text:
\section{Introduction}
In satellite communication, coherent free-space optical communication (FSOC) technology is attractive for its high sensitivity and ability to obtain a high data rate \cite{liu2015new,shemis2017self,zhou2017symbol}. Recently, researchers have focused on designing coherent optical communication systems using fiber-based transmitters and receivers. Because the fiber-based receiver can make full use of the commercial available components from fiber-optic communication systems, such as fiber transmitter and receiver, erbium-doped fiber amplifiers (EDFAs), and fiber multiplexer and demultiplexer units \cite{dikmelik2005fiber, zhang2013fiber}. However, such implementation has its limitations because the overall efficiency (which will be defined in the sequel) is low.

In a coherent optical communication system using fiber-based receiver, when the signal beam reaches the receiver aperture plane, it is first coupled into the fiber and then mixed with the local oscillator (LO) beam to obtain the mixed signal. There are three important parameters associated with this process: coupling efficiency, mixing efficiency, and overall efficiency. The coupling efficiency is defined as the ratio of the average power coupled into the fiber to the average power in the receiver's aperture plane \cite{winzer1998fiber,dikmelik2005fiber,toyoshima2006maximum,grein2014multimode,poliak2016fiber,arisa2017coupling,hu2018fiber}. The mixing efficiency is defined as the ratio of the amplitude of the obtained mixed signal to the amplitude of theoretically mixed signal \cite{mark1992comparison,duncan1993performance,jacob1995heterodyne,leeb1998aperture,ren2012heterodyne}. The overall efficiency is defined as the product of the coupling efficiency and the mixing efficiency, which embodies the extent to which the signal beam can be fully utilized. Low overall efficiency can typically degrade signal-to-noise ratio (SNR) \cite{duncan1993performance,jacob1995heterodyne}.

There are two commonly used fiber-based receiver schemes for coherent optical communication systems. The first receiver scheme is the single-mode fiber (SMF) receiver with SMF mixing. The SMF only propagates one field mode. Because the received signal beam and the LO beam propagate in the same SMF, their field modes are the same, i.e., the received signal beam and LO beam fields are matched both spatially and temporally at the detector. Then the mixing efficiency between the LO beam and the signal beam approaches 100\% \cite{jacob1995heterodyne}. However, the core diameter of SMF is small ($\sim 10 \ \mu m$), which limits achievable fiber coupling efficiency, especially in the presence of atmosphere turbulence in free-space channels \cite{mark1992comparison,winzer1998fiber,dikmelik2005fiber,toyoshima2006maximum}. For example, the maximum coupling efficiency is 81\% in the absence of atmosphere turbulence \cite{winzer1998fiber}. For a moderate strength turbulence ($C_n^2=10^{-13}$ m$^{-\frac 2 3}$, where $C_n^2$ is refractive-index structure constant), the coupling efficiency is less than 5\% \cite{dikmelik2005fiber}. The second receiver scheme is the multimode fiber (MMF) receiver with MMF mixing \cite{duncan1993performance, poliak2016fiber, grein2014multimode}. The coupling efficiency of MMF, whose core diameter is $\sim 50 \ \mu m$ \cite{poliak2016fiber}, is much higher than that of the SMF \cite{duncan1993performance,niu2007coupling,takayama2011studies,zheng2016free,arisa2017coupling}. However, only the portion of the signal beam that is in the same temporal and spatial mode of the LO beam can produce high mixing efficiency \cite{duncan1993performance}. The MMF contains not only the fundamental mode component, but also high-order mode components. Then the mixing efficiency between the LO beam and the signal beam will be degraded \cite{duncan1993performance, ozdur2015photonic, zheng2016free}. For example, the coupling efficiency of MMFs tested in \cite{poliak2016fiber} is greater than 95\%; for asymmetric square waveguide supporting seventy-five distinct modes tested in \cite{duncan1993performance}, the coupling efficiency for MMF receiver is 75-78\%; the mixing efficiency is 21-23\%, and the overall efficiency becomes only 11-17\%. The properties of the SMF receiver and MMF receiver are summarized in Table \ref{tab:my_label}. From Table \ref{tab:my_label}, we can conclude that both SMF receiver and MMF receiver have low overall efficiency.

\begin{table}
\centering
\caption{Comparison between SMF receiver and MMF receiver.}
\begin{tabular}{c c c}
\hline
& SMF receiver& MMF receiver\\
\hline
The coupling efficiencies & low & high \\
The mixing efficiencies & high & low \\
The overall efficiencies & low & low \\
\hline
\end{tabular}
\label{tab:my_label}
\end{table}

Recently, a non-mode-selective photonic lantern (PL) based coherent optical receiver has been proposed, and the overall efficiency of this receiver can be improved \cite{ozdur2013free,ozdur2014performance,ryf2014photonic,ozdur2015photonic,zhang2019study}. Fig. \ref{fig: PL} shows the schematic diagram of a PL \cite{leon2005multimode,birks2015photonic}. In this diagram, one end of the PL is a relatively-large multimode core, and the other end is an array of several relatively-small single-mode cores. In between is a transition region \footnote{There are two types of PLs \cite{huang2015all,birks2015photonic}: mode-selective PL \cite{leon2014mode,li2019mode} and non-mode-selective PL \cite{ozdur2013free,ozdur2014performance,ryf2014photonic,ozdur2015photonic}. In a mode-selective PL, the single-mode cores are designed for transmitting light with different electromagnetic wave modes. While in a non-mode-selective PL, the single-mode cores are designed for transmitting light having the same electromagnetic wave mode. This paper focuses on the non-mode-selective PL.}. Fig. \ref{fig: PL_based_FSOC} shows a structural diagram of a complete coherent FSOC system based on non-mode-selective PL. The signal beam is transmitted from the transmitter, and is coupled into the receiver after passing through the atmosphere turbulence. In the receiver, the large-core MMF end of the PL is placed behind the receiver len to collect the multimode signal beam. Then the PL converts the multimode signal beam into $N$ single-mode signal beams. The single-mode LO beam is split into $N$ equal parts by a fiber beam splitter (FBS). Each single-mode signal beam of the PL is mixed with a single-mode LO beam in an optical hybrid. After that, each mixed signal is converted into an electrical signal by the corresponding balanced photodetector. All the electrical signals are sent to the combiner and the demodulator for processing. The system can fully take advantage of the MMF, which has higher coupling efficiency compared with the SMF, and can take advantage of the SMF that has nearly 100\% mixing efficiency with the single-mode LO beam \cite{ozdur2013free}.

\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth, draft=false]{img//PL.eps}
\caption{Structural diagram of a PL \cite{leon2005multimode}}
\label{fig: PL}
\end{center}
\end{figure}

\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth, draft=false]{img//PL_based_FSOC.pdf}
\caption{Structural diagram of a coherent FSOC system based on non-mode-selective PL}
\label{fig: PL_based_FSOC}
\end{center}
\end{figure*}

A recent work \cite{zheng2018performance} investigated the performance of coherent FSOC receiver under moderate-to-strong turbulence. However, the effect of the power distribution at SMF end of the PL on SNR was not studied. From \cite{birks2015photonic}, we know that the power distribution at SMF end of non-mode-selective PL varies according to input mode profiles, input pointing errors, and temperature or pressure variations on the MMF section of the PL. In FSOC, the signal beam impaired by atmospheric turbulence contains not only the fundamental mode component but also the higher-order mode components, and the influence of atmospheric turbulence on signal beam changes with time and space. Then the mode profile of the signal beam coupled into MMF end of the PL will change with time, resulting in the power distribution variation at SMF end of a non-mode-selective PL. As a result, the SNR of the coherent optical receiver based on PL will change \cite{zhang2019study}. Therefore, it is necessary to study the power distribution at SMF end of the PL. In \cite{zhang2019study}, we proposed two different distributions: the multivariate Gaussian distribution over a simplex for small power fluctuation case and the uniform distribution over a simplex for large power fluctuation case, to describe the power distribution at SMF end of non-mode selective PL. It was found that different power distributions can have different effects on SNR, and when the number of single-mode fibers of the PL is equal to the number of guided modes at multimode end of the PL, the average SNR attains its maximum value \cite{zhang2019study}.

Different from \cite{zhang2019study}, this paper proposes a more accurate power distribution: truncated Gaussian distribution over a simplex, and this proposal is based on the simulation results of photon distribution in Section \ref{section: Power Distribution }. In addition, the SNRs of the communication system using PL based receiver are analyzed using different combining techniques, including selection combining (SC), equal-gain combining (EGC), and maximal-ratio combining (MRC); and they are compared with traditional SMF and MMF receivers in Section \ref{section: SNR}. The bit-error rate (BER) performance of a binary phase-shift keying (BPSK) system is analyzed. The integral solution, lower bound series solution and asymptotic solution of the BER are presented. The BER performance of the system using different combining techniques are studied and compared with the traditional SMF and MMF receivers over the Gamma-Gamma atmosphere turbulence channels in Section \ref{section: BER Analysis}. We demonstrate that the power distribution of the PL has no effect on the SNR and BER performance of the PL based receiver when MRC is used. Simulation results in Section \ref{section: Simulation} show that the power distribution of the PL has only limited influence on the BER performance of PL based receiver when EGC is used; and the power distribution can greatly affect the BER performance when SC is used. Besides, the SNR gains of the PL based receiver using EGC over the SMF receiver and MMF receivers are numerically calculated under different imperfect device parameters; and the scope of application of the communication system is further provided. To the best of the authors' knowledge, this is the first analytical study on the influences of the power distribution of the PL on the performance of different combining techniques. Our findings can provide some useful guidelines for the design of PL based receiver for FSOC systems.

\section{System Model}
\label{section: SystemModel}
\subsection{Free-Space Atmosphere Channel}
\label{section: Free-space Atmosphere Channel}
In FSOC, atmospheric turbulence introduces fluctuation of irradiance, which results in fluctuation of SNR. The probability density function (PDF) of the received signal irradiance can be modeled as a Gamma-Gamma distribution \cite{al2001mathematical, chatzidiamantis2011distribution}, which emerges as a useful turbulence model as it has excellent fit with measurement data over a wide range of turbulence conditions \cite{al2001mathematical}. The PDF of the received signal irradiance $I \ (I>0)$ is given by
\begin{equation}
f \left ({I} \right )=\frac{2{{ \left (\alpha \beta \right )}^{\frac{\alpha +\beta}{2}}}}{\Gamma \left (\alpha \right )\Gamma \left (\beta \right )}{ {I} ^{\frac{\alpha +\beta}{2}-1}}{{K}_{\alpha -\beta}} \left (2\sqrt{\alpha \beta {{I}}} \right ),
\label{eq: f}
\end{equation}
\noindent where $\Gamma \left (\cdot \right )$ is the Gamma function; ${K_{\alpha -\beta}} \left (\cdot \right )$ is the modified Bessel function of the second kind with order $\alpha -\beta $. The parameters $\alpha $ and $\beta $ are directly related to the atmospheric conditions \cite{al2001mathematical}, and they respectively denote the effective numbers of large-scale and small-scale cells of the scattering process, respectively. Without loss of generality, the received signal irradiance $I$ is normalized, i.e., $E[I]=1$, where $E[\cdot]$ denotes the mathematical expectation.

\subsection{Power Distribution in PL}
\label{section: Power Distribution }
When the signal beam transmitted from the transmitter reaches the receiver system after passing through the atmosphere turbulence, it is coupled into the MMF end of the PL. We assume the power received at MMF end of the PL is ${{P}_{M}}$, then we have \cite{li2016performance}
\begin{equation}
P_M={{\zeta}_{M}} {A} {I},
\label{P_M}
\end{equation}
\noindent where ${\zeta}_{M}$ is the coupling efficiency of MMF, and $A$ is the area of receiving aperture of the len. When the PL converts the multimode signal beam into $N$ single-mode signal beams, loss will be introduced \cite{birks2015photonic}. If we denote the loss factor of the PL by $\xi_{PL} \ (0 < \xi_{PL} \leq 1)$, then the output optical power of the PL is ${{P}_{S}}={{\xi}_{PL}}{{P}_M}$.

