optical irradiance, the power at each SMF can also fluctuate due to the instantaneous coupling efficiency and the mode demultiplexer, we regard $b_i\ (i=1,2,\ldots ,N)$ as random variables (RVs) that satisfy

$$b_1+b_2+\cdots+b_N=\zeta a_{tot},\quad 0\leq b_i< 1.$$
The RV $b_i$ can be characterized by its mathematical expectation $E[b_i]=\zeta a_i$ and variance $VAR[b_i]=\sigma _i^{2}$, the variance might vary with manufacturing parameters of mode de-multiplexer, different modes and turbulence intensities. If we denote the power fluctuation part of $b_i$ by $x_i$ for $i=1,2,\ldots ,N$, we have
$$b_i=\zeta a_i+x_i.$$

As $a_i\ (i=1,2,\ldots ,N)$ is a set of constants, according to Eqs. (7) and (8), RVs $x_{i}\ (i=1,2,\ldots ,N)$ satisfy

$$x_1+x_2+\cdots+x_N=0.$$
Then the power distribution at SMFs output is equivalently characterized by a joint PDF $f_{\boldsymbol{\chi}} (x_1,x_2,\ldots ,x_N)$, where $\boldsymbol{\chi}=[x_1,x_2,\ldots ,x_N]^{T}$. According to Eq. (9), we know that covariance matrix $\Sigma _{\boldsymbol{\chi}}$ is a singular matrix, and the rank is $N-1$. We obtain $f_{\boldsymbol{\chi}}(x_1,x_2,\ldots ,x_N)=f_{\boldsymbol{\chi^{*}}}(x_1,x_2,\ldots ,x_{N-1})\delta (x_1+x_2+\cdots +x_N)$, where ${\boldsymbol{\chi^{*}}}=[x_1,x_2,\ldots ,x_{N-1}]^{T}$; $\delta (\cdot )$ is the Dirac delta function. Then we can obtain the joint PDF of $x_1,x_2,\ldots ,x_N$ as
$$f_{\boldsymbol{\chi}}(x_1,x_2,\ldots,x_N)=(2\pi)^{\frac{1-N}{2}}|\Sigma_{\boldsymbol{chi^{*}}}|^{-\frac{1}{2}}g(\boldsymbol{\chi})\delta\textstyle\left(\sum\nolimits^{N}_{i=1}x_i\right)$$
where $g(\boldsymbol{\chi})=\mathrm {exp}\left (-\frac {1}{2}\boldsymbol{\chi}^{\boldsymbol {*}T}\Sigma ^{-1}_{\boldsymbol{\chi^{*}}}{\boldsymbol{\chi^{*}}}\right )$. The covariance matrix $\Sigma _{\boldsymbol{\chi^{*}}}$for$\boldsymbol{\chi^{*}}$ is the first $(N-1)\times (N-1)$dimensional submatrix of $\Sigma _{\boldsymbol{\chi}}$, $\Sigma _{\boldsymbol{\chi^{*}}}$ can be obtained as
$$\Sigma_{\boldsymbol{\chi^{*}}}= \begin{bmatrix} \sigma_1^{2} & \kappa_{1,2} & \cdots\ & \kappa_{1,N-1}\\ \kappa_{2,1} & \sigma_2^{2} & \cdots\ & \kappa_{2,N-1}\\ \vdots & \vdots & \ddots & \vdots \\ \kappa_{N-1,1} & \kappa_{N-1,2} & \cdots\ & \sigma_{N-1}^{2}\\ \end{bmatrix}$$
where$\kappa _{i,j}=E[x_ix_j]$ is the correlation coefficient between$x_i$and$x_j\ (j\neq i,\ i,j=1,2,\ldots ,N)$. The variance $\sigma _i^{2}$ is used to describe the fluctuation amplitude of the optical power of the SMF corresponding to the $i$-th mode of the FMF; the covariance$\kappa _{i,j}$, which satisfies $\kappa _{i,j}=\kappa _{j,i}$, is used to measure the degree of joint variation of the power distribution of the SMFs corresponding to the $i$-th and$j$-th mode of the FMF. Thus, the RVs $x_i$ can be described by a multivariate Gaussian distribution with constraint Eq. (9). However, the exact distribution should be formed as truncated multivariate Gaussian distribution [16] due to the Gaussian RVs $x_i$ take values from $[-\zeta a_i, 1-\zeta a_i]$ instead of $(-\infty , \infty )$. The joint PDF can be obtained as
$$f_{\boldsymbol{\chi}}(x_1,x_2,\ldots,x_N)=\frac{ g(\boldsymbol{\chi})\delta\textstyle\left(\sum\nolimits^{N}_{i=1}x_i\right)}{\displaystyle\int_{-\zeta\textbf{a}}^{\textbf{1}-\zeta \textbf{a}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\scriptscriptstyle{N}\displaystyle\quad g(\boldsymbol{\chi})\delta\textstyle\left(\sum\nolimits^{N}_{i=1}x_i\right)\mathop{}\!\mathrm{d}\boldsymbol{\chi}},\qquad -\zeta\textbf{a}\leqslant\boldsymbol{\chi}\leqslant\textbf{1}-\zeta\textbf{a}$$
where $\textstyle \int _{-\zeta \textbf {a}}^{\textbf {1}-\zeta \textbf {a}}\!\!\!\!\!\!\!\!\!\!\!\!\scriptscriptstyle {N}\qquad$ is an $N$-dimensional Riemann integral from $-\zeta \textbf {a}$ to $\textbf {1}-\zeta \textbf {a}$, and where $\textbf {a}=[a_1,a_2,\ldots ,a_N]^{T}$ and $\textbf {1}=[1,1,\ldots ,1]^{T}$.

2.3 IM/DD FSO receiver

In most practical systems, the noise is mainly determined by shot noise and thermal noise. Both kinds of noises can be modeled as an additional white Gaussian noise (AWGN) that is statistically independent of the desired signal [17]. For a channel with turbulence, the current at the receiver output with OOK is given by

$$I=R_dP_{in}+n=R_dA_sI_s+n$$
where $R_d$ is the photodetector responsivity; $P_{in}$ is the incident optical power on the photodetector, which is the product of photodetector area $A_s$ and irradiance of the received optical signal $I_s$. The noise current $n$ is modeled as AWGN having mean zero and variance $\sigma _n^{2}=N_0/2$. Then, we can obtain the instantaneous SNR for IM/DD FSO system at the input of the electrical demodulator as
$$\gamma=\frac{R_d^{2}A_s^{2}I_s^{2}}{\sigma_n^{2}}.$$
For simplicity, we normalize the first moment of the irradiance $I_s$ to unity. The average electrical SNR for a direct detection system is defined as [18,19]
$$\bar{\gamma}=E\left[\gamma\right]=\frac{R_d^{2}A_s^{2}}{\sigma_n^{2}}\left(E\left[I_s\right]\right)^{2}=\frac{R_d^{2}A_s^{2}}{\sigma_n^{2}}$$
where $E[\cdot ]$ denotes the mathematical expectation.

2.4 Atmospheric turbulence model

In the FSO channel, the atmospheric turbulence causes irradiance fluctuations, known as scintillation, which is mainly caused by small temperature variations in the atmosphere, resulting in refraction-index random variations. The optical irradiance $I_s$ is assumed to be the Gamma-Gamma model, which describes a wide range of irradiance fluctuations from weak to strong turbulence regimes. The PDF of the received signal irradiance is given by [11]

$$f_{I_s}\left (I_s \right )=\frac{2{{ \left (\alpha \beta \right )}^{\frac{\alpha +\beta}{2}}}}{\Gamma \left (\alpha \right )\Gamma \left (\beta \right )}I_s^{\frac{\alpha +\beta}{2}-1}{{K}_{\alpha -\beta}} \left(2\sqrt{\alpha \beta I_s} \right)$$
where $\Gamma (\cdot )$ is the gamma function; ${K}_{\alpha -\beta }(\cdot )$ is the modified Bessel function of the second kind of order $\alpha -\beta$. The positive parameters $\alpha$ and $\beta$ are the effective numbers of small scale and large scale eddies of the scattering environment. They can be directly linked to physical parameters as [11,14]
$$\begin{aligned}\alpha&=\left[\exp{\left(\frac{0.49\sigma_R^{2}}{\left(1+1.11\sigma_R^{12/5}\right)^{7/6}}\right)-1}\right]^{{-}1}\\ \beta&=\left[\exp{\left(\frac{0.51\sigma_R^{2}}{\left(1+0.69\sigma_R^{12/5}\right)^{7/6}}\right)-1}\right]^{{-}1} \end{aligned}$$
where $\sigma _R^{2}=1.23C_n^{2}k^{7/6}L^{11/6}$ is the Rytov variance. We can use the Rytov variance to connect the parameters of coupling efficiency and the Gamma-Gamma model. The Rytov variance can unify both turbulence parameters with plane wave incidence. To obtain series BER expressions later, we use a series expansion of the modified Bessel function [20], we can obtain
$$f_{I_s}\left (I_s \right )=\pi\sum_{p=0}^{\infty}\left(a_p\left(\alpha,\beta \right)I_s^{p+\beta-1}+a_p\left(\beta,\alpha \right)I_s^{p+\alpha-1}\right), \qquad (\alpha-\beta)\notin\mathbb{Z}$$
where $a_p\left (\alpha ,\beta \right )=\left (\alpha \beta \right )^{p+\beta }/[\sin {\left [\pi \left (\alpha -\beta \right )\right ]}\Gamma (\alpha )\Gamma (\beta )p!\Gamma (p-\alpha +\beta +1)]$.