For a PL with $N$ SMFs, if we denote the power distributed at each SMF end by ${{P}_{S, i}} \ (i=1, 2, \cdots, N) $ and denote the ratio of ${{P}_{S, i}}$ to ${{P}_S}$ by $a_i$, then we have
\begin{equation}
{P}_{S, i}=a_i P_S = a_i \xi_{PL} \zeta_M A I,
\label{P_S_i}
\end{equation}
\noindent where random variables (RVs) $a_i \ (i=1, 2, \cdots, N)$ satisfy
\begin{equation}
a_1+a_2+\cdots+a_N=1, \quad 0 \leq a_i \leq 1, \quad i=1,2, \cdots, N,
\label{Simplex}
\end{equation}
\noindent where the set of $\{a_1, a_2, \cdots, a_N\}$ that satisfies \eqref{Simplex} is called a standard unit simplex \cite{onn2011generating}.

The exact power distribution at SMF end of the PL is not known. Because the optical power is proportional to the photon number, the ratios $\{a_1, a_2, \cdots, a_N\}$ for the optical power is identical to the ratios for the photon numbers. Therefore, we can simulate the photon distribution to obtain the power distribution.

\subsubsection{Simulation Model For Photon Distribution}
\label{section: Photon Distribution}
Here, we use a Monte-Carlo method to simulate the photon distribution at SMF end of the PL. We denote the number of SMF of the PL by $N$. Because the loss of PL has no effect on the power distribution at SMF end of the PL, we do not consider the loss of PL in the simulation of photon distribution. Because this work assumes non-mode-selective PL, it is reasonable to assume that each SMF of a PL is exactly the same. Then the probability of each photon at MMF end assigned to any SMF of PL is assumed the same. Therefore, the explicit Monte-Carlo process can be summarized as follows: Step 1, we first generate $M$ photons and assign each photon into one SMF end randomly; Step 2, we calculate and record the ratio of the photon number $m_i$ of $i$th SMF end to the total photon number $M$ as $a_i=m_i/M$, where $i=1, 2, \cdots, N$; Step 3, repeat Step 1 and Step 2 $L$ times. Then we can obtain the distribution of $a_i$ from its $L$ samples for the $i$th SMF and obtain the correlation coefficient between $a_i$ and $a_j$ for $i \neq j$.

\begin{figure}
\begin{center}
\includegraphics [width=0.6\textwidth, draft=false] {img//Gaussian_Fitting.eps}
\caption{The obtained ratio distribution for one SMF end of the PL with $N=5$,  $M=500$, and $L=10^7$ (The range of $a_i$ is between 0 and 1. For simplicity, we only plot the range of $a_i$ from 0 to 0.4.)}
\label{fig: Gau_shape}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics [width=0.6\textwidth, draft=false] {img//Correlation_Coefficient.eps}
\caption{The obtained correlation coefficient between the ratios of two distinct SMF with $M=100\times N$ and $L=10^7$}
\label{fig: Correlation_Coefficient}
\end{center}
\end{figure}

The obtained distribution of the ratio $a_i$ for some SMF end is shown in Fig. \ref{fig: Gau_shape}. We find that the photon number distribution at the $i$th SMF end of the PL has excellent fit with the truncated Gaussian distribution with mean value $1/N$ \footnote{We remark that the obtained variance of $a_i$ can vary as the number of simulation repeating times varies due to the converging property of the Monte-Carlo method. A large number of repeating times results in a small variance. However the correlation coefficient between $a_i$ and $a_j$ is independent of the number of repeating times}. The obtained correlation coefficients between the ratios of two distinct SMF over the number of SMF ends are shown in Fig. \ref{fig: Correlation_Coefficient}. We can see that the correlation coefficients between the ratios of two distinct SMFs are always negative, which is due to the constraint \eqref{Simplex}. Besides, we can see that the correlation coefficient between two SMFs increases as $N$ increases. For example, when $N=2$, according to the constraint \eqref{Simplex}, the correlation coefficient between two SMFs is $-1$. As $N$ approaches $\infty$, the correlation coefficient between two SMFs should approach 0. We also perform the curve fitting on the simulation results and find that the correlation coefficients can be fitted as $-\frac{1}{N-1}$, which coincides to the analytical result obtained in Section \ref{TruncatedGaussian}.

\subsubsection{Truncated Multivariate Gaussian Model For Power Distribution}\label{TruncatedGaussian}
According to above simulation results, it is reasonable to assume that the ratios $\bm{a}=[a_1, a_2, \cdots, a_N]^\text{T}$ for the optical power satisfies a truncated multivariate Gaussian distribution \cite{Wilhelm2010} over the simplex defined in \eqref{Simplex}. The mathematical expectation of this truncated multivariate Gaussian distribution is $E[\bm{a}]=\bm{\mu}_{\bm{a}}=[\frac{1}{N}, \frac{1}{N}, \cdots, \frac{1}{N}]^{\text{T}}$, where $[\cdot]^{\text{T}}$ represents the transpose operator \footnote{Our analysis can be easily extended to the cases where different SMF ends have different mean values by replacing $\bm{\mu}_{\bm{a}}$ with the actual mean values.}. Here we derive the PDF of this truncated multivariate Gaussian distribution analytically.

We first remove the constraint $a_1+a_2+\cdots+a_N=1$, then the joint PDF of the truncated multivariate Gaussian distribution has the following form:
\begin{equation}
\begin{aligned}
f(\bm{a})&=\frac{1}{C_1} \exp \left\{-\frac{1} {2} [\bm{a}-\bm{\mu}_{\bm{a}}]^\text{T} {\bm{\Sigma}}_{\bm{a}} ^{-1} [\bm{a}-\bm{\mu}_{\bm{a}}] \right\},\\
&\quad \quad  0 \leq a_i \leq 1, \quad i=1, 2, \cdots, N,
\label{eq: fbmx}
\end{aligned}
\end{equation}
\noindent where $C_1=\int_V \exp \left\{-\frac{1} {2} [\bm{a}-\bm{\mu}_{\bm{a}}]^\text{T} {\bm{\Sigma}}_{\bm{a}} ^{-1} [\bm{a}-\bm{\mu}_{\bm{a}}] \right\} \mathrm{d} V$ is a constant number for normalization; $V$ is the domain defined as $V=\{0 \leq a_i \leq 1,\ i=1, 2, \cdots, N \}$; $\bm{\Sigma}_{\bm{a}}$ is the covariance matrix of $\bm{a}$. Because this work assumes non-mode-selective PL, it is reasonable to assume that $a_1, a_2, \cdots, a_{N}$ have the same Gaussian variance \footnote{Note that the Gaussian variance $var(a_i)$ here is not the actual variance of $a_i$. This is because the multivariate Gaussian distribution characterized is truncated by the definition domain $V$. Then the actual variance $var_{Actual}(a_i)$ is defined as $var_{Actual}(a_i)\triangleq \int_V (a_i-1/N)^2 f(\bm{a})\mathrm{d}V $, which is smaller than the Gaussian variance $var(a_i)$.} $var(a_i)=\sigma^2, \ i=1, 2, \cdots, N$; and the Gaussian covariances $cov(a_i, a_j)$ for any $a_i$ and $a_j$, when $i \neq j, \ i, j=1, 2, \cdots, N$, are the same. Then the $N \times N$ dimensional Gaussian covariance matrix $\bm{\Sigma}_{\bm{a}}$ can be written as
\begin{equation}
\begin{aligned}
\bm{\Sigma}_{\bm{a}} = \sigma ^2
\left[
\begin{array}{ccccc}
1 & \rho & \cdots & \rho \\
\rho & 1 & \cdots & \rho \\
\vdots & \vdots & \vdots & \vdots\\
\rho & \rho & \cdots & 1 \\
\end{array}
\right],
\label{eq: bmSigma}
\end{aligned}
\end{equation}
\noindent where $\rho=\frac{cov(a_i, a_j)}{\sigma^2}$ is the correlation coefficient between $a_i$ and $a_j$ when $i \neq j, \ i, j=1, 2, \cdots, N$. Then inverse matrix $\bm{\Sigma}_{\bm{a}}^{-1}$ in \eqref{eq: fbmx} can be obtained as
\begin{equation}
\begin{aligned}
\bm{\Sigma}_{\bm{a}} ^{-1}&= \frac{1}{[1+(N-1)\rho](1-\rho)\sigma^2}\\
&\times
\left[ \begin{array}{cccccc}
1+(N-2)\rho & -\rho & \cdots & -\rho\\
-\rho & 1+(N-2)\rho & \cdots & -\rho\\
\vdots & \vdots & \vdots & \vdots\\
-\rho & -\rho & \cdots & 1+(N-2)\rho
\end{array}
\right].
\label{eq: AbmSigmainv}
\end{aligned}
\end{equation}

However, when the constraint $a_1+a_2+\cdots+a_N=1$ is considered, the covariance matrix $\bm{\Sigma}_{\bm{a}}$ becomes a rank-deficient matrix and it has no inverse matrix. We first derive the correlation coefficient $\rho$. The constraint $a_1+a_2+\cdots+a_N=1$ can be rewritten as $[\bm{a}-\bm{\mu}_{\bm{a}}]^\text{T} \bm{1} =0$, where $\bm{1}=[1, 1, \cdots, 1]^\text{T}$ is an $N\times 1$ dimensional vector. Then we have \cite{zhang2019study}
\begin{equation}
\begin{aligned}
\label{MidEq}
E\left[[\bm{a}-\bm{\mu}_{\bm{a}}][\bm{a}-\bm{\mu}_{\bm{a}}]^\text{T}\bm{1}\right]&=\bm{\Sigma}_{\bm{a}}\bm{1}\\
&=\sigma^2(1+(N-1)\rho)\bm{1}\\
&=\bm{0},
\end{aligned}
\end{equation}
\noindent where $\bm{0}=[0, 0, \cdots, 0]^\text{T}$ is an $N\times 1$ dimensional zero vector. Therefore, the correlation coefficient $\rho$ can be obtained from \eqref{MidEq} as $\rho=-\frac{1}{N-1}$, which is the same as the correlation coefficient obtained from the simulation result in Fig. \ref{fig: Correlation_Coefficient}. This correlation coefficient is also consistent with the inverse matrix in \eqref{eq: AbmSigmainv} because the numerator of $\bm{\Sigma}_{\bm{a}}^{-1}$ becomes zero when $\rho=-\frac{1}{N-1}$, and thus the inverse matrix does not exist.