3. SNR with EGC and MRC

We employ OOK modulation for FMF-based IM/DD system with EGC and MRC. Here EGC (or MRC) scheme is used to combine all corresponding SMF signals from the FMF signal. Because there is only one receiving port (i.e., the FMF-based receiver) and no diversity technique is introduced, the name “EGC (or MRC)” should not be confused with the diversity combining technique EGC (or MRC) in wireless communications. Besides, we propose a method to estimate the SNR of MRC by replacing the corresponding weight coefficient with the coupling efficiency of each specific mode.

3.1 EGC combiner output SNRs

For EGC scheme, the combiner sums all the electrical signals with equal weights. The output of the synthesized signal current is $I_{\scriptscriptstyle EGC}=\sum _{i=1}^{N} I_i$, where $I_i$ is the signal current of the $i$-th SMF output generated by the photodetector. Therefore, we can obtain the instantaneous SNR of the FMF-based IM/DD optical receiver with EGC as

$$\gamma_{\scriptscriptstyle EGC}=\frac{R_d^{2}}{N\sigma_n^{2}}\left(\sum_{i=1}^{N} P_{S,i}\right)^{2}=C_{\scriptscriptstyle EGC}I^{2}_s\frac{R_d^{2}A_R^{2}}{\sigma_n^{2}}=C_{\scriptscriptstyle EGC}I^{2}_s\bar{\gamma}_{\scriptscriptstyle R}$$
where $C_{\scriptscriptstyle EGC}=a_{tot}^{2}\zeta ^{2}/N$. The parameter $\bar {\gamma }_{\scriptscriptstyle R}=R_d^{2}A_R^{2}/\sigma _n^{2}$ is introduced to facilitate the following calculation, and it can be seen as the average electrical SNR of the received signal with the aperture area $A_R$. The aperture area $A_R$ can be regarded as the equivalent area of FMF aperture plane which satisfies $P_R=I_sA_R$ (See Section 4.3 for further discuss). We can obtain the average electrical SNR of the FMF-based IM/DD optical receiver with EGC as
$$\bar{\gamma}_{\scriptscriptstyle EGC}=C_{\scriptscriptstyle EGC}\bar{\gamma}_{\scriptscriptstyle R}.$$

3.2 MRC combiner output SNRs

For MRC scheme, the combiner sums all the electrical signals with different weights $m_i\ (i=1,2,\ldots ,N)$. The output of the synthesized signal current is

$$I_{\scriptscriptstyle MRC}=\sum_{i=1}^{N} m_iI_i=\frac{R_dP_S}{\zeta a_{tot}}\sum_{i=1}^{N} m_ib_i$$
where $m_i$ represents the weighting coefficient for the $i$-th SMF output of the mode demultiplexer. The noise variance of the synthesized signal is $\sigma ^{2}_{\scriptscriptstyle MRC}=\sum _{i=1}^{N}m_i^{2}\sigma _n^{2}$. Therefore, we can obtain the instantaneous SNR of the FMF-based IM/DD optical receiver with MRC as
$$\gamma_{\scriptscriptstyle MRC}=\frac{\left(\sum_{i=1}^{N} m_iI_i\right)^{2}}{\sigma^{2}_{\scriptscriptstyle MRC}}=\frac{R_d^{2}A_R^{2}I^{2}_s\left(\sum_{i=1}^{N} m_ib_i\right)^{2}}{\sum_{i=1}^{N}m_i^{2}\sigma_n^{2}}.$$
According to the properties of Cauchy’s inequality, we can obtain $\left (\sum\limits _{i=1}^{N} m_ib_i\right )^{2}\leq \sum\limits_{i=1}^{N}m_i^{2}\sum\limits_{i=1}^{N}b_i^{2}$, and if it satisfies the condition that the weights $m_i\ (i=1,2,\ldots ,N)$ are proportional to $b_i\ (i=1,2,\ldots ,N)$, the inequality takes the equal sign to obtain the maximum value of $\gamma _{\scriptscriptstyle MRC}$. Then Eq. (22) can be rewritten as
$$\gamma_{\scriptscriptstyle MRC}=C_{\scriptscriptstyle MRC}I^{2}_s\bar{\gamma}_{\scriptscriptstyle R}$$
where $C_{\scriptscriptstyle MRC}=\sum _{i=1}^{N} b_i^{2}$. Similarly, we can obtain the average electrical SNR of the FMF-based IM/DD optical receiver with MRC as
$$\bar{\gamma}_{\scriptscriptstyle MRC}=E\left[\sum_{i=1}^{N} b_i^{2}\right]\bar{\gamma}_{\scriptscriptstyle R}$$
where $E\left [\sum _{i=1}^{N} b_i^{2}\right ]=\zeta ^{2}\sum _{i=1}^{N}a_i^{2}+\sum _{i=1}^{N}\sigma _i^{2}$.

These ideal results assume perfect knowledge of $b_i\ (i=1,2,\ldots ,N)$ to obtain the weights $m_i\ (i=1,2,\ldots ,N)$. However, $b_i\ (i=1,2,\ldots ,N)$ are RVs, practical receivers must estimate the weights $m_i\ (i=1,2,\ldots ,N)$, thereby incurring estimation error that needs to be accounted for in the performance analysis. We find that FMF-based MRC for FSO system can make full use of the inherent physical characteristics that a specific mode of FMF can correspond to a specific SMF. Therefore, the weighting coefficient of a specific SMF output can be estimated through the coupling efficiency of the corresponding mode of FMF input. Let $m_i=E[b_i]=\zeta a_i$ and substitute this value into Eq. (22), the estimation of the instantaneous SNR of the FMF-based IM/DD optical receiver with MRC can be represented as

$$\gamma_{esti}=C_{esti}I^{2}_s\bar{\gamma}_{\scriptscriptstyle R}$$
where $C_{esti}=\left (\sum _{i=1}^{N} a_ib_i\right )^{2}/\sum _{i=1}^{N}a_i^{2}$. Similarly, we can obtain the estimation of the average electrical SNR of the FMF-based IM/DD optical receiver with MRC as
$$\bar{\gamma}_{esti}=\frac{E\left[\left(\sum_{i=1}^{N} a_ib_i\right)^{2}\right]}{\sum_{i=1}^{N}a_i^{2}}\bar{\gamma}_{\scriptscriptstyle R}$$
where $E\left [\left (\sum _{i=1}^{N} a_ib_i\right )^{2}\right ]=\zeta ^{2}\sum _{i=1}^{N}a_i^{4}+\sum _{i=1}^{N}a_i^{2}\sigma _i^{2}+2\sum _{i=1}^{N-1}\sum _{j=i+1}^{N}(\zeta ^{2}a_i^{2}a_j^{2}+a_ia_j\kappa _{i,j})$. The ratio of average electrical SNR with and without estimation from Eqs. (24) and (26) is
$$\xi_{SNR}=10\log\left(\frac{\bar{\gamma}_{esti}}{\bar{\gamma}_{\scriptscriptstyle MRC}}\right).$$
The ratio $\xi _{SNR}$ is used to evaluate the accuracy of the estimation.

4. Error rate analysis with EGC and MRC

In the presence of atmospheric turbulence, the unconditional average BER of FSO with OOK modulation can be expressed as

$$P_e=\int_0^{\infty} p_e(I_s)f_{I_s}\left (I_s \right )\mathop{}\!\mathrm{d} I_s$$
where $p_e=Q\left (\sqrt {\frac {\gamma }{2}}\right )$ and where $Q(\cdot )$ is the Gaussian $Q$-function.

4.1 BER with EGC and MRC

To continue the error rate derivation of EGC, one needs to substitute Eqs. (18) and (19) into Eq. (28) for $P_e$ and solve the integral. Starting with the definition of Beta function $\mathrm {B}(x,y)$ [21, eq. (8.380.2)], we can obtain a series solution to the unconditional BER of the FMF-based IM/DD FSO system with EGC as

$$\begin{aligned} P_{e,EGC}=\sum_{p=0}^{\infty}\Bigg[&b_p(\alpha,\beta)\left(\frac{C_{\scriptscriptstyle EGC}\bar{\gamma}_{\scriptscriptstyle R}}{4}\right)^{-\frac{p+\beta }{2}}\!\!\mathrm{B}\left(\frac{1}{2}, \frac{p+\beta+1}{2}\right)\\ &\left.+b_p\left(\beta,\alpha \right)\left(\frac{C_{\scriptscriptstyle EGC}\bar{\gamma}_{\scriptscriptstyle R}}{4}\right)^{-\frac{p+\alpha}{2}}\!\!\mathrm{B}\left(\frac{1}{2}, \frac{p+\alpha+1}{2}\right)\right] \end{aligned}$$
where $b_p\left (\alpha ,\beta \right )=\frac {1}{4}\Gamma \left (\frac {p+\beta }{2}\right )a_p\left (\alpha ,\beta \right )$.

Similarly, substituting Eqs. (18) and (23) into Eq. (28) for $P_e$, we can obtain a series solution to the unconditional BER of the FMF-based IM/DD FSO system with MRC as

$$\begin{aligned} P_{e,MRC}=\!\sum_{p=0}^{\infty}\!\Bigg[&b_p(\alpha,\beta )\Upsilon(N,p,\beta)\!\!\left(\frac{\bar{\gamma}_{\scriptscriptstyle R}}{4}\right)^{\frac{-p-\beta}{2}}\!\!\!\!\!\mathrm{B}\!\left(\!\frac{1}{2}, \frac{p+\beta+1}{2}\!\right)\\ &+b_p\left(\beta,\alpha \right)\Upsilon(N,p,\alpha)\!\!\left(\frac{\bar{\gamma}_{\scriptscriptstyle R}}{4}\right)^{-\frac{p+\alpha}{2}}\!\!\!\!\!\mathrm{B}\left(\!\frac{1}{2}, \frac{p+\alpha+1}{2}\!\right)\Bigg] \end{aligned}$$
where $\Upsilon (N,p,\beta )=\int \left (C_{\scriptscriptstyle MRC}\right )^{-\frac {p+\beta }{2}}f_{\scriptscriptstyle C_{MRC}}(C_{\scriptscriptstyle MRC})\mathop {}\!\mathrm {d} C_{\scriptscriptstyle MRC}$, and $f_{\scriptscriptstyle C_{MRC}}(C_{\scriptscriptstyle MRC})$ is the PDF of $C_{\scriptscriptstyle MRC}$. However, it is challenging to obtain an analytical expression from Eq. (12). On the one hand, we take the exact solution when $N=2$ as an approximation of $\Upsilon (N,p,\beta )$ according to Appendix A.. It follows that we can obtain an approximation of $\Upsilon (N,p,\beta )$ as
$$\Upsilon_1(N,p,\beta)=\sum_{i=1}^{n}\frac{\Omega \sqrt{2\pi}}{4n\sigma_1}\left(\frac{(x_i+1-b)^{2}+b^{2}}{2}\right)^{-\frac{p+\beta}{2}}\mathrm{exp}\left[-\frac{\left(x_i+1-2\zeta a_1\right)^{2}}{8\sigma_1^{2}}\right]\sqrt{1-x_i^{2}}$$
where $\Omega =\left [\Phi (\frac {1-\zeta a_1}{\sigma _1})-\Phi (\frac {-\zeta a_1}{\sigma _1})\right ]^{-1}$; $\Phi (\zeta )=\frac {1}{2}(1+\mathrm {erf}(\zeta /\sqrt {2}))$ is the cumulative distribution function (CDF) of the standard Gaussian distribution; $x_i=\cos {\frac {(2i-1)\pi }{2n}}$; $n$ is a complexity-accuracy trade-off parameter; $b=\zeta (a_1+a_2)$.

On the other hand, as shown in Appendix B, let $\sigma_{\mathrm{min}}=\mathrm{min}(\sigma_i)$, when the variance of the system $\sigma_{\mathrm{min}}^{2}$ is small, we can obtain a lower bound of $\Upsilon(N,p,\beta)$ as

$$\Upsilon_2(N,p,\beta)= \left\{\begin{array}{lr} \sum\limits_{i=1}^{m}\frac{\lambda^{\frac{2-N}{4}}\pi}{4m\sigma_{\mathrm{min}}^{2}}\left(\frac{y_i+1}{2}\right)^{\frac{N-2-2(p+\beta)}{4}}e^{-\frac{y_i+1+2\lambda}{4\sigma_{\mathrm{min}}^{2}}} I_{{\frac{N}{2}-1}}\left(\sqrt{\frac{\lambda y_i+\lambda}{2\sigma_{\mathrm{min}}^{4}}}\right)\sqrt{1-y_i^{2}},\qquad N\leq p+\beta;\\ \lambda^{-N/4}(2\sigma_{\mathrm{min}}^{2})^{\frac{N-2(p+\beta)}{4}}\frac{\Gamma\left(\frac{N-p-\beta}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}e^{-\frac{\lambda}{4\sigma_{\mathrm{min}}^{2}}} M_{\frac{2(p+\beta)-N}{4},\frac{N-2}{4}}\left(\frac{\lambda}{2\sigma_{\mathrm{min}}^{2}}\right), \qquad N>p+\beta; \end{array}\right. \label{upsilon2}$$
where $\lambda=\zeta ^{2}\sum_{i=1}^{N} a_i^{2}$; $I_{l}$ denotes the $l$th-order modified Bessel function of the first kind; $y_i=\cos{\frac{(2i-1)\pi}{2m}}$; $m$ is a complexity-accuracy trade-off parameter; $M_{a,b}(\cdot)$ is the Whittaker equation.

The approximation $\Upsilon _1(N,p,\beta )$ is the exact solution for $N=2$, while the approximation error increases with $N.$ Correspondingly, the lower bound $\Upsilon _2(N,p,\beta )$ can provide a more accurate approximation as $N$ increases. Besides, the truncation error analysis for both approximations is shown in Appendix C.

4.2 Asymptotic BER analysis

Based on the conclusion of the truncation error analysis, we can examine the BER behavior in a large SNR regime. Consequently, the leading term of the series in Eq. (29) dominates. Moreover, the condition $\alpha >\beta >0$ is usually satisfied in Gamma-Gamma turbulence [22], so the term $(\frac {C_{EGC}\bar {\gamma }_{\scriptscriptstyle R}}{4})^{-\frac {p+\alpha }{2}}$ can be ignored for a large $\bar {\gamma }_{\scriptscriptstyle R}$. Therefore, the BER of OOK modulation for FMF-based IM/DD FSO system with EGC in high SNR regimes can be approximated by

$$P_{e,EGC}^{\infty}=\frac{(\alpha\beta)^{\beta}\Gamma\left(\alpha-\beta\right)(\frac{C_{EGC}\bar{\gamma}_{\scriptscriptstyle R}}{4})^{-\frac{\beta }{2}}}{2^{\beta+1}\Gamma(\alpha)\Gamma\left(\frac{\beta}{2}+1\right)}$$
where we have used the Euler’s reflection formula [21, eq. (8.334.3)] and the doubling formula [21, eq. (8.335.1)].