To obtain the explicit form of the joint PDF, we generalize a generalized inverse matrix of $\bm{\Sigma}_{\bm{a}}$, and let ${\rho \to -\frac{1}{N-1}}$ when the constraint $a_1+a_2+\cdots+a_N=1$ is considered. Then the joint PDF of $\bm{a}$ can be obtained by substituting \eqref{eq: AbmSigmainv} into \eqref{eq: fbmx} and letting ${\rho \to -\frac{1}{N-1}}$. After some algebra (see Appendix \ref{Appendix A}), the joint PDF can be obtained as
\begin{equation}
\begin{aligned}
f(\bm{a})&=\frac{1}{C_2}
\exp \left\{ - \frac{1}{2} [\bm{a^*}-\bm{\mu}_{\bm{a^*}}]^\text{T} \bm{\Sigma}_{\bm{a^*}}^{-1} [\bm{a^*}-\bm{\mu}_{\bm{a^*}}]\right\}\\
& \quad \times \delta \left({{a}_{1}}+{{a}_{2}}+\cdots+{{a}_{N}}-1 \right),\\
& \quad 0 \leq a_i \leq 1, \ i=1, 2,\cdots, N,
\label{eq: fbmxF}
\end{aligned}
\end{equation}
\noindent where
\begin{equation}\label{C_2}
\begin{aligned}
C_2&=\int_V \exp \left\{ - \frac{1}{2} [\bm{a^*}-\bm{\mu}_{\bm{a^*}}]^\text{T} \bm{\Sigma}_{\bm{a^*}}^{-1} [\bm{a^*}-\bm{\mu}_{\bm{a^*}}]\right\}\\
&\quad \times
\delta \left({{a}_{1}}+{{a}_{2}}+\cdots+{{a}_{N}}-1 \right) \mathrm{d}V
\end{aligned}
\end{equation}
\noindent is a constant normalization factor; $\bm{a^{*}}=[a_1, a_2, \cdots, a_{N-1}]^\text{T}$ is an $(N-1) \times 1$ dimensional vector; $\bm{\mu}_{\bm{a^*}}=[\frac{1}{N}, \frac{1}{N}, \cdots, \frac{1}{N}]^\text{T}$ is an $(N-1) \times 1$ dimensional vector; and the covariance matrix $\bm{\Sigma}_{\bm{a^{*}}}$ for $\bm{a^*}$ is the first $(N-1)\times(N-1)$ dimensional submatrix of $\bm{\Sigma}_{\bm{a}}$. Then the inverse of $\bm{\Sigma}_{\bm{a^{*}}}$ can be obtained as
\begin{equation}\label{Inverse_a_star}
\begin{aligned}
\bm{\Sigma}_{\bm{a^*}}^{-1}&= \frac{N-1}{N\sigma^2}
\left[ \begin{array}{cccccc}
2 & 1 & \cdots & 1\\
1 & 2 & \cdots & 1\\
\vdots & \vdots & \vdots & \vdots\\
1 & 1 & \cdots & 2
\end{array}
\right].
\end{aligned}
\end{equation}
\subsection{Two Extreme Cases}
\label{TwoCases}
Here we present two extreme cases of the truncated Gaussian distribution: the (joint) degenerate distribution and the (joint) uniform distribution, corresponding to the cases of minimum Gaussian variance $\sigma^2=0$ and maximum Gaussian variance \footnote{The actual variance of $a_i$ is $\frac{1}{12}$, because $a_i$ is uniformly distributed on $[0,1]$.} $\sigma^2=\infty$, respectively.

\subsubsection{Degenerate Distribution Case}
For a degenerate distribution, $a_i=\frac{1}{N}, i=1, 2, \cdots, N$ with probability one. Therefore, the joint PDF can be expressed as
\begin{equation}
\begin{aligned}
f(\bm{a})&=\Pi_{i=1}^{N}\delta(a_i-\frac{1}{N}).
\label{eq: degeneratePDF}
\end{aligned}
\end{equation}

\subsubsection{Uniform Distribution Case}
For an uniform distribution, $a_i, i=1, 2, \cdots, N$ is uniformly distributed on $[0, 1]$. The explicit joint PDF can be obtained by letting $\sigma^2\to\infty$ in \eqref{eq: fbmxF}, i.e.,
\begin{equation}
\begin{aligned}
f(\bm{a})&=\lim_{\sigma^2 \to \infty}\frac{\exp \left\{ - \frac{1}{2} [\bm{a^*}-\bm{\mu}_{\bm{a^*}}]^\text{T} \bm{\Sigma}_{\bm{a^*}}^{-1} [\bm{a^*}-\bm{\mu}_{\bm{a^*}}]\right\}\delta \left({{a}_{1}}+{{a}_{2}}+\cdots+{{a}_{N}}-1 \right)}{C_2}.
\label{eq: uniformPDF_1}
\end{aligned}
\end{equation}

From the expression of $\bm{\Sigma}_{\bm{a^*}}^{-1}$ in \eqref{Inverse_a_star}, we can find that the exponential term in \eqref{eq: uniformPDF_1} approaches one when $\sigma^2 \to \infty$. Similarly, for the denominator $C_2$, we have
\begin{equation}
\begin{aligned}
\lim_{\sigma^2 \to \infty}C_2&= \int_V \delta \left({{a}_{1}}+{{a}_{2}}+\cdots+{{a}_{N}}-1 \right) \mathrm{d}V\\
&=\int_{V_{s}} \mathrm{d}V_s\\
&= \frac{1}{(N-1)!},
\label{eq: C_2_uniform}
\end{aligned}
\end{equation}
\noindent where $V_s=\frac{1}{(N-1)!}$ is the volume of the standard simplex defined in \eqref{Simplex}. Substituting \eqref{eq: C_2_uniform} into \eqref{eq: uniformPDF_1}, we can obtain the joint PDF as
\begin{equation}
\begin{aligned}
f(\bm{a})&=(N-1)!\delta \left({{a}_{1}}+{{a}_{2}}+\cdots+{{a}_{N}}-1 \right).
\label{eq: uniformPDF_2}
\end{aligned}
\end{equation}

\section{Signal-to-Noise Ratio}
\label{section: SNR}
We assume that the shot noise is the dominated noise source in the coherent receiver, which is reasonable due to the presence of high intensity LO beams in balanced photodetectors. When the output signals of the SMF ends are combined using EGC \footnote{Here EGC method is used to combine the output signals of all SMF ends of the PL. Because there is only one receiving port and no diversity technique is introduced, the name ``EGC" should not be confused with the diversity combining technique EGC in wireless communications.}, the instantaneous SNR can be obtained as \cite{zhang2019study}
\begin{equation}
\gamma^{EGC}_{PL}=\frac{R\eta_S\left(\sum_{i=1}^{N}\sqrt{P_{S,i}}\right)^2}{NqB},
\end{equation}
\noindent where $R$ is the responsivity of the photodiode; $\eta_S$ is the mixing efficiency of SMF; $q$ is the electronic charge and $B$ is the noise equivalent bandwidth of the detector. Substituting ${{P}_{S, i}}$ into $\gamma_{PL}$, we can obtain
\begin{equation}\label{SNR_PL}
\gamma^{EGC}_{PL}=K\left(\sum_{i=1}^{N}\sqrt{a_i}\right)^2 I,
\end{equation}
\noindent where $K=\frac{R A \zeta_M\xi_{PL}\eta_s}{NqB}$.

Then the average SNR of coherent FSOC system using PL based receiver with EGC is
\begin{equation}
\begin{aligned}
\bar{\gamma}^{EGC}_{PL}&=E[{\gamma_{PL}}]\\
&=K E\left[\left(\sum_{i=1}^{N}{\sqrt{a_i}}\right)^{2}\right]E[I]\\
&=K \left[1+N(N-1)E \left[\sqrt{a_1 a_2}\right]\right],
\end{aligned}
\label{eq: gammaEGCave}
\end{equation}
\noindent where we have used the assumption $E[I]=1$ and the equality $E\left[\left(\sum_{i=1}^{N}{\sqrt{a_i}}\right)^{2}\right]=1+N(N-1)E \left[\sqrt{a_1 a_2}\right]$.

Similarly, the instantaneous and average SNR for SC can be respectively obtained as
\begin{equation}\label{SNR_SC}
\begin{aligned}
\gamma^{SC}_{PL}&=KN\max_{i}\{a_i\} I
\end{aligned}
\end{equation}
\noindent and
\begin{equation}\label{Average_SNR_SC}
\begin{aligned}
\bar{\gamma}^{SC}_{PL}&=KNE\left[\max_{i}\{a_i\}\right].
\end{aligned}
\end{equation}
The instantaneous and average SNR for MRC can be respectively obtained as
\begin{equation}\label{SNR_MRC}
\begin{aligned}
\gamma^{MRC}_{PL}&=KN\sum_{i=1}^{N}a_i I\\
&=KNI
\end{aligned}
\end{equation}
\noindent and
\begin{equation}
\begin{aligned}
\bar{\gamma}^{MRC}_{PL}&=KN.
\end{aligned}
\end{equation}

An important observation is that the instantaneous and average SNR for MRC are irrelevant to the power distribution of the PL due to the relation $\sum_{i=1}^{N}a_i=1$. Since the BER is determined by the instantaneous SNR, we can also conclude that the power distribution of the PL has no effect on the BER performance of the PL based receiver when MRC is used. This is an unique feature of the MRC for PL based receiver. However, the MRC requires the measurements of both the amplitude the phase of the signal in each branch, which is more complex compared with other combining techniques. In practical implementation, EGC and SC are two widely used combining techniques.

For the average SNR of EGC or SC, it is challenging to obtain an analytical expression for $E \left[\sqrt{a_1 a_2}\right]$ in \eqref{eq: gammaEGCave} or $E\left[\max_{i}\{a_i\}\right]$ in \eqref{Average_SNR_SC} when a general truncated multivariate Gaussian distribution is considered. However, it is still meaningful to consider the extreme cases defined in \ref{TwoCases}, because the degenerate case and the uniform case correspond to the smallest and the largest variance of the signal strength in each branch, respectively.

\subsection{Degenerate Distribution Case}
For the degenerate distribution, $a_i=\frac{1}{N}, (i=1, 2, \cdots, N)$ and we have $E\left[\left(\sum_{i=1}^{N}{\sqrt{a_i}}\right)^{2}\right]=N$ and $E\left[\max_{i}\{a_i\}\right]=\frac{1}{N}$. Then the average SNR for EGC and SC become
\begin{equation}
\label{AverageSNR_Degenerate}
\bar{\gamma}^{EGC}_{PL,Deg}=K N
\end{equation}
\noindent and
\begin{equation}
\label{AverageSNR_SC_Degenerate}
\bar{\gamma}^{SC}_{PL,Deg}=K.
\end{equation}
\noindent The average SNR of the EGC in degenerate distribution case equals the average SNR of the MRC. This is because all the branches have the same signal strength, then the EGC becomes an MRC.

\subsection{Uniform Distribution Case}
For the uniform distribution, $a_i, (i=1, 2, \cdots, N)$ is uniformed distributed in $[0,1]$ and we can obtain $E \left[\sqrt{a_1 a_2}\right]=\frac{\pi}{4 N}$ (see Appendix \ref{Appendix B}). Then the average SNR for EGC becomes
\begin{equation}
\label{AverageSNR_Uniform}
\bar{\gamma}^{EGC}_{PL,Uni}=K \frac{\pi N+4-\pi}{4}.
\end{equation}
However, it is still challenging to obtain an analytical form of average SNR of SC for uniform distribution case, except for the case with $N=2$. When $N=2$, we have $E\left[\max_{i}\{a_i\}\right]=\frac{3}{4}$. When $N>2$, we can use the Monte-Carlo method to numerically obtain $E\left[\max_{i}\{a_i\}\right]$ and then use the curve fitting method to approximate $E\left[\max_{i}\{a_i\}\right]$ as $\frac{4.45}{N+4.33}$. Then the average SNR for SC can be obtained as
\begin{equation}
\label{AverageSNR_SC_Uniform}
\bar{\gamma}^{SC}_{PL,Uni}\approx K \frac{4.45N}{N+4.33}.
\end{equation}

\subsection{General Distribution Case}
For a general truncated multivariate Gaussian distribution, i.e., $0<\sigma^2<\infty$, the average SNR is between the SNR of degenerate distribution and the SNR of uniform distribution. Now we consider the average SNR ratio of the degenerate distribution over the uniform distribution.

For the EGC, we have
\begin{equation}
\frac{\bar{\gamma}^{EGC}_{PL,Deg}}{\bar{\gamma}^{EGC}_{PL,Uni}}=\frac{4N}{\pi N+4-\pi},
\end{equation}
\noindent which is between $\frac{8}{4+\pi}\approx 1.12$ when $N=2$ and $\frac{4}{\pi} \approx 1.27$ when $N=\infty$. This implies that the influence of the power distribution of PL on the average SNR is relatively small when EGC method is used for signal combining.

For the SC, we have
\begin{equation}
\frac{\bar{\gamma}^{SC}_{PL,Deg}}{\bar{\gamma}^{SC}_{PL,Uni}}\approx \frac{N+4.33}{4.45 N},
\end{equation}
which is between $0.667$ when $N=2$ and $0.225$ when $N=\infty$. Since $\frac{\bar{\gamma}^{SC}_{PL,Deg}}{\bar{\gamma}^{SC}_{PL,Uni}}$ is always smaller than one, an interesting observation is that the SC prefers a large variance $\sigma^2$ than a small one. This is because the SC selects the largest $a_i$ as the output, then a larger variance of $a_i$ can have a larger possibility of obtaining a large $a_i$.

\subsection{Signal-to-Noise Ratios for SMF and MMF receivers}
For comparison, we also present the SNR of the SMF receiver and MMF receiver here. When shot noise is the dominated noise, the instantaneous SNR of the SMF receiver is
\begin{equation}
\gamma_{SMF}=\frac{\zeta_S \eta_S R A}{qB} I,
\end{equation}
\noindent where $\zeta_S$ is the coupling efficiency of SMF; and the average SNR of SMF is
\begin{equation}
\bar{\gamma}_{SMF}=E[\gamma_{SMF}]=\frac{\zeta_S \eta_S R A}{qB}.
\end{equation}

Similarly, the instantaneous SNR of the MMF receiver is
\begin{equation}
\gamma_{MMF}=\frac{\zeta_M \eta_M R A}{qB} I,
\end{equation}
\noindent where $\eta_M$ is the mixing efficiency of MMF mixer; and the average SNR of MMF is
\begin{equation}
\bar{\gamma}_{MMF}=E[\gamma_{MMF}]=\frac{\zeta_M \eta_M R A}{qB}.
\end{equation}

\section{Bit-Error Rate}
\label{section: BER Analysis}
\subsection{Integral Expression of BER}
The BER conditioned on received signal irradiance $I$ and power distribution $\bm{a}$ for an FSOC BPSK system \footnote{Although we only present the BER for BPSK scheme here, the BER and symbol error rate (SER) for other coherent modulation schemes can be easily found in a similar way.} using PL based receiver is given by \cite{song2012error}
\begin{equation}
P_{e,PL}(I,\bm{a})=Q(\sqrt{\gamma_{PL}}),
\end{equation}
\noindent where $Q(\cdot)$ is the Gaussian $Q$-function; and $\gamma_{PL}$ is the instantaneous SNR, which can be obtained in \eqref{SNR_PL}, \eqref{SNR_SC}, and \eqref{SNR_MRC} for EGC, SC, and MRC, respectively. Then the unconditional BER for PL based receiver can be obtained as the following integral form
\begin{equation}
P_{e,PL}=\int_{0}^{\infty}\int_V f(I) f(\bm{a}) Q(\sqrt{\gamma_{PL}})\mathrm{d}\bm{a}\mathrm{d}I,
\label{eq: Pe_EGC}
\end{equation}
\noindent where $f(I)$ is the PDF of the signal irradiance $I$ given in \eqref{eq: f}, and $f(\bm{a})$ is the joint PDF of power ratios $\bm{a}$ given in \eqref{eq: fbmxF}.

Similarly, the unconditional BERs for SMF receiver and MMF receiver are obtained as
\begin{equation}
P_{e,SMF}=\int_{0}^{\infty}f(I)Q(\sqrt{\gamma_{SMF}})\mathrm{d}I
\end{equation}
\noindent and
\begin{equation}
P_{e,MMF}=\int_{0}^{\infty}f(I)Q(\sqrt{\gamma_{MMF}})\mathrm{d}I,
\end{equation}
\noindent respectively.

\subsection{Analytical Lower Bound For BER}
\label{section: Series Solution}

Because $\left(\sum_{i=1}^{N}\sqrt{\hat{a}_i}\right)^2 \leq N$, where the equal sign is obtained when $a_i=\frac{1}{N}, i=1,2,\cdots,N$, we have $\gamma^{EGC}_{PL}\leq KNI=\gamma^{MRC}_{PL}$. Besides, noting that $\max_{i}\{a_i\}\leq 1$, we have $\gamma^{SC}_{PL}\leq KNI=\gamma^{MRC}_{PL}$. Therefore, the SNR of PL based receiver is bounded by the SNR of MRC \footnote{This coincides with the fact that MRC is the optimal combining regarding the SNR performance.}. Then we can obtain a lower bound for $P_{e,PL}$ as
\begin{equation}
\begin{aligned}
P_{e,PL}^{lower}&=\int_{0}^{\infty}\int_V f(I) f(\bm{a})Q\left(\sqrt{KN I}\right)\mathrm{d}\bm{a}\mathrm{d}I\\
&=\int_{0}^{\infty}f(I) Q(\sqrt{\bar{\gamma}^{MRC}_{PL}I})\mathrm{d}I,
\label{P_e_lowerbound}
\end{aligned}
\end{equation}
\noindent which is also the unconditional BER of PL based receiver using MRC.

Then we can obtain an analytical expression of the lower bound \eqref{P_e_lowerbound} by using a series expansion of the modified Bessel function of the second kind in \eqref{eq: f} as \cite{song2012error}
\begin{equation}
\begin{aligned}
K_{v}(x)=&\frac{\pi}{2 \sin (\pi v)}\sum\limits_{p=0}^{\infty}{\left[\frac{(x/2)^{2p-v}}{\Gamma(p-v+1)p!}-\frac{(x/2)^{2p+v}}{\Gamma(p+v+1)p!}\right]}, \\
&\quad \quad \quad v\notin Z, \left| x \right|<\infty
\label{eq: KvX}
\end{aligned}
\end{equation}
\noindent and an alternative expression of the $Q$-function \cite{park2010average}
\begin{equation}
Q(x)=\frac{1}{\pi}\int_{0}^{\pi/2} \exp \left(-\frac{x^2}{2 \sin^2 \theta}\right)\mathrm{d}\theta.
\label{Qfunc}
\end{equation}

Substituting \eqref{eq: f}, \eqref{eq: KvX}, and \eqref{Qfunc} into \eqref{P_e_lowerbound}, and after some algebra (see Appendix \ref{Appendix C}), we can obtain an analytical lower bound in series form as
\begin{equation}
\begin{aligned}
P_{e, PL}^{lower}&=\frac{\Lambda\left(\alpha, \beta\right)}{2}
\sum_{p=0}^{\infty}\left\{{a_p\left(\alpha, \beta\right)}\left(\frac{{\bar{\gamma}^{MRC}_{PL}}}{2}\right)^{-\left({p+\beta}\right)}
B\left(\frac{1}{2}, p+\beta+\frac{1}{2}\right)\right. \\
&\quad \quad \quad -\left.{a_p\left(\beta,\alpha\right)}
\left(\frac{{\bar{\gamma}^{MRC}_{PL}}}{2}\right)^{-\left({p+\alpha}\right)}
B\left(\frac{1}{2}, p+\alpha+\frac{1}{2}\right)\right\},
\label{eq: PeEGC2}
\end{aligned}
\end{equation}
\noindent where $B(x,y)=\int_0^1t^{x-1}(1-t)^{y-1}\mathrm{d}t$ is the Beta function, and
\begin{equation}\label{LambdaAnda_p}
\begin{aligned}
&\Lambda(\alpha, \beta)=\frac{1}{\Gamma{(\alpha)}\Gamma{(\beta)} \sin [(\alpha-\beta)\pi]};\\
&a_p(x,y)=\frac{(xy)^{p+y}{\Gamma\left({p+y}\right)}}{\Gamma(p-x+y+1)p!}.
\end{aligned}
\end{equation}

In addition, by replacing $\bar{\gamma}^{MRC}_{PL}$ in \eqref{eq: PeEGC2} with $\bar{\gamma}_{SMF}$ and $\bar{\gamma}_{MMF}$, we can obtain the unconditional BER for SMF receiver and MMF receiver, respectively.

\subsection{Truncation Error Analysis}
To implement the series form lower bound BER in \eqref{eq: PeEGC2}, we have to truncate the summation of infinite terms into a summation of finite terms. Therefore, it is necessary to analyze the truncation error. For simplicity, in the following we use $\bar{\gamma}$ to represent $\bar{\gamma}^{MRC}_{PL}$, $\bar{\gamma}_{SMF}$, and $\bar{\gamma}_{MMF}$. Substituting $B\left(x, y\right)=\frac{\Gamma\left(x\right)\Gamma\left(y\right)}{\Gamma\left(x+y\right)}$
\cite[ 8.384(1)]{gradshteyn2014table} and $\Gamma(x+1)=x\Gamma(x)$ \cite[ 8.331(1)]{gradshteyn2014table} into \eqref{eq: PeEGC2}, we obtain the error probability as
\begin{equation}
\begin{aligned}
P_{e}= \frac{\sqrt{\pi}}{2}\Lambda\left(\alpha, \beta\right)
\sum_{p=0}^{\infty}\frac{1}{p!}\left(\frac{2\alpha\beta}{\bar{\gamma}}\right)^p
\left\{G_p{\left(\alpha, \beta\right)}-G_p{\left(\beta, \alpha\right)}\right\},
\label{eq: PeEGC3}
\end{aligned}
\end{equation}
\noindent where $G_p{(x, y)}$ is defined as
\begin{equation}
G_p{(x,y)}=\frac{\Gamma(p+y+\frac{1}{2})}{(p+y)\Gamma(p-x+y+1)}\left(\frac{2xy}{\bar{\gamma}}\right)^y.
\end{equation}

Now we can estimate the truncation error caused by eliminating the infinite terms after the first $J$ terms in \eqref{eq: PeEGC3}. This truncation error can be defined as
\begin{equation}
\begin{aligned}
{\epsilon}_{J}=\frac{\sqrt{\pi}}{2}\Lambda\left(\alpha, \beta\right)
\sum_{p=J}^{\infty}\frac{1}{p!}\left(\frac{2\alpha\beta}{\bar{\gamma}}\right)^p
\left\{G_p{\left(\alpha, \beta\right)}-G_p{\left(\beta, \alpha\right)}\right\}.
\end{aligned}
\end{equation}

When $ p\to \infty$, we have $G_p{\left(\alpha, \beta\right)} \to 0$ and $G_p{\left(\beta, \alpha\right)} \to 0$. Then we can obtain an upper bound of the truncation error as
\begin{equation}
\begin{aligned}
{\epsilon}_{J}&<\frac{\sqrt{\pi}}{2}\Lambda\left(\alpha, \beta\right)
\sum_{p=J}^{\infty}\frac{1}{p!}\left(\frac{2\alpha\beta}{\bar{\gamma}}\right)^p
\max_{p\geq L}\left\{G_p{\left(\alpha, \beta\right)}-G_p{\left(\beta, \alpha\right)}\right\}\\
&\quad \quad <\frac{\sqrt{\pi}}{2}\frac{\Lambda\left(\alpha, \beta\right)}{J!}\left(\frac{2\alpha\beta}{\bar{\gamma}}\right)^J
\exp\left(\frac{2\alpha\beta}{\bar{\gamma}}\right)\\
&\quad \quad \quad \quad \times \max_{p\geq L}\left\{G_p{\left(\alpha, \beta\right)}-G_p{\left(\beta, \alpha\right)}\right\},
\end{aligned}
\end{equation}
\noindent where in the last inequality we have used the Lagrange form for the remainder term of Taylor series expansion for the exponential function. Note that when $J$ approaches $\infty$, the term $\frac{1}{J!}\left(\frac{2\alpha\beta}{\bar{\gamma}}\right)^J$ approaches zero. Therefore, truncation error ${\epsilon}_{J}$ diminishes to zero with increasing index $J$. Besides, we can also observe that ${\epsilon}_{J}$ diminishes rapidly with the average SNR $\bar{\gamma}$. This suggests that the series lower bound solution is highly accurate in the large SNR regimes. We can therefore perform an asymptotic BER analysis.