Similarly, the BER of OOK modulation for the FMF-based IM/DD FSO system with MRC in high SNR regimes can be approximated by

$$P_{e,MRC}^{\infty}=\frac{(\alpha\beta)^{\beta}\Gamma\left(\alpha-\beta\right)\Upsilon(N,p,\beta)(\frac{\bar{\gamma}_{\scriptscriptstyle R}}{4})^{-\frac{\beta }{2}}}{2^{\beta+1}\Gamma(\alpha)\Gamma\left(\frac{\beta}{2}+1\right)}.$$

4.3 Performance comparison with transmitted optical power constraints

To evaluate the error rate performance, we need to link the equivalent average electrical SNR of FMF input with the average received optical power first. By recalling the average electrical SNR definition, we have

$$\bar{\gamma}_{\scriptscriptstyle R}=\frac{R_d^{2}A_R^{2}}{\sigma_n^{2}}\left(E\left[I_s\right]\right)^{2}=\frac{R_d^{2}}{\sigma_n^{2}}{\bar{P}_R}^{2}.$$
We assume that the same transmitted optical power $P_t$ will result in the same average received optical power $\bar {P}_R$, i.e., $\bar {P}_R=hP_t$, where $h$ denotes a constant path loss factor. The influence of geometric spread and pointing errors on path loss factor are ignored. Equation (35) can be rewritten as
$$\bar{\gamma}_{\scriptscriptstyle R}=\frac{h^{2}R_d^{2}}{\sigma_n^{2}}P_t^{2}.$$
By substituting Eq. (36) into Eqs. (29), (30), (33) and (34), we can readily evaluate system performance in terms of transmitted optical power for EGC and MRC receivers.

5. Numerical results and discussions

In this section, We first use a six-mode FMF (holding $\mathrm {LP_{01}}$, $\mathrm {LP_{11}}$, $\mathrm {LP_{21}}$ and $\mathrm {LP_{02}}$ modes) as a representative case to list turbulence and system parameters. We also present an intuitive comparison of the approximation and a lower bound of $\Upsilon (N,p,\beta )$. Besides, we compare the approximate error rate with the exact error rates calculated from Eq. (28), which has been confirmed by Monte Carlo simulations. Then we present numerical case studies on the SNR estimate factor for MRC.

5.1 Turbulence and system parameters

The typical values for photodetector responsivity is $R_d=0.5$, the noise standard deviation is $\sigma _n=10^{-7}\mathrm {A/Hz}$ [23], the insertion loss factor $\zeta =0.9$ [24], the wavenumber $k=2\pi /1550$ $\mathrm {nm^{-1}}$, the communication link distance $L=1$ $\mathrm {km}$, and coupling lens diameter $d_R=2$ $\mathrm {cm}$. Experimental measurements show that the path loss factor is empirically expressed in terms of visibility. Therefore, we can obtain $h=0.903$ when the weather is clear and the visibility is $10$ $\mathrm {km}$ [25]. Based on the current research, it is difficult to quantitatively characterize the variances $\sigma _i$, we assume that $\sigma =\sigma _1=\sigma _2=\cdots =\sigma _N$ and qualitatively analyze the impact of $\sigma$ on system performance. Besides, considering that the $\mathrm {LP}_{02}$ mode is excited less efficiently than the $\mathrm {LP_{01}}$ mode due to the field vectors reverse [26]. Based on our previous work [6], the coupling geometry parameter $\tau =1.12$ is obtained to minimize the initial coupling efficiency (in the absence of turbulence where $A_R/A_C=0$) of higher-order modes. All series error rates are calculated using the first $30$ terms, and the two complexity-accuracy trade-off parameters are set to $n=40$ and $m=40$.

Experimental measurements show that $C_n^{2}$ varies from $10^{-15}$ to $10^{-12}$ as the turbulence strength varies from weak to strong conditions [27]. In Table 1, we present different parameters calculated from Eqs. (2), (4) and (17), as well as their corresponding values for $C_n^{2}$. Two cases are considered: weak turbulence where $\sigma _R^{2}=1.095$, and strong turbulence where $\sigma _R^{2}=6.769$. We can find that the coupling efficiency of the $\mathrm {LP_{01}}$ mode decreases rapidly with $A_R/A_C$, while the other modes decrease slightly or even increase. Since the $\mathrm {LP_{11a}}$ mode and the $\mathrm {LP_{11b}}$ mode have phase antisymmetry in the absence of turbulence [28]. When they are superimposed, for a fiber waveguide energized by a normally incident plane wave, the coupling efficiency of the $\mathrm {LP_{11}}$ mode is zero. However, atmospheric turbulence that causes wavefront distortion destroys this phase antisymmetry, which in turn leads to a transient rise in the coupling efficiency of the $\mathrm {LP_{11}}$ mode. The same as the $\mathrm {LP_{21}}$ mode, but its trend lags behind the $\mathrm {LP_{11}}$ mode. Due to the artificial setting of $\tau =1.12$, the initial coupling efficiency of the $\mathrm {LP_{02}}$ mode approaches zero. Therefore, the coupling efficiency of the $\mathrm {LP}_{02}$ mode increases slightly. When $C_n^{2}$ is set to indicate stronger atmospheric turbulence than $C_n^{2}=3.40\times 10^{-13}$, all modes’ coupling efficiencies decrease.

Tables Icon

Table 1. SYSTEM PARAMETERS THAT VARY WITH $C_n^{2}$

View Table

Without loss of generality, we assume that the coupling efficiencies of both $\mathrm {LP_{11a}}\ (\mathrm {LP_{21a}})$ mode and $\mathrm {LP_{11b}}\ (\mathrm {LP_{21b}})$ mode are equal to half of the coupling efficiency of the $\mathrm {LP_{11}}\ (\mathrm {LP_{21}})$ mode in our following work. Besides, as the correlation terms between modes are ignored in this work, we assume that the same mode in three-mode FMF or six-mode FMF has the same coupling efficiency. Therefore, we can use the parameters list in Table 1 to study the three-mode FMF-based mode demultiplexer.

5.2 BER versus transmitted optical power

In Fig. 1, the series solutions calculated from the approximation Eq. (31) and the lower bound Eq. (32) are compared with the exact solution of BER of FMF-based IM/DD FSO system with MRC. Two cases are considered: three-mode FMF-based mode demultiplexer over weak turbulence channels where $\sigma _R^{2}=1.095$, and six-mode FMF-based mode demultiplexer over strong turbulence channels where $\sigma _R^{2}=6.769$. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ for both cases. It is seen that for the three-mode FMF case, the exact BER is closer to the approximation result from $\Upsilon _1(N,p,\beta )$. Specifically, at a transmitted optical power of $-6.0$ dBm for a three-mode FMF-based system over the weak turbulence channel, the difference between the approximation result from $\Upsilon _1(N,p,\beta )$ and the exact value is $0.10\,\%$. In contrast, $\Upsilon _2(N,p,\beta )$ presents a larger difference of $1.08\,\%$. Because the actual situation of three-mode FMF-based system is closer to the assumption of $\Upsilon _1(N,p,\beta )$, which only considers two random variables. Therefore, as shown in Eq. (12), it is reasonable that the independence among random variables increases with a six-mode FMF-based mode demultiplexer. As a comparison, at a transmitted optical power of $-1.5$ dBm for six-mode FMF-based system over the strong turbulence channel, the difference between the approximation result from $\Upsilon _1(N,p,\beta )$ and the exact value is $37.88\,\%$, while $\Upsilon _2(N,p,\beta )$ presents a smaller difference of $0.46\,\%$. Comparing these two cases, we find that the difference between the approximation result from $\Upsilon _1(N,p,\beta )$ and the exact value increases with the number of holding modes of mode demultiplexer. In addition to the independence among random variables, another reason is that $\Upsilon _1(N,p,\beta )$ only considers the first two coupling efficiencies $a_1$ and $a_2$, while the coupling efficiencies of the remaining modes are ignored in the calculation.

 figure: Fig. 1.

Fig. 1. BER comparison among the two approximate: $\Upsilon _1(N,p,\beta )$, $\Upsilon _2(N,p,\beta )$ and the exact value subject to the same transmitted optical power with MRC. Two cases are considered: three-mode FMF-based mode demultiplexer over weak ($\sigma _R^{2}=1.095$) turbulence channels and six-mode FMF-based mode demultiplexer over strong ($\sigma _R^{2}=6.769$) turbulence channels. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ for both cases.