\subsection{Asymptotic Lower Bound For BER}
We now examine the lower bound BER behavior in the large SNR regimes. When $\bar{\gamma} \to \infty$, we have $G_p(\alpha,\beta) \to 0$ and $G_p(\beta,\alpha) \to 0$. From \eqref{eq: PeEGC3} we know that the first term $(p=0)$ of the series summation becomes the dominant term in the large SNR regimes. Therefore, the unconditional lower bound BER in high SNR regimes can be approximated by
\begin{equation}
\begin{aligned}
P_{e} \approx \frac{\sqrt{\pi}}{2}\Lambda\left(\alpha, \beta\right) \left[G_0{\left(\alpha, \beta\right)}-G_0{\left(\beta, \alpha\right)}\right].
\label{eq: PeEGCA}
\end{aligned}
\end{equation}

For typical turbulence conditions, we have $\alpha>\beta$. Then in high SNR regimes, we have
\begin{equation}
\begin{aligned}
\frac{G_0{\left(\beta, \alpha\right)}}{G_0{\left(\alpha, \beta\right)}}
=\frac{\beta \Gamma(\alpha+\frac{1}{2})\Gamma(-\alpha+\beta+1)}{\alpha \Gamma(\beta+\frac{1}{2})\Gamma(-\beta+\alpha+1)}
\left(\frac{2\alpha\beta}{\bar{\gamma}}\right)^{\alpha-\beta} \ll 1.
\end{aligned}
\end{equation}

Therefore, we can omit the second term in \eqref{eq: PeEGCA} and obtain
\begin{equation}
\begin{aligned}
P_{e} \approx H(\alpha, \beta) \left(\frac{1}{\bar{\gamma}}\right)^{\beta},
\label{eq: PeAsymError}
\end{aligned}
\end{equation}
\noindent where
\begin{equation}
H(\alpha, \beta)=\frac{\sqrt{\pi}}{2}\frac{\left({2\alpha\beta}\right)^{\beta}{\Gamma\left(\beta+\frac{1}{2}\right)}}{{\Gamma{(\alpha)}}{\Gamma\left(\beta+1\right)}
{\Gamma\left(-\alpha+\beta+1\right)} \sin [(\alpha-\beta)\pi]}.
\end{equation}

This indicates that the asymptotic lower bound BERs in high SNR regimes of the coherent optical communication system based on PL receiver, SMF receiver, and MMF receiver are decayed exponentially by the average SNR with an exponential decay constant $\beta$.

\section{Numerical Results}
\label{section: Simulation}

When the PL power distribution satisfies a general truncated multivariate Gaussian distribution and $N$ is large, we have to count on the stochastic numerical integration methods to calculate the normalization constant $C_2$, the average SNR and unconditional BER. However, the generation of random numbers satisfying truncated multivariate Gaussian distribution is not trivial. Here we use the Monte-Carlo integration (MCI) method (see Appendix \ref{Appendix D}) to calculate them. We set the number of SMF ends as $N=10$ in the following simulations. The turbulence parameters are $\left(\alpha =2.23, \ \beta =1.54\right)$ for moderate turbulence condition and $\left(\alpha =2.34, \ \beta =1.02\right)$ for strong turbulence condition \cite{al2001mathematical}. For the PL based receiver, we consider three different Gaussian variances $\sigma^2=0$, $\sigma^2=0.01$, and $\sigma^2=\infty$, which corresponds to the multivariate degenerate distribution, general truncated multivariate Gaussian distribution, and multivariate uniform distribution, respectively.

\begin{figure}
\begin{center}
\includegraphics [width=0.6\textwidth, draft=false] {img//BER_Comparison_different_combining.eps}
\caption{The BER comparison between PL based receiver, SMF receiver, and MMF receiver for coherent FSOC system ($\xi_{PL}=0.8$, $\frac{\eta_S}{\eta_M}=5$, $\frac{\zeta_M}{\zeta_S}=6$)}
\label{BERComparison_different combining}
\end{center}
\end{figure}

We first present the BERs of SMF receiver, MMF receiver, and PL based receiver using different combining techniques under moderate turbulence, shown in Fig. \ref{BERComparison_different combining}. We take the SMF receiver as the reference and the horizontal axis is the average SNR of the SMF receiver, i.e., $\bar{\gamma}_0=\bar{\gamma}_{SMF}$. As we have demonstrated in Sections \ref{section: SNR} and \ref{section: BER Analysis}, the BER for MRC is irrelevant with the power distribution of the PL and it equals to the BER for EGC with degenerate distribution. From Fig. \ref{BERComparison_different combining}, we can see that the BER performance for EGC is much better than that for SC. Besides, the BERs for EGC under different PL power distributions are close to each other. The ratio of BER of $\sigma^2=\infty$ over BER when $\sigma^2=0$ is around $1.3$. This indicates that the PL power distribution has limited influence on the BER performance of the PL based receiver when EGC is used. However, the BER for SC of the uniform distribution ($\sigma^2=\infty$) is much lower than that of the degenerate distribution ($\sigma^2=0$). The ratio of BER when $\sigma^2=0$ over BER when $\sigma^2=\infty$ is around $4.8$. This indicates that the PL power distribution can greatly affect the BER performance of the PL based receiver when SC is used.

Because the BERs for EGC under different PL power distributions are close to the BER for MRC and they are much better than the BER for SC under different PL power distributions, next we will focus on the performance comparison between the PL based receiver with EGC, the SMF receiver, and the MMF receiver.

\begin{figure}
\begin{center}
\includegraphics [width=0.6\textwidth, draft=false] {img//BER_Comparison.eps}
\caption{The BER comparison between PL based receiver, SMF receiver, and MMF receiver for coherent FSOC system ($\xi_{PL}=0.8$, $\frac{\eta_S}{\eta_M}=5$, $\frac{\zeta_M}{\zeta_S}=6$)}
\label{BERComparison}
\end{center}
\end{figure}

The BER comparison between the PL based receiver with EGC, the SMF receiver, and the MMF receiver for coherent FSOC systems under the moderate and strong turbulence conditions is shown in Fig. \ref{BERComparison}. From Fig. \ref{BERComparison}, we can find that, when the BER is $10^{-6}$, the ${\bar{\gamma}_{0}}$ for the PL based receiver with EGC, the MMF receiver, and the SMF under moderate turbulence are about $37$ dB, $42.5 $ dB and $43$ dB, respectively; and under strong turbulence are about $53$ dB, $57.5$ dB and $58$ dB, respectively. This suggests that SMF receiver and MMF receiver require an additional $6$ dB and $5.5$ dB SNR to achieve the same BER as the PL based receiver with EGC under moderate turbulence; and require an additional $5$ dB and $4.5$ dB SNR to achieve the same BER as the PL based receiver with EGC under strong turbulence.

Then we present the average SNR gains of the PL based receiver with EGC over the SMF receiver
$\bar{\gamma}_{PL}/\bar{\gamma}_{SMF}=\frac{\xi_{PL}}{N}\frac{\zeta_M}{\zeta_S}E\left[\left(\sum\limits_{i=1}^{N}\sqrt{a_i}\right )^{2}\right]$, and over the MMF receiver $\bar{\gamma}_{PL}/\bar{\gamma}_{MMF} =\frac{\xi_{PL}}{N}\frac{\eta_S}{\eta_M}E\left[\left (\sum\limits_{i=1}^{N}\sqrt{a_i}\right)^{2}\right]$ under various imperfect device parameters, including the coupling efficiency, the mixing efficiency, and the PL loss. We analyze the value of the coupling efficiencies of MMF, few-mode fiber and SMF in the literature \cite{winzer1998fiber,dikmelik2005fiber,toyoshima2006maximum,grein2014multimode,poliak2016fiber,arisa2017coupling,hu2018fiber,ozdur2015photonic,zheng2016free}, and set the coupling efficiency gain of MMF over SMF as $\frac{\zeta_M}{\zeta_S} \in [0,20]$. We analyze the value of the mixing efficiency of SMF and MMF in the literature \cite{mark1992comparison,duncan1993performance,jacob1995heterodyne,Marshalek1996,leeb1998aperture,ren2012heterodyne}, and set the mixing efficiency gain of SMF over MMF as $\frac {{\eta}_{S}} {{\eta}_{M}} \in [4,8]$. The range of the PL loss is set as $\xi_{PL} \in [0,1]$.

\begin{figure}
\begin{center}
\includegraphics [width=0.6\textwidth, draft=false] {{img//SNR_PLvsSMF.eps}}
\caption{The average SNR gain of PL based receiver over SMF receiver for coherent FSOC system}
\label{fig: loss_coupling_S}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics [width=0.6\textwidth, draft=false] {{img//SNR_PLvsMMF.eps}}
\caption{The average SNR gain of PL based receiver over MMF receiver for coherent FSOC system}
\label{fig: loss_coupling_M}
\end{center}
\end{figure}

Figures \ref{fig: loss_coupling_S} and \ref{fig: loss_coupling_M} show the obtained average SNR gain $\bar{\gamma}^{EGC}_{PL}/\bar{\gamma}_{SMF}$ and $\bar{\gamma}^{EGC}_{PL}/\bar{\gamma}_{MMF}$, respectively. The scope of application of the PL based receiver for FSOC systems can be obtained from Figs. \ref{fig: loss_coupling_S} and \ref{fig: loss_coupling_M}: when $\bar{\gamma}^{EGC}_{PL}/\bar{\gamma}_{SMF}>1$, we can choose to use the PL based receiver instead of SMF receiver for coherent FSOC systems; when $\bar{\gamma}^{EGC}_{PL}/\bar{\gamma}_{MMF}>1$, we can choose to use the PL based receiver instead of MMF receiver for coherent FSOC systems. Besides, from Figs. \ref{fig: loss_coupling_S} and \ref{fig: loss_coupling_M}, we can observe that the difference of the average SNR gain among three PL power distributions increases as average SNR gain increases. This indicates that the influence of the PL power distribution on the average SNR gain becomes significant in high SNR gain.

\begin{figure}
\begin{center}
\includegraphics [width=0.6\textwidth, draft=false] {{img//Closed-form_vs_integral.eps}}
\caption{Comparison between integral solution, series solution, and asymptotic solution of coherent FSOC system using PL based receiver.}
\label{fig: BERaym}
\end{center}
\end{figure}

At last, we present the integral solution, series lower bound solution and the asymptotic solution of the unconditional lower bound BER in Fig. \ref{fig: BERaym}. The series lower bound solution is calculated by \eqref{eq: PeEGC2} with $J=30$. We can see that the series lower bound solution is consistent with the integral solution, and the asymptotic lower bound BER approaches the exact BER curve in high SNR regimes ($\bar{\gamma} >30$ dB).

\section{Conclusion}
\label{section: Conclusion}
This paper proposed a truncated multivariate Gaussian distribution over a simplex for the power distribution at SMF ends of the PL. The SNR and BER for PL based receiver are analyzed using different combining techniques, including SC, EGC, and MRC; and they are compared with the SMF and MMF receivers for FSOC systems. We demonstrated that the power distribution of the PL has no effect on the SNR and BER performance of PL based receiver when MRC is used. Simulation results showed that the power distribution of the PL has limited influence on the BER performance of PL based receiver when EGC is used; and it can greatly affect the BER performance of the PL based receiver when SC is used. Besides, we quantified the SNR gains of the PL based receiver using EGC over the SMF and MMF receivers under different imperfect devices parameters; and provided the scope of application of the considered communication system. These findings can provide some useful guidelines for the design of PL based receiver for FSOC systems.

We have to remark that the PL with $N$ SMF ends requires $N$ balanced photodetectors to detect the received beams. Therefore, the cost and the complexity of the PL based receiver are higher than those of the SMF receiver. To reduce the number of balanced photodetectors and lower the complexity of the receiver, in the future work we will combine hybrid combining techniques, e.g., the hybrid-selection/equal-gain combining \cite{ma2007unified}, with PL based receiver.

\section*{Acknowledgement}
Bo Zhang acknowledges the support from the UCAS Joint PhD Training Program that allows her to conduct this research while she was visiting The University of British Columbia, Canada.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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text:
CaptureAudioOptions
===================

> Encapsula opciones personalizables para la captura de audio.

Atributos
---------

- __limit:__ El numero máximo de clips de audio que el dispositivo puede grabar en la misma operación. Debe ser mayor o igual a 1 (Por defecto: 1).
- __duration:__ La duración máxima de un clip de audio, en segundos.
- __mode:__ El modo de audio seleccionado. Este valor debe coincidir con uno de los elementos en `capture.supportedAudioModes`.

Ejemplo Rápido
--------------

    // Limita la operación de captura a 3 ficheros de audio, de no mas de 10 segundos cada uno
    var options = { limit: 3, duration: 10 };

    navigator.device.capture.captureAudio(captureSuccess, captureError, options);

Peculiaridades Android
----------------------

- El argumento __duration__ no esta soportado. La duración de la grabación no puede limitarse programaticamente.
- El argumento __mode__ no esta soportado. El formato de grabación de audio no puede ser controlado programaticamente.  Las grabaciones son codificadas usando el formato 'Adaptive Multi-Rate' (AMR) (audio/amr).

Peculiaridades BlackBerry WebWorks
----------------------------------

- El argumento __duration__ no esta soportado. La duración de la grabación no puede limitarse programaticamente.
- El argumento __mode__ no esta soportado. El formato de grabación de audio no puede ser controlado programaticamente.  Las grabaciones son codificadas usando el formato 'Adaptive Multi-Rate' (AMR) (audio/amr).

Peculiaridades iOS
------------------

- El argumento __limit__ no esta soportado. Solo se puede grabar un clip de audio por cada operación.
- El argumento __mode__ no esta soportado. El formato de grabación de audio no puede ser controlado programaticamente. El formato de grabación de audio no puede ser alterado programaticamente.  Las grabaciones son codificadas usando el formato 'Waveform Audio' (WAV) (audio/wav).

redpajama_set_name: RedPajamaGithub


At least one record was broken at the 144th Run for the Roses — and no one was celebrating. The National Weather Service declared it the wettest Derby on record, with an official rainfall total exceeding 2.83 inches. Rain was still pouring as horses took their positions, and it was still coming down hard when Justify crossed the line for the win. Ankle-deep puddles were scattered around Churchill Downs, and the downpours sent Derby goers cowering for cover all day long.


Synergy Formwork offers the most complete and up-to-date solutions in the formworks industry. The company has demonstrated leadership and set itself apart through the quality of its work and continues to leave its mark through partnerships and collaborations on major projects, for key construction sites throughout Eastern Canada.
Our workforce is made up of genuinely passionate industry professionals and people who understand the importance of a job well done, of optimal site safety and of the synergy between the various trades present on site. Synergy Formwork is driven by the constant need to innovate in order to offer services that meet the challenges with which we have entrusted.
The use of the latest technologies and state of the art solutions are key components that set our company apart and make us leaders in the concrete formwork industry. Our clients trust our know-how and commitment to quality, which is why they continue to use our services for their commercial, institutional, civil and industrial projects and applications. Synergy Formwork's wide-ranging expertise and our commitment to provide outstanding quality of service enable our company to continually raise the formwork industry's quality standards.


text:
Legislator calls for use of telehealth tools to expand
by Ashley Gold |
Aug 20, 2013 1:06pm
State Representative Klint Kesto (R) of Michigan's 39th district made a public call for greater use of telehealth tools in a letter to the editor published today in the Kalamazoo Gazette.
Kesto, who wrote the letter along with registered nurse Sheryl Stone of Decatur, R.I., claims that although healthcare exchanges under the Affordable Care Act dominate the headlines today, technology is just as deserving of attention from constituents and lawmakers.
"One important aspect that is often lost in the political discourse is how best we can utilize technology more effectively to help curb rising healthcare costs while expanding access to care through new innovative delivery systems," Kesto said. "Incorporating more technology into healthcare is something that most people would expect has already happened. Our nation's health care may seem modernized and equipped with the latest technology, but the reality is that we are not yet maximizing the use of even commonplace technologies we use in other areas of our lives to improve treatment and patient outcomes."
Calling the laws governing healthcare "antiquated," Kesto highlighted the many hoops providers must jump through to practice telehealth in their states, and how the rules don't reflect the benefits of telehealth and its innovations.
"As a result, it places barriers between patients and the high-quality care that could be delivered across state lines," Kesto said. "Lawmakers should immediately encourage the use of technology to delivery health care services."
Kesto added that he thinks Congress should begin by focusing on Medicare--by letting providers treat patients across state lines with telehealth.
A report published last monty by Research and Markets highlights the global interest in telemedicine, particularly in the fields of teleradiology, telepathology and teledermatology, with cost seen as the major barrier to many programs.
According to a panel of experts speaking at an Alliance for Health Reform event in late July in Washington, D.C., telehealth will be a vital cog in the rollout of the ACA and is creating a more "level playing field" in the healthcare industry.
- read Kesto's letter
- read this Research and Markets announcement
CMS rule would expand Medicare payments for telehealth
Telehealth expansion bill reintroduced to Congress
ATA head: Government a 'lagging partner' for telehealth

redpajama_set_name: RedPajamaCommonCrawl


Moving your server hardware and network equipment into a new datacenter might seem like a daunting task. This is understandably so, as service interruptions and excessive downtime are something no organization can afford. Consulting with a firm whose sole focus is assuring a successful migration can alleviate those concerns through sound planning and execution that makes sure your transition is as smooth as possible.
The first step in any good plan is identifying what needs to be completed before, during and after you move into your new colocation space. Things to consider are any current agreements for existing colo, Internet bandwidth needs, IPv4/IPv6 addressing, DNS records that would need to be updated. Are you using equipment or buying all new equipment? How will you access your equipment in the datacenter across the Internet? VPN, private fiber, RDP, etc. CyberLynk can help plan every step of your migration if you would like.
Racking equipment is not an everyday event for most people. CyberLynk engineers do it everyday and can rack your equipment very quickly if you would like. Unlike almost every colocation provider you will CyberLynk will provide a PDU to power your equipment along with FREE CAT5 cables and FREE power cables. We also have Cisco, Mac, Linux and Windows engineers on staff that can usually solve any problem you might have if you need help. CyberLynk is here to make sure your move to our datacenter goes smoothly.
Whether you are located within driving distance to the datacenter or thousands of miles away from Milwaukee. Simply ship your equipment to our Milwaukee datacenter, and we handle the rest from there. The CyberLynk team unpacks your server hardware, routers, switches and other equipment, then installs and configures it free of charge as you have directed us to via a support ticket. There is no need for you to even make the trip to our facility but you are always welcome to stop by our datacenter to install your equipment too.
At Milwaukee Colo, our goal is to quickly and efficiently migrate your IT systems while you and your staff continue to tend to the core objectives of your business. By choosing us as your colocation partner, you will be removing a huge part of the datacenter migration hurdles many organizations are facing today.
Have a network design or business model in mind but not interested in spending hundreds of thousands of dollars in hardware/software? Share your requirements with us and we will provide all of the infrastructure to build your business on. Turn your capital expenses into monthly operating expenses.
Too busy? Need additional help? Cyberlynk's highly skilled engineers are always available to schedule the tasks you need help with via our Professional Services. Never again do you have worry about being down for extended periods of time or not having the skill sets required to accomplish your goals.


Araminta Robertson is the creator of Financially Mint, a personal finance blog for university students written by an actual student. She interviews experts, does weird experiments and a ton of research to help her and others graduate financially intelligent.
Whether you're a parent or someone who simply wants to help students out, there are several ways you can help our university students learn how to manage money.
Why is it important? University is exactly the time when students start to get a bit more of a taste of the real world. And yet, no one is taught how to budget, how to understand the inflows and outflows of money, or even how to earn more money. So it's important to teach the basics in university, to make the transition into the real world even smoother.
But… it does seem hard to get them interested – I know, I have a blog about it. In university all you want to do is have fun, enjoy life and not have to worry about money. It's normal. But one of the reasons I write a blog to help students manage their money is this: I meet way too many 30+ year olds who wish they had learnt about money when they were in university. So I'm not giving up, and you shouldn't either.
Honestly, this is the hardest part: convincing them that learning about money is important and that they should get onto it now.
Depending on the personality, different things will work differently. For me, it was a book: Rich Dad Poor Dad. But for another student it may be a documentary: something like Enron. For some it may even be a podcast: try Listen Money Matters.
The first step is showing them that money is important and that they need to learn how to manage it – the rest will come with self-motivation and a bit of guidance.
Another way to get them interested is by using the words 'how to make money'. No one can resist that, least of all, students. Tell your son/daughter/nephew/niece/whoever that you can teach them how to make money, and that you'll help them through the process. If they have a mentor to look up to, they won't feel so lost and confused.
What's the first lesson of any money management course? Budgeting. The simple act of tracking expenses and putting a limit on them. Making a budget is simple, but not easy – one needs a lot of self-discipline and motivation to stick to it.
So how do I make my student suddenly acquire this? You do it with them. Sit down with them and work through the ins and outs of a budget (here's a cool guide). Set them up on an app like Money Dashboard or Cleo and explain how to do each step.
Ask them to do it for the month and report back. It's a challenge, but it's one that they can enjoy: 'Look at me #adulting!'. The most important is to have a conversation about money: this is your income, these are your expenses, you want your income to be higher than your expenses and to save the rest. You want to teach them the systems: Save before spending, adjust the budget.
If it's coming from someone they trust, they'll be more interested in trying it out.
Students love investing the minute they understand it. Compound interest is very much in their favour because they start so young. Explain to them the difference between someone who starts investing at 20 and someone who starts at 30. Explain to them what the stock market it is and how it returns on average 7% every year, what index funds are and how they could be millionaires when they retire.
Yes, investing sounds risky and scary – so start with the concept of saving, of Pay Yourself First. Every time they get their income, tell them to stash away a certain percentage of it. As they get used to the idea of saving, they'll then be more interested in investing. Make sure you teach them how it works and the benefits, then set them up with a simple broker.
Finance is always associated with 'boring' – many times the reason why students don't even want to pretend they're interested. So how do we make finance fun?
If they're college students (like me), apps are a good way to make it interesting. It's on a phone, it's easy to use and many times gamified. I also have list of cool apps to download, but the best ones are apps like Revolut, Chip and Moneybox.
⟵2018 SHOMOs nominations are closed. Who's made the longlists?
Thanks for this information. Very timely. My 18 year old starts university next month and he'll be living at home. This week I sat him down and talked money. He agreed to have a budget for his groceries. We went shopping and he had his own trolley. Previously I found I was buying more and more to meet his appetite. He knows that the food has to last him a week. He seemed to enjoy the process and when I checked last night he was able to tell me that he still had sufficient meals for the rest of the week.
I already use Cleo and had not thought of suggesting he use it too.
I've got him set up on some survey sites and he's already started to generate some money. Yes he likes the make money bit.
I want to give him a good financial start and make it fun too.
I shall check out the rest of your content and of course share your link with my son too.
Wow! Off to a very good start – congrats. I wish the best of luck for your son and don't hesitate to get in touch with me if you want some more information, advice, etc.


text:
UPDATE: Ex arrested after Ohio teacher shot at on way to work
by: Jessica Patterson, Lane Ball
Posted: Aug 23, 2022 / 01:06 PM EDT
Updated: Aug 23, 2022 / 11:50 PM EDT
UPDATE: (2:45 p.m. Aug. 23, 2022) – The Lawrence County Sheriff's Office says a man has been arrested after a teacher was shot at on her way to work this morning.
According to the Lawrence County Sheriff's Office, Jerrino D. Johnson, 52, of South Point, was arrested around 1:40 p.m. this afternoon, Tuesday, Aug. 23 and taken to the Lawrence County Jail on "suspicion of Attempted Murder," a 2nd-degree felony.
Lawrence County Sheriff Jeff Lawless says through the investigation, detectives found evidence that led them to suspect Johnson. Lawless also says Johnson is a former boyfriend of the victim.
According to the sheriff's office, the investigation is ongoing and additional charges could be filed.
Lawless also adds that his office chose to release the victim's employer because the incident put the schools were on a heightened alert and they wanted to be transparent with parents about the incident.
LAWRENCE COUNTY, OH (WOWK) – The Lawrence County Sheriff's Office is investigating after a teacher reported that she had been shot at early this morning on her way to work.
Authorities say the woman, a teacher at Rock Hill Middle School, was not injured in the incident. According to the sheriff's office, the incident happened just before 7:42 a.m. Tuesday, Aug. 23. The teacher told authorities she was driving to work when she began feeling ill, pulled over in the area of the Lawrence County Juvenile Center on State Route 93, and parked in the gravel lot leading to the walking track.
Police: Man breaks into officer's home in St. Albans, cooks himself a meal
The sheriff's office says the woman said as she was sitting in her vehicle, she saw an unknown male approach her vehicle on foot and begin shooting at her with a handgun. She then drove away from the scene as two bullets struck her vehicle, according to authorities. Authorities say one bullet hit the back glass and the second hit the passenger-side rear fender.
According to the sheriff's office, the woman said she did not get a good look at the suspect, but was able to describe him as wearing a camouflage outfit along with a camouflage hat with a netting that covered his face. She said she did not see any vehicle associated with the suspect.
The sheriff's office says due to the woman being employed by Rock Hill Schools and on her way to work, the school district is asked to be on a heightened alert as a precaution.
Lawrence County Sheriff Jeff Lawless says the investigation is still in it's early stages. Anyone with any information regarding the incident is asked to contact the Lawrence County Sheriff's Office.
Republicans in Ohio House infighting for power
WDTN.com Video
Administration | 3 mins ago
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Homeless shelters nearing capacity in the Miami Valley
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Several Amish man answer to citations for buggy lights
Intra-party conflict among Ohio statehouse Republicans …