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The effect of different values of the variance of the system $\sigma ^{2}$ on BER is shown in Fig. 2. Besides, this BER comparison is based on the six-mode FMF-based system with MRC over weak turbulence channels where $\sigma _R^{2}=1.095$ and strong turbulence channels where $\sigma _R^{2}=6.769$. We have compared both approximations based on $\Upsilon _1(N,p,\beta )$ and $\Upsilon _2(N,p,\beta )$. After comparison, only the approximation result from $\Upsilon _1(N,p,\beta )$ when $\sigma ^{2}=5\times 10^{-3}$, and the approximation result from $\Upsilon _2(N,p,\beta )$ when $\sigma ^{2}=1\times 10^{-3}$ are shown in Fig. 2. This system exhibits high sensitivity to the increase of $\sigma ^{2}$ under weak turbulence condition. Intuitively, a larger value of $\sigma ^{2}$ means that the distribution of each mode’s optical power transmission through FMF-based system tends to be more uniform. Besides, considering the restriction of RVs Eq. (7), when the difference of optical power between the modes held by FMF is large (i.e., in the presence of weak turbulence), this trend of "energy uniformity" for each mode transmitted through the FMF-based system becomes more obvious. It is seen that $\Upsilon _2(N,p,\beta )$ can well approximate the exact BER when $\sigma ^{2}=1\times 10^{-3}$, whether over the weak or strong turbulence channel.

 figure: Fig. 2.

Fig. 2. BER comparison of six-mode FMF-based mode demultiplexer between the approximate BER and the exact value subject to the same transmitted optical power with MRC over weak ($\sigma _R^{2}=1.095$) and strong ($\sigma _R^{2}=6.769$) turbulence channels. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ and $5\times 10^{-3}$.

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From Figs. 1 and 2, we can conclude that the approximation based on $\Upsilon _1(N,p,\beta )$ is suitable for the cases when the number of modes held by FMF is small or the variance of the system $\sigma ^{2}$ is large. Correspondingly, the approximation based on the lower bound $\Upsilon _2(N,p,\beta )$ is suitable for the cases when the number of modes held by FMF is large or the variance of the system $\sigma ^{2}$ is small.

In Fig. 3, the BER of six-mode FMF-based system with EGC or MRC is compared over weak turbulence channels where $\sigma _R^{2}=1.095$. Both approximations are based on the lower bound $\Upsilon _2(N,p,\beta )$, and the variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$. It can be found that both asymptotic BER of MRC and EGC are highly accurate especially in the large transmitted optical power regimes.

 figure: Fig. 3.

Fig. 3. BER comparison of six-mode FMF-based mode demultiplexer among the lower bound $\Upsilon _2(N,p,\beta )$, the exact value and the asymptotic analysis subject to the same transmitted optical power with EGC/MRC over weak ($\sigma _R^{2}=1.095$) turbulence channels. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$.

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Figure 4 shows the BER comparison of six-mode FMF-based system between the estimated BER and the exact BER subject to the same transmitted optical power with MRC over the different intensity of turbulence channels. The estimated BER is calculated from Eqs. (25) and (28), which have been confirmed by Monte Carlo simulations. It is found that the estimated BER has an excellent agreement with the exact BER over the weak ($\sigma _R^{2}=1.095$) turbulent channel for both cases that the variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ and $5\times 10^{-3}$. However, the difference between the estimated BER and the exact BER varies with different $\sigma ^{2}$ over strong ($\sigma _R^{2}=6.769$) turbulence channels. For instance, at a transmitted optical power of $-4.5$ dBm in strong turbulence, the difference between the estimated BER and the exact BER is $1.3\times 10^{-6}$ when $\sigma ^{2}=1\times 10^{-3}$, while that value is $5.8\times 10^{-6}$ when $\sigma ^{2}=5\times 10^{-3}$. This observation confirms that the coupling efficiencies of modes held by FMF can be used to estimate the BER of the FMF-based IM/DD FSO system with MRC, especially in weak turbulence.

 figure: Fig. 4.

Fig. 4. BER comparison of six-mode FMF-based system between the estimated BER and the exact BER subject to the same transmitted optical power with MRC over weak ($\sigma _R^{2}=1.095$) and strong ($\sigma _R^{2}=6.769$) turbulence channels. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ and $5\times 10^{-3}$.

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5.3 SNR estimate factor of MRC

The effects of the variance of the system $\sigma ^{2}$ on the ratio of average electrical SNR with and without estimation $\xi _{SNR}$ are shown in Fig. 5. It is seen that $\xi _{SNR}$ of the three-mode FMF-based system attains $-0.07$ dB when $\sigma ^{2}=2\times 10^{-3}$ in the presence of weak turbulence. However, $\xi _{SNR}$ of the three-mode FMF-based system drops to $-0.89$ dB in a strong turbulence channel, while $\xi _{SNR}$ of the six-mode FMF-based system drops to $-0.17$ dB in a weak turbulence channel. For a certain value of $\sigma ^{2}$, the turbulence intensity has a stronger effect on $\xi _{SNR}$ than the number of modes held by FMF.

 figure: Fig. 5.

Fig. 5. SNR estimate factor $\xi _{SNR}$ of MRC with three-mode/six-mode FMF as a function of $\sigma ^{2}$ over weak ($\sigma _R^{2}=1.095$) and strong ($\sigma _R^{2}=6.769$) turbulence channels.

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As expected, FMF’s coupling efficiency can be used to estimate the average electrical SNR of the FMF-based IM/DD optical receiver with MRC, especially in weak turbulence. For instance, When $\sigma ^{2}=2.5\times 10^{-3}$, $\xi _{SNR}$ is higher than $-0.20$ dB for both three-mode and six-mode FMF-based system with weak turbulence.

From Figs. 4 and 5, we find that the turbulence intensity can significantly increase the estimation error caused by $\sigma ^{2}$. The estimation error caused by the system on communication performance is acceptable especially in weak turbulence.

6. Conclusion

We evaluated the performance of the FMF-based IM/DD FSO system in the presence of turbulence. We used the coupling efficiency of the FMF to characterize the performance of the FMF-based FSO system. The power distribution at SMFs output of the mode demultiplexer is described as a truncated multivariate Gaussian distribution. We presented an approximation and a lower bound to obtain the analytical series solution of BER. The approximation $\Upsilon _1(N,p,\beta )$ is suitable for cases where the number of modes held by FMF is small or $\sigma ^{2}$ is large. Correspondingly, the approximation based on the lower bound $\Upsilon _2(N,p,\beta )$ is suitable for the cases where the number of modes held by FMF is large or $\sigma ^{2}$ is small. Besides, we used the coupling efficiency of each mode of FMF to estimate weighting coefficients of the FMF-based FSO system with MRC scheme. Through the analytical studies, we find that the turbulence intensity can significantly increase the estimation error caused by the system on communication performance. Moreover, the BERs versus transmitted optical power has been shown and compared for MRC and EGC. The asymptotic BER of MRC and EGC are highly accurate in the large transmitted optical power regimes.

A. Derivation of the approximation of $\Upsilon (N, {p},\beta )$

The PDF of $C_{\scriptscriptstyle MRC}$ can be obtained from the joint PDF Eq. (12). Due to formidable mathematical complexity, the value of $\Upsilon (N,p,\beta )$ should be approximated. Considering the constraint Eq. (9), random variables $b_i\ (i=1,2,\ldots ,N)$ are correlated. As a truncated Gaussian distribution with mean $\zeta a_1$ and variance $\sigma _1^{2}$, we can obtain the PDF of $b_1\ (0\leq b_1 < 1)$ as

$$f(b_1;\zeta a_1,\sigma_1,0,1)=\frac{\phi\left(\frac{b_1-\zeta a_1}{\sigma_1}\right)}{\sigma_1\left[\Phi\left(\frac{1-\zeta a_1}{\sigma_1}\right)-\Phi\left(\frac{-\zeta a_1}{\sigma_1}\right)\right]}=\frac{\Omega}{\sigma_1\sqrt{2\pi}}\mathrm{exp}\left[-\frac{(b_1-\zeta a_1)^{2}}{2\sigma_1^{2}}\right]$$
where $\phi (\zeta )\!=\!\frac {1}{\sqrt {2\pi }}\mathrm {exp}(-\frac {1}{2}\zeta ^{2})$ is the PDF of the standard Gaussian distribution; $\Phi (\zeta )\!=\!\frac {1}{2}(1+\mathrm {erf}(\zeta /\sqrt {2}))$ is its cumulative distribution function; $\Omega =\left [\Phi (\frac {1-\zeta a_1}{\sigma _1})-\Phi (\frac {-\zeta a_1}{\sigma _1})\right ]^{-1}$. When $N=2$, $C_{\scriptscriptstyle MRC}=b_1^{2}+(b-b_1)^{2}$ where $b=\zeta (a_1+a_2)$. Using the Gaussian-Chebyshev quadrature [29, eq. (25.4.38)], we can approximate $\Upsilon (N,p,\beta )$ as
$$\Upsilon_1(N,p,\beta)\approx\sum_{i=1}^{n}\frac{\Omega \sqrt{2\pi}}{4n\sigma_1}\left(\frac{(x_i+1-b)^{2}+b^{2}}{2}\right)^{-\frac{p+\beta}{2}}\mathrm{exp}\left[-\frac{\left(x_i+1-2\zeta a_1\right)^{2}}{8\sigma_1^{2}}\right]\sqrt{1-x_i^{2}}$$
where $x_i=\cos {\frac {(2i-1)\pi }{2n}}$ and $n$ is a complexity-accuracy trade-off parameter. The approximation $\Upsilon _1(N,p,\beta )$ is the exact solution for $N=2$, while the approximation error increases with $N$.