Fundamental Law of Conventional Conferences. Dave Winer. 2006.
I was introduced to this idea when reading a 2008* blog post by Ian Usher. *I didn't read it in 2008, though I wish I had – i didn't find it until 2015 when I started chasing up early information about TeachMeet.
firstly in a conversation with a friend that I don't see often enough, who was speaking of her ennui when sitting passively listening to stuff she already knew being explained to her from a stage. Her joy on hearing my profound pronouncement ex-cathedra that there was a Fundamental Law to explain her predicament made me feel very wise.
secondly at a recent conference, INTED (in Valencia Spain), where Winer's Law was certainly true in terms of audience expertise, but the organisers of the conference were very thoughtful with the structure of the conference – of the over 700 attendees, almost all were presenting at some stage (literally the 15 mins of fame!), and the rules of engagement were very clearly drawn up with the Law Of Two Feet in mind – the audience in each of the many rooms was free to some and go as necessary. Coupled with the long siestas coffee and lunch breaks, it made for a conversational, relaxed-but-still-full-on-learning atmosphere. At no stage was an audience member in any 'captive', but there by choice and out of interest. It made for a very dynamic and lively two days. As giant strictly organised conventional conferences go, INTED was the first I've encountered which has, consciously or otherwise, taken Winer's Law into account during the planning process.
thirdly, on Twitter this morning, I read a conversation about TeachMeet between @johnjohnston and @MrM which conjured again for me Winer's Law, and prompted me to reply via this blog post (Hello John, Hello Athole). John summarises neatly here, and others pipe in with comments. As it happens, I was speaking of TeachMeet at the INTED conference mentioned above – it is always interesting to watch the reactions of education academics when TeachMeet – a leaderless organisation, by teachers for teachers – is explained and outlined to them for the first time. There is keen interest and a plethora of questions; a little tension; and a LOT of delight. The image that works best I find is the 'desire line' metaphor here.
So thank you Dave for your law, and thank you Ian for pointing me towards it, and thank you Athole and John and commenters for reminding us of TeachMeet's unconference birth, and thank you to my friend Carmel for the sheer joy of your reaction to it all.
Jano (Peru), Mags (Ireland), Marta (Espana), Pablo (UK), Sung (China), Maria (Espana).


Balangir best free online dating service - Meet singles in Balangir to find best match and Balangir is full of girls and guys seeking for love, friendship and date in Balangir. Join now without any payment and quick search dating profiles of Balangir singles and start dating in Balangir right away!
Have mercy upon me, O God, according to the lovingKindness.


text:
Accidents, crowds mar Mckenzie River's Blue Pool
Zach Urness
This story was originally published on May 14, 2016.
MCKENZIE BRIDGE — One of the most beautiful sights in Oregon has become one of the most problematic spots in Willamette National Forest.
Blue Pool, located east of Eugene, is a phantasmal spring of sapphire water surrounded by the cliffs of a dry waterfall, where the McKenzie River seeps from underground lava fields into a pool of iridescent clarity.
The site's beauty and the two-mile hike to reach it have always made Blue Pool — also known as Tamolitch Falls — a popular spot with hikers, photographers and mountain bikers passing on the McKenzie River Trail.
Yet a common ill of the internet age has reared its head here during the past few years. As pictures and video of the once-tranquil pool spread on social media, larger crowds have been showing up, bringing with them conflict, accidents and even death.
At least five people have been injured and required rescue here since 2012. Two people have died. On summer weekends, hundreds swarm the Blue Pool area, said McKenzie River district ranger Terry Baker.
'Bike bypass' coming to McKenzie River Trail
"It has definitely become one of the main places causing concern for us," Baker said. "The sheer volume of people has brought a number of problems, both for safety and for the resource."
Of all the problems — which include ecological damage and conflict between hikers and mountain bikers — the worst is cliff-jumping.
Search online, and it's easy to find video of people leaping off the cliffs and dropping 70 feet into Blue Pool. While those featured in the video escape unscathed, the reality is that an errant jump from that height can go bad in a moment.
On May 1, John Frank, 37, had to be rescued by helicopter after making the jump and suffering chest and back injuries.
He was able to swim to a rocky shoreline at 4:25 p.m., but due to the remote location and lack of cell phone service, he wasn't airlifted from the pool until after midnight.
That's almost seven hours of suffering in the cold and darkness.
As bad as that sounds, Frank was actually lucky.
In 2013, Alex Rovello, 21, died after making the same jump. He hit the water face first and never surfaced.
Rovello was a four-time state tennis singles champion at Cleveland High and a student of the University of Oregon.
"(The jump) is like hitting a concrete wall at 40 miles per hour," said Todd Shechter, volunteer mission coordinator of the Corvallis Mountain Rescue Unit. "But it's not just that. Because of the water's clarity, it's hard to judge where the rocks are and where to jump. Plus, the water itself is extremely cold, around 37 to 40 degrees, which can cause a real shock to the system."
Booze, trespassing and crowds mar Little North swimming holes
Of the five documented incidents at Blue Pool — and there are likely far more, Shechter said — three involved cliff-jumping and one involved a fall.
In April of 2015, Bend resident Joel Martin fell 45 feet while taking pictures and landed in shallow water, dying shortly thereafter.
Rescue operations at Blue Pool tend to take time.
The lack of cellphone service, remote location and two-mile hike means an injured person will have to wait hours for the first wave of responders.
"It's not like calling 911 from your home and an ambulance shows up 10 minutes later," Shechter said. "It's normally going to take at least four hours just to get to you, and it's a complicated place to conduct a rescue."
Seven deaths at Cape Kiwanda spark inquiry
The growing number of people visiting, and the number of incidents, means the Forest Service is considering changes to the way it manages the site. To start, they're planning to add signs to the trailheads, warning of the danger in jumping. They may also consider signs at the pool itself, Baker said.
Baker and Shechter aren't advocating for people to avoid visiting Blue Pool — just to be smart about the decisions they make when they do visit.
"It's very pretty place," Shechter said. "But it's also important for us to share with people how dangerous it can be."
Note: Shechter emphasized that people who get in trouble shouldn't hesitate to call for help. He said people won't be billed for the rescue effort — it's an all-volunteer force.
"We rather you called for help right away than waiting until midnight, when it might be too late," he said. He added that every hiker should bring the 10 essentials and let people know in advance where they're headed and when they'll be back.
Don't get caught without your outdoor essentials
Zach Urness has been an outdoors writer, photographer and videographer in Oregon for eight years. He is the author of the book "Hiking Southern Oregon" and can be reached at zurness@StatesmanJournal.com or 503-399-6801. Find him on Zach Urness or @ZachsORoutdoors on Twitter.
Documented accidents at Blue Pool
July 22, 2012: Lisa Benthuysen jumped off cliff and was injured. Corvallis Mountain Rescue and Eugene Mountain Rescue did a 65-foot vertical raise and moved her to a landing zone where she was taken by Oregon Army National Guard to hospital.
May 11, 2013: Alex Rovello jumped into the pool and drown. Corvallis Mountain Rescue and Eugene Mountain Rescue assisted with vertical raise for recovery.
September 9, 2014: Corvallis Mountain Rescue rescued an injured hiker. Details unknown.
April 12, 2015: Joel Martin fell off cliff while doing photography. Corvallis Mountain Rescue and Eugene Mountain Rescue assisted with vertical raise for recovery.
May 1, 2016: John Frank jumped into pool and was injured. Corvallis Mountain Rescue and Eugene Mountain Rescue assisted Oregon Army National Guard with helicopter extraction.
(Information courtesy of Linn County Sheriff's Office and Corvallis Mountain Rescue)


Why choose Sri Lanka? From the pristine shores of the south coast to the dense jungles of Kitulgala, this is a country brimming with natural wonders; where leopards, mongeese and purple-faced leaf monkeys roam through the landscape.
Whether you're white water rafting through dense jungles or relishing the fresh breeze from the top of Sigiriya, Sri Lanka will enthral you.


text:
package com.im.config;

import java.util.Properties;
import javax.persistence.EntityManagerFactory;
import javax.sql.DataSource;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.context.annotation.Bean;
import org.springframework.context.annotation.ComponentScan;
import org.springframework.context.annotation.Configuration;
import org.springframework.context.annotation.PropertySource;
import org.springframework.core.env.Environment;
import org.springframework.dao.annotation.PersistenceExceptionTranslationPostProcessor;
import org.springframework.http.converter.HttpMessageConverter;
import org.springframework.http.converter.json.MappingJackson2HttpMessageConverter;
import org.springframework.jdbc.datasource.DriverManagerDataSource;
import org.springframework.orm.jpa.JpaTransactionManager;
import org.springframework.orm.jpa.JpaVendorAdapter;
import org.springframework.orm.jpa.LocalContainerEntityManagerFactoryBean;
import org.springframework.orm.jpa.vendor.HibernateJpaVendorAdapter;
import org.springframework.transaction.PlatformTransactionManager;
import org.springframework.transaction.annotation.EnableTransactionManagement;
import org.springframework.web.multipart.commons.CommonsMultipartResolver;
import org.springframework.web.servlet.config.annotation.WebMvcConfigurerAdapter;
import org.springframework.web.servlet.mvc.annotation.AnnotationMethodHandlerAdapter;

@Configuration
@ComponentScan( basePackages = "com.im" )
@PropertySource( value = { "classpath:application.properties" } )
@EnableTransactionManagement
public class AppConfig extends WebMvcConfigurerAdapter {

	@Autowired
	private Environment environment;

	/*
	 * @Override
	 * public void configureDefaultServletHandling(
	 * DefaultServletHandlerConfigurer configurer )
	 * {
	 * configurer.enable();
	 * }
	 * @Bean
	 * public ServletRegistrationBean dispatcherRegistration(DispatcherServlet
	 * dispatcherServlet) {
	 * ServletRegistrationBean registration = new ServletRegistrationBean(
	 * dispatcherServlet);
	 * registration.addUrlMappings("/im");
	 * return registration;
	 * }
	 * @Bean
	 * public ViewResolver getViewResolver(){
	 * InternalResourceViewResolver resolver = new
	 * InternalResourceViewResolver();
	 * resolver.setPrefix("/");
	 * resolver.setSuffix(".jsp");
	 * return resolver;
	 * }
	 */

	  @Bean
	    public AnnotationMethodHandlerAdapter annotationMethodHandlerAdapter()
	    {
	        final AnnotationMethodHandlerAdapter annotationMethodHandlerAdapter = new AnnotationMethodHandlerAdapter();
	        final MappingJackson2HttpMessageConverter mappingJacksonHttpMessageConverter = new MappingJackson2HttpMessageConverter();

	        HttpMessageConverter<?>[] httpMessageConverter = { mappingJacksonHttpMessageConverter };

	        String[] supportedHttpMethods = { "POST", "GET", "HEAD" };

	        annotationMethodHandlerAdapter.setMessageConverters(httpMessageConverter);
	        annotationMethodHandlerAdapter.setSupportedMethods(supportedHttpMethods);

	        return annotationMethodHandlerAdapter;
	    }
	@Bean
	public LocalContainerEntityManagerFactoryBean entityManagerFactory()
	{
		LocalContainerEntityManagerFactoryBean entityManager = new LocalContainerEntityManagerFactoryBean();
		entityManager.setDataSource(dataSource());
		entityManager.setPackagesToScan(new String[] { "com.im.domain" });