B. Derivation of a lower bound of $\Upsilon (N, {p},\beta )$

As $N$ increases, the independence among random variables $b_i$ will increase and we assume $b_i\sim N(\zeta a_i,\sigma _i^{2})$. Let $\sigma _{\mathrm {min}}=\mathrm {min}(\sigma _i)$, then the PDF of $\sum _{i=1}^{N} b_i^{2}$ can be approximated by the noncentral chi-squared distribution

$$f_X(x;N,\sigma_{\mathrm{min}},\lambda)=\frac{1}{2\sigma_{\mathrm{min}}^{2}}\left(\frac{x}{\lambda}\right)^{\frac{N-2}{4}}\!\!\mathrm{exp}\left(-\frac{x+\lambda}{2\sigma_{\mathrm{min}}^{2}}\right)I_{\frac{N-2}{2}}\!\left(\!\sqrt{\frac{\lambda x}{\sigma_{\mathrm{min}}^{4}}}\right)$$
where $\lambda =\zeta ^{2}\sum _{i=1}^{N} a_i^{2}$; $I_{l}$ denotes the $l$th-order modified Bessel function of the first kind. When $N$ is small, we assume that the upper limit of the integral is one. We can obtain
$$\Upsilon_2(N,p,\beta)\approx\sum_{i=1}^{m}\frac{\lambda^{\frac{2-N}{4}}\pi}{4m\sigma_{\mathrm{min}}^{2}}\left(\frac{y_i+1}{2}\right)^{\frac{N-2-2(p+\beta)}{4}}\!\!\sqrt{1-y_i^{2}}\mathrm{exp}\left(-\frac{y_i+1+2\lambda}{4\sigma_{\mathrm{min}}^{2}}\right)I_{{\frac{N}{2}-1}}\left(\sqrt{\frac{\lambda y_i+\lambda}{2\sigma_{\mathrm{min}}^{4}}}\right)$$
where the subscript $\Upsilon _2$ represents the second approximation; $y_i=\cos {\frac {(2i-1)\pi }{2m}}$; and $m$ is a complexity-accuracy trade-off parameter. As $N$ increases, expand the upper limit of the integral to infinity, and combining the integral of Bessel function [21, eq. (6.643.2)], when $N>p+\beta$ we can obtain
$$\Upsilon_2(N,p,\beta)=\lambda^{{-}N/4}(2\sigma_{\mathrm{min}}^{2})^{\frac{N-2(p+\beta)}{4}}\frac{\Gamma\left(\frac{N-p-\beta}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}\mathrm{exp}\left(-\frac{\lambda}{4\sigma_{\mathrm{min}}^{2}}\right)M_{\frac{2(p+\beta)-N}{4},\frac{N-2}{4}}\left(\frac{\lambda}{2\sigma_{\mathrm{min}}^{2}}\right)$$
where $M_{a,b}(\cdot )$ is the Whittaker equation.

Therefore, we have

$$\Upsilon_2(N,p,\beta)= \left\{ \begin{array}{lr} \sum\limits_{i=1}^{m}\frac{\lambda^{\frac{2-N}{4}}\pi}{4m\sigma_{\mathrm{min}}^{2}}\left(\frac{y_i+1}{2}\right)^{\frac{N-2-2(p+\beta)}{4}}e^{-\frac{y_i+1+2\lambda}{4\sigma_{\mathrm{min}}^{2}}} I_{{\frac{N}{2}-1}}\left(\sqrt{\frac{\lambda y_i+\lambda}{2\sigma_{\mathrm{min}}^{4}}}\right)\sqrt{1-y_i^{2}},\qquad N\leq p+\beta;\\ \lambda^{{-}N/4}(2\sigma_{\mathrm{min}}^{2})^{\frac{N-2(p+\beta)}{4}}\frac{\Gamma\left(\frac{N-p-\beta}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}e^{-\frac{\lambda}{4\sigma_{\mathrm{min}}^{2}}} M_{\frac{2(p+\beta)-N}{4},\frac{N-2}{4}}\left(\frac{\lambda}{2\sigma_{\mathrm{min}}^{2}}\right),\ \quad N>p+\beta. \end{array} \right.$$
The approximation $\Upsilon _2(N,p,\beta )$ can provide a more accurate approximation as $N$ increases. If $\sigma _{\mathrm {min}}$ is small, the truncated Gaussian distribution $f(b_i;\zeta a_i,\sigma _{\mathrm {min}},0,1)$ is approximately equivalent to $b_i\sim N(\zeta a_i,\sigma _{\mathrm {min}}^{2})$. It can be proved that $\Upsilon (N,p,\beta )\geq \Upsilon _2(N,p,\beta )$, and $\Upsilon _2(N,p,\beta )$ can be seen as a lower bound of $\Upsilon (N,p,\beta )$.

C. Truncation error analysis

We first analyze the truncation error of the unconditional BER of the FMF-based IM/DD optical receiver with EGC. To evaluate the truncation error caused by eliminating the infinite terms after the first $J+1$ terms in Eq. (29), we define the truncation error as

$$\varepsilon_J=\sum_{p=J+1}^{\infty}\left(d_p(\alpha,\beta)+d_p(\beta,\alpha)\right)$$
where $d_p(\alpha ,\beta )=b_p\left (\alpha ,\beta \right )(\frac {C_{EGC}\bar {\gamma }_{\scriptscriptstyle R}}{4})^{-\frac {p+\beta }{2}}\mathrm {B}\left (\frac {1}{2}, \frac {p+\beta +1}{2}\right )$. The value for $\frac {d_{p+1}(\alpha ,\beta )}{d_p(\alpha ,\beta )}$ is
$$\frac{d_{p+1}(\alpha,\beta)}{d_p(\alpha,\beta)}=\frac{\alpha\beta\sqrt{\frac{4}{C_{EGC}\bar{\gamma}_{\scriptscriptstyle R}}}(p+\beta)\Gamma(\frac{p+\beta+2}{2})}{(p-\alpha+\beta+1)(p+1)(p+\beta+1)\Gamma(\frac{p+\beta+1}{2})}.$$
When $p\to \infty$, $\frac {d_{p+1}(\alpha ,\beta )}{d_p(\alpha ,\beta )}\approx \frac {\alpha \beta \sqrt {\frac {4}{C_{EGC}\bar {\gamma }_{\scriptscriptstyle R}}}}{(p-\alpha +\beta +1)(p+1)}\to 0$. Then we know that there exists $J$, when $p>J$, $\frac {d_{p+1}(\alpha ,\beta )}{d_p(\alpha ,\beta )}<1$. Similarly, there exists $J$, when $p>J$, $\frac {d_{p+1}(\beta ,\alpha )}{d_p(\beta ,\alpha )}<1$. Therefore, the series solution converges, and the truncation error exists and has a maximum value. We can then obtain an upper bound of the truncation error. Furthermore, from Eq. (44), the truncation error diminishes rapidly with increasing average SNR $\bar {\gamma }_{\scriptscriptstyle R}$. This suggests that our series solution is highly accurate in large SNR regime.

As $n$ is a complexity-accuracy trade-off parameter, which can be seen as a constant. There is a maximum value for $\Upsilon _1(N,p,\beta )$ as

$$\Upsilon_1(N,p,\beta)\leq \frac{\Omega\sqrt{2\pi}}{4\sigma_1}\max _{1\leq i\leq n}\left\{\sqrt{1-x_i^{2}}e^{-\frac{\left(x_i+1-2\zeta a_1\right)^{2}}{8\sigma_1^{2}}}\right\}\max _{1\leq i\leq n}\left\{\frac{(x_i+1-b)^{2}+b^{2}}{2a_{tot}^{2}}\right\}^{-\frac{p+\beta}{2}}.$$
Therefore, $\Upsilon _1(N,p,\beta )$ can be seen as an exponential function, involving the base with a maximum value and the exponent $-\frac {p+\beta }{2}$. This means that $\Upsilon _1(N,p,\beta )$ can be combined with the term $(\frac {\bar {\gamma }_{\scriptscriptstyle R}}{4})^{-\frac {p+\beta }{2}}$ in Eq. (30). When $N\leq p+\beta$, $\Upsilon _2(N,p,\beta )$ can be constructed as the similar form as Eq. (45). When $N>p+\beta$, using the Gaussian-Chebyshev quadrature and making a change of variable $x=\frac {1+y}{1-y}$, we can approximate $\Upsilon _2(N,p,\beta )$ as
$$\begin{aligned}\Upsilon_2(N,p,\beta)\!\approx&\!\frac{\lambda^{\frac{2-N}{4}}\pi}{m\sigma_{\mathrm{min}}^{2}}\sum_{i=1}^{m}\left(1+y_i\right)^{\frac{N-2(p+\beta)}{4}}\!\!\left(1-y_i\right)^{\frac{2(p+\beta)-2-N}{4}}\\ &\times\mathrm{exp}\left(-\frac{1+y_i+\lambda(1-y_i)}{2\sigma_{\mathrm{min}}^{2}(1-y_i)}\right)I_{\frac{N-2}{2}}\left(\sqrt{\frac{\lambda (1+y_i)}{\sigma_{\mathrm{min}}^{4}(1-y_i)}}\right) \end{aligned}$$
where $y_i=\cos {\frac {(2i-1)\pi }{2m}}$; and $m$ is a complexity-accuracy trade-off parameter. Similarly, there is a maximum value for $\Upsilon _2(N,p,\beta )$ as
$$\begin{aligned}\Upsilon_2(N,p,\beta)\leq &\max _{1\leq i\leq n}\left\{\frac{1+y_i}{1-y_i}\right\}^{-\frac{p+\beta}{2}}\frac{\lambda^{\frac{2-N}{4}}\pi}{\sigma_{\mathrm{min}}^{2}}\mathrm{exp}\left(-\frac{\lambda}{2\sigma_{\mathrm{min}}^{2}}\right)\max _{1\leq i\leq n}\left\{\left(1+y_i\right)^{\frac{N}{4}}\left(1-y_i\right)^{\frac{-2-N}{4}}\right\}\\ &\times \max _{1\leq i\leq n}\left\{\mathrm{exp}\left(-\frac{1+y_i}{2\sigma_{\mathrm{min}}^{2}(1-y_i)}\right)I_{\frac{N-2}{2}}\left(\sqrt{\frac{\lambda (1+y_i)}{\sigma_{\mathrm{min}}^{4}(1-y_i)}}\right)\right\}. \end{aligned}$$
$\Upsilon _2(N,p,\beta )$ can also be seen as an exponential function, involving the base with a maximum value and the exponent $-\frac {p+\beta }{2}$.

Both $\Upsilon _1(N,p,\beta )$ and $\Upsilon _2(N,p,\beta )$ can be seen as an exponential function, involving the base with a maximum value and the exponent $-\frac {p+\beta }{2}$, and this term can be combined with the term $(\frac {\bar {\gamma }_{\scriptscriptstyle R}}{4})^{-\frac {p+\beta }{2}}$. Therefore, the truncation error analysis for the FMF-based IM/DD optical receiver with MRC can be simplified to the analysis that similar to the truncation error of $P_{e,EGC}$.

Disclosures

The authors declare no conflicts of interest related to this paper.

Data availability

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

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Data availability

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

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Figures (5)
Fig. 1.
Fig. 1. BER comparison among the two approximate: $\Upsilon _1(N,p,\beta )$ , $\Upsilon _2(N,p,\beta )$ and the exact value subject to the same transmitted optical power with MRC. Two cases are considered: three-mode FMF-based mode demultiplexer over weak ( $\sigma _R^{2}=1.095$ ) turbulence channels and six-mode FMF-based mode demultiplexer over strong ( $\sigma _R^{2}=6.769$ ) turbulence channels. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ for both cases.

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Fig. 2.
Fig. 2. BER comparison of six-mode FMF-based mode demultiplexer between the approximate BER and the exact value subject to the same transmitted optical power with MRC over weak ( $\sigma _R^{2}=1.095$ ) and strong ( $\sigma _R^{2}=6.769$ ) turbulence channels. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ and $5\times 10^{-3}$ .

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Fig. 3.
Fig. 3. BER comparison of six-mode FMF-based mode demultiplexer among the lower bound $\Upsilon _2(N,p,\beta )$ , the exact value and the asymptotic analysis subject to the same transmitted optical power with EGC/MRC over weak ( $\sigma _R^{2}=1.095$ ) turbulence channels. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ .

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Fig. 4.
Fig. 4. BER comparison of six-mode FMF-based system between the estimated BER and the exact BER subject to the same transmitted optical power with MRC over weak ( $\sigma _R^{2}=1.095$ ) and strong ( $\sigma _R^{2}=6.769$ ) turbulence channels. The variance of the system $\sigma ^{2}$ is set to $1\times 10^{-3}$ and $5\times 10^{-3}$ .

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Fig. 5.
Fig. 5. SNR estimate factor $\xi _{SNR}$ of MRC with three-mode/six-mode FMF as a function of $\sigma ^{2}$ over weak ( $\sigma _R^{2}=1.095$ ) and strong ( $\sigma _R^{2}=6.769$ ) turbulence channels.

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Tables (1)

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Table 1. SYSTEM PARAMETERS THAT VARY WITH C n 2

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Equations (47)