		JpaVendorAdapter vendorAdapter = new HibernateJpaVendorAdapter();
		entityManager.setJpaVendorAdapter(vendorAdapter);
		entityManager.setJpaProperties(hibernateProperties());

		return entityManager;
	}

	@Bean
	public DataSource dataSource()
	{
		DriverManagerDataSource dataSource = new DriverManagerDataSource();
		dataSource.setDriverClassName(environment.getRequiredProperty("jdbc.driverClassName"));
		dataSource.setUrl(environment.getRequiredProperty("jdbc.url"));
		dataSource.setUsername(environment.getRequiredProperty("jdbc.username"));
		dataSource.setPassword(environment.getRequiredProperty("jdbc.password"));
		return dataSource;
	}

	@Bean
	public CommonsMultipartResolver multipartResolver()
	{
		CommonsMultipartResolver resolver = new CommonsMultipartResolver();
		resolver.setDefaultEncoding("utf-8");
		return resolver;
	}

	@Bean
	public PlatformTransactionManager transactionManager( EntityManagerFactory entityManager )
	{
		JpaTransactionManager transactionManager = new JpaTransactionManager();
		transactionManager.setEntityManagerFactory(entityManager);
		return transactionManager;
	}

	@Bean
	public PersistenceExceptionTranslationPostProcessor exceptionTranslation()
	{
		return new PersistenceExceptionTranslationPostProcessor();
	}

	@Bean
	public Properties hibernateProperties()
	{
		Properties properties = new Properties();
		properties.put("hibernate.dialect", environment.getRequiredProperty("hibernate.dialect"));
		properties.put("hibernate.show_sql", environment.getRequiredProperty("hibernate.show_sql"));
		properties.put("hibernate.format_sql", environment.getRequiredProperty("hibernate.format_sql"));
		properties.put("hibernate.hbm2ddl.auto", environment.getRequiredProperty("hibernate.hbm2ddl.auto"));
		return properties;
	}

}


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text:
Ian Campbell is a Scottish retired amateur football forward and midfielder who made over 210 appearances in the Scottish League for Queen's Park.

References

Scottish footballers
Scottish Football League players
Queen's Park F.C. players
Association football forwards
Living people
Place of birth missing (living people)
Year of birth missing (living people)
Association football midfielders
Scotland amateur international footballers

redpajama_set_name: RedPajamaWikipedia


was a writer for Education Week and Education Week Digital Directios as well as web producer of the latter. He also contributed to the blog Digital Education. Ian Quillen covered educational technology and the states of Georgia, North Carolina, North Dakota, and South Dakota.
The growing availability of digital video sparked the "flipped classroom" movement, but identifying quality video content is still a challenge.
As digital video becomes a more popular educational tool, the companies and organizations that provide such content are becoming more diverse.
Educators say digital tools used both inside the classroom and outside it in students' personal lives are having positive and negative effects on academic and social development.
Surveys Track Social Networking, Online Ed.
Gates Foundation Turns Attention to Ed. Tech.
Mascot-Name Law On Books in Wis.


text:
58% of California Voters Support Marijuana Legalization
September 15, 2016 by Frank Gormlie
Proposition 64 Enjoys Wide Support Among Most Voter Categories
By Frank Gormlie / OB Rag
A new USC Dornsife/Los Angeles Times poll found that a good majority of California voters support marijuana legalization and back Proposition 64.
58% of voters in the state support Prop 64, which would allow Californians ages 21 or older to possess, transport and use up to an ounce of cannabis for recreational purposes, and would allow individuals to grow as many as six plants. The measure would also impose a 15% tax on retail sales of the drug.
And according to the poll, the favorable attitude toward grass cuts across most lines of age, race, income and gender.
Of the 1,879 respondents to the poll, only a third (34%) reported that they would vote against the ballot measure if the election were held today; 8% said they had no answer to the question.
This new high among California voters who support the herb stands in contrast to what happened just 6 years ago, when Prop. 19 was rejected by 53.5% of voters. (Yet, Prop. 19 garnered more votes than the Republican candidate for governor that year – 2010.)
And in 2010, four of the top 5 precincts in San Diego County that voted in favor of Prop. 19 were in Ocean Beach. Put that in your pipe and smoke it.
Frank Gormlie
A lawyer and grassroots activist, I was finally convinced by Patty Jones to start the OB Rag, a blog of citizen journalists, after she got tired of listening to my rants about the news. Way back during the Dinosaurs in 1970, I founded the original Ocean Beach People's Rag - OB's famous underground newspaper -, and then later during the early Eighties, published The Whole Damn Pie Shop, a progressive alternative to the Reader.
Latest posts by Frank Gormlie (see all)
Mayor's Office Confirms Sale of Liberty Station Leases by McMillin – But Questions Remain - December 5, 2018
Why Would Anyone Want to Flee Beautiful Honduras? - November 29, 2018
A Progressive Looks at the San Diego Mid-Terms – OB Rag Interview With Doug Porter - November 21, 2018
Filed Under: Culture, Government, Marijuana
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County Supervisors D3 Contest: A Whole Lotta Meh »
Wayne says
The 15% retail sales tax is ridiculous and actually corrupt. I feel recreational marijuana should be legal but this bill imposes a very steep tax. It also imposes severe start-up costs for retail businesses. We will end up with a few very wealthy people/corporations owning all the retail shops. So, we end up with a prohibitive, punishing tax and concentrated retail ownership. Seems like this proposition is meant more to discourage both medical and recreational use while at the same time further enriching the wealthy elite.


text:
Members of the North Peninsula Jewish and Muslim communities after Friday afternoon services at Yaseen Foundation in Belmont, March 15, 2019 (Photo/Courtesy Stephanie Levine)
Bay Area Jews show up for Muslim neighbors — how you can help
By David A.M. Wilensky | March 15, 2019
Around the world, Jewish communities are denouncing today's shootings at two mosques in New Zealand, which claimed 49 lives.
In the Bay Area, the Jewish Federations based in San Francisco and the East Bay, the Jewish Community Relations Council and the Anti-Defamation League expressed their horror and urged local Jews to support their Muslim neighbors and attend one of several gatherings at local mosques, including Jummah (Friday afternoon prayers) today.
A large group from the North Peninsula Jewish community attended afternoon prayers in Belmont at Yaseen Foundation, a San Mateo county Muslim community organization. Jewish organizations encouraging their members to attend included Peninsula JCC, Peninsula Temple Beth El, Peninsula Temple Sholom, Peninsula Sinai Congregation and Congregation Beth Jacob.
Fattin Wekselman, who is Muslim and involved in Muslim-Jewish interfaith work, along with Stephanie Levin, chief engagement and innovation officer at PJCC, organized the Jewish visit to services at Yaseen Foundation.
"It was a very tough day, but at the same time it was a warm day, the nicest day I have felt in this community in a long time," said Wekselman.
"I am a Muslim and my husband is Jewish. It is very important to me to see that both of the communities in my neighborhood are stronger together," she said.
About 40 Jews attended, joining some 200 Muslims. "It was packed," said Levin. At one point, with the room filling up, she asked one of the leaders whether some of the Jews should leave to make more room for Muslim worshippers. "But they were adamant that they wanted us in the room with them," she said.
Levin planned for the Jews to stand outside as symbolic guards. "But when I got there, one of the leaders said he'd rather that we came in to be with them during the service. They set up seats for us and everyone was extremely welcoming, and the spiritual leader who spoke talked a lot about the shared history of Muslims and Jews," she said.
"So many Muslims came to Shabbat services after the Tree of Life shooting. So I called Fattin last night to make sure I could show up the same way."
A number of San Francisco Jews, including Rabbi Mychal Copeland of Congregation Sha'ar Zahav, attended afternoon prayers at the Islamic Society of San Francisco, another event Jewish leaders urged people to attend.
In an email sent to Sha'ar Zahav members, Copeland said, "After the Pittsburgh shooting last fall, [Cantor Sharon Bernstein] and I received calls and emails from faith communities around San Francisco, and many came to our vigils…. I encourage you to reach out in solidarity to any Muslims you know to express your grief. I can tell you from experience that it really does make a difference."
She added, "We will be asking our friends at the Mission San Francisco Police Station to be present with us; religious communities of all kinds are encouraged to reach out for extra security for their weekend worship."
Reach out in solidarity to any Muslims you know to express your grief. I can tell you from experience that it makes a difference.
In the South Bay, a handful of Jewish leaders attended afternoon prayers at the Muslim Community Association in Santa Clara. Diane Fisher, director of the JCRC of the Jewish Federation of Silicon Valley, organized the group.
Fisher and Sheldon Gilbert, a member of the board of directors of Addison-Penzak JCC in Los Gatos, "held signs, standing at the doorway as people came and left prayers. My sign said, 'Jews & Muslims stand together against hate, white supremacy and bigotry,'" Fisher told J. in an email. "Fortunately, people feel safe here in Silicon Valley, but the deep appreciation for our solidarity was clear. People spoke about the Pittsburgh shooting with us and how we must stand together."
The example of Pittsburgh, where 11 Jews were murdered during Shabbat services at the Tree of Life synagogue last October, was on many local Jewish minds. Remembering the outpouring of support Jews received from non-Jews, including Muslims, they felt the need to reciprocate.
"Just as we saw at the Pittsburgh synagogue, this horrific massacre continues the trend of targeting houses of worship, using immigration and fears of the 'other' as a pretext for violence," said a joint statement by the Jewish Federation of the East Bay and the S.F.-based Jewish Community Federation, Jewish Community Relations Council and JCC of San Francisco.
"When Pittsburgh happened, our Muslim community was there to support us, and we are here now to support them," the statement continued.
There will be an interfaith vigil at 6 p.m. tonight at the Islamic Center of Mill Valley. Rabbi Paul Steinberg of Congregation Kol Shofar in Tiburon as well as other religious leaders will share messages of peace and hope, according to Khadija Hansia, a volunteer leader at Islamic Center of Mill Valley, who is organizing the vigil. Forty-nine candles will be lit, one for each life taken in the shootings.
"Historically, people think that Muslim and Jewish people don't get along. To come together in the face of hate is one of the most important things we can do," Hansia said. "The Jewish community here has always stood with us. The Osher Marin JCC has shown endless support for us over the years, and we're so grateful. They come for dinner during Ramadan and we've come to Shabbat. The Jewish community has been amazing at showing up for us."
Heidi Sanders, director of the Taube Center for Jewish Peoplehood at the Marin JCC, said, "We have been working with the Muslim community for years now. They've showed up for us. In the wake of the bomb threats, they brought flowers and showed up physically. They are our allies. We've had many conversations with them about rising anti-Muslim and anti-Semitic rhetoric.
"This is a horrible tragedy, but this is what we do for our neighbors, we show up to support them and they show up to support us."
There will be a vigil at 6 p.m. tonight in San Francisco, with the location still yet to be announced. Follow this Facebook event page for more details.
Pacifica Institute, a national nonprofit that promotes interfaith cooperation, will hold vigils on Saturday night, March 16, at 8 p.m. at its Albany center.
Bend the Arc: Jewish Action has a national registry of solidarity events that can be searched by ZIP code.
The Jewish Federation of Greater Pittsburgh has set up a New Zealand Attack Emergency Relief Fund to support the Muslim community there. The Jewish community of Pittsburgh received an outpouring of financial support from Muslims following the shooting at Tree of Life – Or L'Simcha Congregation.
Donations can also be made directly to the New Zealand Muslim community through the New Zealand Islamic Information Centre.
The San Jose-based Islamic Networks Group has created a list of educational resources and ways to support the Bay Area Muslim community.
David A.M. Wilensky
David A.M. Wilensky is the online editor of J. and "Jew in the Pew" columnist. He can be reached at david@jweekly.com.
Follow davidamwilensky
Tags: mosque, Muslim-Jewish relations
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