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a i = 4 π d R 2 ∫ ∫ A exp ⁡ ( − | r → 1 − r → 2 | 2 / ρ 2 ) E i ∗ ( r → 1 ) E i ( r → 2 ) d r → 1 d r → 2
ρ = ( 1.46 C n 2 k 2 L ) − 3 / 5
L G p l ( r , ϕ ) = B p l ω ( 2 r ω ) l L p l ( 2 r 2 ω 2 ) exp ⁡ ( − r 2 ω 2 ) e − j l ϕ
a i = 2 π B p l 2 ∫ 0 1 ∫ 0 1 I l ( A R A C 2 x 1 x 2 ) exp ⁡ ( − A R A C ( x 1 2 + x 2 2 ) ) × L p l ( 2 τ 2 x 1 2 ) L p l ( 2 τ 2 x 2 2 ) ( 2 τ 2 x 1 x 2 ) l + 1 exp ⁡ ( − τ 2 ( x 1 2 + x 2 2 ) ) d x 1 d x 2 .
a t o t = ∑ i = 1 N a i
P F , i = a i P R = a i a t o t P F , 0 ≤ a i < 1
b 1 + b 2 + ⋯ + b N = ζ a t o t , 0 ≤ b i < 1.
b i = ζ a i + x i .
x 1 + x 2 + ⋯ + x N = 0.
f χ ( x 1 , x 2 , … , x N ) = ( 2 π ) 1 − N 2 | Σ c h i ∗ | − 1 2 g ( χ ) δ ( ∑ i = 1 N x i )
Σ χ ∗ = [ σ 1 2 κ 1 , 2 ⋯   κ 1 , N − 1 κ 2 , 1 σ 2 2 ⋯   κ 2 , N − 1 ⋮ ⋮ ⋱ ⋮ κ N − 1 , 1 κ N − 1 , 2 ⋯   σ N − 1 2 ]
f χ ( x 1 , x 2 , … , x N ) = g ( χ ) δ ( ∑ i = 1 N x i ) ∫ − ζ a 1 − ζ a N g ( χ ) δ ( ∑ i = 1 N x i ) d χ , − ζ a ⩽ χ ⩽ 1 − ζ a
I = R d P i n + n = R d A s I s + n
γ = R d 2 A s 2 I s 2 σ n 2 .
γ ¯ = E [ γ ] = R d 2 A s 2 σ n 2 ( E [ I s ] ) 2 = R d 2 A s 2 σ n 2
f I s ( I s ) = 2 ( α β ) α + β 2 Γ ( α ) Γ ( β ) I s α + β 2 − 1 K α − β ( 2 α β I s )
α = [ exp ⁡ ( 0.49 σ R 2 ( 1 + 1.11 σ R 12 / 5 ) 7 / 6 ) − 1 ] − 1 β = [ exp ⁡ ( 0.51 σ R 2 ( 1 + 0.69 σ R 12 / 5 ) 7 / 6 ) − 1 ] − 1
f I s ( I s ) = π ∑ p = 0 ∞ ( a p ( α , β ) I s p + β − 1 + a p ( β , α ) I s p + α − 1 ) , ( α − β ) ∉ Z
γ E G C = R d 2 N σ n 2 ( ∑ i = 1 N P S , i ) 2 = C E G C I s 2 R d 2 A R 2 σ n 2 = C E G C I s 2 γ ¯ R
γ ¯ E G C = C E G C γ ¯ R .
I M R C = ∑ i = 1 N m i I i = R d P S ζ a t o t ∑ i = 1 N m i b i
γ M R C = ( ∑ i = 1 N m i I i ) 2 σ M R C 2 = R d 2 A R 2 I s 2 ( ∑ i = 1 N m i b i ) 2 ∑ i = 1 N m i 2 σ n 2 .
γ M R C = C M R C I s 2 γ ¯ R
γ ¯ M R C = E [ ∑ i = 1 N b i 2 ] γ ¯ R
γ e s t i = C e s t i I s 2 γ ¯ R
γ ¯ e s t i = E [ ( ∑ i = 1 N a i b i ) 2 ] ∑ i = 1 N a i 2 γ ¯ R
ξ S N R = 10 log ⁡ ( γ ¯ e s t i γ ¯ M R C ) .
P e = ∫ 0 ∞ p e ( I s ) f I s ( I s ) d I s
P e , E G C = ∑ p = 0 ∞ [ b p ( α , β ) ( C E G C γ ¯ R 4 ) − p + β 2 B ( 1 2 , p + β + 1 2 ) + b p ( β , α ) ( C E G C γ ¯ R 4 ) − p + α 2 B ( 1 2 , p + α + 1 2 ) ]
P e , M R C = ∑ p = 0 ∞ [ b p ( α , β ) Υ ( N , p , β ) ( γ ¯ R 4 ) − p − β 2 B ( 1 2 , p + β + 1 2 ) + b p ( β , α ) Υ ( N , p , α ) ( γ ¯ R 4 ) − p + α 2 B ( 1 2 , p + α + 1 2 ) ]
Υ 1 ( N , p , β ) = ∑ i = 1 n Ω 2 π 4 n σ 1 ( ( x i + 1 − b ) 2 + b 2 2 ) − p + β 2 e x p [ − ( x i + 1 − 2 ζ a 1 ) 2 8 σ 1 2 ] 1 − x i 2
Υ 2 ( N , p , β ) = { ∑ i = 1 m λ 2 − N 4 π 4 m σ m i n 2 ( y i + 1 2 ) N − 2 − 2 ( p + β ) 4 e − y i + 1 + 2 λ 4 σ m i n 2 I N 2 − 1 ( λ y i + λ 2 σ m i n 4 ) 1 − y i 2 , N ≤ p + β ; λ − N / 4 ( 2 σ m i n 2 ) N − 2 ( p + β ) 4 Γ ( N − p − β 2 ) Γ ( N 2 ) e − λ 4 σ m i n 2 M 2 ( p + β ) − N 4 , N − 2 4 ( λ 2 σ m i n 2 ) , N > p + β ;
P e , E G C ∞ = ( α β ) β Γ ( α − β ) ( C E G C γ ¯ R 4 ) − β 2 2 β + 1 Γ ( α ) Γ ( β 2 + 1 )
P e , M R C ∞ = ( α β ) β Γ ( α − β ) Υ ( N , p , β ) ( γ ¯ R 4 ) − β 2 2 β + 1 Γ ( α ) Γ ( β 2 + 1 ) .
γ ¯ R = R d 2 A R 2 σ n 2 ( E [ I s ] ) 2 = R d 2 σ n 2 P ¯ R 2 .
γ ¯ R = h 2 R d 2 σ n 2 P t 2 .
f ( b 1 ; ζ a 1 , σ 1 , 0 , 1 ) = ϕ ( b 1 − ζ a 1 σ 1 ) σ 1 [ Φ ( 1 − ζ a 1 σ 1 ) − Φ ( − ζ a 1 σ 1 ) ] = Ω σ 1 2 π e x p [ − ( b 1 − ζ a 1 ) 2 2 σ 1 2 ]
Υ 1 ( N , p , β ) ≈ ∑ i = 1 n Ω 2 π 4 n σ 1 ( ( x i + 1 − b ) 2 + b 2 2 ) − p + β 2 e x p [ − ( x i + 1 − 2 ζ a 1 ) 2 8 σ 1 2 ] 1 − x i 2
f X ( x ; N , σ m i n , λ ) = 1 2 σ m i n 2 ( x λ ) N − 2 4 e x p ( − x + λ 2 σ m i n 2 ) I N − 2 2 ( λ x σ m i n 4 )
Υ 2 ( N , p , β ) ≈ ∑ i = 1 m λ 2 − N 4 π 4 m σ m i n 2 ( y i + 1 2 ) N − 2 − 2 ( p + β ) 4 1 − y i 2 e x p ( − y i + 1 + 2 λ 4 σ m i n 2 ) I N 2 − 1 ( λ y i + λ 2 σ m i n 4 )
Υ 2 ( N , p , β ) = λ − N / 4 ( 2 σ m i n 2 ) N − 2 ( p + β ) 4 Γ ( N − p − β 2 ) Γ ( N 2 ) e x p ( − λ 4 σ m i n 2 ) M 2 ( p + β ) − N 4 , N − 2 4 ( λ 2 σ m i n 2 )
Υ 2 ( N , p , β ) = { ∑ i = 1 m λ 2 − N 4 π 4 m σ m i n 2 ( y i + 1 2 ) N − 2 − 2 ( p + β ) 4 e − y i + 1 + 2 λ 4 σ m i n 2 I N 2 − 1 ( λ y i + λ 2 σ m i n 4 ) 1 − y i 2 , N ≤ p + β ; λ − N / 4 ( 2 σ m i n 2 ) N − 2 ( p + β ) 4 Γ ( N − p − β 2 ) Γ ( N 2 ) e − λ 4 σ m i n 2 M 2 ( p + β ) − N 4 , N − 2 4 ( λ 2 σ m i n 2 ) ,   N > p + β .
ε J = ∑ p = J + 1 ∞ ( d p ( α , β ) + d p ( β , α ) )
d p + 1 ( α , β ) d p ( α , β ) = α β 4 C E G C γ ¯ R ( p + β ) Γ ( p + β + 2 2 ) ( p − α + β + 1 ) ( p + 1 ) ( p + β + 1 ) Γ ( p + β + 1 2 ) .
Υ 1 ( N , p , β ) ≤ Ω 2 π 4 σ 1 max 1 ≤ i ≤ n { 1 − x i 2 e − ( x i + 1 − 2 ζ a 1 ) 2 8 σ 1 2 } max 1 ≤ i ≤ n { ( x i + 1 − b ) 2 + b 2 2 a t o t 2 } − p + β 2 .
Υ 2 ( N , p , β ) ≈ λ 2 − N 4 π m σ m i n 2 ∑ i = 1 m ( 1 + y i ) N − 2 ( p + β ) 4 ( 1 − y i ) 2 ( p + β ) − 2 − N 4 × e x p ( − 1 + y i + λ ( 1 − y i ) 2 σ m i n 2 ( 1 − y i ) ) I N − 2 2 ( λ ( 1 + y i ) σ m i n 4 ( 1 − y i ) )
Υ 2 ( N , p , β ) ≤ max 1 ≤ i ≤ n { 1 + y i 1 − y i } − p + β 2 λ 2 − N 4 π σ m i n 2 e x p ( − λ 2 σ m i n 2 ) max 1 ≤ i ≤ n { ( 1 + y i ) N 4 ( 1 − y i ) − 2 − N 4 } × max 1 ≤ i ≤ n { e x p ( − 1 + y i 2 σ m i n 2 ( 1 − y i ) ) I N − 2 2 ( λ ( 1 + y i ) σ m i n 4 ( 1 − y i ) ) } .
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